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  <title>`std::quantity` as a numeric type</title>
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l.CharacterMap("AMSsymbols-mathchar0mi",u.default.mathchar0mi,{digamma:"\u03dd",varkappa:"\u03f0",varGamma:["\u0393",{mathvariant:c.TexConstant.Variant.ITALIC}],varDelta:["\u0394",{mathvariant:c.TexConstant.Variant.ITALIC}],varTheta:["\u0398",{mathvariant:c.TexConstant.Variant.ITALIC}],varLambda:["\u039b",{mathvariant:c.TexConstant.Variant.ITALIC}],varXi:["\u039e",{mathvariant:c.TexConstant.Variant.ITALIC}],varPi:["\u03a0",{mathvariant:c.TexConstant.Variant.ITALIC}],varSigma:["\u03a3",{mathvariant:c.TexConstant.Variant.ITALIC}],varUpsilon:["\u03a5",{mathvariant:c.TexConstant.Variant.ITALIC}],varPhi:["\u03a6",{mathvariant:c.TexConstant.Variant.ITALIC}],varPsi:["\u03a8",{mathvariant:c.TexConstant.Variant.ITALIC}],varOmega:["\u03a9",{mathvariant:c.TexConstant.Variant.ITALIC}],beth:"\u2136",gimel:"\u2137",daleth:"\u2138",backprime:["\u2035",{variantForm:!0}],hslash:"\u210f",varnothing:["\u2205",{variantForm:!0}],blacktriangle:"\u25b4",triangledown:["\u25bd",{variantForm:!0}],blacktriangledown:"\u25be",square:"\u25fb",Box:"\u25fb",blacksquare:"\u25fc",lozenge:"\u25ca",Diamond:"\u25ca",blacklozenge:"\u29eb",circledS:["\u24c8",{mathvariant:c.TexConstant.Variant.NORMAL}],bigstar:"\u2605",sphericalangle:"\u2222",measuredangle:"\u2221",nexists:"\u2204",complement:"\u2201",mho:"\u2127",eth:["\xf0",{mathvariant:c.TexConstant.Variant.NORMAL}],Finv:"\u2132",diagup:"\u2571",Game:"\u2141",diagdown:"\u2572",Bbbk:["k",{mathvariant:c.TexConstant.Variant.DOUBLESTRUCK}],yen:"\xa5",circledR:"\xae",checkmark:"\u2713",maltese:"\u2720"}),new l.CharacterMap("AMSsymbols-mathchar0mo",u.default.mathchar0mo,{dotplus:"\u2214",ltimes:"\u22c9",smallsetminus:["\u2216",{variantForm:!0}],rtimes:"\u22ca",Cap:"\u22d2",doublecap:"\u22d2",leftthreetimes:"\u22cb",Cup:"\u22d3",doublecup:"\u22d3",rightthreetimes:"\u22cc",barwedge:"\u22bc",curlywedge:"\u22cf",veebar:"\u22bb",curlyvee:"\u22ce",doublebarwedge:"\u2a5e",boxminus:"\u229f",circleddash:"\u229d",boxtimes:"\u22a0",circledast:"\u229b",boxdot:"\u22a1",circledcirc:"\u229a",boxplus:"\u229e",centerdot:["\u22c5",{variantForm:!0}],divideontimes:"\u22c7",intercal:"\u22ba",leqq:"\u2266",geqq:"\u2267",leqslant:"\u2a7d",geqslant:"\u2a7e",eqslantless:"\u2a95",eqslantgtr:"\u2a96",lesssim:"\u2272",gtrsim:"\u2273",lessapprox:"\u2a85",gtrapprox:"\u2a86",approxeq:"\u224a",lessdot:"\u22d6",gtrdot:"\u22d7",lll:"\u22d8",llless:"\u22d8",ggg:"\u22d9",gggtr:"\u22d9",lessgtr:"\u2276",gtrless:"\u2277",lesseqgtr:"\u22da",gtreqless:"\u22db",lesseqqgtr:"\u2a8b",gtreqqless:"\u2a8c",doteqdot:"\u2251",Doteq:"\u2251",eqcirc:"\u2256",risingdotseq:"\u2253",circeq:"\u2257",fallingdotseq:"\u2252",triangleq:"\u225c",backsim:"\u223d",thicksim:["\u223c",{variantForm:!0}],backsimeq:"\u22cd",thickapprox:["\u2248",{variantForm:!0}],subseteqq:"\u2ac5",supseteqq:"\u2ac6",Subset:"\u22d0",Supset:"\u22d1",sqsubset:"\u228f",sqsupset:"\u2290",preccurlyeq:"\u227c",succcurlyeq:"\u227d",curlyeqprec:"\u22de",curlyeqsucc:"\u22df",precsim:"\u227e",succsim:"\u227f",precapprox:"\u2ab7",succapprox:"\u2ab8",vartriangleleft:"\u22b2",lhd:"\u22b2",vartriangleright:"\u22b3",rhd:"\u22b3",trianglelefteq:"\u22b4",unlhd:"\u22b4",trianglerighteq:"\u22b5",unrhd:"\u22b5",vDash:["\u22a8",{variantForm:!0}],Vdash:"\u22a9",Vvdash:"\u22aa",smallsmile:["\u2323",{variantForm:!0}],shortmid:["\u2223",{variantForm:!0}],smallfrown:["\u2322",{variantForm:!0}],shortparallel:["\u2225",{variantForm:!0}],bumpeq:"\u224f",between:"\u226c",Bumpeq:"\u224e",pitchfork:"\u22d4",varpropto:["\u221d",{variantForm:!0}],backepsilon:"\u220d",blacktriangleleft:"\u25c2",blacktriangleright:"\u25b8",therefore:"\u2234",because:"\u2235",eqsim:"\u2242",vartriangle:["\u25b3",{variantForm:!0}],Join:"\u22c8",nless:"\u226e",ngtr:"\u226f",nleq:"\u2270",ngeq:"\u2271",nleqslant:["\u2a87",{variantForm:!0}],ngeqslant:["\u2a88",{variantForm:!0}],nleqq:["\u2270",{variantForm:!0}],ngeqq:["\u2271",{variantForm:!0}],lneq:"\u2a87",gneq:"\u2a88",lneqq:"\u2268",gneqq:"\u2269",lvertneqq:["\u2268",{variantForm:!0}],gvertneqq:["\u2269",{variantForm:!0}],lnsim:"\u22e6",gnsim:"\u22e7",lnapprox:"\u2a89",gnapprox:"\u2a8a",nprec:"\u2280",nsucc:"\u2281",npreceq:["\u22e0",{variantForm:!0}],nsucceq:["\u22e1",{variantForm:!0}],precneqq:"\u2ab5",succneqq:"\u2ab6",precnsim:"\u22e8",succnsim:"\u22e9",precnapprox:"\u2ab9",succnapprox:"\u2aba",nsim:"\u2241",ncong:"\u2247",nshortmid:["\u2224",{variantForm:!0}],nshortparallel:["\u2226",{variantForm:!0}],nmid:"\u2224",nparallel:"\u2226",nvdash:"\u22ac",nvDash:"\u22ad",nVdash:"\u22ae",nVDash:"\u22af",ntriangleleft:"\u22ea",ntriangleright:"\u22eb",ntrianglelefteq:"\u22ec",ntrianglerighteq:"\u22ed",nsubseteq:"\u2288",nsupseteq:"\u2289",nsubseteqq:["\u2288",{variantForm:!0}],nsupseteqq:["\u2289",{variantForm:!0}],subsetneq:"\u228a",supsetneq:"\u228b",varsubsetneq:["\u228a",{variantForm:!0}],varsupsetneq:["\u228b",{variantForm:!0}],subsetneqq:"\u2acb",supsetneqq:"\u2acc",varsubsetneqq:["\u2acb",{variantForm:!0}],varsupsetneqq:["\u2acc",{variantForm:!0}],leftleftarrows:"\u21c7",rightrightarrows:"\u21c9",leftrightarrows:"\u21c6",rightleftarrows:"\u21c4",Lleftarrow:"\u21da",Rrightarrow:"\u21db",twoheadleftarrow:"\u219e",twoheadrightarrow:"\u21a0",leftarrowtail:"\u21a2",rightarrowtail:"\u21a3",looparrowleft:"\u21ab",looparrowright:"\u21ac",leftrightharpoons:"\u21cb",rightleftharpoons:["\u21cc",{variantForm:!0}],curvearrowleft:"\u21b6",curvearrowright:"\u21b7",circlearrowleft:"\u21ba",circlearrowright:"\u21bb",Lsh:"\u21b0",Rsh:"\u21b1",upuparrows:"\u21c8",downdownarrows:"\u21ca",upharpoonleft:"\u21bf",upharpoonright:"\u21be",downharpoonleft:"\u21c3",restriction:"\u21be",multimap:"\u22b8",downharpoonright:"\u21c2",leftrightsquigarrow:"\u21ad",rightsquigarrow:"\u21dd",leadsto:"\u21dd",dashrightarrow:"\u21e2",dashleftarrow:"\u21e0",nleftarrow:"\u219a",nrightarrow:"\u219b",nLeftarrow:"\u21cd",nRightarrow:"\u21cf",nleftrightarrow:"\u21ae",nLeftrightarrow:"\u21ce"}),new l.DelimiterMap("AMSsymbols-delimiter",u.default.delimiter,{"\\ulcorner":"\u231c","\\urcorner":"\u231d","\\llcorner":"\u231e","\\lrcorner":"\u231f"}),new l.CommandMap("AMSsymbols-macros",{implies:["Macro","\\;\\Longrightarrow\\;"],impliedby:["Macro","\\;\\Longleftarrow\\;"]},s.AmsMethods)},4387:function(t,e,r){var n=this&&this.__assign||function(){return n=Object.assign||function(t){for(var e,r=1,n=arguments.length;r<n;r++)for(var o in e=arguments[r])Object.prototype.hasOwnProperty.call(e,o)&&(t[o]=e[o]);return t},n.apply(this,arguments)},o=this&&this.__read||function(t,e){var r="function"==typeof Symbol&&t[Symbol.iterator];if(!r)return t;var n,o,i=r.call(t),a=[];try{for(;(void 0===e||e-- >0)&&!(n=i.next()).done;)a.push(n.value)}catch(t){o={error:t}}finally{try{n&&!n.done&&(r=i.return)&&r.call(i)}finally{if(o)throw o.error}}return a},i=this&&this.__importDefault||function(t){return t&&t.__esModule?t:{default:t}};Object.defineProperty(e,"__esModule",{value:!0}),e.NEW_OPS=e.AmsMethods=void 0;var a=i(r(1130)),s=i(r(5450)),l=i(r(1256)),c=r(8317),u=i(r(8417)),p=i(r(3971)),h=r(8803),d=i(r(7693)),f=r(9007);function m(t){if(!t||t.isInferred&&0===t.childNodes.length)return[null,null];if(t.isKind("msubsup")&&y(t))return[t,null];var e=l.default.getChildAt(t,0);return t.isInferred&&e&&y(e)?(t.childNodes.splice(0,1),[e,t]):[null,t]}function y(t){var e=t.childNodes[0];return e&&e.isKind("mi")&&""===e.getText()}e.AmsMethods={},e.AmsMethods.AmsEqnArray=function(t,e,r,n,o,i,s){var l=t.GetBrackets("\\begin{"+e.getName()+"}"),c=d.default.EqnArray(t,e,r,n,o,i,s);return a.default.setArrayAlign(c,l)},e.AmsMethods.AlignAt=function(t,r,n,o){var i,s,l=r.getName(),c="",u=[];if(o||(s=t.GetBrackets("\\begin{"+l+"}")),(i=t.GetArgument("\\begin{"+l+"}")).match(/[^0-9]/))throw new p.default("PositiveIntegerArg","Argument to %1 must me a positive integer","\\begin{"+l+"}");for(var h=parseInt(i,10);h>0;)c+="rl",u.push("0em 0em"),h--;var d=u.join(" ");if(o)return e.AmsMethods.EqnArray(t,r,n,o,c,d);var f=e.AmsMethods.EqnArray(t,r,n,o,c,d);return a.default.setArrayAlign(f,s)},e.AmsMethods.Multline=function(t,e,r){t.Push(e),a.default.checkEqnEnv(t);var n=t.itemFactory.create("multline",r,t.stack);return n.arraydef={displaystyle:!0,rowspacing:".5em",columnspacing:"100%",width:t.options.ams.multlineWidth,side:t.options.tagSide,minlabelspacing:t.options.tagIndent,framespacing:t.options.ams.multlineIndent+" 0",frame:"","data-width-includes-label":!0},n},e.AmsMethods.XalignAt=function(t,r,n,o){var i=t.GetArgument("\\begin{"+r.getName()+"}");if(i.match(/[^0-9]/))throw new p.default("PositiveIntegerArg","Argument to %1 must me a positive integer","\\begin{"+r.getName()+"}");var a=o?"crl":"rlc",s=o?"fit auto auto":"auto auto fit",l=e.AmsMethods.FlalignArray(t,r,n,o,!1,a,s,!0);return l.setProperty("xalignat",2*parseInt(i)),l},e.AmsMethods.FlalignArray=function(t,e,r,n,o,i,s,l){void 0===l&&(l=!1),t.Push(e),a.default.checkEqnEnv(t),i=i.split("").join(" ").replace(/r/g,"right").replace(/l/g,"left").replace(/c/g,"center");var c=t.itemFactory.create("flalign",e.getName(),r,n,o,t.stack);return c.arraydef={width:"100%",displaystyle:!0,columnalign:i,columnspacing:"0em",columnwidth:s,rowspacing:"3pt",side:t.options.tagSide,minlabelspacing:l?"0":t.options.tagIndent,"data-width-includes-label":!0},c.setProperty("zeroWidthLabel",l),c},e.NEW_OPS="ams-declare-ops",e.AmsMethods.HandleDeclareOp=function(t,r){var n=t.GetStar()?"*":"",o=a.default.trimSpaces(t.GetArgument(r));"\\"===o.charAt(0)&&(o=o.substr(1));var i=t.GetArgument(r);t.configuration.handlers.retrieve(e.NEW_OPS).add(o,new h.Macro(o,e.AmsMethods.Macro,["\\operatorname".concat(n,"{").concat(i,"}")]))},e.AmsMethods.HandleOperatorName=function(t,e){var r=t.GetStar(),o=a.default.trimSpaces(t.GetArgument(e)),i=new u.default(o,n(n({},t.stack.env),{font:c.TexConstant.Variant.NORMAL,multiLetterIdentifiers:/^[-*a-z]+/i,operatorLetters:!0}),t.configuration).mml();if(i.isKind("mi")||(i=t.create("node","TeXAtom",[i])),l.default.setProperties(i,{movesupsub:r,movablelimits:!0,texClass:f.TEXCLASS.OP}),!r){var s=t.GetNext(),p=t.i;"\\"===s&&++t.i&&"limits"!==t.GetCS()&&(t.i=p)}t.Push(i)},e.AmsMethods.SideSet=function(t,e){var r=o(m(t.ParseArg(e)),2),n=r[0],i=r[1],s=o(m(t.ParseArg(e)),2),c=s[0],u=s[1],p=t.ParseArg(e),h=p;n&&(i?n.replaceChild(t.create("node","mphantom",[t.create("node","mpadded",[a.default.copyNode(p,t)],{width:0})]),l.default.getChildAt(n,0)):(h=t.create("node","mmultiscripts",[p]),c&&l.default.appendChildren(h,[l.default.getChildAt(c,1)||t.create("node","none"),l.default.getChildAt(c,2)||t.create("node","none")]),l.default.setProperty(h,"scriptalign","left"),l.default.appendChildren(h,[t.create("node","mprescripts"),l.default.getChildAt(n,1)||t.create("node","none"),l.default.getChildAt(n,2)||t.create("node","none")]))),c&&h===p&&(c.replaceChild(p,l.default.getChildAt(c,0)),h=c);var d=t.create("node","TeXAtom",[],{texClass:f.TEXCLASS.OP,movesupsub:!0,movablelimits:!0});i&&(n&&d.appendChild(n),d.appendChild(i)),d.appendChild(h),u&&d.appendChild(u),t.Push(d)},e.AmsMethods.operatorLetter=function(t,e){return!!t.stack.env.operatorLetters&&s.default.variable(t,e)},e.AmsMethods.MultiIntegral=function(t,e,r){var n=t.GetNext();if("\\"===n){var o=t.i;n=t.GetArgument(e),t.i=o,"\\limits"===n&&(r="\\idotsint"===e?"\\!\\!\\mathop{\\,\\,"+r+"}":"\\!\\!\\!\\mathop{\\,\\,\\,"+r+"}")}t.string=r+" "+t.string.slice(t.i),t.i=0},e.AmsMethods.xArrow=function(t,e,r,n,o){var i={width:"+"+a.default.Em((n+o)/18),lspace:a.default.Em(n/18)},s=t.GetBrackets(e),c=t.ParseArg(e),p=t.create("node","mspace",[],{depth:".25em"}),h=t.create("token","mo",{stretchy:!0,texClass:f.TEXCLASS.REL},String.fromCodePoint(r));h=t.create("node","mstyle",[h],{scriptlevel:0});var d=t.create("node","munderover",[h]),m=t.create("node","mpadded",[c,p],i);if(l.default.setAttribute(m,"voffset","-.2em"),l.default.setAttribute(m,"height","-.2em"),l.default.setChild(d,d.over,m),s){var y=new u.default(s,t.stack.env,t.configuration).mml(),g=t.create("node","mspace",[],{height:".75em"});m=t.create("node","mpadded",[y,g],i),l.default.setAttribute(m,"voffset",".15em"),l.default.setAttribute(m,"depth","-.15em"),l.default.setChild(d,d.under,m)}l.default.setProperty(d,"subsupOK",!0),t.Push(d)},e.AmsMethods.HandleShove=function(t,e,r){var n=t.stack.Top();if("multline"!==n.kind)throw new p.default("CommandOnlyAllowedInEnv","%1 only allowed in %2 environment",t.currentCS,"multline");if(n.Size())throw new p.default("CommandAtTheBeginingOfLine","%1 must come at the beginning of the line",t.currentCS);n.setProperty("shove",r)},e.AmsMethods.CFrac=function(t,e){var r=a.default.trimSpaces(t.GetBrackets(e,"")),n=t.GetArgument(e),o=t.GetArgument(e),i={l:c.TexConstant.Align.LEFT,r:c.TexConstant.Align.RIGHT,"":""},s=new u.default("\\strut\\textstyle{"+n+"}",t.stack.env,t.configuration).mml(),h=new u.default("\\strut\\textstyle{"+o+"}",t.stack.env,t.configuration).mml(),d=t.create("node","mfrac",[s,h]);if(null==(r=i[r]))throw new p.default("IllegalAlign","Illegal alignment specified in %1",t.currentCS);r&&l.default.setProperties(d,{numalign:r,denomalign:r}),t.Push(d)},e.AmsMethods.Genfrac=function(t,e,r,n,o,i){null==r&&(r=t.GetDelimiterArg(e)),null==n&&(n=t.GetDelimiterArg(e)),null==o&&(o=t.GetArgument(e)),null==i&&(i=a.default.trimSpaces(t.GetArgument(e)));var s=t.ParseArg(e),c=t.ParseArg(e),u=t.create("node","mfrac",[s,c]);if(""!==o&&l.default.setAttribute(u,"linethickness",o),(r||n)&&(l.default.setProperty(u,"withDelims",!0),u=a.default.fixedFence(t.configuration,r,u,n)),""!==i){var h=parseInt(i,10),d=["D","T","S","SS"][h];if(null==d)throw new p.default("BadMathStyleFor","Bad math style for %1",t.currentCS);u=t.create("node","mstyle",[u]),"D"===d?l.default.setProperties(u,{displaystyle:!0,scriptlevel:0}):l.default.setProperties(u,{displaystyle:!1,scriptlevel:h-1})}t.Push(u)},e.AmsMethods.HandleTag=function(t,e){if(!t.tags.currentTag.taggable&&t.tags.env)throw new p.default("CommandNotAllowedInEnv","%1 not allowed in %2 environment",t.currentCS,t.tags.env);if(t.tags.currentTag.tag)throw new p.default("MultipleCommand","Multiple %1",t.currentCS);var r=t.GetStar(),n=a.default.trimSpaces(t.GetArgument(e));t.tags.tag(n,r)},e.AmsMethods.HandleNoTag=d.default.HandleNoTag,e.AmsMethods.HandleRef=d.default.HandleRef,e.AmsMethods.Macro=d.default.Macro,e.AmsMethods.Accent=d.default.Accent,e.AmsMethods.Tilde=d.default.Tilde,e.AmsMethods.Array=d.default.Array,e.AmsMethods.Spacer=d.default.Spacer,e.AmsMethods.NamedOp=d.default.NamedOp,e.AmsMethods.EqnArray=d.default.EqnArray,e.AmsMethods.Equation=d.default.Equation},583:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.AmsCdConfiguration=void 0;var n=r(9899);r(8289),e.AmsCdConfiguration=n.Configuration.create("amscd",{handler:{character:["amscd_special"],macro:["amscd_macros"],environment:["amscd_environment"]},options:{amscd:{colspace:"5pt",rowspace:"5pt",harrowsize:"2.75em",varrowsize:"1.75em",hideHorizontalLabels:!1}}})},8289:function(t,e,r){var n=this&&this.__createBinding||(Object.create?function(t,e,r,n){void 0===n&&(n=r);var 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e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"style"},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(e){if(!e.isClose)return t.prototype.checkItem.call(this,e);var r=this.create("node","mstyle",this.nodes,this.getProperty("styles"));return[[this.factory.create("mml",r),e],!0]},e}(f.BaseItem);e.StyleItem=C;var w=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"position"},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(e){if(e.isClose)throw new p.default("MissingBoxFor","Missing box for %1",this.getName());if(e.isFinal){var r=e.toMml();switch(this.getProperty("move")){case"vertical":return r=this.create("node","mpadded",[r],{height:this.getProperty("dh"),depth:this.getProperty("dd"),voffset:this.getProperty("dh")}),[[this.factory.create("mml",r)],!0];case"horizontal":return[[this.factory.create("mml",this.getProperty("left")),e,this.factory.create("mml",this.getProperty("right"))],!0]}}return t.prototype.checkItem.call(this,e)},e}(f.BaseItem);e.PositionItem=w;var T=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"cell"},enumerable:!1,configurable:!0}),Object.defineProperty(e.prototype,"isClose",{get:function(){return!0},enumerable:!1,configurable:!0}),e}(f.BaseItem);e.CellItem=T;var N=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),Object.defineProperty(e.prototype,"isFinal",{get:function(){return!0},enumerable:!1,configurable:!0}),Object.defineProperty(e.prototype,"kind",{get:function(){return"mml"},enumerable:!1,configurable:!0}),e}(f.BaseItem);e.MmlItem=N;var L=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"fn"},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(e){var r=this.First;if(r){if(e.isOpen)return f.BaseItem.success;if(!e.isKind("fn")){var n=e.First;if(!e.isKind("mml")||!n)return[[r,e],!0];if(d.default.isType(n,"mstyle")&&n.childNodes.length&&d.default.isType(n.childNodes[0].childNodes[0],"mspace")||d.default.isType(n,"mspace"))return[[r,e],!0];d.default.isEmbellished(n)&&(n=d.default.getCoreMO(n));var o=d.default.getForm(n);if(null!=o&&[0,0,1,1,0,1,1,0,0,0][o[2]])return[[r,e],!0]}var i=this.create("token","mo",{texClass:u.TEXCLASS.NONE},c.entities.ApplyFunction);return[[r,i,e],!0]}return t.prototype.checkItem.apply(this,arguments)},e}(f.BaseItem);e.FnItem=L;var P=function(t){function e(){var e=null!==t&&t.apply(this,arguments)||this;return e.remap=l.MapHandler.getMap("not_remap"),e}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"not"},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(t){var e,r,n;if(t.isKind("open")||t.isKind("left"))return f.BaseItem.success;if(t.isKind("mml")&&(d.default.isType(t.First,"mo")||d.default.isType(t.First,"mi")||d.default.isType(t.First,"mtext"))&&(e=t.First,1===(r=d.default.getText(e)).length&&!d.default.getProperty(e,"movesupsub")&&1===d.default.getChildren(e).length))return this.remap.contains(r)?(n=this.create("text",this.remap.lookup(r).char),d.default.setChild(e,0,n)):(n=this.create("text","\u0338"),d.default.appendChildren(e,[n])),[[t],!0];n=this.create("text","\u29f8");var o=this.create("node","mtext",[],{},n),i=this.create("node","mpadded",[o],{width:0});return[[e=this.create("node","TeXAtom",[i],{texClass:u.TEXCLASS.REL}),t],!0]},e}(f.BaseItem);e.NotItem=P;var I=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"nonscript"},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(t){if(t.isKind("mml")&&1===t.Size()){var e=t.First;if(e.isKind("mstyle")&&e.notParent&&(e=d.default.getChildren(d.default.getChildren(e)[0])[0]),e.isKind("mspace")){if(e!==t.First){var r=this.create("node","mrow",[t.Pop()]);t.Push(r)}this.factory.configuration.addNode("nonscript",t.First)}}return[[t],!0]},e}(f.BaseItem);e.NonscriptItem=I;var k=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"dots"},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(t){if(t.isKind("open")||t.isKind("left"))return f.BaseItem.success;var e=this.getProperty("ldots"),r=t.First;if(t.isKind("mml")&&d.default.isEmbellished(r)){var n=d.default.getTexClass(d.default.getCoreMO(r));n!==u.TEXCLASS.BIN&&n!==u.TEXCLASS.REL||(e=this.getProperty("cdots"))}return[[e,t],!0]},e}(f.BaseItem);e.DotsItem=k;var R=function(t){function e(){var e=null!==t&&t.apply(this,arguments)||this;return e.table=[],e.row=[],e.frame=[],e.hfill=[],e.arraydef={},e.dashed=!1,e}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"array"},enumerable:!1,configurable:!0}),Object.defineProperty(e.prototype,"isOpen",{get:function(){return!0},enumerable:!1,configurable:!0}),Object.defineProperty(e.prototype,"copyEnv",{get:function(){return!1},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(e){if(e.isClose&&!e.isKind("over")){if(e.getProperty("isEntry"))return this.EndEntry(),this.clearEnv(),f.BaseItem.fail;if(e.getProperty("isCR"))return this.EndEntry(),this.EndRow(),this.clearEnv(),f.BaseItem.fail;this.EndTable(),this.clearEnv();var r=this.factory.create("mml",this.createMml());if(this.getProperty("requireClose")){if(e.isKind("close"))return[[r],!0];throw new p.default("MissingCloseBrace","Missing close brace")}return[[r,e],!0]}return t.prototype.checkItem.call(this,e)},e.prototype.createMml=function(){var t=this.arraydef.scriptlevel;delete this.arraydef.scriptlevel;var e=this.create("node","mtable",this.table,this.arraydef);return t&&e.setProperty("scriptlevel",t),4===this.frame.length?d.default.setAttribute(e,"frame",this.dashed?"dashed":"solid"):this.frame.length&&(this.arraydef.rowlines&&(this.arraydef.rowlines=this.arraydef.rowlines.replace(/none( none)+$/,"none")),d.default.setAttribute(e,"frame",""),e=this.create("node","menclose",[e],{notation:this.frame.join(" ")}),"none"===(this.arraydef.columnlines||"none")&&"none"===(this.arraydef.rowlines||"none")||d.default.setAttribute(e,"data-padding",0)),(this.getProperty("open")||this.getProperty("close"))&&(e=h.default.fenced(this.factory.configuration,this.getProperty("open"),e,this.getProperty("close"))),e},e.prototype.EndEntry=function(){var t=this.create("node","mtd",this.nodes);this.hfill.length&&(0===this.hfill[0]&&d.default.setAttribute(t,"columnalign","right"),this.hfill[this.hfill.length-1]===this.Size()&&d.default.setAttribute(t,"columnalign",d.default.getAttribute(t,"columnalign")?"center":"left")),this.row.push(t),this.Clear(),this.hfill=[]},e.prototype.EndRow=function(){var t;this.getProperty("isNumbered")&&3===this.row.length?(this.row.unshift(this.row.pop()),t=this.create("node","mlabeledtr",this.row)):t=this.create("node","mtr",this.row),this.table.push(t),this.row=[]},e.prototype.EndTable=function(){(this.Size()||this.row.length)&&(this.EndEntry(),this.EndRow()),this.checkLines()},e.prototype.checkLines=function(){if(this.arraydef.rowlines){var t=this.arraydef.rowlines.split(/ /);t.length===this.table.length?(this.frame.push("bottom"),t.pop(),this.arraydef.rowlines=t.join(" ")):t.length<this.table.length-1&&(this.arraydef.rowlines+=" none")}if(this.getProperty("rowspacing")){for(var e=this.arraydef.rowspacing.split(/ /);e.length<this.table.length;)e.push(this.getProperty("rowspacing")+"em");this.arraydef.rowspacing=e.join(" ")}},e.prototype.addRowSpacing=function(t){if(this.arraydef.rowspacing){var e=this.arraydef.rowspacing.split(/ /);if(!this.getProperty("rowspacing")){var r=h.default.dimen2em(e[0]);this.setProperty("rowspacing",r)}for(var n=this.getProperty("rowspacing");e.length<this.table.length;)e.push(h.default.Em(n));e[this.table.length-1]=h.default.Em(Math.max(0,n+h.default.dimen2em(t))),this.arraydef.rowspacing=e.join(" ")}},e}(f.BaseItem);e.ArrayItem=R;var j=function(t){function e(e){for(var r=[],n=1;n<arguments.length;n++)r[n-1]=arguments[n];var o=t.call(this,e)||this;return o.maxrow=0,o.factory.configuration.tags.start(r[0],r[2],r[1]),o}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"eqnarray"},enumerable:!1,configurable:!0}),e.prototype.EndEntry=function(){this.row.length&&h.default.fixInitialMO(this.factory.configuration,this.nodes);var t=this.create("node","mtd",this.nodes);this.row.push(t),this.Clear()},e.prototype.EndRow=function(){this.row.length>this.maxrow&&(this.maxrow=this.row.length);var t="mtr",e=this.factory.configuration.tags.getTag();e&&(this.row=[e].concat(this.row),t="mlabeledtr"),this.factory.configuration.tags.clearTag();var r=this.create("node",t,this.row);this.table.push(r),this.row=[]},e.prototype.EndTable=function(){t.prototype.EndTable.call(this),this.factory.configuration.tags.end(),this.extendArray("columnalign",this.maxrow),this.extendArray("columnwidth",this.maxrow),this.extendArray("columnspacing",this.maxrow-1)},e.prototype.extendArray=function(t,e){if(this.arraydef[t]){var r=this.arraydef[t].split(/ /),n=a([],i(r),!1);if(n.length>1){for(;n.length<e;)n.push.apply(n,a([],i(r),!1));this.arraydef[t]=n.slice(0,e).join(" ")}}},e}(R);e.EqnArrayItem=j;var B=function(t){function e(e){for(var r=[],n=1;n<arguments.length;n++)r[n-1]=arguments[n];var o=t.call(this,e)||this;return o.factory.configuration.tags.start("equation",!0,r[0]),o}return o(e,t),Object.defineProperty(e.prototype,"kind",{get:function(){return"equation"},enumerable:!1,configurable:!0}),Object.defineProperty(e.prototype,"isOpen",{get:function(){return!0},enumerable:!1,configurable:!0}),e.prototype.checkItem=function(e){if(e.isKind("end")){var r=this.toMml(),n=this.factory.configuration.tags.getTag();return this.factory.configuration.tags.end(),[[n?this.factory.configuration.tags.enTag(r,n):r,e],!0]}if(e.isKind("stop"))throw new p.default("EnvMissingEnd","Missing \\end{%1}",this.getName());return t.prototype.checkItem.call(this,e)},e}(f.BaseItem);e.EquationItem=B},1267:function(t,e,r){var n=this&&this.__createBinding||(Object.create?function(t,e,r,n){void 0===n&&(n=r);var o=Object.getOwnPropertyDescriptor(e,r);o&&!("get"in o?!e.__esModule:o.writable||o.configurable)||(o={enumerable:!0,get:function(){return e[r]}}),Object.defineProperty(t,n,o)}:function(t,e,r,n){void 0===n&&(n=r),t[n]=e[r]}),o=this&&this.__setModuleDefault||(Object.create?function(t,e){Object.defineProperty(t,"default",{enumerable:!0,value:e})}:function(t,e){t.default=e}),i=this&&this.__importStar||function(t){if(t&&t.__esModule)return t;var e={};if(null!=t)for(var r in t)"default"!==r&&Object.prototype.hasOwnProperty.call(t,r)&&n(e,t,r);return o(e,t),e},a=this&&this.__importDefault||function(t){return t&&t.__esModule?t:{default:t}};Object.defineProperty(e,"__esModule",{value:!0});var s=i(r(9140)),l=r(8317),c=a(r(7693)),u=a(r(5450)),p=a(r(1130)),h=r(9007),d=r(6010);new s.RegExpMap("letter",u.default.variable,/[a-z]/i),new s.RegExpMap("digit",u.default.digit,/[0-9.,]/),new s.RegExpMap("command",u.default.controlSequence,/^\\/),new s.MacroMap("special",{"{":"Open","}":"Close","~":"Tilde","^":"Superscript",_:"Subscript"," ":"Space","\t":"Space","\r":"Space","\n":"Space","'":"Prime","%":"Comment","&":"Entry","#":"Hash","\xa0":"Space","\u2019":"Prime"},c.default),new s.CharacterMap("mathchar0mi",u.default.mathchar0mi,{alpha:"\u03b1",beta:"\u03b2",gamma:"\u03b3",delta:"\u03b4",epsilon:"\u03f5",zeta:"\u03b6",eta:"\u03b7",theta:"\u03b8",iota:"\u03b9",kappa:"\u03ba",lambda:"\u03bb",mu:"\u03bc",nu:"\u03bd",xi:"\u03be",omicron:"\u03bf",pi:"\u03c0",rho:"\u03c1",sigma:"\u03c3",tau:"\u03c4",upsilon:"\u03c5",phi:"\u03d5",chi:"\u03c7",psi:"\u03c8",omega:"\u03c9",varepsilon:"\u03b5",vartheta:"\u03d1",varpi:"\u03d6",varrho:"\u03f1",varsigma:"\u03c2",varphi:"\u03c6",S:["\xa7",{mathvariant:l.TexConstant.Variant.NORMAL}],aleph:["\u2135",{mathvariant:l.TexConstant.Variant.NORMAL}],hbar:["\u210f",{variantForm:!0}],imath:"\u0131",jmath:"\u0237",ell:"\u2113",wp:["\u2118",{mathvariant:l.TexConstant.Variant.NORMAL}],Re:["\u211c",{mathvariant:l.TexConstant.Variant.NORMAL}],Im:["\u2111",{mathvariant:l.TexConstant.Variant.NORMAL}],partial:["\u2202",{mathvariant:l.TexConstant.Variant.ITALIC}],infty:["\u221e",{mathvariant:l.TexConstant.Variant.NORMAL}],prime:["\u2032",{variantForm:!0}],emptyset:["\u2205",{mathvariant:l.TexConstant.Variant.NORMAL}],nabla:["\u2207",{mathvariant:l.TexConstant.Variant.NORMAL}],top:["\u22a4",{mathvariant:l.TexConstant.Variant.NORMAL}],bot:["\u22a5",{mathvariant:l.TexConstant.Variant.NORMAL}],angle:["\u2220",{mathvariant:l.TexConstant.Variant.NORMAL}],triangle:["\u25b3",{mathvariant:l.TexConstant.Variant.NORMAL}],backslash:["\u2216",{mathvariant:l.TexConstant.Variant.NORMAL}],forall:["\u2200",{mathvariant:l.TexConstant.Variant.NORMAL}],exists:["\u2203",{mathvariant:l.TexConstant.Variant.NORMAL}],neg:["\xac",{mathvariant:l.TexConstant.Variant.NORMAL}],lnot:["\xac",{mathvariant:l.TexConstant.Variant.NORMAL}],flat:["\u266d",{mathvariant:l.TexConstant.Variant.NORMAL}],natural:["\u266e",{mathvariant:l.TexConstant.Variant.NORMAL}],sharp:["\u266f",{mathvariant:l.TexConstant.Variant.NORMAL}],clubsuit:["\u2663",{mathvariant:l.TexConstant.Variant.NORMAL}],diamondsuit:["\u2662",{mathvariant:l.TexConstant.Variant.NORMAL}],heartsuit:["\u2661",{mathvariant:l.TexConstant.Variant.NORMAL}],spadesuit:["\u2660",{mathvariant:l.TexConstant.Variant.NORMAL}]}),new s.CharacterMap("mathchar0mo",u.default.mathchar0mo,{surd:"\u221a",coprod:["\u2210",{texClass:h.TEXCLASS.OP,movesupsub:!0}],bigvee:["\u22c1",{texClass:h.TEXCLASS.OP,movesupsub:!0}],bigwedge:["\u22c0",{texClass:h.TEXCLASS.OP,movesupsub:!0}],biguplus:["\u2a04",{texClass:h.TEXCLASS.OP,movesupsub:!0}],bigcap:["\u22c2",{texClass:h.TEXCLASS.OP,movesupsub:!0}],bigcup:["\u22c3",{texClass:h.TEXCLASS.OP,movesupsub:!0}],int:["\u222b",{texClass:h.TEXCLASS.OP}],intop:["\u222b",{texClass:h.TEXCLASS.OP,movesupsub:!0,movablelimits:!0}],iint:["\u222c",{texClass:h.TEXCLASS.OP}],iiint:["\u222d",{texClass:h.TEXCLASS.OP}],prod:["\u220f",{texClass:h.TEXCLASS.OP,movesupsub:!0}],sum:["\u2211",{texClass:h.TEXCLASS.OP,movesupsub:!0}],bigotimes:["\u2a02",{texClass:h.TEXCLASS.OP,movesupsub:!0}],bigoplus:["\u2a01",{texClass:h.TEXCLASS.OP,movesupsub:!0}],bigodot:["\u2a00",{texClass:h.TEXCLASS.OP,movesupsub:!0}],oint:["\u222e",{texClass:h.TEXCLASS.OP}],bigsqcup:["\u2a06",{texClass:h.TEXCLASS.OP,movesupsub:!0}],smallint:["\u222b",{largeop:!1}],triangleleft:"\u25c3",triangleright:"\u25b9",bigtriangleup:"\u25b3",bigtriangledown:"\u25bd",wedge:"\u2227",land:"\u2227",vee:"\u2228",lor:"\u2228",cap:"\u2229",cup:"\u222a",ddagger:"\u2021",dagger:"\u2020",sqcap:"\u2293",sqcup:"\u2294",uplus:"\u228e",amalg:"\u2a3f",diamond:"\u22c4",bullet:"\u2219",wr:"\u2240",div:"\xf7",divsymbol:"\xf7",odot:["\u2299",{largeop:!1}],oslash:["\u2298",{largeop:!1}],otimes:["\u2297",{largeop:!1}],ominus:["\u2296",{largeop:!1}],oplus:["\u2295",{largeop:!1}],mp:"\u2213",pm:"\xb1",circ:"\u2218",bigcirc:"\u25ef",setminus:"\u2216",cdot:"\u22c5",ast:"\u2217",times:"\xd7",star:"\u22c6",propto:"\u221d",sqsubseteq:"\u2291",sqsupseteq:"\u2292",parallel:"\u2225",mid:"\u2223",dashv:"\u22a3",vdash:"\u22a2",leq:"\u2264",le:"\u2264",geq:"\u2265",ge:"\u2265",lt:"<",gt:">",succ:"\u227b",prec:"\u227a",approx:"\u2248",succeq:"\u2ab0",preceq:"\u2aaf",supset:"\u2283",subset:"\u2282",supseteq:"\u2287",subseteq:"\u2286",in:"\u2208",ni:"\u220b",notin:"\u2209",owns:"\u220b",gg:"\u226b",ll:"\u226a",sim:"\u223c",simeq:"\u2243",perp:"\u22a5",equiv:"\u2261",asymp:"\u224d",smile:"\u2323",frown:"\u2322",ne:"\u2260",neq:"\u2260",cong:"\u2245",doteq:"\u2250",bowtie:"\u22c8",models:"\u22a8",notChar:"\u29f8",Leftrightarrow:"\u21d4",Leftarrow:"\u21d0",Rightarrow:"\u21d2",leftrightarrow:"\u2194",leftarrow:"\u2190",gets:"\u2190",rightarrow:"\u2192",to:["\u2192",{accent:!1}],mapsto:"\u21a6",leftharpoonup:"\u21bc",leftharpoondown:"\u21bd",rightharpoonup:"\u21c0",rightharpoondown:"\u21c1",nearrow:"\u2197",searrow:"\u2198",nwarrow:"\u2196",swarrow:"\u2199",rightleftharpoons:"\u21cc",hookrightarrow:"\u21aa",hookleftarrow:"\u21a9",longleftarrow:"\u27f5",Longleftarrow:"\u27f8",longrightarrow:"\u27f6",Longrightarrow:"\u27f9",Longleftrightarrow:"\u27fa",longleftrightarrow:"\u27f7",longmapsto:"\u27fc",ldots:"\u2026",cdots:"\u22ef",vdots:"\u22ee",ddots:"\u22f1",dotsc:"\u2026",dotsb:"\u22ef",dotsm:"\u22ef",dotsi:"\u22ef",dotso:"\u2026",ldotp:[".",{texClass:h.TEXCLASS.PUNCT}],cdotp:["\u22c5",{texClass:h.TEXCLASS.PUNCT}],colon:[":",{texClass:h.TEXCLASS.PUNCT}]}),new 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e=t.padding/2,r=t.thickness;return[2*e,e,e+r,3*e+r]},border:function(t){return[0,0,t.thickness,0]},remove:"bottom"}],h.Arrow("up"),h.Arrow("down"),h.Arrow("left"),h.Arrow("right"),h.Arrow("updown"),h.Arrow("leftright"),h.DiagonalArrow("updiagonal"),h.DiagonalArrow("northeast"),h.DiagonalArrow("southeast"),h.DiagonalArrow("northwest"),h.DiagonalArrow("southwest"),h.DiagonalArrow("northeastsouthwest"),h.DiagonalArrow("northwestsoutheast"),["longdiv",{renderer:function(t,e){var r=t.adaptor;r.setStyle(e,"border-top",t.em(t.thickness)+" solid");var n=r.append(t.chtml,t.html("dbox")),o=t.thickness,i=t.padding;o!==h.THICKNESS&&r.setStyle(n,"border-width",t.em(o)),i!==h.PADDING&&(r.setStyle(n,"left",t.em(-1.5*i)),r.setStyle(n,"width",t.em(3*i)),r.setStyle(n,"clip-path","inset(0 0 0 "+t.em(1.5*i)+")"))},bbox:function(t){var e=t.padding,r=t.thickness;return[e+r,e,e,2*e+r/2]}}],["radical",{renderer:function(t,e){t.msqrt.toCHTML(e);var 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mjx-cell":{"text-align":"right"},'[script-align="left"] > mjx-row > mjx-cell':{"text-align":"left"},'[script-align="center"] > mjx-row > mjx-cell':{"text-align":"center"},'[script-align="right"] > mjx-row > mjx-cell':{"text-align":"right"}},e}((0,s.CommonMmultiscriptsMixin)(a.CHTMLmsubsup));e.CHTMLmmultiscripts=u},6460:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},n(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.CHTMLmn=void 0;var i=r(5355),a=r(5023),s=r(3050),l=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return 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l=i(this.tableRows),c=l.next();!c.done;c=l.next()){var u=c.value,p=0;try{for(var h=(r=void 0,i(u.tableCells)),d=h.next();!d.done;d=h.next()){var f=d.value,m=a[p++],y=a[p],g=f?f.chtml:this.adaptor.childNodes(u.chtml)[p];(p>1&&"0.4em"!==m||s&&1===p)&&this.adaptor.setStyle(g,"paddingLeft",m),(p<this.numCols&&"0.4em"!==y||s&&p===this.numCols)&&this.adaptor.setStyle(g,"paddingRight",y)}}catch(t){r={error:t}}finally{try{d&&!d.done&&(n=h.return)&&n.call(h)}finally{if(r)throw r.error}}}}catch(e){t={error:e}}finally{try{c&&!c.done&&(e=l.return)&&e.call(l)}finally{if(t)throw t.error}}},e.prototype.handleColumnLines=function(){var t,e,r,n;if("none"!==this.node.attributes.get("columnlines")){var o=this.getColumnAttributes("columnlines");try{for(var a=i(this.childNodes),s=a.next();!s.done;s=a.next()){var l=s.value,c=0;try{for(var u=(r=void 0,i(this.adaptor.childNodes(l.chtml).slice(1))),p=u.next();!p.done;p=u.next()){var h=p.value,d=o[c++];"none"!==d&&this.adaptor.setStyle(h,"borderLeft",".07em "+d)}}catch(t){r={error:t}}finally{try{p&&!p.done&&(n=u.return)&&n.call(u)}finally{if(r)throw r.error}}}}catch(e){t={error:e}}finally{try{s&&!s.done&&(e=a.return)&&e.call(a)}finally{if(t)throw t.error}}}},e.prototype.handleColumnWidths=function(){var t,e,r,n;try{for(var o=i(this.childNodes),a=o.next();!a.done;a=o.next()){var s=a.value,l=0;try{for(var c=(r=void 0,i(this.adaptor.childNodes(s.chtml))),u=c.next();!u.done;u=c.next()){var p=u.value,h=this.cWidths[l++];if(null!==h){var d="number"==typeof h?this.em(h):h;this.adaptor.setStyle(p,"width",d),this.adaptor.setStyle(p,"maxWidth",d),this.adaptor.setStyle(p,"minWidth",d)}}}catch(t){r={error:t}}finally{try{u&&!u.done&&(n=c.return)&&n.call(c)}finally{if(r)throw r.error}}}}catch(e){t={error:e}}finally{try{a&&!a.done&&(e=o.return)&&e.call(o)}finally{if(t)throw t.error}}},e.prototype.handleRowSpacing=function(){var t,e,r,n,o=this.childNodes[0]?1/this.childNodes[0].getBBox().rscale:1,a=this.getEmHalfSpacing(this.fSpace[1],this.rSpace,o),s=this.frame,l=0;try{for(var c=i(this.childNodes),u=c.next();!u.done;u=c.next()){var p=u.value,h=a[l++],d=a[l];try{for(var f=(r=void 0,i(p.childNodes)),m=f.next();!m.done;m=f.next()){var y=m.value;(l>1&&"0.215em"!==h||s&&1===l)&&this.adaptor.setStyle(y.chtml,"paddingTop",h),(l<this.numRows&&"0.215em"!==d||s&&l===this.numRows)&&this.adaptor.setStyle(y.chtml,"paddingBottom",d)}}catch(t){r={error:t}}finally{try{m&&!m.done&&(n=f.return)&&n.call(f)}finally{if(r)throw r.error}}}}catch(e){t={error:e}}finally{try{u&&!u.done&&(e=c.return)&&e.call(c)}finally{if(t)throw t.error}}},e.prototype.handleRowLines=function(){var t,e,r,n;if("none"!==this.node.attributes.get("rowlines")){var o=this.getRowAttributes("rowlines"),a=0;try{for(var s=i(this.childNodes.slice(1)),l=s.next();!l.done;l=s.next()){var c=l.value,u=o[a++];if("none"!==u)try{for(var p=(r=void 0,i(this.adaptor.childNodes(c.chtml))),h=p.next();!h.done;h=p.next()){var d=h.value;this.adaptor.setStyle(d,"borderTop",".07em "+u)}}catch(t){r={error:t}}finally{try{h&&!h.done&&(n=p.return)&&n.call(p)}finally{if(r)throw r.error}}}}catch(e){t={error:e}}finally{try{l&&!l.done&&(e=s.return)&&e.call(s)}finally{if(t)throw t.error}}}},e.prototype.handleRowHeights=function(){this.node.attributes.get("equalrows")&&this.handleEqualRows()},e.prototype.handleEqualRows=function(){for(var t=this.getRowHalfSpacing(),e=this.getTableData(),r=e.H,n=e.D,o=e.NH,i=e.ND,a=this.getEqualRowHeight(),s=0;s<this.numRows;s++){var l=this.childNodes[s];this.setRowHeight(l,a+t[s]+t[s+1]+this.rLines[s]),a!==o[s]+i[s]&&this.setRowBaseline(l,a,(a-r[s]+n[s])/2)}},e.prototype.setRowHeight=function(t,e){this.adaptor.setStyle(t.chtml,"height",this.em(e))},e.prototype.setRowBaseline=function(t,e,r){var n,o,a=t.node.attributes.get("rowalign");try{for(var s=i(t.childNodes),l=s.next();!l.done;l=s.next()){var c=l.value;if(this.setCellBaseline(c,a,e,r))break}}catch(t){n={error:t}}finally{try{l&&!l.done&&(o=s.return)&&o.call(s)}finally{if(n)throw n.error}}},e.prototype.setCellBaseline=function(t,e,r,n){var o=t.node.attributes.get("rowalign");if("baseline"===o||"axis"===o){var i=this.adaptor,a=i.lastChild(t.chtml);i.setStyle(a,"height",this.em(r)),i.setStyle(a,"verticalAlign",this.em(-n));var s=t.parent;if(!(s.node.isKind("mlabeledtr")&&t===s.childNodes[0]||"baseline"!==e&&"axis"!==e))return!0}return!1},e.prototype.handleFrame=function(){this.frame&&this.fLine&&this.adaptor.setStyle(this.itable,"border",".07em "+this.node.attributes.get("frame"))},e.prototype.handleWidth=function(){var t=this.adaptor,e=this.getBBox(),r=e.w,n=e.L,o=e.R;t.setStyle(this.chtml,"minWidth",this.em(n+r+o));var i=this.node.attributes.get("width");if((0,u.isPercent)(i))t.setStyle(this.chtml,"width",""),t.setAttribute(this.chtml,"width","full");else if(!this.hasLabels){if("auto"===i)return;i=this.em(this.length2em(i)+2*this.fLine)}var a=t.firstChild(this.chtml);if(t.setStyle(a,"width",i),t.setStyle(a,"minWidth",this.em(r)),n||o){t.setStyle(this.chtml,"margin","");var s=this.node.attributes.get("data-width-includes-label")?"padding":"margin";n===o?t.setStyle(a,s,"0 "+this.em(o)):t.setStyle(a,s,"0 "+this.em(o)+" 0 "+this.em(n))}t.setAttribute(this.itable,"width","full")},e.prototype.handleAlign=function(){var t=a(this.getAlignmentRow(),2),e=t[0],r=t[1];if(null===r)"axis"!==e&&this.adaptor.setAttribute(this.chtml,"align",e);else{var n=this.getVerticalPosition(r,e);this.adaptor.setAttribute(this.chtml,"align","top"),this.adaptor.setStyle(this.chtml,"verticalAlign",this.em(n))}},e.prototype.handleJustify=function(){var t=this.getAlignShift()[0];"center"!==t&&this.adaptor.setAttribute(this.chtml,"justify",t)},e.prototype.handleLabels=function(){if(this.hasLabels){var t=this.labels,e=this.node.attributes,r=this.adaptor,n=e.get("side");r.setAttribute(this.chtml,"side",n),r.setAttribute(t,"align",n),r.setStyle(t,n,"0");var o=a(this.addLabelPadding(n),2),i=o[0],s=o[1];if(s){var l=r.firstChild(this.chtml);this.setIndent(l,i,s)}this.updateRowHeights(),this.addLabelSpacing()}},e.prototype.addLabelPadding=function(t){var e=a(this.getPadAlignShift(t),3),r=e[1],n=e[2],o={};if("right"===t&&!this.node.attributes.get("data-width-includes-label")){var i=this.node.attributes.get("width"),s=this.getBBox(),l=s.w,c=s.L,p=s.R;o.style={width:(0,u.isPercent)(i)?"calc("+i+" + "+this.em(c+p)+")":this.em(c+l+p)}}return this.adaptor.append(this.chtml,this.html("mjx-labels",o,[this.labels])),[r,n]},e.prototype.updateRowHeights=function(){for(var t=this.getTableData(),e=t.H,r=t.D,n=t.NH,o=t.ND,i=this.getRowHalfSpacing(),a=0;a<this.numRows;a++){var s=this.childNodes[a];this.setRowHeight(s,e[a]+r[a]+i[a]+i[a+1]+this.rLines[a]),e[a]!==n[a]||r[a]!==o[a]?this.setRowBaseline(s,e[a]+r[a],r[a]):s.node.isKind("mlabeledtr")&&this.setCellBaseline(s.childNodes[0],"",e[a]+r[a],r[a])}},e.prototype.addLabelSpacing=function(){for(var t=this.adaptor,e=this.node.attributes.get("equalrows"),r=this.getTableData(),n=r.H,o=r.D,i=e?this.getEqualRowHeight():0,a=this.getRowHalfSpacing(),s=this.fLine,l=t.firstChild(this.labels),c=0;c<this.numRows;c++){this.childNodes[c].node.isKind("mlabeledtr")?(s&&t.insert(this.html("mjx-mtr",{style:{height:this.em(s)}}),l),t.setStyle(l,"height",this.em((e?i:n[c]+o[c])+a[c]+a[c+1])),l=t.next(l),s=this.rLines[c]):s+=a[c]+(e?i:n[c]+o[c])+a[c+1]+this.rLines[c]}},e.kind=c.MmlMtable.prototype.kind,e.styles={"mjx-mtable":{"vertical-align":".25em","text-align":"center",position:"relative","box-sizing":"border-box","border-spacing":0,"border-collapse":"collapse"},'mjx-mstyle[size="s"] mjx-mtable':{"vertical-align":".354em"},"mjx-labels":{position:"absolute",left:0,top:0},"mjx-table":{display:"inline-block","vertical-align":"-.5ex","box-sizing":"border-box"},"mjx-table > mjx-itable":{"vertical-align":"middle","text-align":"left","box-sizing":"border-box"},"mjx-labels > mjx-itable":{position:"absolute",top:0},'mjx-mtable[justify="left"]':{"text-align":"left"},'mjx-mtable[justify="right"]':{"text-align":"right"},'mjx-mtable[justify="left"][side="left"]':{"padding-right":"0 ! important"},'mjx-mtable[justify="left"][side="right"]':{"padding-left":"0 ! important"},'mjx-mtable[justify="right"][side="left"]':{"padding-right":"0 ! important"},'mjx-mtable[justify="right"][side="right"]':{"padding-left":"0 ! important"},"mjx-mtable[align]":{"vertical-align":"baseline"},'mjx-mtable[align="top"] > mjx-table':{"vertical-align":"top"},'mjx-mtable[align="bottom"] > mjx-table':{"vertical-align":"bottom"},'mjx-mtable[side="right"] mjx-labels':{"min-width":"100%"}},e}((0,l.CommonMtableMixin)(s.CHTMLWrapper));e.CHTMLmtable=p},7056:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},n(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.CHTMLmtd=void 0;var i=r(5355),a=r(5164),s=r(4359),l=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),e.prototype.toCHTML=function(e){t.prototype.toCHTML.call(this,e);var r=this.node.attributes.get("rowalign"),n=this.node.attributes.get("columnalign");r!==this.parent.node.attributes.get("rowalign")&&this.adaptor.setAttribute(this.chtml,"rowalign",r),"center"===n||"mlabeledtr"===this.parent.kind&&this===this.parent.childNodes[0]&&n===this.parent.parent.node.attributes.get("side")||this.adaptor.setStyle(this.chtml,"textAlign",n),this.parent.parent.node.getProperty("useHeight")&&this.adaptor.append(this.chtml,this.html("mjx-tstrut"))},e.kind=s.MmlMtd.prototype.kind,e.styles={"mjx-mtd":{display:"table-cell","text-align":"center",padding:".215em .4em"},"mjx-mtd:first-child":{"padding-left":0},"mjx-mtd:last-child":{"padding-right":0},"mjx-mtable > * > mjx-itable > *:first-child > mjx-mtd":{"padding-top":0},"mjx-mtable > * > mjx-itable > *:last-child > mjx-mtd":{"padding-bottom":0},"mjx-tstrut":{display:"inline-block",height:"1em","vertical-align":"-.25em"},'mjx-labels[align="left"] > mjx-mtr > mjx-mtd':{"text-align":"left"},'mjx-labels[align="right"] > mjx-mtr > mjx-mtd':{"text-align":"right"},"mjx-mtd[extra]":{padding:0},'mjx-mtd[rowalign="top"]':{"vertical-align":"top"},'mjx-mtd[rowalign="center"]':{"vertical-align":"middle"},'mjx-mtd[rowalign="bottom"]':{"vertical-align":"bottom"},'mjx-mtd[rowalign="baseline"]':{"vertical-align":"baseline"},'mjx-mtd[rowalign="axis"]':{"vertical-align":".25em"}},e}((0,a.CommonMtdMixin)(i.CHTMLWrapper));e.CHTMLmtd=l},1259:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},n(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.CHTMLmtext=void 0;var i=r(5355),a=r(6319),s=r(4770),l=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),e.kind=s.MmlMtext.prototype.kind,e}((0,a.CommonMtextMixin)(i.CHTMLWrapper));e.CHTMLmtext=l},3571:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},n(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.CHTMLmlabeledtr=e.CHTMLmtr=void 0;var i=r(5355),a=r(5766),s=r(5766),l=r(5022),c=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),e.prototype.toCHTML=function(e){t.prototype.toCHTML.call(this,e);var r=this.node.attributes.get("rowalign");"baseline"!==r&&this.adaptor.setAttribute(this.chtml,"rowalign",r)},e.kind=l.MmlMtr.prototype.kind,e.styles={"mjx-mtr":{display:"table-row"},'mjx-mtr[rowalign="top"] > mjx-mtd':{"vertical-align":"top"},'mjx-mtr[rowalign="center"] > mjx-mtd':{"vertical-align":"middle"},'mjx-mtr[rowalign="bottom"] > mjx-mtd':{"vertical-align":"bottom"},'mjx-mtr[rowalign="baseline"] > mjx-mtd':{"vertical-align":"baseline"},'mjx-mtr[rowalign="axis"] > mjx-mtd':{"vertical-align":".25em"}},e}((0,a.CommonMtrMixin)(i.CHTMLWrapper));e.CHTMLmtr=c;var u=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),e.prototype.toCHTML=function(e){t.prototype.toCHTML.call(this,e);var r=this.adaptor.firstChild(this.chtml);if(r){this.adaptor.remove(r);var n=this.node.attributes.get("rowalign"),o="baseline"!==n&&"axis"!==n?{rowalign:n}:{},i=this.html("mjx-mtr",o,[r]);this.adaptor.append(this.parent.labels,i)}},e.prototype.markUsed=function(){t.prototype.markUsed.call(this),this.jax.wrapperUsage.add(c.kind)},e.kind=l.MmlMlabeledtr.prototype.kind,e.styles={"mjx-mlabeledtr":{display:"table-row"},'mjx-mlabeledtr[rowalign="top"] > mjx-mtd':{"vertical-align":"top"},'mjx-mlabeledtr[rowalign="center"] > mjx-mtd':{"vertical-align":"middle"},'mjx-mlabeledtr[rowalign="bottom"] > mjx-mtd':{"vertical-align":"bottom"},'mjx-mlabeledtr[rowalign="baseline"] > mjx-mtd':{"vertical-align":"baseline"},'mjx-mlabeledtr[rowalign="axis"] > mjx-mtd':{"vertical-align":".25em"}},e}((0,s.CommonMlabeledtrMixin)(c));e.CHTMLmlabeledtr=u},6590:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},n(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.CHTMLmunderover=e.CHTMLmover=e.CHTMLmunder=void 0;var i=r(4300),a=r(1971),s=r(1971),l=r(1971),c=r(5184),u=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),e.prototype.toCHTML=function(e){if(this.hasMovableLimits())return t.prototype.toCHTML.call(this,e),void this.adaptor.setAttribute(this.chtml,"limits","false");this.chtml=this.standardCHTMLnode(e);var r=this.adaptor.append(this.adaptor.append(this.chtml,this.html("mjx-row")),this.html("mjx-base")),n=this.adaptor.append(this.adaptor.append(this.chtml,this.html("mjx-row")),this.html("mjx-under"));this.baseChild.toCHTML(r),this.scriptChild.toCHTML(n);var o=this.baseChild.getOuterBBox(),i=this.scriptChild.getOuterBBox(),a=this.getUnderKV(o,i)[0],s=this.isLineBelow?0:this.getDelta(!0);this.adaptor.setStyle(n,"paddingTop",this.em(a)),this.setDeltaW([r,n],this.getDeltaW([o,i],[0,-s])),this.adjustUnderDepth(n,i)},e.kind=c.MmlMunder.prototype.kind,e.styles={"mjx-over":{"text-align":"left"},'mjx-munder:not([limits="false"])':{display:"inline-table"},"mjx-munder > mjx-row":{"text-align":"left"},"mjx-under":{"padding-bottom":".1em"}},e}((0,a.CommonMunderMixin)(i.CHTMLmsub));e.CHTMLmunder=u;var p=function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return o(e,t),e.prototype.toCHTML=function(e){if(this.hasMovableLimits())return t.prototype.toCHTML.call(this,e),void 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e=[],r=0;r<arguments.length;r++)e[r]=arguments[r];var n=t.apply(this,i([],o(e),!1))||this;if(n.bevel=null,n.pad=n.node.getProperty("withDelims")?0:n.font.params.nulldelimiterspace,n.node.attributes.get("bevelled")){var a=n.getBevelData(n.isDisplay()).H,s=n.bevel=n.createMo("/");s.node.attributes.set("symmetric",!0),s.canStretch(1),s.getStretchedVariant([a],!0)}return n}return n(e,t),e.prototype.computeBBox=function(t,e){void 0===e&&(e=!1),t.empty();var r=this.node.attributes.getList("linethickness","bevelled"),n=r.linethickness,o=r.bevelled,i=this.isDisplay(),a=null;if(o)this.getBevelledBBox(t,i);else{var s=this.length2em(String(n),.06);a=-2*this.pad,0===s?this.getAtopBBox(t,i):(this.getFractionBBox(t,i,s),a-=.2),a+=t.w}t.clean(),this.setChildPWidths(e,a)},e.prototype.getFractionBBox=function(t,e,r){var n=this.childNodes[0].getOuterBBox(),o=this.childNodes[1].getOuterBBox(),i=this.font.params.axis_height,a=this.getTUV(e,r),s=a.T,l=a.u,c=a.v;t.combine(n,0,i+s+Math.max(n.d*n.rscale,l)),t.combine(o,0,i-s-Math.max(o.h*o.rscale,c)),t.w+=2*this.pad+.2},e.prototype.getTUV=function(t,e){var r=this.font.params,n=r.axis_height,o=(t?3.5:1.5)*e;return{T:(t?3.5:1.5)*e,u:(t?r.num1:r.num2)-n-o,v:(t?r.denom1:r.denom2)+n-o}},e.prototype.getAtopBBox=function(t,e){var r=this.getUVQ(e),n=r.u,o=r.v,i=r.nbox,a=r.dbox;t.combine(i,0,n),t.combine(a,0,-o),t.w+=2*this.pad},e.prototype.getUVQ=function(t){var e=this.childNodes[0].getOuterBBox(),r=this.childNodes[1].getOuterBBox(),n=this.font.params,i=o(t?[n.num1,n.denom1]:[n.num3,n.denom2],2),a=i[0],s=i[1],l=(t?7:3)*n.rule_thickness,c=a-e.d*e.scale-(r.h*r.scale-s);return c<l&&(a+=(l-c)/2,s+=(l-c)/2,c=l),{u:a,v:s,q:c,nbox:e,dbox:r}},e.prototype.getBevelledBBox=function(t,e){var r=this.getBevelData(e),n=r.u,o=r.v,i=r.delta,a=r.nbox,s=r.dbox,l=this.bevel.getOuterBBox();t.combine(a,0,n),t.combine(l,t.w-i/2,0),t.combine(s,t.w-i/2,o)},e.prototype.getBevelData=function(t){var e=this.childNodes[0].getOuterBBox(),r=this.childNodes[1].getOuterBBox(),n=t?.4:.15,o=Math.max(e.scale*(e.h+e.d),r.scale*(r.h+r.d))+2*n,i=this.font.params.axis_height;return{H:o,delta:n,u:e.scale*(e.d-e.h)/2+i+n,v:r.scale*(r.d-r.h)/2+i-n,nbox:e,dbox:r}},e.prototype.canStretch=function(t){return!1},e.prototype.isDisplay=function(){var t=this.node.attributes.getList("displaystyle","scriptlevel"),e=t.displaystyle,r=t.scriptlevel;return e&&0===r},e.prototype.getWrapWidth=function(t){var e=this.node.attributes;return e.get("bevelled")?this.childNodes[t].getOuterBBox().w:this.getBBox().w-(this.length2em(e.get("linethickness"))?.2:0)-2*this.pad},e.prototype.getChildAlign=function(t){var e=this.node.attributes;return e.get("bevelled")?"left":e.get(["numalign","denomalign"][t])},e}(t)}},5636:function(t,e){var r,n=this&&this.__extends||(r=function(t,e){return r=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},r(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function n(){this.constructor=t}r(t,e),t.prototype=null===e?Object.create(e):(n.prototype=e.prototype,new n)}),o=this&&this.__read||function(t,e){var r="function"==typeof Symbol&&t[Symbol.iterator];if(!r)return t;var n,o,i=r.call(t),a=[];try{for(;(void 0===e||e-- >0)&&!(n=i.next()).done;)a.push(n.value)}catch(t){o={error:t}}finally{try{n&&!n.done&&(r=i.return)&&r.call(i)}finally{if(o)throw o.error}}return a},i=this&&this.__spreadArray||function(t,e,r){if(r||2===arguments.length)for(var n,o=0,i=e.length;o<i;o++)!n&&o in e||(n||(n=Array.prototype.slice.call(e,0,o)),n[o]=e[o]);return t.concat(n||Array.prototype.slice.call(e))};Object.defineProperty(e,"__esModule",{value:!0}),e.CommonMglyphMixin=void 0,e.CommonMglyphMixin=function(t){return function(t){function e(){for(var e=[],r=0;r<arguments.length;r++)e[r]=arguments[r];var n=t.apply(this,i([],o(e),!1))||this;return n.getParameters(),n}return n(e,t),e.prototype.getParameters=function(){var t=this.node.attributes.getList("width","height","valign","src","index"),e=t.width,r=t.height,n=t.valign,o=t.src,i=t.index;if(o)this.width="auto"===e?1:this.length2em(e),this.height="auto"===r?1:this.length2em(r),this.valign=this.length2em(n||"0");else{var a=String.fromCodePoint(parseInt(i)),s=this.node.factory;this.charWrapper=this.wrap(s.create("text").setText(a)),this.charWrapper.parent=this}},e.prototype.computeBBox=function(t,e){void 0===e&&(e=!1),this.charWrapper?t.updateFrom(this.charWrapper.getBBox()):(t.w=this.width,t.h=this.height+this.valign,t.d=-this.valign)},e}(t)}},5723:function(t,e){var r,n=this&&this.__extends||(r=function(t,e){return r=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},r(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function n(){this.constructor=t}r(t,e),t.prototype=null===e?Object.create(e):(n.prototype=e.prototype,new n)});Object.defineProperty(e,"__esModule",{value:!0}),e.CommonMiMixin=void 0,e.CommonMiMixin=function(t){return function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return n(e,t),e.prototype.computeBBox=function(e,r){void 0===r&&(r=!1),t.prototype.computeBBox.call(this,e),this.copySkewIC(e)},e}(t)}},8009:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},n(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)}),i=this&&this.__read||function(t,e){var r="function"==typeof Symbol&&t[Symbol.iterator];if(!r)return t;var n,o,i=r.call(t),a=[];try{for(;(void 0===e||e-- >0)&&!(n=i.next()).done;)a.push(n.value)}catch(t){o={error:t}}finally{try{n&&!n.done&&(r=i.return)&&r.call(i)}finally{if(o)throw o.error}}return a},a=this&&this.__spreadArray||function(t,e,r){if(r||2===arguments.length)for(var n,o=0,i=e.length;o<i;o++)!n&&o in e||(n||(n=Array.prototype.slice.call(e,0,o)),n[o]=e[o]);return t.concat(n||Array.prototype.slice.call(e))},s=this&&this.__values||function(t){var e="function"==typeof Symbol&&Symbol.iterator,r=e&&t[e],n=0;if(r)return r.call(t);if(t&&"number"==typeof t.length)return{next:function(){return t&&n>=t.length&&(t=void 0),{value:t&&t[n++],done:!t}}};throw new TypeError(e?"Object is not iterable.":"Symbol.iterator is not defined.")};Object.defineProperty(e,"__esModule",{value:!0}),e.CommonMmultiscriptsMixin=e.ScriptNames=e.NextScript=void 0;var l=r(6469);e.NextScript={base:"subList",subList:"supList",supList:"subList",psubList:"psupList",psupList:"psubList"},e.ScriptNames=["sup","sup","psup","psub"],e.CommonMmultiscriptsMixin=function(t){return function(t){function r(){for(var e=[],r=0;r<arguments.length;r++)e[r]=arguments[r];var n=t.apply(this,a([],i(e),!1))||this;return n.scriptData=null,n.firstPrescript=0,n.getScriptData(),n}return o(r,t),r.prototype.combinePrePost=function(t,e){var r=new l.BBox(t);return r.combine(e,0,0),r},r.prototype.computeBBox=function(t,e){void 0===e&&(e=!1);var r=this.font.params.scriptspace,n=this.scriptData,o=this.combinePrePost(n.sub,n.psub),a=this.combinePrePost(n.sup,n.psup),s=i(this.getUVQ(o,a),2),l=s[0],c=s[1];if(t.empty(),n.numPrescripts&&(t.combine(n.psup,r,l),t.combine(n.psub,r,c)),t.append(n.base),n.numScripts){var u=t.w;t.combine(n.sup,u,l),t.combine(n.sub,u,c),t.w+=r}t.clean(),this.setChildPWidths(e)},r.prototype.getScriptData=function(){var t=this.scriptData={base:null,sub:l.BBox.empty(),sup:l.BBox.empty(),psub:l.BBox.empty(),psup:l.BBox.empty(),numPrescripts:0,numScripts:0},e=this.getScriptBBoxLists();this.combineBBoxLists(t.sub,t.sup,e.subList,e.supList),this.combineBBoxLists(t.psub,t.psup,e.psubList,e.psupList),t.base=e.base[0],t.numPrescripts=e.psubList.length,t.numScripts=e.subList.length},r.prototype.getScriptBBoxLists=function(){var t,r,n={base:[],subList:[],supList:[],psubList:[],psupList:[]},o="base";try{for(var i=s(this.childNodes),a=i.next();!a.done;a=i.next()){var l=a.value;l.node.isKind("mprescripts")?o="psubList":(n[o].push(l.getOuterBBox()),o=e.NextScript[o])}}catch(e){t={error:e}}finally{try{a&&!a.done&&(r=i.return)&&r.call(i)}finally{if(t)throw t.error}}return this.firstPrescript=n.subList.length+n.supList.length+2,this.padLists(n.subList,n.supList),this.padLists(n.psubList,n.psupList),n},r.prototype.padLists=function(t,e){t.length>e.length&&e.push(l.BBox.empty())},r.prototype.combineBBoxLists=function(t,e,r,n){for(var o=0;o<r.length;o++){var a=i(this.getScaledWHD(r[o]),3),s=a[0],l=a[1],c=a[2],u=i(this.getScaledWHD(n[o]),3),p=u[0],h=u[1],d=u[2],f=Math.max(s,p);t.w+=f,e.w+=f,l>t.h&&(t.h=l),c>t.d&&(t.d=c),h>e.h&&(e.h=h),d>e.d&&(e.d=d)}},r.prototype.getScaledWHD=function(t){var e=t.w,r=t.h,n=t.d,o=t.rscale;return[e*o,r*o,n*o]},r.prototype.getUVQ=function(e,r){var n;if(!this.UVQ){var o=i([0,0,0],3),a=o[0],s=o[1],l=o[2];0===e.h&&0===e.d?a=this.getU():0===r.h&&0===r.d?a=-this.getV():(a=(n=i(t.prototype.getUVQ.call(this,e,r),3))[0],s=n[1],l=n[2]),this.UVQ=[a,s,l]}return this.UVQ},r}(t)}},5023:function(t,e){var r,n=this&&this.__extends||(r=function(t,e){return r=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)Object.prototype.hasOwnProperty.call(e,r)&&(t[r]=e[r])},r(t,e)},function(t,e){if("function"!=typeof e&&null!==e)throw new TypeError("Class extends value "+String(e)+" is not a constructor or null");function n(){this.constructor=t}r(t,e),t.prototype=null===e?Object.create(e):(n.prototype=e.prototype,new n)});Object.defineProperty(e,"__esModule",{value:!0}),e.CommonMnMixin=void 0,e.CommonMnMixin=function(t){return function(t){function e(){return null!==t&&t.apply(this,arguments)||this}return n(e,t),e.prototype.remapChars=function(t){if(t.length){var e=this.font.getRemappedChar("mn",t[0]);if(e){var 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n=t.apply(this,l([],s(e),!1))||this;return n.size=null,n.isAccent=n.node.isAccent,n}return i(e,t),e.prototype.computeBBox=function(t,e){if(void 0===e&&(e=!1),this.protoBBox(t),this.node.attributes.get("symmetric")&&2!==this.stretch.dir){var r=this.getCenterOffset(t);t.h+=r,t.d-=r}this.node.getProperty("mathaccent")&&(0===this.stretch.dir||this.size>=0)&&(t.w=0)},e.prototype.protoBBox=function(e){var r=0!==this.stretch.dir;r&&null===this.size&&this.getStretchedVariant([0]),r&&this.size<0||(t.prototype.computeBBox.call(this,e),this.copySkewIC(e))},e.prototype.getAccentOffset=function(){var t=u.BBox.empty();return this.protoBBox(t),-t.w/2},e.prototype.getCenterOffset=function(e){return void 0===e&&(e=null),e||(e=u.BBox.empty(),t.prototype.computeBBox.call(this,e)),(e.h+e.d)/2+this.font.params.axis_height-e.h},e.prototype.getVariant=function(){this.node.attributes.get("largeop")?this.variant=this.node.attributes.get("displaystyle")?"-largeop":"-smallop":this.node.attributes.getExplicit("mathvariant")||!1!==this.node.getProperty("pseudoscript")?t.prototype.getVariant.call(this):this.variant="-tex-variant"},e.prototype.canStretch=function(t){if(0!==this.stretch.dir)return this.stretch.dir===t;if(!this.node.attributes.get("stretchy"))return!1;var e=this.getText();if(1!==Array.from(e).length)return!1;var r=this.font.getDelimiter(e.codePointAt(0));return this.stretch=r&&r.dir===t?r:h.NOSTRETCH,0!==this.stretch.dir},e.prototype.getStretchedVariant=function(t,e){var r,n;if(void 0===e&&(e=!1),0!==this.stretch.dir){var o=this.getWH(t),i=this.getSize("minsize",0),s=this.getSize("maxsize",1/0),l=this.node.getProperty("mathaccent");o=Math.max(i,Math.min(s,o));var 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t.concat(n||Array.prototype.slice.call(e))},s=this&&this.__values||function(t){var e="function"==typeof Symbol&&Symbol.iterator,r=e&&t[e],n=0;if(r)return r.call(t);if(t&&"number"==typeof t.length)return{next:function(){return t&&n>=t.length&&(t=void 0),{value:t&&t[n++],done:!t}}};throw new TypeError(e?"Object is not iterable.":"Symbol.iterator is not defined.")};Object.defineProperty(e,"__esModule",{value:!0}),e.CommonMtableMixin=void 0;var l=r(6469),c=r(505),u=r(7875);e.CommonMtableMixin=function(t){return function(t){function e(){for(var e=[],r=0;r<arguments.length;r++)e[r]=arguments[r];var n=t.apply(this,a([],i(e),!1))||this;n.numCols=0,n.numRows=0,n.data=null,n.pwidthCells=[],n.pWidth=0,n.numCols=(0,u.max)(n.tableRows.map((function(t){return t.numCells}))),n.numRows=n.childNodes.length,n.hasLabels=n.childNodes.reduce((function(t,e){return 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o=(n-(e[t]+r[t]))/2;o<1e-5||(e[t]+=o,r[t]+=o)},e.prototype.recordPWidthCell=function(t,e){t.childNodes[0]&&t.childNodes[0].getBBox().pwidth&&this.pwidthCells.push([t,e])},e.prototype.computeBBox=function(t,e){void 0===e&&(e=!1);var r,n,o=this.getTableData(),a=o.H,s=o.D;if(this.node.attributes.get("equalrows")){var l=this.getEqualRowHeight();r=(0,u.sum)([].concat(this.rLines,this.rSpace))+l*this.numRows}else r=(0,u.sum)(a.concat(s,this.rLines,this.rSpace));r+=2*(this.fLine+this.fSpace[1]);var p=this.getComputedWidths();n=(0,u.sum)(p.concat(this.cLines,this.cSpace))+2*(this.fLine+this.fSpace[0]);var h=this.node.attributes.get("width");"auto"!==h&&(n=Math.max(this.length2em(h,0)+2*this.fLine,n));var d=i(this.getBBoxHD(r),2),f=d[0],m=d[1];t.h=f,t.d=m,t.w=n;var y=i(this.getBBoxLR(),2),g=y[0],b=y[1];t.L=g,t.R=b,(0,c.isPercent)(h)||this.setColumnPWidths()},e.prototype.setChildPWidths=function(t,e,r){var n=this.node.attributes.get("width");if(!(0,c.isPercent)(n))return!1;this.hasLabels||(this.bbox.pwidth="",this.container.bbox.pwidth="");var o=this.bbox,i=o.w,a=o.L,s=o.R,l=this.node.attributes.get("data-width-includes-label"),p=Math.max(i,this.length2em(n,Math.max(e,a+i+s)))-(l?a+s:0),h=this.node.attributes.get("equalcolumns")?Array(this.numCols).fill(this.percent(1/Math.max(1,this.numCols))):this.getColumnAttributes("columnwidth",0);this.cWidths=this.getColumnWidthsFixed(h,p);var d=this.getComputedWidths();return this.pWidth=(0,u.sum)(d.concat(this.cLines,this.cSpace))+2*(this.fLine+this.fSpace[0]),this.isTop&&(this.bbox.w=this.pWidth),this.setColumnPWidths(),this.pWidth!==i&&this.parent.invalidateBBox(),this.pWidth!==i},e.prototype.setColumnPWidths=function(){var t,e,r=this.cWidths;try{for(var n=s(this.pwidthCells),o=n.next();!o.done;o=n.next()){var a=i(o.value,2),l=a[0],c=a[1];l.setChildPWidths(!1,r[c])&&(l.invalidateBBox(),l.getBBox())}}catch(e){t={error:e}}finally{try{o&&!o.done&&(e=n.return)&&e.call(n)}finally{if(t)throw t.error}}},e.prototype.getBBoxHD=function(t){var e=i(this.getAlignmentRow(),2),r=e[0],n=e[1];if(null===n){var o=this.font.params.axis_height,a=t/2;return{top:[0,t],center:[a,a],bottom:[t,0],baseline:[a,a],axis:[a+o,a-o]}[r]||[a,a]}var s=this.getVerticalPosition(n,r);return[s,t-s]},e.prototype.getBBoxLR=function(){if(this.hasLabels){var t=this.node.attributes,e=t.get("side"),r=i(this.getPadAlignShift(e),2),n=r[0],o=r[1],a=this.hasLabels&&!!t.get("data-width-includes-label");return a&&this.frame&&this.fSpace[0]&&(n-=this.fSpace[0]),"center"!==o||a?"left"===e?[n,0]:[0,n]:[n,n]}return[0,0]},e.prototype.getPadAlignShift=function(t){var 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this.node.attributes.get("equalcolumns")&&(r=Array(r.length).fill((0,u.max)(r))),r},e.prototype.getColumnWidths=function(){var t=this.node.attributes.get("width");if(this.node.attributes.get("equalcolumns"))return this.getEqualColumns(t);var e=this.getColumnAttributes("columnwidth",0);return"auto"===t?this.getColumnWidthsAuto(e):(0,c.isPercent)(t)?this.getColumnWidthsPercent(e):this.getColumnWidthsFixed(e,this.length2em(t))},e.prototype.getEqualColumns=function(t){var e,r=Math.max(1,this.numCols);if("auto"===t){var n=this.getTableData().W;e=(0,u.max)(n)}else if((0,c.isPercent)(t))e=this.percent(1/r);else{var o=(0,u.sum)([].concat(this.cLines,this.cSpace))+2*this.fSpace[0];e=Math.max(0,this.length2em(t)-o)/r}return Array(this.numCols).fill(e)},e.prototype.getColumnWidthsAuto=function(t){var e=this;return t.map((function(t){return"auto"===t||"fit"===t?null:(0,c.isPercent)(t)?t:e.length2em(t)}))},e.prototype.getColumnWidthsPercent=function(t){var 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/*!
 *************************************************************************
 *
 *  mhchemParser.ts
 *  4.1.1
 *
 *  Parser for the \ce command and \pu command for MathJax and Co.
 *
 *  mhchem's \ce is a tool for writing beautiful chemical equations easily.
 *  mhchem's \pu is a tool for writing physical units easily.
 *
 *  ----------------------------------------------------------------------
 *
 *  Copyright (c) 2015-2021 Martin Hensel
 *
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *  you may not use this file except in compliance with the License.
 *  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 *
 *  ----------------------------------------------------------------------
 *
 *  https://github.com/mhchem/mhchemParser
 *
 */
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e=this.html;e.appendChild(this.generateTitle()),e.appendChild(this.contentDiv),e.appendChild(this.generateSignature()),e.appendChild(this.close.html),e.setAttribute("tabindex","0")},e.prototype.post=function(){t.prototype.post.call(this);var e=document.documentElement,r=this.html,n=window.innerHeight||e.clientHeight||e.scrollHeight||0,o=Math.floor(-r.offsetWidth/2),i=Math.floor((n-r.offsetHeight)/3);r.setAttribute("style","margin-left: "+o+"px; top: "+i+"px;"),window.event instanceof MouseEvent&&r.classList.add(a.HtmlClasses.MOUSEPOST),r.focus()},e.prototype.display=function(){this.menu.registerWidget(this),this.contentDiv.innerHTML=this.content();var t=this.menu.html;t.parentNode&&t.parentNode.removeChild(t),this.menu.frame.appendChild(this.html)},e.prototype.click=function(t){},e.prototype.keydown=function(e){this.bubbleKey(),t.prototype.keydown.call(this,e)},e.prototype.escape=function(t){this.unpost()},e.prototype.unpost=function(){t.prototype.unpost.call(this),this.html.classList.remove(a.HtmlClasses.MOUSEPOST),this.menu.unregisterWidget(this)},e.prototype.generateClose=function(){var t=new i.CloseButton(this),e=t.html;return e.classList.add(a.HtmlClasses.INFOCLOSE),e.setAttribute("aria-label","Close Dialog Box"),t},e.prototype.generateTitle=function(){var t=document.createElement("span");return t.innerHTML=this.title,t.classList.add(a.HtmlClasses.INFOTITLE),t},e.prototype.generateContent=function(){var t=document.createElement("div");return t.classList.add(a.HtmlClasses.INFOCONTENT),t.setAttribute("tabindex","0"),t},e.prototype.generateSignature=function(){var t=document.createElement("span");return t.innerHTML=this.signature,t.classList.add(a.HtmlClasses.INFOSIGNATURE),t},e.prototype.toJson=function(){return{type:""}},e}(r(8372).AbstractPostable);e.Info=s},1409:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])},n(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Checkbox=void 0;var i=r(6765),a=r(2556),s=r(2165),l=function(t){function e(e,r,n,o){var i=t.call(this,e,"checkbox",r,o)||this;return i.role="menuitemcheckbox",i.variable=e.pool.lookup(n),i.register(),i}return o(e,t),e.fromJson=function(t,e,r){return new 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this(r,e.content,e.variable,e.id)},e.prototype.executeAction=function(){this.variable.setValue(this.input.value,a.MenuUtil.getActiveElement(this))},e.prototype.space=function(e){t.prototype.space.call(this,e),a.MenuUtil.close(this)},e.prototype.focus=function(){t.prototype.focus.call(this),this.input.focus()},e.prototype.unfocus=function(){t.prototype.unfocus.call(this),this.updateSpan()},e.prototype.generateHtml=function(){t.prototype.generateHtml.call(this),this.html.classList.add(s.HtmlClasses.MENUCOMBOBOX)},e.prototype.generateSpan=function(){this.span=document.createElement("span"),this.span.classList.add(s.HtmlClasses.MENUINPUTBOX),this.input=document.createElement("input"),this.input.addEventListener("keydown",this.inputKey.bind(this)),this.input.setAttribute("size","10em"),this.input.setAttribute("type","text"),this.input.setAttribute("tabindex","-1"),this.span.appendChild(this.input)},e.prototype.inputKey=function(t){this.bubbleKey(),this.inputEvent=!0},e.prototype.keydown=function(e){if(this.inputEvent&&e.keyCode!==l.KEY.ESCAPE&&e.keyCode!==l.KEY.RETURN)return this.inputEvent=!1,void e.stopPropagation();t.prototype.keydown.call(this,e),e.stopPropagation()},e.prototype.updateAria=function(){},e.prototype.updateSpan=function(){var t;try{t=this.variable.getValue(a.MenuUtil.getActiveElement(this))}catch(e){t=""}this.input.value=t},e.prototype.toJson=function(){return{type:""}},e}(i.AbstractVariableItem);e.Combo=c},3467:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])},n(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Command=void 0;var i=r(1340),a=r(2556),s=function(t){function e(e,r,n,o){var i=t.call(this,e,"command",r,o)||this;return i.command=n,i}return o(e,t),e.fromJson=function(t,e,r){return new 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r,n=1;r=t.childNodes[n];n++)e.push(c.walkTree(r));t=t.childNodes[0]}const a=n.createElement("dummy"),s=o.childNodes[0];n.replaceNode(o,a),n.replaceNode(r,o),n.replaceNode(o.childNodes[0],r),n.replaceNode(a,s),e||(e=o)}return c.walkTree(this.ofence),c.walkTree(this.cfence),this.cleanupParents_(),e||r}specialCase_(t,e){const r=n.tagName(e);let o,i=null;if("MSUBSUP"===r?(i=t.childNodes[0],o=a.CaseDoubleScript):"MMULTISCRIPTS"===r&&("superscript"===t.type||"subscript"===t.type?o=s.CaseMultiscripts:"tensor"===t.type&&(o=l.CaseTensor),i=o&&t.childNodes[0]&&"subsup"===t.childNodes[0].role?t.childNodes[0]:t),!i)return null;const c=i.childNodes[0],u=p.makeEmptyNode_(c.id);return i.childNodes[0]=u,e=new o(t).getMathml(),i.childNodes[0]=c,this.parentCleanup.push(e),i.childNodes[0]}introduceNewLayer_(){const t=this.fullFence(this.ofenceMml),e=this.fullFence(this.cfenceMml);let r=n.createElement("mrow");if(n.replaceNode(this.fencedMml,r),this.fencedMmlNodes.forEach((t=>r.appendChild(t))),r.insertBefore(t,this.fencedMml),r.appendChild(e),!r.parentNode){const t=n.createElement("mrow");for(;r.childNodes.length>0;)t.appendChild(r.childNodes[0]);r.appendChild(t),r=t}return r}fullFence(t){const e=this.fencedMml.parentNode;let r=t;for(;r.parentNode&&r.parentNode!==e;)r=r.parentNode;return r}cleanupParents_(){this.parentCleanup.forEach((function(t){const e=t.childNodes[1].getAttribute(u.Attribute.PARENT);t.childNodes[0].setAttribute(u.Attribute.PARENT,e)}))}}e.CaseEmbellished=p},6265:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseLimit=void 0;const n=r(5740),o=r(9268),i=r(5452),a=r(2298);class s extends o.AbstractEnrichCase{constructor(t){super(t),this.mml=t.mathmlTree}static test(t){if(!t.mathmlTree||!t.childNodes.length)return!1;const e=n.tagName(t.mathmlTree),r=t.type;return("limupper"===r||"limlower"===r)&&("MSUBSUP"===e||"MUNDEROVER"===e)||"limboth"===r&&("MSUB"===e||"MUNDER"===e||"MSUP"===e||"MOVER"===e)}static walkTree_(t){t&&i.walkTree(t)}getMathml(){const t=this.semantic.childNodes;return"limboth"!==this.semantic.type&&this.mml.childNodes.length>=3&&(this.mml=i.introduceNewLayer([this.mml],this.semantic)),(0,a.setAttributes)(this.mml,this.semantic),t[0].mathmlTree||(t[0].mathmlTree=this.semantic.mathmlTree),t.forEach(s.walkTree_),this.mml}}e.CaseLimit=s},6514:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseLine=void 0;const n=r(9268),o=r(5452),i=r(2298);class a extends n.AbstractEnrichCase{constructor(t){super(t),this.mml=t.mathmlTree}static test(t){return!!t.mathmlTree&&"line"===t.type}getMathml(){return this.semantic.contentNodes.length&&o.walkTree(this.semantic.contentNodes[0]),this.semantic.childNodes.length&&o.walkTree(this.semantic.childNodes[0]),(0,i.setAttributes)(this.mml,this.semantic),this.mml}}e.CaseLine=a},6839:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseMultiindex=void 0;const n=r(5740),o=r(9268),i=r(5452),a=r(2298);class s extends o.AbstractEnrichCase{constructor(t){super(t),this.mml=t.mathmlTree}static multiscriptIndex(t){return"punctuated"===t.type&&"dummy"===t.contentNodes[0].role?i.collapsePunctuated(t):(i.walkTree(t),t.id)}static createNone_(t){const e=n.createElement("none");return t&&(0,a.setAttributes)(e,t),e.setAttribute(a.Attribute.ADDED,"true"),e}completeMultiscript(t,e){const r=n.toArray(this.mml.childNodes).slice(1);let o=0;const l=t=>{for(let e,n=0;e=t[n];n++){const t=r[o];if(t&&e===parseInt(i.getInnerNode(t).getAttribute(a.Attribute.ID)))i.getInnerNode(t).setAttribute(a.Attribute.PARENT,this.semantic.id.toString()),o++;else{const r=this.semantic.querySelectorAll((t=>t.id===e));this.mml.insertBefore(s.createNone_(r[0]),t||null)}}};l(t),r[o]&&"MPRESCRIPTS"!==n.tagName(r[o])?this.mml.insertBefore(r[o],n.createElement("mprescripts")):o++,l(e)}}e.CaseMultiindex=s},7014:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseMultiscripts=void 0;const n=r(5740),o=r(5656),i=r(6839),a=r(5452),s=r(2298);class l extends i.CaseMultiindex{static test(t){if(!t.mathmlTree)return!1;return"MMULTISCRIPTS"===n.tagName(t.mathmlTree)&&("superscript"===t.type||"subscript"===t.type)}constructor(t){super(t)}getMathml(){let t,e,r;if((0,s.setAttributes)(this.mml,this.semantic),this.semantic.childNodes[0]&&"subsup"===this.semantic.childNodes[0].role){const n=this.semantic.childNodes[0];t=n.childNodes[0],e=i.CaseMultiindex.multiscriptIndex(this.semantic.childNodes[1]),r=i.CaseMultiindex.multiscriptIndex(n.childNodes[1]);const l=[this.semantic.id,[n.id,t.id,r],e];a.addCollapsedAttribute(this.mml,l),this.mml.setAttribute(s.Attribute.TYPE,n.role),this.completeMultiscript(o.SemanticSkeleton.interleaveIds(r,e),[])}else{t=this.semantic.childNodes[0],e=i.CaseMultiindex.multiscriptIndex(this.semantic.childNodes[1]);const r=[this.semantic.id,t.id,e];a.addCollapsedAttribute(this.mml,r)}const n=o.SemanticSkeleton.collapsedLeafs(r||[],e),l=a.walkTree(t);return a.getInnerNode(l).setAttribute(s.Attribute.PARENT,this.semantic.id.toString()),n.unshift(t.id),this.mml.setAttribute(s.Attribute.CHILDREN,n.join(",")),this.mml}}e.CaseMultiscripts=l},3416:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseProof=void 0;const n=r(9268),o=r(5452),i=r(2298);class a extends n.AbstractEnrichCase{constructor(t){super(t),this.mml=t.mathmlTree}static test(t){return!!t.mathmlTree&&("inference"===t.type||"premises"===t.type)}getMathml(){return this.semantic.childNodes.length?(this.semantic.contentNodes.forEach((function(t){o.walkTree(t),(0,i.setAttributes)(t.mathmlTree,t)})),this.semantic.childNodes.forEach((function(t){o.walkTree(t)})),(0,i.setAttributes)(this.mml,this.semantic),this.mml.getAttribute("data-semantic-id")===this.mml.getAttribute("data-semantic-parent")&&this.mml.removeAttribute("data-semantic-parent"),this.mml):this.mml}}e.CaseProof=a},5699:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseTable=void 0;const n=r(5740),o=r(9268),i=r(5452),a=r(2298);class s extends o.AbstractEnrichCase{constructor(t){super(t),this.inner=[],this.mml=t.mathmlTree}static test(t){return"matrix"===t.type||"vector"===t.type||"cases"===t.type}getMathml(){const t=i.cloneContentNode(this.semantic.contentNodes[0]),e=this.semantic.contentNodes[1]?i.cloneContentNode(this.semantic.contentNodes[1]):null;if(this.inner=this.semantic.childNodes.map(i.walkTree),this.mml)if("MFENCED"===n.tagName(this.mml)){const r=this.mml.childNodes;this.mml.insertBefore(t,r[0]||null),e&&this.mml.appendChild(e),this.mml=i.rewriteMfenced(this.mml)}else{const r=[t,this.mml];e&&r.push(e),this.mml=i.introduceNewLayer(r,this.semantic)}else this.mml=i.introduceNewLayer([t].concat(this.inner,[e]),this.semantic);return(0,a.setAttributes)(this.mml,this.semantic),this.mml}}e.CaseTable=s},6887:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseTensor=void 0;const n=r(5656),o=r(6839),i=r(5452),a=r(2298);class s extends o.CaseMultiindex{static test(t){return!!t.mathmlTree&&"tensor"===t.type}constructor(t){super(t)}getMathml(){i.walkTree(this.semantic.childNodes[0]);const t=o.CaseMultiindex.multiscriptIndex(this.semantic.childNodes[1]),e=o.CaseMultiindex.multiscriptIndex(this.semantic.childNodes[2]),r=o.CaseMultiindex.multiscriptIndex(this.semantic.childNodes[3]),s=o.CaseMultiindex.multiscriptIndex(this.semantic.childNodes[4]);(0,a.setAttributes)(this.mml,this.semantic);const l=[this.semantic.id,this.semantic.childNodes[0].id,t,e,r,s];i.addCollapsedAttribute(this.mml,l);const c=n.SemanticSkeleton.collapsedLeafs(t,e,r,s);return c.unshift(this.semantic.childNodes[0].id),this.mml.setAttribute(a.Attribute.CHILDREN,c.join(",")),this.completeMultiscript(n.SemanticSkeleton.interleaveIds(r,s),n.SemanticSkeleton.interleaveIds(t,e)),this.mml}}e.CaseTensor=s},9236:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CaseText=void 0;const n=r(9268),o=r(5452),i=r(2298);class a extends n.AbstractEnrichCase{constructor(t){super(t),this.mml=t.mathmlTree}static test(t){return"punctuated"===t.type&&("text"===t.role||t.contentNodes.every((t=>"dummy"===t.role)))}getMathml(){const t=[],e=o.collapsePunctuated(this.semantic,t);return this.mml=o.introduceNewLayer(t,this.semantic),(0,i.setAttributes)(this.mml,this.semantic),this.mml.removeAttribute(i.Attribute.CONTENT),o.addCollapsedAttribute(this.mml,e),this.mml}}e.CaseText=a},5714:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.prepareMmlString=e.testTranslation__=e.semanticMathml=e.semanticMathmlSync=e.semanticMathmlNode=void 0;const n=r(2057),o=r(5740),i=r(5897),a=r(1414),s=r(5452),l=r(2298);function c(t){const e=o.cloneNode(t),r=a.getTree(e);return s.enrich(e,r)}function u(t){return c(o.parseInput(t))}function p(t){return t.match(/^<math/)||(t="<math>"+t),t.match(/\/math>$/)||(t+="</math>"),t}r(1513),e.semanticMathmlNode=c,e.semanticMathmlSync=u,e.semanticMathml=function(t,e){i.EnginePromise.getall().then((()=>{const r=o.parseInput(t);e(c(r))}))},e.testTranslation__=function(t){n.Debugger.getInstance().init();const e=u(p(t)).toString();return(0,l.removeAttributePrefix)(e),n.Debugger.getInstance().exit(),e},e.prepareMmlString=p},2298:function(t,e){var r;function n(t){return t.map((function(t){return t.id})).join(",")}function o(t,e){const n=[];"mglyph"===e.role&&n.push("image"),e.attributes.href&&n.push("link"),n.length&&t.setAttribute(r.POSTFIX,n.join(" "))}Object.defineProperty(e,"__esModule",{value:!0}),e.addPrefix=e.removeAttributePrefix=e.setPostfix=e.setAttributes=e.makeIdList=e.EnrichAttributes=e.Attribute=e.Prefix=void 0,e.Prefix="data-semantic-",function(t){t.ADDED="data-semantic-added",t.ALTERNATIVE="data-semantic-alternative",t.CHILDREN="data-semantic-children",t.COLLAPSED="data-semantic-collapsed",t.CONTENT="data-semantic-content",t.EMBELLISHED="data-semantic-embellished",t.FENCEPOINTER="data-semantic-fencepointer",t.FONT="data-semantic-font",t.ID="data-semantic-id",t.ANNOTATION="data-semantic-annotation",t.OPERATOR="data-semantic-operator",t.OWNS="data-semantic-owns",t.PARENT="data-semantic-parent",t.POSTFIX="data-semantic-postfix",t.PREFIX="data-semantic-prefix",t.ROLE="data-semantic-role",t.SPEECH="data-semantic-speech",t.STRUCTURE="data-semantic-structure",t.TYPE="data-semantic-type"}(r=e.Attribute||(e.Attribute={})),e.EnrichAttributes=[r.ADDED,r.ALTERNATIVE,r.CHILDREN,r.COLLAPSED,r.CONTENT,r.EMBELLISHED,r.FENCEPOINTER,r.FONT,r.ID,r.ANNOTATION,r.OPERATOR,r.OWNS,r.PARENT,r.POSTFIX,r.PREFIX,r.ROLE,r.SPEECH,r.STRUCTURE,r.TYPE],e.makeIdList=n,e.setAttributes=function(t,i){t.setAttribute(r.TYPE,i.type);const a=i.allAttributes();for(let r,n=0;r=a[n];n++)t.setAttribute(e.Prefix+r[0].toLowerCase(),r[1]);i.childNodes.length&&t.setAttribute(r.CHILDREN,n(i.childNodes)),i.contentNodes.length&&t.setAttribute(r.CONTENT,n(i.contentNodes)),i.parent&&t.setAttribute(r.PARENT,i.parent.id.toString()),o(t,i)},e.setPostfix=o,e.removeAttributePrefix=function(t){return t.toString().replace(new RegExp(e.Prefix,"g"),"")},e.addPrefix=function(t){return e.Prefix+t}},3532:function(t,e){Object.defineProperty(e,"__esModule",{value:!0}),e.factory=e.getCase=void 0,e.getCase=function(t){for(let r,n=0;r=e.factory[n];n++)if(r.test(t))return r.constr(t);return null},e.factory=[]},1513:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0});const n=r(6061),o=r(5765),i=r(7251),a=r(6265),s=r(6514),l=r(7014),c=r(3416),u=r(5699),p=r(6887),h=r(9236);r(3532).factory.push({test:a.CaseLimit.test,constr:t=>new a.CaseLimit(t)},{test:i.CaseEmbellished.test,constr:t=>new i.CaseEmbellished(t)},{test:o.CaseDoubleScript.test,constr:t=>new o.CaseDoubleScript(t)},{test:p.CaseTensor.test,constr:t=>new p.CaseTensor(t)},{test:l.CaseMultiscripts.test,constr:t=>new l.CaseMultiscripts(t)},{test:s.CaseLine.test,constr:t=>new s.CaseLine(t)},{test:n.CaseBinomial.test,constr:t=>new n.CaseBinomial(t)},{test:c.CaseProof.test,constr:t=>new c.CaseProof(t)},{test:u.CaseTable.test,constr:t=>new u.CaseTable(t)},{test:h.CaseText.test,constr:t=>new h.CaseText(t)})},5452:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.printNodeList__=e.collapsePunctuated=e.formattedOutput_=e.formattedOutput=e.getInnerNode=e.setOperatorAttribute_=e.createInvisibleOperator_=e.rewriteMfenced=e.cloneContentNode=e.addCollapsedAttribute=e.parentNode_=e.isIgnorable_=e.unitChild_=e.descendNode_=e.ascendNewNode=e.validLca_=e.pathToRoot_=e.attachedElement_=e.prunePath_=e.mathmlLca_=e.lcaType=e.functionApplication_=e.isDescendant_=e.insertNewChild_=e.mergeChildren_=e.collectChildNodes_=e.collateChildNodes_=e.childrenSubset_=e.moveSemanticAttributes_=e.introduceLayerAboveLca=e.introduceNewLayer=e.walkTree=e.enrich=e.SETTINGS=void 0;const n=r(2057),o=r(5740),i=r(5897),a=r(3588),s=r(7516),l=r(5656),c=r(4795),u=r(2298),p=r(3532);function h(t){const e=(0,p.getCase)(t);let r;if(e)return r=e.getMathml(),T(r);if(1===t.mathml.length)return n.Debugger.getInstance().output("Walktree Case 0"),r=t.mathml[0],u.setAttributes(r,t),t.childNodes.length&&(n.Debugger.getInstance().output("Walktree Case 0.1"),t.childNodes.forEach((function(t){"empty"===t.type&&r.appendChild(h(t))}))),T(r);const o=t.contentNodes.map(k);B(t,o);const i=t.childNodes.map(h),a=l.SemanticSkeleton.combineContentChildren(t,o,i);if(r=t.mathmlTree,null===r)n.Debugger.getInstance().output("Walktree Case 1"),r=d(a,t);else{const t=A(a);n.Debugger.getInstance().output("Walktree Case 2"),t?(n.Debugger.getInstance().output("Walktree Case 2.1"),r=t.parentNode):(n.Debugger.getInstance().output("Walktree Case 2.2"),r=D(r))}return r=R(r),v(r,a,t),u.setAttributes(r,t),T(r)}function d(t,e){const r=O(t);let i=r.node;const a=r.type;if(a!==M.VALID||!c.hasEmptyTag(i))if(n.Debugger.getInstance().output("Walktree Case 1.1"),i=o.createElement("mrow"),a===M.PRUNED)n.Debugger.getInstance().output("Walktree Case 1.1.0"),i=f(i,r.node,t);else if(t[0]){n.Debugger.getInstance().output("Walktree Case 1.1.1");const e=A(t),r=y(e.parentNode,t);o.replaceNode(e,i),r.forEach((function(t){i.appendChild(t)}))}return e.mathmlTree||(e.mathmlTree=i),i}function f(t,e,r){let i=N(e);if(c.hasMathTag(i)){n.Debugger.getInstance().output("Walktree Case 1.1.0.0"),m(i,t),o.toArray(i.childNodes).forEach((function(e){t.appendChild(e)}));const e=t;t=i,i=e}const a=r.indexOf(e);return r[a]=i,o.replaceNode(i,t),t.appendChild(i),r.forEach((function(e){t.appendChild(e)})),t}function m(t,e){for(const r of u.EnrichAttributes)t.hasAttribute(r)&&(e.setAttribute(r,t.getAttribute(r)),t.removeAttribute(r))}function y(t,e){const r=o.toArray(t.childNodes);let n=1/0,i=-1/0;return e.forEach((function(t){const e=r.indexOf(t);-1!==e&&(n=Math.min(n,e),i=Math.max(i,e))})),r.slice(n,i+1)}function g(t,e,r){const n=[];let i=o.toArray(t.childNodes),a=!1;for(;i.length;){const t=i.shift();if(t.hasAttribute(u.Attribute.TYPE)){n.push(t);continue}const 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${r}`}:{convertible:!0,enumerator:i,denominator:o}}Object.defineProperty(e,"__esModule",{value:!0}),e.vulgarFractionSmall=e.convertVulgarFraction=e.Combiners=e.siCombiner=e.identityTransformer=e.pluralCase=void 0,e.pluralCase=function(t,e){return t.toString()},e.identityTransformer=function(t){return t.toString()},e.siCombiner=function(t,e){return t+e.toLowerCase()},e.Combiners={},e.Combiners.identityCombiner=function(t,e,r){return t+e+r},e.Combiners.prefixCombiner=function(t,e,r){return t=r?r+" "+t:t,e?e+" "+t:t},e.Combiners.postfixCombiner=function(t,e,r){return t=r?r+" "+t:t,e?t+" "+e:t},e.Combiners.romanceCombiner=function(t,e,r){return t=r?t+" "+r:t,e?t+" "+e:t},e.convertVulgarFraction=r,e.vulgarFractionSmall=function(t,e,n){const o=r(t);if(o.convertible){const t=o.enumerator,r=o.denominator;return t>0&&t<e&&r>0&&r<n}return!1}},4504:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.Condition=e.BaseRuleStore=void 0;const n=r(7052),o=r(1676),i=r(4650),a=r(4106);class s{constructor(){this.context=new a.SpeechRuleContext,this.parseOrder=o.DynamicCstr.DEFAULT_ORDER,this.parser=new o.DynamicCstrParser(this.parseOrder),this.locale=o.DynamicCstr.DEFAULT_VALUES[o.Axis.LOCALE],this.modality=o.DynamicCstr.DEFAULT_VALUES[o.Axis.MODALITY],this.domain="",this.initialized=!1,this.inherits=null,this.kind="standard",this.customTranscriptions={},this.preconditions=new Map,this.speechRules_=[],this.rank=0,this.parseMethods={Rule:this.defineRule,Generator:this.generateRules,Action:this.defineAction,Precondition:this.definePrecondition,Ignore:this.ignoreRules}}static compareStaticConstraints_(t,e){if(t.length!==e.length)return!1;for(let r,n=0;r=t[n];n++)if(-1===e.indexOf(r))return!1;return!0}static comparePreconditions_(t,e){const r=t.precondition,n=e.precondition;return r.query===n.query&&s.compareStaticConstraints_(r.constraints,n.constraints)}defineRule(t,e,r,n,...o){const a=this.parseAction(r),s=this.parsePrecondition(n,o),l=this.parseCstr(e);if(!(a&&s&&l))return console.error(`Rule Error: ${n}, (${e}): ${r}`),null;const c=new i.SpeechRule(t,l,s,a);return c.precondition.rank=this.rank++,this.addRule(c),c}addRule(t){t.context=this.context,this.speechRules_.unshift(t)}deleteRule(t){const e=this.speechRules_.indexOf(t);-1!==e&&this.speechRules_.splice(e,1)}findRule(t){for(let e,r=0;e=this.speechRules_[r];r++)if(t(e))return e;return null}findAllRules(t){return this.speechRules_.filter(t)}evaluateDefault(t){const e=t.textContent.slice(0);return e.match(/^\s+$/)?this.evaluateWhitespace(e):this.evaluateString(e)}evaluateWhitespace(t){return[]}evaluateCustom(t){const e=this.customTranscriptions[t];return void 0!==e?n.AuditoryDescription.create({text:e},{adjust:!0,translate:!1}):null}evaluateCharacter(t){return this.evaluateCustom(t)||n.AuditoryDescription.create({text:t},{adjust:!0,translate:!0})}removeDuplicates(t){for(let e,r=this.speechRules_.length-1;e=this.speechRules_[r];r--)e!==t&&t.dynamicCstr.equal(e.dynamicCstr)&&s.comparePreconditions_(e,t)&&this.speechRules_.splice(r,1)}getSpeechRules(){return this.speechRules_}setSpeechRules(t){this.speechRules_=t}getPreconditions(){return this.preconditions}parseCstr(t){try{return this.parser.parse(this.locale+"."+this.modality+(this.domain?"."+this.domain:"")+"."+t)}catch(e){if("RuleError"===e.name)return console.error("Rule Error ",`Illegal Dynamic Constraint: ${t}.`,e.message),null;throw e}}parsePrecondition(t,e){try{const r=this.parsePrecondition_(t);t=r[0];let n=r.slice(1);for(const t of e)n=n.concat(this.parsePrecondition_(t));return new i.Precondition(t,...n)}catch(r){if("RuleError"===r.name)return console.error("Rule Error ",`Illegal preconditions: ${t}, ${e}.`,r.message),null;throw r}}parseAction(t){try{return i.Action.fromString(t)}catch(e){if("RuleError"===e.name)return console.error("Rule Error ",`Illegal action: ${t}.`,e.message),null;throw e}}parse(t){this.modality=t.modality||this.modality,this.locale=t.locale||this.locale,this.domain=t.domain||this.domain,this.context.parse(t.functions||[]),"actions"!==t.kind&&(this.kind=t.kind||this.kind,this.inheritRules()),this.parseRules(t.rules||[])}parseRules(t){for(let e,r=0;e=t[r];r++){const t=e[0],r=this.parseMethods[t];t&&r&&r.apply(this,e.slice(1))}}generateRules(t){const e=this.context.customGenerators.lookup(t);e&&e(this)}defineAction(t,e){let r;try{r=i.Action.fromString(e)}catch(t){if("RuleError"===t.name)return void console.error("Action Error ",e,t.message);throw t}const n=this.getFullPreconditions(t);if(!n)return void console.error(`Action Error: No precondition for action ${t}`);this.ignoreRules(t);const o=new RegExp("^\\w+\\.\\w+\\."+(this.domain?"\\w+\\.":""));n.conditions.forEach((([e,n])=>{const a=this.parseCstr(e.toString().replace(o,""));this.addRule(new i.SpeechRule(t,a,n,r))}))}getFullPreconditions(t){const e=this.preconditions.get(t);return e||!this.inherits?e:this.inherits.getFullPreconditions(t)}definePrecondition(t,e,r,...n){const o=this.parsePrecondition(r,n),i=this.parseCstr(e);o&&i?(o.rank=this.rank++,this.preconditions.set(t,new l(i,o))):console.error(`Precondition Error: ${r}, (${e})`)}inheritRules(){if(!this.inherits||!this.inherits.getSpeechRules().length)return;const t=new RegExp("^\\w+\\.\\w+\\."+(this.domain?"\\w+\\.":""));this.inherits.getSpeechRules().forEach((e=>{const r=this.parseCstr(e.dynamicCstr.toString().replace(t,""));this.addRule(new i.SpeechRule(e.name,r,e.precondition,e.action))}))}ignoreRules(t,...e){let r=this.findAllRules((e=>e.name===t));if(!e.length)return void r.forEach(this.deleteRule.bind(this));let n=[];for(const t of e){const e=this.parseCstr(t);for(const t of r)e.equal(t.dynamicCstr)?this.deleteRule(t):n.push(t);r=n,n=[]}}parsePrecondition_(t){const e=this.context.customGenerators.lookup(t);return e?e():[t]}}e.BaseRuleStore=s;class l{constructor(t,e){this.base=t,this._conditions=[],this.constraints=[],this.allCstr={},this.constraints.push(t),this.addCondition(t,e)}get conditions(){return this._conditions}addConstraint(t){if(this.constraints.filter((e=>e.equal(t))).length)return;this.constraints.push(t);const e=[];for(const[r,n]of this.conditions)this.base.equal(r)&&e.push([t,n]);this._conditions=this._conditions.concat(e)}addBaseCondition(t){this.addCondition(this.base,t)}addFullCondition(t){this.constraints.forEach((e=>this.addCondition(e,t)))}addCondition(t,e){const r=t.toString()+" "+e.toString();this.allCstr.condStr||(this.allCstr[r]=!0,this._conditions.push([t,e]))}}e.Condition=l},2469:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.BrailleStore=void 0;const n=r(7630),o=r(9935);class i extends o.MathStore{constructor(){super(...arguments),this.modality="braille",this.customTranscriptions={"\u22ca":"\u2808\u2821\u2833"}}evaluateString(t){const e=[],r=Array.from(t);for(let t=0;t<r.length;t++)e.push(this.evaluateCharacter(r[t]));return e}annotations(){for(let t,e=0;t=this.annotators[e];e++)(0,n.activate)(this.locale,t)}}e.BrailleStore=i},1676:function(t,e){var r;Object.defineProperty(e,"__esModule",{value:!0}),e.DefaultComparator=e.DynamicCstrParser=e.DynamicCstr=e.DynamicProperties=e.Axis=void 0,function(t){t.DOMAIN="domain",t.STYLE="style",t.LOCALE="locale",t.TOPIC="topic",t.MODALITY="modality"}(r=e.Axis||(e.Axis={}));class n{constructor(t,e=Object.keys(t)){this.properties=t,this.order=e}static createProp(...t){const e=o.DEFAULT_ORDER,r={};for(let n=0,o=t.length,i=e.length;n<o&&n<i;n++)r[e[n]]=t[n];return new n(r)}getProperties(){return this.properties}getOrder(){return this.order}getAxes(){return this.order}getProperty(t){return this.properties[t]}updateProperties(t){this.properties=t}allProperties(){const t=[];return this.order.forEach((e=>t.push(this.getProperty(e).slice()))),t}toString(){const t=[];return this.order.forEach((e=>t.push(e+": "+this.getProperty(e).toString()))),t.join("\n")}}e.DynamicProperties=n;class o extends n{constructor(t,e){const r={};for(const[e,n]of Object.entries(t))r[e]=[n];super(r,e),this.components=t}static createCstr(...t){const e=o.DEFAULT_ORDER,r={};for(let n=0,o=t.length,i=e.length;n<o&&n<i;n++)r[e[n]]=t[n];return new o(r)}static defaultCstr(){return o.createCstr.apply(null,o.DEFAULT_ORDER.map((function(t){return o.DEFAULT_VALUES[t]})))}static validOrder(t){const e=o.DEFAULT_ORDER.slice();return t.every((t=>{const r=e.indexOf(t);return-1!==r&&e.splice(r,1)}))}getComponents(){return this.components}getValue(t){return this.components[t]}getValues(){return this.order.map((t=>this.getValue(t)))}allProperties(){const t=super.allProperties();for(let e,r,n=0;e=t[n],r=this.order[n];n++){const t=this.getValue(r);-1===e.indexOf(t)&&e.unshift(t)}return t}toString(){return this.getValues().join(".")}equal(t){const e=t.getAxes();if(this.order.length!==e.length)return!1;for(let r,n=0;r=e[n];n++){const e=this.getValue(r);if(!e||t.getValue(r)!==e)return!1}return!0}}e.DynamicCstr=o,o.DEFAULT_ORDER=[r.LOCALE,r.MODALITY,r.DOMAIN,r.STYLE,r.TOPIC],o.BASE_LOCALE="base",o.DEFAULT_VALUE="default",o.DEFAULT_VALUES={[r.LOCALE]:"en",[r.DOMAIN]:o.DEFAULT_VALUE,[r.STYLE]:o.DEFAULT_VALUE,[r.TOPIC]:o.DEFAULT_VALUE,[r.MODALITY]:"speech"};e.DynamicCstrParser=class{constructor(t){this.order=t}parse(t){const e=t.split("."),r={};if(e.length>this.order.length)throw new Error("Invalid dynamic constraint: "+r);let n=0;for(let t,o=0;t=this.order[o],e.length;o++,n++){const n=e.shift();r[t]=n}return new o(r,this.order.slice(0,n))}};e.DefaultComparator=class{constructor(t,e=new n(t.getProperties(),t.getOrder())){this.reference=t,this.fallback=e,this.order=this.reference.getOrder()}getReference(){return this.reference}setReference(t,e){this.reference=t,this.fallback=e||new n(t.getProperties(),t.getOrder()),this.order=this.reference.getOrder()}match(t){const e=t.getAxes();return e.length===this.reference.getAxes().length&&e.every((e=>{const r=t.getValue(e);return r===this.reference.getValue(e)||-1!==this.fallback.getProperty(e).indexOf(r)}))}compare(t,e){let r=!1;for(let n,o=0;n=this.order[o];o++){const o=t.getValue(n),i=e.getValue(n);if(!r){const t=this.reference.getValue(n);if(t===o&&t!==i)return-1;if(t===i&&t!==o)return 1;if(t===o&&t===i)continue;t!==o&&t!==i&&(r=!0)}const a=this.fallback.getProperty(n),s=a.indexOf(o),l=a.indexOf(i);if(s<l)return-1;if(l<s)return 1}return 0}toString(){return this.reference.toString()+"\n"+this.fallback.toString()}}},2105:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.numbersToAlpha=e.Grammar=e.ATTRIBUTE=void 0;const n=r(5740),o=r(5897),i=r(2536),a=r(4356);e.ATTRIBUTE="grammar";class s{constructor(){this.currentFlags={},this.parameters_={},this.corrections_={},this.preprocessors_={},this.stateStack_=[]}static getInstance(){return s.instance=s.instance||new s,s.instance}static parseInput(t){const e={},r=t.split(":");for(let t=0,n=r.length;t<n;t++){const n=r[t].split("="),o=n[0].trim();n[1]?e[o]=n[1].trim():o.match(/^!/)?e[o.slice(1)]=!1:e[o]=!0}return e}static parseState(t){const e={},r=t.split(" ");for(let t=0,n=r.length;t<n;t++){const n=r[t].split(":"),o=n[0],i=n[1];e[o]=i||!0}return e}static translateString_(t){if(t.match(/:unit$/))return s.translateUnit_(t);const e=o.default.getInstance();let r=e.evaluator(t,e.dynamicCstr);return r=null===r?t:r,s.getInstance().getParameter("plural")&&(r=a.LOCALE.FUNCTIONS.plural(r)),r}static translateUnit_(t){t=s.prepareUnit_(t);const e=o.default.getInstance(),r=s.getInstance().getParameter("plural"),n=e.strict,i=`${e.locale}.${e.modality}.default`;let l,c;return e.strict=!0,r&&(l=e.defaultParser.parse(i+".plural"),c=e.evaluator(t,l)),c?(e.strict=n,c):(l=e.defaultParser.parse(i+".default"),c=e.evaluator(t,l),e.strict=n,c?(r&&(c=a.LOCALE.FUNCTIONS.plural(c)),c):s.cleanUnit_(t))}static prepareUnit_(t){const e=t.match(/:unit$/);return e?t.slice(0,e.index).replace(/\s+/g," ")+t.slice(e.index):t}static cleanUnit_(t){return t.match(/:unit$/)?t.replace(/:unit$/,""):t}clear(){this.parameters_={},this.stateStack_=[]}setParameter(t,e){const r=this.parameters_[t];return e?this.parameters_[t]=e:delete this.parameters_[t],r}getParameter(t){return this.parameters_[t]}setCorrection(t,e){this.corrections_[t]=e}setPreprocessor(t,e){this.preprocessors_[t]=e}getCorrection(t){return this.corrections_[t]}getState(){const t=[];for(const e in this.parameters_){const r=this.parameters_[e];t.push("string"==typeof r?e+":"+r:e)}return t.join(" ")}pushState(t){for(const e in t)t[e]=this.setParameter(e,t[e]);this.stateStack_.push(t)}popState(){const t=this.stateStack_.pop();for(const e in t)this.setParameter(e,t[e])}setAttribute(t){if(t&&t.nodeType===n.NodeType.ELEMENT_NODE){const r=this.getState();r&&t.setAttribute(e.ATTRIBUTE,r)}}preprocess(t){return this.runProcessors_(t,this.preprocessors_)}correct(t){return this.runProcessors_(t,this.corrections_)}apply(t,e){return this.currentFlags=e||{},t=this.currentFlags.adjust||this.currentFlags.preprocess?s.getInstance().preprocess(t):t,(this.parameters_.translate||this.currentFlags.translate)&&(t=s.translateString_(t)),t=this.currentFlags.adjust||this.currentFlags.correct?s.getInstance().correct(t):t,this.currentFlags={},t}runProcessors_(t,e){for(const r in this.parameters_){const n=e[r];if(!n)continue;const o=this.parameters_[r];t=!0===o?n(t):n(t,o)}return t}}function l(t,e){if(!e||!t)return t;const r=a.LOCALE.FUNCTIONS.fontRegexp(i.localFont(e));return t.replace(r,"")}function c(t){return t.match(/\d+/)?a.LOCALE.NUMBERS.numberToWords(parseInt(t,10)):t}e.Grammar=s,e.numbersToAlpha=c,s.getInstance().setCorrection("localFont",i.localFont),s.getInstance().setCorrection("localRole",i.localRole),s.getInstance().setCorrection("localEnclose",i.localEnclose),s.getInstance().setCorrection("ignoreFont",l),s.getInstance().setPreprocessor("annotation",(function(t,e){return t+":"+e})),s.getInstance().setPreprocessor("noTranslateText",(function(t){return t.match(new RegExp("^["+a.LOCALE.MESSAGES.regexp.TEXT+"]+$"))&&(s.getInstance().currentFlags.translate=!1),t})),s.getInstance().setCorrection("ignoreCaps",(function(t){let e=a.LOCALE.ALPHABETS.capPrefix[o.default.getInstance().domain];return void 0===e&&(e=a.LOCALE.ALPHABETS.capPrefix.default),l(t,e)})),s.getInstance().setPreprocessor("numbers2alpha",c)},2780:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.enumerate=e.lookupString=e.lookupCategory=e.lookupRule=e.addSiUnitRules=e.addUnitRules=e.addFunctionRules=e.addSymbolRules=e.defineRule=e.defineRules=e.setSiPrefixes=void 0;const n=r(2057),o=r(5897),i=r(7491),a=r(4658),s=r(1676);let l=s.DynamicCstr.DEFAULT_VALUES[s.Axis.LOCALE],c=s.DynamicCstr.DEFAULT_VALUES[s.Axis.MODALITY],u={};e.setSiPrefixes=function(t){u=t};const p={};function h(t,e,r,n){const o=_(e);S(o,r),o.defineRulesFromMappings(t,l,c,e,n)}function d(t){if(v(t))return;const e=t.names,r=t.mappings,n=t.category;for(let t,o=0;t=e[o];o++)h(t,t,n,r)}function f(t){for(const e of Object.keys(u)){const r=Object.assign({},t);r.mappings={};const n=u[e];r.key=e+r.key,r.names=r.names.map((function(t){return e+t}));for(const e of Object.keys(t.mappings)){r.mappings[e]={};for(const o of Object.keys(t.mappings[e]))r.mappings[e][o]=i.locales[l]().FUNCTIONS.si(n,t.mappings[e][o])}b(r)}b(t)}function m(t,e){const r=p[t];return r?r.lookupRule(null,e):null}function y(t,e){const r=m(t,e);return r?r.action:null}function g(t,e){return e=e||{},t.length?(e[t[0]]=g(t.slice(1),e[t[0]]),e):e}function b(t){const e=t.names;e&&(t.names=e.map((function(t){return t+":unit"}))),d(t)}function v(t){return!(!t.locale&&!t.modality)&&(l=t.locale||l,c=t.modality||c,!0)}function _(t){let e=p[t];return e?(n.Debugger.getInstance().output("Store exists! "+t),e):(e=new a.MathSimpleStore,p[t]=e,e)}function S(t,e){e&&(t.category=e)}e.defineRules=h,e.defineRule=function(t,e,r,n,o,i){const a=_(o);S(a,n),a.defineRuleFromStrings(t,l,c,e,r,o,i)},e.addSymbolRules=function(t){if(v(t))return;const e=a.MathSimpleStore.parseUnicode(t.key);h(t.key,e,t.category,t.mappings)},e.addFunctionRules=d,e.addUnitRules=function(t){v(t)||(t.si?f(t):b(t))},e.addSiUnitRules=f,e.lookupRule=m,e.lookupCategory=function(t){const e=p[t];return e?e.category:""},e.lookupString=y,o.default.getInstance().evaluator=y,e.enumerate=function(t={}){for(const e of Object.values(p))for(const[,r]of e.rules.entries())for(const{cstr:e}of r)t=g(e.getValues(),t);return t}},4658:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.MathSimpleStore=void 0;const n=r(5897),o=r(1676);class i{constructor(){this.category="",this.rules=new Map}static parseUnicode(t){const e=parseInt(t,16);return String.fromCodePoint(e)}static testDynamicConstraints_(t,e){return n.default.getInstance().strict?e.cstr.equal(t):n.default.getInstance().comparator.match(e.cstr)}defineRulesFromMappings(t,e,r,n,o){for(const i in o)for(const a in o[i]){const s=o[i][a];this.defineRuleFromStrings(t,e,r,i,a,n,s)}}getRules(t){let e=this.rules.get(t);return e||(e=[],this.rules.set(t,e)),e}defineRuleFromStrings(t,e,r,o,i,a,s){let l=this.getRules(e);const c=n.default.getInstance().parsers[o]||n.default.getInstance().defaultParser,u=n.default.getInstance().comparators[o],p=`${e}.${r}.${o}.${i}`,h=c.parse(p),d=u?u():n.default.getInstance().comparator,f=d.getReference();d.setReference(h);const m={cstr:h,action:s};l=l.filter((t=>!h.equal(t.cstr))),l.push(m),this.rules.set(e,l),d.setReference(f)}lookupRule(t,e){let r=this.getRules(e.getValue(o.Axis.LOCALE));return r=r.filter((function(t){return i.testDynamicConstraints_(e,t)})),1===r.length?r[0]:r.length?r.sort(((t,e)=>n.default.getInstance().comparator.compare(t.cstr,e.cstr)))[0]:null}}e.MathSimpleStore=i},9935:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.MathStore=void 0;const n=r(707),o=r(4356),i=r(7630),a=r(4504),s=r(4650);class l extends a.BaseRuleStore{constructor(){super(),this.annotators=[],this.parseMethods.Alias=this.defineAlias,this.parseMethods.SpecializedRule=this.defineSpecializedRule,this.parseMethods.Specialized=this.defineSpecialized}initialize(){this.initialized||(this.annotations(),this.initialized=!0)}annotations(){for(let t,e=0;t=this.annotators[e];e++)(0,i.activate)(this.domain,t)}defineAlias(t,e,...r){const n=this.parsePrecondition(e,r);if(!n)return void console.error(`Precondition Error: ${e} ${r}`);const o=this.preconditions.get(t);o?o.addFullCondition(n):console.error(`Alias Error: No precondition by the name of ${t}`)}defineRulesAlias(t,e,...r){const n=this.findAllRules((function(e){return e.name===t}));if(0===n.length)throw new s.OutputError("Rule with name "+t+" does not exist.");const o=[];n.forEach((t=>{(t=>{const e=t.dynamicCstr.toString(),r=t.action.toString();for(let t,n=0;t=o[n];n++)if(t.action===r&&t.cstr===e)return!1;return o.push({cstr:e,action:r}),!0})(t)&&this.addAlias_(t,e,r)}))}defineSpecializedRule(t,e,r,n){const o=this.parseCstr(e),i=this.findRule((e=>e.name===t&&o.equal(e.dynamicCstr))),a=this.parseCstr(r);if(!i&&n)throw new s.OutputError("Rule named "+t+" with style "+e+" does not exist.");const l=n?s.Action.fromString(n):i.action,c=new s.SpeechRule(i.name,a,i.precondition,l);this.addRule(c)}defineSpecialized(t,e,r){const n=this.parseCstr(r);if(!n)return void console.error(`Dynamic Constraint Error: ${r}`);const o=this.preconditions.get(t);o?o.addConstraint(n):console.error(`Alias Error: No precondition by the name of ${t}`)}evaluateString(t){const e=[];if(t.match(/^\s+$/))return e;let r=this.matchNumber_(t);if(r&&r.length===t.length)return e.push(this.evaluateCharacter(r.number)),e;const i=n.removeEmpty(t.replace(/\s/g," ").split(" "));for(let t,n=0;t=i[n];n++)if(1===t.length)e.push(this.evaluateCharacter(t));else if(t.match(new RegExp("^["+o.LOCALE.MESSAGES.regexp.TEXT+"]+$")))e.push(this.evaluateCharacter(t));else{let n=t;for(;n;){r=this.matchNumber_(n);const t=n.match(new RegExp("^["+o.LOCALE.MESSAGES.regexp.TEXT+"]+"));if(r)e.push(this.evaluateCharacter(r.number)),n=n.substring(r.length);else if(t)e.push(this.evaluateCharacter(t[0])),n=n.substring(t[0].length);else{const t=Array.from(n),r=t[0];e.push(this.evaluateCharacter(r)),n=t.slice(1).join("")}}}return e}parse(t){super.parse(t),this.annotators=t.annotators||[]}addAlias_(t,e,r){const n=this.parsePrecondition(e,r),o=new s.SpeechRule(t.name,t.dynamicCstr,n,t.action);o.name=t.name,this.addRule(o)}matchNumber_(t){const e=t.match(new RegExp("^"+o.LOCALE.MESSAGES.regexp.NUMBER)),r=t.match(new RegExp("^"+l.regexp.NUMBER));if(!e&&!r)return null;const n=r&&r[0]===t;if(e&&e[0]===t||!n)return e?{number:e[0],length:e[0].length}:null;return{number:r[0].replace(new RegExp(l.regexp.DIGIT_GROUP,"g"),"X").replace(new RegExp(l.regexp.DECIMAL_MARK,"g"),o.LOCALE.MESSAGES.regexp.DECIMAL_MARK).replace(/X/g,o.LOCALE.MESSAGES.regexp.DIGIT_GROUP.replace(/\\/g,"")),length:r[0].length}}}e.MathStore=l,l.regexp={NUMBER:"((\\d{1,3})(?=(,| ))((,| )\\d{3})*(\\.\\d+)?)|^\\d*\\.\\d+|^\\d+",DECIMAL_MARK:"\\.",DIGIT_GROUP:","}},4650:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.OutputError=e.Precondition=e.Action=e.Component=e.ActionType=e.SpeechRule=void 0;const n=r(5897),o=r(2105);var i;function a(t){switch(t){case"[n]":return i.NODE;case"[m]":return i.MULTI;case"[t]":return i.TEXT;case"[p]":return i.PERSONALITY;default:throw"Parse error: "+t}}e.SpeechRule=class{constructor(t,e,r,n){this.name=t,this.dynamicCstr=e,this.precondition=r,this.action=n,this.context=null}toString(){return this.name+" | "+this.dynamicCstr.toString()+" | "+this.precondition.toString()+" ==> "+this.action.toString()}},function(t){t.NODE="NODE",t.MULTI="MULTI",t.TEXT="TEXT",t.PERSONALITY="PERSONALITY"}(i=e.ActionType||(e.ActionType={}));class s{constructor({type:t,content:e,attributes:r,grammar:n}){this.type=t,this.content=e,this.attributes=r,this.grammar=n}static grammarFromString(t){return o.Grammar.parseInput(t)}static fromString(t){const e={type:a(t.substring(0,3))};let r=t.slice(3).trim();if(!r)throw new u("Missing content.");switch(e.type){case i.TEXT:if('"'===r[0]){const t=p(r,"\\(")[0].trim();if('"'!==t.slice(-1))throw new u("Invalid string syntax.");e.content=t,r=r.slice(t.length).trim(),-1===r.indexOf("(")&&(r="");break}case i.NODE:case i.MULTI:{const t=r.indexOf(" (");if(-1===t){e.content=r.trim(),r="";break}e.content=r.substring(0,t).trim(),r=r.slice(t).trim()}}if(r){const t=s.attributesFromString(r);t.grammar&&(e.grammar=t.grammar,delete t.grammar),Object.keys(t).length&&(e.attributes=t)}return new s(e)}static attributesFromString(t){if("("!==t[0]||")"!==t.slice(-1))throw new u("Invalid attribute expression: "+t);const e={},r=p(t.slice(1,-1),",");for(let t=0,n=r.length;t<n;t++){const n=r[t],i=n.indexOf(":");if(-1===i)e[n.trim()]="true";else{const t=n.substring(0,i).trim(),r=n.slice(i+1).trim();e[t]=t===o.ATTRIBUTE?s.grammarFromString(r):r}}return e}toString(){let t="";t+=function(t){switch(t){case i.NODE:return"[n]";case i.MULTI:return"[m]";case i.TEXT:return"[t]";case i.PERSONALITY:return"[p]";default:throw"Unknown type error: "+t}}(this.type),t+=this.content?" "+this.content:"";const e=this.attributesToString();return t+=e?" "+e:"",t}grammarToString(){return this.getGrammar().join(":")}getGrammar(){const t=[];for(const e in this.grammar)!0===this.grammar[e]?t.push(e):!1===this.grammar[e]?t.push("!"+e):t.push(e+"="+this.grammar[e]);return t}attributesToString(){const t=this.getAttributes(),e=this.grammarToString();return e&&t.push("grammar:"+e),t.length>0?"("+t.join(", ")+")":""}getAttributes(){const t=[];for(const e in this.attributes){const r=this.attributes[e];"true"===r?t.push(e):t.push(e+":"+r)}return t}}e.Component=s;class l{constructor(t){this.components=t}static fromString(t){const e=p(t,";").filter((function(t){return t.match(/\S/)})).map((function(t){return t.trim()})),r=[];for(let t=0,n=e.length;t<n;t++){const n=s.fromString(e[t]);n&&r.push(n)}return new l(r)}toString(){return this.components.map((function(t){return t.toString()})).join("; ")}}e.Action=l;class c{constructor(t,...e){this.query=t,this.constraints=e;const[r,n]=this.presetPriority();this.priority=r?n:this.calculatePriority()}static constraintValue(t,e){for(let r,n=0;r=e[n];n++)if(t.match(r))return++n;return 0}toString(){const t=this.constraints.join(", ");return`${this.query}, ${t} (${this.priority}, ${this.rank})`}calculatePriority(){const t=c.constraintValue(this.query,c.queryPriorities);if(!t)return 0;const e=this.query.match(/^self::.+\[(.+)\]/)[1];return 100*t+10*c.constraintValue(e,c.attributePriorities)}presetPriority(){if(!this.constraints.length)return[!1,0];const t=this.constraints[this.constraints.length-1].match(/^priority=(.*$)/);if(!t)return[!1,0];this.constraints.pop();const e=parseFloat(t[1]);return[!0,isNaN(e)?0:e]}}e.Precondition=c,c.queryPriorities=[/^self::\*\[.+\]$/,/^self::[\w-]+\[.+\]$/],c.attributePriorities=[/^@[\w-]+$/,/^@[\w-]+!=".+"$/,/^not\(contains\(@[\w-]+,\s*".+"\)\)$/,/^contains\(@[\w-]+,".+"\)$/,/^@[\w-]+=".+"$/];class u extends n.SREError{constructor(t){super(t),this.name="RuleError"}}function p(t,e){const r=[];let n="";for(;""!==t;){const o=t.search(e);if(-1===o){if((t.match(/"/g)||[]).length%2!=0)throw new u("Invalid string in expression: "+t);r.push(n+t),n="",t=""}else if((t.substring(0,o).match(/"/g)||[]).length%2==0)r.push(n+t.substring(0,o)),n="",t=t.substring(o+1);else{const e=t.substring(o).search('"');if(-1===e)throw new u("Invalid string in expression: "+t);n+=t.substring(0,o+e+1),t=t.substring(o+e+1)}}return n&&r.push(n),r}e.OutputError=u},4106:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SpeechRuleContext=void 0;const n=r(5274),o=r(5662);e.SpeechRuleContext=class{constructor(){this.customQueries=new o.CustomQueries,this.customStrings=new o.CustomStrings,this.contextFunctions=new o.ContextFunctions,this.customGenerators=new o.CustomGenerators}applyCustomQuery(t,e){const r=this.customQueries.lookup(e);return r?r(t):null}applySelector(t,e){return this.applyCustomQuery(t,e)||n.evalXPath(e,t)}applyQuery(t,e){const r=this.applySelector(t,e);return r.length>0?r[0]:null}applyConstraint(t,e){return!!this.applyQuery(t,e)||n.evaluateBoolean(e,t)}constructString(t,e){if(!e)return"";if('"'===e.charAt(0))return e.slice(1,-1);const r=this.customStrings.lookup(e);return r?r(t):n.evaluateString(e,t)}parse(t){const e=Array.isArray(t)?t:Object.entries(t);for(let t,r=0;t=e[r];r++){switch(t[0].slice(0,3)){case"CQF":this.customQueries.add(t[0],t[1]);break;case"CSF":this.customStrings.add(t[0],t[1]);break;case"CTF":this.contextFunctions.add(t[0],t[1]);break;case"CGF":this.customGenerators.add(t[0],t[1]);break;default:console.error("FunctionError: Invalid function name "+t[0])}}}}},2362:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.storeFactory=e.SpeechRuleEngine=void 0;const n=r(7052),o=r(2057),i=r(5740),a=r(5897),s=r(4440),l=r(5274),c=r(7283),u=r(7599),p=r(2469),h=r(1676),d=r(2105),f=r(9935),m=r(4650),y=r(4508);class g{constructor(){this.trie=null,this.evaluators_={},this.trie=new y.Trie}static getInstance(){return g.instance=g.instance||new g,g.instance}static debugSpeechRule(t,e){const r=t.precondition,n=t.context.applyQuery(e,r.query);o.Debugger.getInstance().output(r.query,n?n.toString():n),r.constraints.forEach((r=>o.Debugger.getInstance().output(r,t.context.applyConstraint(e,r))))}static debugNamedSpeechRule(t,e){const r=g.getInstance().trie.collectRules().filter((e=>e.name==t));for(let n,i=0;n=r[i];i++)o.Debugger.getInstance().output("Rule",t,"DynamicCstr:",n.dynamicCstr.toString(),"number",i),g.debugSpeechRule(n,e)}evaluateNode(t){(0,l.updateEvaluator)(t);const e=(new Date).getTime();let r=[];try{r=this.evaluateNode_(t)}catch(t){console.error("Something went wrong computing speech."),o.Debugger.getInstance().output(t)}const n=(new Date).getTime();return o.Debugger.getInstance().output("Time:",n-e),r}toString(){return this.trie.collectRules().map((t=>t.toString())).join("\n")}runInSetting(t,e){const r=a.default.getInstance(),n={};for(const e in t)n[e]=r[e],r[e]=t[e];r.setDynamicCstr();const o=e();for(const t in n)r[t]=n[t];return r.setDynamicCstr(),o}addStore(t){const e=v(t);"abstract"!==e.kind&&e.getSpeechRules().forEach((t=>this.trie.addRule(t))),this.addEvaluator(e)}processGrammar(t,e,r){const n={};for(const o in r){const i=r[o];n[o]="string"==typeof i?t.constructString(e,i):i}d.Grammar.getInstance().pushState(n)}addEvaluator(t){const e=t.evaluateDefault.bind(t),r=this.evaluators_[t.locale];if(r)return void(r[t.modality]=e);const n={};n[t.modality]=e,this.evaluators_[t.locale]=n}getEvaluator(t,e){const r=this.evaluators_[t]||this.evaluators_[h.DynamicCstr.DEFAULT_VALUES[h.Axis.LOCALE]];return r[e]||r[h.DynamicCstr.DEFAULT_VALUES[h.Axis.MODALITY]]}enumerate(t){return this.trie.enumerate(t)}evaluateNode_(t){return t?(this.updateConstraint_(),this.evaluateTree_(t)):[]}evaluateTree_(t){const e=a.default.getInstance();let r;o.Debugger.getInstance().output(e.mode!==s.Mode.HTTP?t.toString():t),d.Grammar.getInstance().setAttribute(t);const i=this.lookupRule(t,e.dynamicCstr);if(!i)return e.strict?[]:(r=this.getEvaluator(e.locale,e.modality)(t),t.attributes&&this.addPersonality_(r,{},!1,t),r);o.Debugger.getInstance().generateOutput((()=>["Apply Rule:",i.name,i.dynamicCstr.toString(),(e.mode,s.Mode.HTTP,t).toString()]));const c=i.context,u=i.action.components;r=[];for(let e,o=0;e=u[o];o++){let o=[];const i=e.content||"",s=e.attributes||{};let u=!1;e.grammar&&this.processGrammar(c,t,e.grammar);let p=null;if(s.engine){p=a.default.getInstance().dynamicCstr.getComponents();const t=d.Grammar.parseInput(s.engine);a.default.getInstance().setDynamicCstr(t)}switch(e.type){case m.ActionType.NODE:{const e=c.applyQuery(t,i);e&&(o=this.evaluateTree_(e))}break;case m.ActionType.MULTI:{u=!0;const e=c.applySelector(t,i);e.length>0&&(o=this.evaluateNodeList_(c,e,s.sepFunc,c.constructString(t,s.separator),s.ctxtFunc,c.constructString(t,s.context)))}break;case m.ActionType.TEXT:{const e=s.span,r={};if(e){const n=(0,l.evalXPath)(e,t);n.length&&(r.extid=n[0].getAttribute("extid"))}const a=c.constructString(t,i);(a||""===a)&&(o=Array.isArray(a)?a.map((function(t){return n.AuditoryDescription.create({text:t.speech,attributes:t.attributes},{adjust:!0})})):[n.AuditoryDescription.create({text:a,attributes:r},{adjust:!0})])}break;case m.ActionType.PERSONALITY:default:o=[n.AuditoryDescription.create({text:i})]}o[0]&&!u&&(s.context&&(o[0].context=c.constructString(t,s.context)+(o[0].context||"")),s.annotation&&(o[0].annotation=s.annotation)),this.addLayout(o,s,u),e.grammar&&d.Grammar.getInstance().popState(),r=r.concat(this.addPersonality_(o,s,u,t)),p&&a.default.getInstance().setDynamicCstr(p)}return r}evaluateNodeList_(t,e,r,o,i,a){if(!e.length)return[];const s=o||"",l=a||"",c=t.contextFunctions.lookup(i),u=c?c(e,l):function(){return l},p=t.contextFunctions.lookup(r),h=p?p(e,s):function(){return[n.AuditoryDescription.create({text:s},{translate:!0})]};let d=[];for(let t,r=0;t=e[r];r++){const n=this.evaluateTree_(t);if(n.length>0&&(n[0].context=u()+(n[0].context||""),d=d.concat(n),r<e.length-1)){const t=h();d=d.concat(t)}}return d}addLayout(t,e,r){const o=e.layout;o&&(o.match(/^begin/)?t.unshift(new n.AuditoryDescription({text:"",layout:o})):o.match(/^end/)?t.push(new n.AuditoryDescription({text:"",layout:o})):(t.unshift(new n.AuditoryDescription({text:"",layout:`begin${o}`})),t.push(new n.AuditoryDescription({text:"",layout:`end${o}`}))))}addPersonality_(t,e,r,o){const i={};let a=null;for(const t of s.personalityPropList){const r=e[t];if(void 0===r)continue;const n=parseFloat(r),o=isNaN(n)?'"'===r.charAt(0)?r.slice(1,-1):r:n;t===s.personalityProps.PAUSE?a=o:i[t]=o}for(let e,r=0;e=t[r];r++)this.addRelativePersonality_(e,i),this.addExternalAttributes_(e,o);if(r&&t.length&&delete t[t.length-1].personality[s.personalityProps.JOIN],a&&t.length){const e=t[t.length-1];e.text||Object.keys(e.personality).length?t.push(n.AuditoryDescription.create({text:"",personality:{pause:a}})):e.personality[s.personalityProps.PAUSE]=a}return t}addExternalAttributes_(t,e){if(e.hasAttributes()){const r=e.attributes;for(let e=r.length-1;e>=0;e--){const n=r[e].name;!t.attributes[n]&&n.match(/^ext/)&&(t.attributes[n]=r[e].value)}}}addRelativePersonality_(t,e){if(!t.personality)return t.personality=e,t;const r=t.personality;for(const t in e)r[t]&&"number"==typeof r[t]&&"number"==typeof e[t]?r[t]=r[t]+e[t]:r[t]||(r[t]=e[t]);return t}updateConstraint_(){const t=a.default.getInstance().dynamicCstr,e=a.default.getInstance().strict,r=this.trie,n={};let o=t.getValue(h.Axis.LOCALE),i=t.getValue(h.Axis.MODALITY),s=t.getValue(h.Axis.DOMAIN);r.hasSubtrie([o,i,s])||(s=h.DynamicCstr.DEFAULT_VALUES[h.Axis.DOMAIN],r.hasSubtrie([o,i,s])||(i=h.DynamicCstr.DEFAULT_VALUES[h.Axis.MODALITY],r.hasSubtrie([o,i,s])||(o=h.DynamicCstr.DEFAULT_VALUES[h.Axis.LOCALE]))),n[h.Axis.LOCALE]=[o],n[h.Axis.MODALITY]=["summary"!==i?i:h.DynamicCstr.DEFAULT_VALUES[h.Axis.MODALITY]],n[h.Axis.DOMAIN]=["speech"!==i?h.DynamicCstr.DEFAULT_VALUES[h.Axis.DOMAIN]:s];const l=t.getOrder();for(let r,o=0;r=l[o];o++)if(!n[r]){const o=t.getValue(r),i=this.makeSet_(o,t.preference),a=h.DynamicCstr.DEFAULT_VALUES[r];e||o===a||i.push(a),n[r]=i}t.updateProperties(n)}makeSet_(t,e){return e&&Object.keys(e).length?t.split(":"):[t]}lookupRule(t,e){if(!t||t.nodeType!==i.NodeType.ELEMENT_NODE&&t.nodeType!==i.NodeType.TEXT_NODE)return null;const r=this.lookupRules(t,e);return r.length>0?this.pickMostConstraint_(e,r):null}lookupRules(t,e){return this.trie.lookupRules(t,e.allProperties())}pickMostConstraint_(t,e){const r=a.default.getInstance().comparator;return e.sort((function(t,e){return r.compare(t.dynamicCstr,e.dynamicCstr)||e.precondition.priority-t.precondition.priority||e.precondition.constraints.length-t.precondition.constraints.length||e.precondition.rank-t.precondition.rank})),o.Debugger.getInstance().generateOutput((()=>e.map((t=>t.name+"("+t.dynamicCstr.toString()+")"))).bind(this)),e[0]}}e.SpeechRuleEngine=g;const b=new Map;function v(t){const e=`${t.locale}.${t.modality}.${t.domain}`;if("actions"===t.kind){const r=b.get(e);return r.parse(t),r}u.init(),t&&!t.functions&&(t.functions=c.getStore(t.locale,t.modality,t.domain));const r="braille"===t.modality?new p.BrailleStore:new f.MathStore;return b.set(e,r),t.inherits&&(r.inherits=b.get(`${t.inherits}.${t.modality}.${t.domain}`)),r.parse(t),r.initialize(),r}e.storeFactory=v,a.default.nodeEvaluator=g.getInstance().evaluateNode.bind(g.getInstance())},5662:function(t,e){Object.defineProperty(e,"__esModule",{value:!0}),e.CustomGenerators=e.ContextFunctions=e.CustomStrings=e.CustomQueries=void 0;class r{constructor(t,e){this.prefix=t,this.store=e}add(t,e){this.checkCustomFunctionSyntax_(t)&&(this.store[t]=e)}addStore(t){const e=Object.keys(t.store);for(let r,n=0;r=e[n];n++)this.add(r,t.store[r])}lookup(t){return this.store[t]}checkCustomFunctionSyntax_(t){const e=new RegExp("^"+this.prefix);return!!t.match(e)||(console.error("FunctionError: Invalid function name. 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e.type="row",e.role="cycle",e.textContent="",e.contentNodes=[],t}),(t=>"fenced"===t.type&&"infixop"===t.childNodes[0].type&&"implicit"===t.childNodes[0].role&&t.childNodes[0].childNodes.every((function(t){return"number"===t.type}))&&t.childNodes[0].contentNodes.every((function(t){return"space"===t.role})))))},7228:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticMathml=void 0;const n=r(5740),o=r(5250),i=r(5609),a=r(3308),s=r(4795);class l extends o.SemanticAbstractParser{constructor(){super("MathML"),this.parseMap_={SEMANTICS:this.semantics_.bind(this),MATH:this.rows_.bind(this),MROW:this.rows_.bind(this),MPADDED:this.rows_.bind(this),MSTYLE:this.rows_.bind(this),MFRAC:this.fraction_.bind(this),MSUB:this.limits_.bind(this),MSUP:this.limits_.bind(this),MSUBSUP:this.limits_.bind(this),MOVER:this.limits_.bind(this),MUNDER:this.limits_.bind(this),MUNDEROVER:this.limits_.bind(this),MROOT:this.root_.bind(this),MSQRT:this.sqrt_.bind(this),MTABLE:this.table_.bind(this),MLABELEDTR:this.tableLabeledRow_.bind(this),MTR:this.tableRow_.bind(this),MTD:this.tableCell_.bind(this),MS:this.text_.bind(this),MTEXT:this.text_.bind(this),MSPACE:this.space_.bind(this),"ANNOTATION-XML":this.text_.bind(this),MI:this.identifier_.bind(this),MN:this.number_.bind(this),MO:this.operator_.bind(this),MFENCED:this.fenced_.bind(this),MENCLOSE:this.enclosed_.bind(this),MMULTISCRIPTS:this.multiscripts_.bind(this),ANNOTATION:this.empty_.bind(this),NONE:this.empty_.bind(this),MACTION:this.action_.bind(this)};const t={type:"identifier",role:"numbersetletter",font:"double-struck"};["C","H","N","P","Q","R","Z","\u2102","\u210d","\u2115","\u2119","\u211a","\u211d","\u2124"].forEach((e=>this.getFactory().defaultMap.add(e,t)).bind(this))}static getAttribute_(t,e,r){if(!t.hasAttribute(e))return r;const n=t.getAttribute(e);return n.match(/^\s*$/)?null:n}parse(t){a.default.getInstance().setNodeFactory(this.getFactory());const e=n.toArray(t.childNodes),r=n.tagName(t),o=this.parseMap_[r],i=(o||this.dummy_.bind(this))(t,e);return s.addAttributes(i,t),-1!==["MATH","MROW","MPADDED","MSTYLE","SEMANTICS"].indexOf(r)||(i.mathml.unshift(t),i.mathmlTree=t),i}semantics_(t,e){return e.length?this.parse(e[0]):this.getFactory().makeEmptyNode()}rows_(t,e){const r=t.getAttribute("semantics");if(r&&r.match("bspr_"))return a.default.proof(t,r,this.parseList.bind(this));let n;return 1===(e=s.purgeNodes(e)).length?(n=this.parse(e[0]),"empty"!==n.type||n.mathmlTree||(n.mathmlTree=t)):n=a.default.getInstance().row(this.parseList(e)),n.mathml.unshift(t),n}fraction_(t,e){if(!e.length)return this.getFactory().makeEmptyNode();const r=this.parse(e[0]),n=e[1]?this.parse(e[1]):this.getFactory().makeEmptyNode();return a.default.getInstance().fractionLikeNode(r,n,t.getAttribute("linethickness"),"true"===t.getAttribute("bevelled"))}limits_(t,e){return a.default.getInstance().limitNode(n.tagName(t),this.parseList(e))}root_(t,e){return e[1]?this.getFactory().makeBranchNode("root",[this.parse(e[1]),this.parse(e[0])],[]):this.sqrt_(t,e)}sqrt_(t,e){const r=this.parseList(s.purgeNodes(e));return this.getFactory().makeBranchNode("sqrt",[a.default.getInstance().row(r)],[])}table_(t,e){const r=t.getAttribute("semantics");if(r&&r.match("bspr_"))return a.default.proof(t,r,this.parseList.bind(this));const n=this.getFactory().makeBranchNode("table",this.parseList(e),[]);return n.mathmlTree=t,a.default.tableToMultiline(n),n}tableRow_(t,e){const r=this.getFactory().makeBranchNode("row",this.parseList(e),[]);return r.role="table",r}tableLabeledRow_(t,e){if(!e.length)return this.tableRow_(t,e);const r=this.parse(e[0]);r.role="label";const n=this.getFactory().makeBranchNode("row",this.parseList(e.slice(1)),[r]);return n.role="table",n}tableCell_(t,e){const r=this.parseList(s.purgeNodes(e));let n;n=r.length?1===r.length&&i.isType(r[0],"empty")?r:[a.default.getInstance().row(r)]:[];const o=this.getFactory().makeBranchNode("cell",n,[]);return o.role="table",o}space_(t,e){const r=t.getAttribute("width"),o=r&&r.match(/[a-z]*$/);if(!o)return this.empty_(t,e);const i=o[0],s=parseFloat(r.slice(0,o.index)),l={cm:.4,pc:.5,em:.5,ex:1,in:.15,pt:5,mm:5}[i];if(!l||isNaN(s)||s<l)return this.empty_(t,e);const c=this.getFactory().makeUnprocessed(t);return a.default.getInstance().text(c,n.tagName(t))}text_(t,e){const r=this.leaf_(t,e);return t.textContent?(r.updateContent(t.textContent,!0),a.default.getInstance().text(r,n.tagName(t))):r}identifier_(t,e){const r=this.leaf_(t,e);return a.default.getInstance().identifierNode(r,a.default.getInstance().font(t.getAttribute("mathvariant")),t.getAttribute("class"))}number_(t,e){const r=this.leaf_(t,e);return a.default.number(r),r}operator_(t,e){const r=this.leaf_(t,e);return a.default.getInstance().operatorNode(r),r}fenced_(t,e){const r=this.parseList(s.purgeNodes(e)),n=l.getAttribute_(t,"separators",","),o=l.getAttribute_(t,"open","("),i=l.getAttribute_(t,"close",")"),c=a.default.getInstance().mfenced(o,i,n,r);return a.default.getInstance().tablesInRow([c])[0]}enclosed_(t,e){const r=this.parseList(s.purgeNodes(e)),n=this.getFactory().makeBranchNode("enclose",[a.default.getInstance().row(r)],[]);return n.role=t.getAttribute("notation")||"unknown",n}multiscripts_(t,e){if(!e.length)return this.getFactory().makeEmptyNode();const r=this.parse(e.shift());if(!e.length)return r;const o=[],i=[],l=[],c=[];let u=!1,p=0;for(let t,r=0;t=e[r];r++)"MPRESCRIPTS"!==n.tagName(t)?(u?1&p?o.push(t):i.push(t):1&p?l.push(t):c.push(t),p++):(u=!0,p=0);return s.purgeNodes(o).length||s.purgeNodes(i).length?a.default.getInstance().tensor(r,this.parseList(i),this.parseList(o),this.parseList(c),this.parseList(l)):a.default.getInstance().pseudoTensor(r,this.parseList(c),this.parseList(l))}empty_(t,e){return this.getFactory().makeEmptyNode()}action_(t,e){return e.length>1?this.parse(e[1]):this.getFactory().makeUnprocessed(t)}dummy_(t,e){const r=this.getFactory().makeUnprocessed(t);return r.role=t.tagName,r.textContent=t.textContent,r}leaf_(t,e){if(1===e.length&&e[0].nodeType!==n.NodeType.TEXT_NODE){const r=this.getFactory().makeUnprocessed(t);return r.role=e[0].tagName,s.addAttributes(r,e[0]),r}return this.getFactory().makeLeafNode(t.textContent,a.default.getInstance().font(t.getAttribute("mathvariant")))}}e.SemanticMathml=l},5952:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticNode=void 0;const n=r(5740),o=r(3588),i=r(4795);class a{constructor(t){this.id=t,this.mathml=[],this.parent=null,this.type="unknown",this.role="unknown",this.font="unknown",this.embellished=null,this.fencePointer="",this.childNodes=[],this.textContent="",this.mathmlTree=null,this.contentNodes=[],this.annotation={},this.attributes={},this.nobreaking=!1}static fromXml(t){const e=parseInt(t.getAttribute("id"),10),r=new a(e);return r.type=t.tagName,a.setAttribute(r,t,"role"),a.setAttribute(r,t,"font"),a.setAttribute(r,t,"embellished"),a.setAttribute(r,t,"fencepointer","fencePointer"),t.getAttribute("annotation")&&r.parseAnnotation(t.getAttribute("annotation")),i.addAttributes(r,t),a.processChildren(r,t),r}static setAttribute(t,e,r,n){n=n||r;const o=e.getAttribute(r);o&&(t[n]=o)}static processChildren(t,e){for(const r of n.toArray(e.childNodes)){if(r.nodeType===n.NodeType.TEXT_NODE){t.textContent=r.textContent;continue}const e=n.toArray(r.childNodes).map(a.fromXml);e.forEach((e=>e.parent=t)),"CONTENT"===n.tagName(r)?t.contentNodes=e:t.childNodes=e}}querySelectorAll(t){let e=[];for(let r,n=0;r=this.childNodes[n];n++)e=e.concat(r.querySelectorAll(t));for(let r,n=0;r=this.contentNodes[n];n++)e=e.concat(r.querySelectorAll(t));return t(this)&&e.unshift(this),e}xml(t,e){const r=function(r,n){const o=n.map((function(r){return r.xml(t,e)})),i=t.createElementNS("",r);for(let t,e=0;t=o[e];e++)i.appendChild(t);return i},n=t.createElementNS("",this.type);return e||this.xmlAttributes(n),n.textContent=this.textContent,this.contentNodes.length>0&&n.appendChild(r("content",this.contentNodes)),this.childNodes.length>0&&n.appendChild(r("children",this.childNodes)),n}toString(t=!1){const e=n.parseInput("<snode/>");return n.serializeXml(this.xml(e,t))}allAttributes(){const t=[];return t.push(["role",this.role]),"unknown"!==this.font&&t.push(["font",this.font]),Object.keys(this.annotation).length&&t.push(["annotation",this.xmlAnnotation()]),this.embellished&&t.push(["embellished",this.embellished]),this.fencePointer&&t.push(["fencepointer",this.fencePointer]),t.push(["id",this.id.toString()]),t}xmlAnnotation(){const t=[];for(const e in this.annotation)this.annotation[e].forEach((function(r){t.push(e+":"+r)}));return t.join(";")}toJson(){const t={};t.type=this.type;const e=this.allAttributes();for(let r,n=0;r=e[n];n++)t[r[0]]=r[1].toString();return this.textContent&&(t.$t=this.textContent),this.childNodes.length&&(t.children=this.childNodes.map((function(t){return t.toJson()}))),this.contentNodes.length&&(t.content=this.contentNodes.map((function(t){return t.toJson()}))),t}updateContent(t,e){const r=e?t.replace(/^[ \f\n\r\t\v\u200b]*/,"").replace(/[ \f\n\r\t\v\u200b]*$/,""):t.trim();if(t=t&&!r?t:r,this.textContent===t)return;const n=(0,o.lookupMeaning)(t);this.textContent=t,this.role=n.role,this.type=n.type,this.font=n.font}addMathmlNodes(t){for(let e,r=0;e=t[r];r++)-1===this.mathml.indexOf(e)&&this.mathml.push(e)}appendChild(t){this.childNodes.push(t),this.addMathmlNodes(t.mathml),t.parent=this}replaceChild(t,e){const r=this.childNodes.indexOf(t);if(-1===r)return;t.parent=null,e.parent=this,this.childNodes[r]=e;const n=t.mathml.filter((function(t){return-1===e.mathml.indexOf(t)})),o=e.mathml.filter((function(e){return-1===t.mathml.indexOf(e)}));this.removeMathmlNodes(n),this.addMathmlNodes(o)}appendContentNode(t){t&&(this.contentNodes.push(t),this.addMathmlNodes(t.mathml),t.parent=this)}removeContentNode(t){if(t){const e=this.contentNodes.indexOf(t);-1!==e&&this.contentNodes.slice(e,1)}}equals(t){if(!t)return!1;if(this.type!==t.type||this.role!==t.role||this.textContent!==t.textContent||this.childNodes.length!==t.childNodes.length||this.contentNodes.length!==t.contentNodes.length)return!1;for(let e,r,n=0;e=this.childNodes[n],r=t.childNodes[n];n++)if(!e.equals(r))return!1;for(let e,r,n=0;e=this.contentNodes[n],r=t.contentNodes[n];n++)if(!e.equals(r))return!1;return!0}displayTree(){console.info(this.displayTree_(0))}addAnnotation(t,e){e&&this.addAnnotation_(t,e)}getAnnotation(t){const e=this.annotation[t];return e||[]}hasAnnotation(t,e){const r=this.annotation[t];return!!r&&-1!==r.indexOf(e)}parseAnnotation(t){const e=t.split(";");for(let t=0,r=e.length;t<r;t++){const r=e[t].split(":");this.addAnnotation(r[0],r[1])}}meaning(){return{type:this.type,role:this.role,font:this.font}}xmlAttributes(t){const e=this.allAttributes();for(let r,n=0;r=e[n];n++)t.setAttribute(r[0],r[1]);this.addExternalAttributes(t)}addExternalAttributes(t){for(const e in this.attributes)t.setAttribute(e,this.attributes[e])}removeMathmlNodes(t){const e=this.mathml;for(let r,n=0;r=t[n];n++){const t=e.indexOf(r);-1!==t&&e.splice(t,1)}this.mathml=e}displayTree_(t){t++;const e=Array(t).join("  ");let r="";r+="\n"+e+this.toString(),r+="\n"+e+"MathmlTree:",r+="\n"+e+this.mathmlTreeString(),r+="\n"+e+"MathML:";for(let t,n=0;t=this.mathml[n];n++)r+="\n"+e+t.toString();return r+="\n"+e+"Begin Content",this.contentNodes.forEach((function(e){r+=e.displayTree_(t)})),r+="\n"+e+"End Content",r+="\n"+e+"Begin Children",this.childNodes.forEach((function(e){r+=e.displayTree_(t)})),r+="\n"+e+"End Children",r}mathmlTreeString(){return this.mathmlTree?this.mathmlTree.toString():"EMPTY"}addAnnotation_(t,e){const r=this.annotation[t];r?r.push(e):this.annotation[t]=[e]}}e.SemanticNode=a},6537:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticNodeFactory=void 0;const n=r(8158),o=r(8158),i=r(5952);e.SemanticNodeFactory=class{constructor(){this.leafMap=new o.SemanticNodeCollator,this.defaultMap=new n.SemanticDefault,this.idCounter_=-1}makeNode(t){return this.createNode_(t)}makeUnprocessed(t){const e=this.createNode_();return e.mathml=[t],e.mathmlTree=t,e}makeEmptyNode(){const t=this.createNode_();return t.type="empty",t}makeContentNode(t){const e=this.createNode_();return e.updateContent(t),e}makeMultipleContentNodes(t,e){const r=[];for(let n=0;n<t;n++)r.push(this.makeContentNode(e));return r}makeLeafNode(t,e){if(!t)return this.makeEmptyNode();const r=this.makeContentNode(t);r.font=e||r.font;const n=this.defaultMap.retrieveNode(r);return n&&(r.type=n.type,r.role=n.role,r.font=n.font),this.leafMap.addNode(r),r}makeBranchNode(t,e,r,n){const o=this.createNode_();return n&&o.updateContent(n),o.type=t,o.childNodes=e,o.contentNodes=r,e.concat(r).forEach((function(t){t.parent=o,o.addMathmlNodes(t.mathml)})),o}createNode_(t){return void 0!==t?this.idCounter_=Math.max(this.idCounter_,t):t=++this.idCounter_,new i.SemanticNode(t)}}},3882:function(t,e){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticComparator=e.reduce=e.sort=e.apply=e.add=void 0;const r=[];function n(t){r.push(t)}function o(t,e){for(let n,o=0;n=r[o];o++){const r=n.compare(t,e);if(0!==r)return r}return 0}function i(t){t.sort(o)}e.add=n,e.apply=o,e.sort=i,e.reduce=function(t){if(t.length<=1)return t;const e=t.slice();i(e);const r=[];let n;do{n=e.pop(),r.push(n)}while(n&&e.length&&0===o(e[e.length-1],n));return r};class a{constructor(t,e=null){this.comparator=t,this.type=e,n(this)}compare(t,e){return this.type&&this.type===t.type&&this.type===e.type?this.comparator(t,e):0}}e.SemanticComparator=a,new a((function(t,e){return"simple function"===t.role?1:"simple function"===e.role?-1:0}),"identifier")},5250:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticAbstractParser=void 0;const n=r(6537);e.SemanticAbstractParser=class{constructor(t){this.type=t,this.factory_=new n.SemanticNodeFactory}getFactory(){return this.factory_}setFactory(t){this.factory_=t}getType(){return this.type}parseList(t){const e=[];for(let r,n=0;r=t[n];n++)e.push(this.parse(r));return e}}},5609:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.isMembership=e.elligibleRightNeutral=e.elligibleLeftNeutral=e.compareNeutralFences=e.isNeutralFence=e.isImplicitOp=e.isImplicit=e.isPureUnit=e.isUnitCounter=e.isNumber=e.isSingletonSetContent=e.scriptedElement_=e.illegalSingleton_=e.isSetNode=e.isRightBrace=e.isLeftBrace=e.isSimpleFunction=e.singlePunctAtPosition=e.isSimpleFunctionHead=e.isLimitBase=e.isBinomial=e.lineIsLabelled=e.tableIsMultiline=e.tableIsCases=e.isFencedElement=e.tableIsMatrixOrVector=e.isTableOrMultiline=e.isElligibleEmbellishedFence=e.isFence=e.isPunctuation=e.isRelation=e.isOperator=e.isEmbellished=e.isGeneralFunctionBoundary=e.isIntegralDxBoundarySingle=e.isIntegralDxBoundary=e.isBigOpBoundary=e.isPrefixFunctionBoundary=e.isSimpleFunctionScope=e.isAccent=e.isRole=e.embellishedType=e.isType=void 0;const n=r(3588),o=r(4795);function i(t,e){return t.type===e}function a(t,e){return t.embellished===e}function s(t,e){return t.role===e}function l(t){return u(t)||p(t)}function c(t){return i(t,"operator")||a(t,"operator")}function u(t){return i(t,"relation")||a(t,"relation")}function p(t){return i(t,"punctuation")||a(t,"punctuation")}function h(t){return i(t,"fence")||a(t,"fence")}function d(t){return!t.embellished||!function(t){return i(t,"tensor")&&(!i(t.childNodes[1],"empty")||!i(t.childNodes[2],"empty"))&&(!i(t.childNodes[3],"empty")||!i(t.childNodes[4],"empty"))}(t)&&((!s(t,"close")||!i(t,"tensor"))&&((!s(t,"open")||!i(t,"subscript")&&!i(t,"superscript"))&&d(t.childNodes[0])))}function f(t){return!!t&&(i(t,"table")||i(t,"multiline"))}function m(t){return!!t&&i(t,"fenced")&&(s(t,"leftright")||v(t))&&1===t.childNodes.length}function y(t){return!!t&&-1!==["{","\ufe5b","\uff5b"].indexOf(t.textContent)}function g(t){return!!t&&-1!==["}","\ufe5c","\uff5d"].indexOf(t.textContent)}function b(t){return"number"===t.type&&("integer"===t.role||"float"===t.role)}function v(t){return"neutral"===t.role||"metric"===t.role}e.isType=i,e.embellishedType=a,e.isRole=s,e.isAccent=function(t){const e=new RegExp("\u221e|\u1ab2");return i(t,"fence")||i(t,"punctuation")||i(t,"operator")&&!t.textContent.match(e)||i(t,"relation")||i(t,"identifier")&&s(t,"unknown")&&!t.textContent.match(n.allLettersRegExp)&&!t.textContent.match(e)},e.isSimpleFunctionScope=function(t){const e=t.childNodes;if(0===e.length)return!0;if(e.length>1)return!1;const r=e[0];if("infixop"===r.type){if("implicit"!==r.role)return!1;if(r.childNodes.some((t=>i(t,"infixop"))))return!1}return!0},e.isPrefixFunctionBoundary=function(t){return c(t)&&!s(t,"division")||i(t,"appl")||l(t)},e.isBigOpBoundary=function(t){return c(t)||l(t)},e.isIntegralDxBoundary=function(t,e){return!!e&&i(e,"identifier")&&n.lookupSecondary("d",t.textContent)},e.isIntegralDxBoundarySingle=function(t){if(i(t,"identifier")){const e=t.textContent[0];return e&&t.textContent[1]&&n.lookupSecondary("d",e)}return!1},e.isGeneralFunctionBoundary=l,e.isEmbellished=function(t){return t.embellished?t.embellished:n.isEmbellishedType(t.type)?t.type:null},e.isOperator=c,e.isRelation=u,e.isPunctuation=p,e.isFence=h,e.isElligibleEmbellishedFence=function(t){return!(!t||!h(t))&&(!t.embellished||d(t))},e.isTableOrMultiline=f,e.tableIsMatrixOrVector=function(t){return!!t&&m(t)&&f(t.childNodes[0])},e.isFencedElement=m,e.tableIsCases=function(t,e){return e.length>0&&s(e[e.length-1],"openfence")},e.tableIsMultiline=function(t){return t.childNodes.every((function(t){return t.childNodes.length<=1}))},e.lineIsLabelled=function(t){return i(t,"line")&&t.contentNodes.length&&s(t.contentNodes[0],"label")},e.isBinomial=function(t){return 2===t.childNodes.length},e.isLimitBase=function t(e){return i(e,"largeop")||i(e,"limboth")||i(e,"limlower")||i(e,"limupper")||i(e,"function")&&s(e,"limit function")||(i(e,"overscore")||i(e,"underscore"))&&t(e.childNodes[0])},e.isSimpleFunctionHead=function(t){return"identifier"===t.type||"latinletter"===t.role||"greekletter"===t.role||"otherletter"===t.role},e.singlePunctAtPosition=function(t,e,r){return 1===e.length&&("punctuation"===t[r].type||"punctuation"===t[r].embellished)&&t[r]===e[0]},e.isSimpleFunction=function(t){return i(t,"identifier")&&s(t,"simple function")},e.isLeftBrace=y,e.isRightBrace=g,e.isSetNode=function(t){return y(t.contentNodes[0])&&g(t.contentNodes[1])},e.illegalSingleton_=["punctuation","punctuated","relseq","multirel","table","multiline","cases","inference"],e.scriptedElement_=["limupper","limlower","limboth","subscript","superscript","underscore","overscore","tensor"],e.isSingletonSetContent=function t(r){const n=r.type;return-1===e.illegalSingleton_.indexOf(n)&&("infixop"!==n||"implicit"===r.role)&&("fenced"===n?"leftright"!==r.role||t(r.childNodes[0]):-1===e.scriptedElement_.indexOf(n)||t(r.childNodes[0]))},e.isNumber=b,e.isUnitCounter=function(t){return b(t)||"vulgar"===t.role||"mixed"===t.role},e.isPureUnit=function(t){const e=t.childNodes;return"unit"===t.role&&(!e.length||"unit"===e[0].role)},e.isImplicit=function(t){return"implicit"===t.role||"unit"===t.role&&!!t.contentNodes.length&&t.contentNodes[0].textContent===n.invisibleTimes()},e.isImplicitOp=function(t){return"infixop"===t.type&&"implicit"===t.role},e.isNeutralFence=v,e.compareNeutralFences=function(t,e){return v(t)&&v(e)&&(0,o.getEmbellishedInner)(t).textContent===(0,o.getEmbellishedInner)(e).textContent},e.elligibleLeftNeutral=function(t){return!!v(t)&&(!t.embellished||"superscript"!==t.type&&"subscript"!==t.type&&("tensor"!==t.type||"empty"===t.childNodes[3].type&&"empty"===t.childNodes[4].type))},e.elligibleRightNeutral=function(t){return!!v(t)&&(!t.embellished||("tensor"!==t.type||"empty"===t.childNodes[1].type&&"empty"===t.childNodes[2].type))},e.isMembership=function(t){return["element","nonelement","reelement","renonelement"].includes(t.role)}},3308:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0});const n=r(5740),o=r(3588),i=r(7516),a=r(6537),s=r(5609),l=r(4795);class c{constructor(){this.funcAppls={},this.factory_=new a.SemanticNodeFactory,i.updateFactory(this.factory_)}static getInstance(){return c.instance=c.instance||new c,c.instance}static tableToMultiline(t){if(s.tableIsMultiline(t)){t.type="multiline";for(let e,r=0;e=t.childNodes[r];r++)c.rowToLine_(e,"multiline");1===t.childNodes.length&&!s.lineIsLabelled(t.childNodes[0])&&s.isFencedElement(t.childNodes[0].childNodes[0])&&c.tableToMatrixOrVector_(c.rewriteFencedLine_(t)),c.binomialForm_(t),c.classifyMultiline(t)}else c.classifyTable(t)}static number(t){"unknown"!==t.type&&"identifier"!==t.type||(t.type="number"),c.numberRole_(t),c.exprFont_(t)}static classifyMultiline(t){let e=0;const r=t.childNodes.length;let n;for(;e<r&&(!(n=t.childNodes[e])||!n.childNodes.length);)e++;if(e>=r)return;const o=n.childNodes[0].role;"unknown"!==o&&t.childNodes.every((function(t){const e=t.childNodes[0];return!e||e.role===o&&(s.isType(e,"relation")||s.isType(e,"relseq"))}))&&(t.role=o)}static classifyTable(t){const e=c.computeColumns_(t);c.classifyByColumns_(t,e,"equality")||c.classifyByColumns_(t,e,"inequality",["equality"])||c.classifyByColumns_(t,e,"arrow")||c.detectCaleyTable(t)}static detectCaleyTable(t){if(!t.mathmlTree)return!1;const e=t.mathmlTree,r=e.getAttribute("columnlines"),n=e.getAttribute("rowlines");return!(!r||!n)&&(!(!c.cayleySpacing(r)||!c.cayleySpacing(n))&&(t.role="cayley",!0))}static cayleySpacing(t){const e=t.split(" ");return("solid"===e[0]||"dashed"===e[0])&&e.slice(1).every((t=>"none"===t))}static proof(t,e,r){const n=c.separateSemantics(e);return c.getInstance().proof(t,n,r)}static findSemantics(t,e,r){const n=null==r?null:r,o=c.getSemantics(t);return!!o&&(!!o[e]&&(null==n||o[e]===n))}static getSemantics(t){const e=t.getAttribute("semantics");return e?c.separateSemantics(e):null}static removePrefix(t){const[,...e]=t.split("_");return e.join("_")}static separateSemantics(t){const e={};return t.split(";").forEach((function(t){const[r,n]=t.split(":");e[c.removePrefix(r)]=n})),e}static matchSpaces_(t,e){for(let r,n=0;r=e[n];n++){const e=t[n].mathmlTree,o=t[n+1].mathmlTree;if(!e||!o)continue;const i=e.nextSibling;if(!i||i===o)continue;const a=c.getSpacer_(i);a&&(r.mathml.push(a),r.mathmlTree=a,r.role="space")}}static getSpacer_(t){if("MSPACE"===n.tagName(t))return t;for(;l.hasEmptyTag(t)&&1===t.childNodes.length;)if(t=t.childNodes[0],"MSPACE"===n.tagName(t))return t;return null}static fenceToPunct_(t){const e=c.FENCE_TO_PUNCT_[t.role];if(e){for(;t.embellished;)t.embellished="punctuation",s.isRole(t,"subsup")||s.isRole(t,"underover")||(t.role=e),t=t.childNodes[0];t.type="punctuation",t.role=e}}static classifyFunction_(t,e){if("appl"===t.type||"bigop"===t.type||"integral"===t.type)return"";if(e[0]&&e[0].textContent===o.functionApplication()){c.getInstance().funcAppls[t.id]=e.shift();let r="simple function";return i.run("simple2prefix",t),"prefix function"!==t.role&&"limit function"!==t.role||(r=t.role),c.propagateFunctionRole_(t,r),"prefix"}const r=c.CLASSIFY_FUNCTION_[t.role];return r||(s.isSimpleFunctionHead(t)?"simple":"")}static propagateFunctionRole_(t,e){if(t){if("infixop"===t.type)return;s.isRole(t,"subsup")||s.isRole(t,"underover")||(t.role=e),c.propagateFunctionRole_(t.childNodes[0],e)}}static getFunctionOp_(t,e){if(e(t))return t;for(let r,n=0;r=t.childNodes[n];n++){const t=c.getFunctionOp_(r,e);if(t)return t}return null}static tableToMatrixOrVector_(t){const e=t.childNodes[0];s.isType(e,"multiline")?c.tableToVector_(t):c.tableToMatrix_(t),t.contentNodes.forEach(e.appendContentNode.bind(e));for(let t,r=0;t=e.childNodes[r];r++)c.assignRoleToRow_(t,c.getComponentRoles_(e));return e.parent=null,e}static tableToVector_(t){const e=t.childNodes[0];e.type="vector",1!==e.childNodes.length?c.binomialForm_(e):c.tableToSquare_(t)}static binomialForm_(t){s.isBinomial(t)&&(t.role="binomial",t.childNodes[0].role="binomial",t.childNodes[1].role="binomial")}static tableToMatrix_(t){const e=t.childNodes[0];e.type="matrix",e.childNodes&&e.childNodes.length>0&&e.childNodes[0].childNodes&&e.childNodes.length===e.childNodes[0].childNodes.length?c.tableToSquare_(t):e.childNodes&&1===e.childNodes.length&&(e.role="rowvector")}static tableToSquare_(t){const e=t.childNodes[0];s.isNeutralFence(t)?e.role="determinant":e.role="squarematrix"}static getComponentRoles_(t){const e=t.role;return e&&"unknown"!==e?e:t.type.toLowerCase()||"unknown"}static tableToCases_(t,e){for(let e,r=0;e=t.childNodes[r];r++)c.assignRoleToRow_(e,"cases");return t.type="cases",t.appendContentNode(e),s.tableIsMultiline(t)&&c.binomialForm_(t),t}static rewriteFencedLine_(t){const e=t.childNodes[0],r=t.childNodes[0].childNodes[0],n=t.childNodes[0].childNodes[0].childNodes[0];return r.parent=t.parent,t.parent=r,n.parent=e,r.childNodes=[t],e.childNodes=[n],r}static rowToLine_(t,e){const r=e||"unknown";s.isType(t,"row")&&(t.type="line",t.role=r,1===t.childNodes.length&&s.isType(t.childNodes[0],"cell")&&(t.childNodes=t.childNodes[0].childNodes,t.childNodes.forEach((function(e){e.parent=t}))))}static assignRoleToRow_(t,e){s.isType(t,"line")?t.role=e:s.isType(t,"row")&&(t.role=e,t.childNodes.forEach((function(t){s.isType(t,"cell")&&(t.role=e)})))}static nextSeparatorFunction_(t){let e;if(t){if(t.match(/^\s+$/))return null;e=t.replace(/\s/g,"").split("").filter((function(t){return t}))}else e=[","];return function(){return e.length>1?e.shift():e[0]}}static numberRole_(t){if("unknown"!==t.role)return;const e=[...t.textContent].filter((t=>t.match(/[^\s]/))),r=e.map(o.lookupMeaning);if(r.every((function(t){return"number"===t.type&&"integer"===t.role||"punctuation"===t.type&&"comma"===t.role})))return t.role="integer",void("0"===e[0]&&t.addAnnotation("general","basenumber"));r.every((function(t){return"number"===t.type&&"integer"===t.role||"punctuation"===t.type}))?t.role="float":t.role="othernumber"}static exprFont_(t){if("unknown"!==t.font)return;const e=[...t.textContent].map(o.lookupMeaning).reduce((function(t,e){return t&&e.font&&"unknown"!==e.font&&e.font!==t?"unknown"===t?e.font:null:t}),"unknown");e&&(t.font=e)}static purgeFences_(t){const e=t.rel,r=t.comp,n=[],o=[];for(;e.length>0;){const t=e.shift();let i=r.shift();s.isElligibleEmbellishedFence(t)?(n.push(t),o.push(i)):(c.fenceToPunct_(t),i.push(t),i=i.concat(r.shift()),r.unshift(i))}return o.push(r.shift()),{rel:n,comp:o}}static rewriteFencedNode_(t){const e=t.contentNodes[0],r=t.contentNodes[1];let n=c.rewriteFence_(t,e);return t.contentNodes[0]=n.fence,n=c.rewriteFence_(n.node,r),t.contentNodes[1]=n.fence,t.contentNodes[0].parent=t,t.contentNodes[1].parent=t,n.node.parent=null,n.node}static rewriteFence_(t,e){if(!e.embellished)return{node:t,fence:e};const r=e.childNodes[0],n=c.rewriteFence_(t,r);return s.isType(e,"superscript")||s.isType(e,"subscript")||s.isType(e,"tensor")?(s.isRole(e,"subsup")||(e.role=t.role),r!==n.node&&(e.replaceChild(r,n.node),r.parent=t),c.propagateFencePointer_(e,r),{node:e,fence:n.fence}):(e.replaceChild(r,n.fence),e.mathmlTree&&-1===e.mathml.indexOf(e.mathmlTree)&&e.mathml.push(e.mathmlTree),{node:n.node,fence:e})}static propagateFencePointer_(t,e){t.fencePointer=e.fencePointer||e.id.toString(),t.embellished=null}static classifyByColumns_(t,e,r,n){return!!(3===e.length&&c.testColumns_(e,1,(t=>c.isPureRelation_(t,r)))||2===e.length&&(c.testColumns_(e,1,(t=>c.isEndRelation_(t,r)||c.isPureRelation_(t,r)))||c.testColumns_(e,0,(t=>c.isEndRelation_(t,r,!0)||c.isPureRelation_(t,r)))))&&(t.role=r,!0)}static isEndRelation_(t,e,r){const n=r?t.childNodes.length-1:0;return s.isType(t,"relseq")&&s.isRole(t,e)&&s.isType(t.childNodes[n],"empty")}static isPureRelation_(t,e){return s.isType(t,"relation")&&s.isRole(t,e)}static computeColumns_(t){const e=[];for(let r,n=0;r=t.childNodes[n];n++)for(let t,n=0;t=r.childNodes[n];n++){e[n]?e[n].push(t):e[n]=[t]}return e}static testColumns_(t,e,r){const n=t[e];return!!n&&(n.some((function(t){return t.childNodes.length&&r(t.childNodes[0])}))&&n.every((function(t){return!t.childNodes.length||r(t.childNodes[0])})))}setNodeFactory(t){c.getInstance().factory_=t,i.updateFactory(c.getInstance().factory_)}getNodeFactory(){return c.getInstance().factory_}identifierNode(t,e,r){if("MathML-Unit"===r)t.type="identifier",t.role="unit";else if(!e&&1===t.textContent.length&&("integer"===t.role||"latinletter"===t.role||"greekletter"===t.role)&&"normal"===t.font)return t.font="italic",i.run("simpleNamedFunction",t);return"unknown"===t.type&&(t.type="identifier"),c.exprFont_(t),i.run("simpleNamedFunction",t)}implicitNode(t){if(t=c.getInstance().getMixedNumbers_(t),1===(t=c.getInstance().combineUnits_(t)).length)return t[0];const e=c.getInstance().implicitNode_(t);return i.run("combine_juxtaposition",e)}text(t,e){return c.exprFont_(t),t.type="text","MS"===e?(t.role="string",t):"MSPACE"===e||t.textContent.match(/^\s*$/)?(t.role="space",t):t}row(t){return 0===(t=t.filter((function(t){return!s.isType(t,"empty")}))).length?c.getInstance().factory_.makeEmptyNode():(t=c.getInstance().getFencesInRow_(t),t=c.getInstance().tablesInRow(t),t=c.getInstance().getPunctuationInRow_(t),t=c.getInstance().getTextInRow_(t),t=c.getInstance().getFunctionsInRow_(t),c.getInstance().relationsInRow_(t))}limitNode(t,e){if(!e.length)return c.getInstance().factory_.makeEmptyNode();let r,n=e[0],o="unknown";if(!e[1])return n;if(s.isLimitBase(n)){r=c.MML_TO_LIMIT_[t];const i=r.length;if(o=r.type,e=e.slice(0,r.length+1),1===i&&s.isAccent(e[1])||2===i&&s.isAccent(e[1])&&s.isAccent(e[2]))return r=c.MML_TO_BOUNDS_[t],c.getInstance().accentNode_(n,e,r.type,r.length,r.accent);if(2===i){if(s.isAccent(e[1]))return n=c.getInstance().accentNode_(n,[n,e[1]],{MSUBSUP:"subscript",MUNDEROVER:"underscore"}[t],1,!0),e[2]?c.getInstance().makeLimitNode_(n,[n,e[2]],null,"limupper"):n;if(e[2]&&s.isAccent(e[2]))return n=c.getInstance().accentNode_(n,[n,e[2]],{MSUBSUP:"superscript",MUNDEROVER:"overscore"}[t],1,!0),c.getInstance().makeLimitNode_(n,[n,e[1]],null,"limlower");e[i]||(o="limlower")}return c.getInstance().makeLimitNode_(n,e,null,o)}return r=c.MML_TO_BOUNDS_[t],c.getInstance().accentNode_(n,e,r.type,r.length,r.accent)}tablesInRow(t){let e=l.partitionNodes(t,s.tableIsMatrixOrVector),r=[];for(let t,n=0;t=e.rel[n];n++)r=r.concat(e.comp.shift()),r.push(c.tableToMatrixOrVector_(t));r=r.concat(e.comp.shift()),e=l.partitionNodes(r,s.isTableOrMultiline),r=[];for(let t,n=0;t=e.rel[n];n++){const n=e.comp.shift();s.tableIsCases(t,n)&&c.tableToCases_(t,n.pop()),r=r.concat(n),r.push(t)}return r.concat(e.comp.shift())}mfenced(t,e,r,n){if(r&&n.length>0){const t=c.nextSeparatorFunction_(r),e=[n.shift()];n.forEach((r=>{e.push(c.getInstance().factory_.makeContentNode(t())),e.push(r)})),n=e}return t&&e?c.getInstance().horizontalFencedNode_(c.getInstance().factory_.makeContentNode(t),c.getInstance().factory_.makeContentNode(e),n):(t&&n.unshift(c.getInstance().factory_.makeContentNode(t)),e&&n.push(c.getInstance().factory_.makeContentNode(e)),c.getInstance().row(n))}fractionLikeNode(t,e,r,n){let o;if(!n&&l.isZeroLength(r)){const r=c.getInstance().factory_.makeBranchNode("line",[t],[]),n=c.getInstance().factory_.makeBranchNode("line",[e],[]);return o=c.getInstance().factory_.makeBranchNode("multiline",[r,n],[]),c.binomialForm_(o),c.classifyMultiline(o),o}return o=c.getInstance().fractionNode_(t,e),n&&o.addAnnotation("general","bevelled"),o}tensor(t,e,r,n,o){const i=c.getInstance().factory_.makeBranchNode("tensor",[t,c.getInstance().scriptNode_(e,"leftsub"),c.getInstance().scriptNode_(r,"leftsuper"),c.getInstance().scriptNode_(n,"rightsub"),c.getInstance().scriptNode_(o,"rightsuper")],[]);return i.role=t.role,i.embellished=s.isEmbellished(t),i}pseudoTensor(t,e,r){const n=t=>!s.isType(t,"empty"),o=e.filter(n).length,i=r.filter(n).length;if(!o&&!i)return t;const a=o?i?"MSUBSUP":"MSUB":"MSUP",l=[t];return o&&l.push(c.getInstance().scriptNode_(e,"rightsub",!0)),i&&l.push(c.getInstance().scriptNode_(r,"rightsuper",!0)),c.getInstance().limitNode(a,l)}font(t){const e=c.MATHJAX_FONTS[t];return e||t}proof(t,e,r){if(e.inference||e.axiom||console.log("Noise"),e.axiom){const e=c.getInstance().cleanInference(t.childNodes),n=e.length?c.getInstance().factory_.makeBranchNode("inference",r(e),[]):c.getInstance().factory_.makeEmptyNode();return n.role="axiom",n.mathmlTree=t,n}const n=c.getInstance().inference(t,e,r);return e.proof&&(n.role="proof",n.childNodes[0].role="final"),n}inference(t,e,r){if(e.inferenceRule){const e=c.getInstance().getFormulas(t,[],r);return c.getInstance().factory_.makeBranchNode("inference",[e.conclusion,e.premises],[])}const o=e.labelledRule,i=n.toArray(t.childNodes),a=[];"left"!==o&&"both"!==o||a.push(c.getInstance().getLabel(t,i,r,"left")),"right"!==o&&"both"!==o||a.push(c.getInstance().getLabel(t,i,r,"right"));const s=c.getInstance().getFormulas(t,i,r),l=c.getInstance().factory_.makeBranchNode("inference",[s.conclusion,s.premises],a);return l.mathmlTree=t,l}getLabel(t,e,r,o){const i=c.getInstance().findNestedRow(e,"prooflabel",o),a=c.getInstance().factory_.makeBranchNode("rulelabel",r(n.toArray(i.childNodes)),[]);return a.role=o,a.mathmlTree=i,a}getFormulas(t,e,r){const o=e.length?c.getInstance().findNestedRow(e,"inferenceRule"):t,i="up"===c.getSemantics(o).inferenceRule,a=i?o.childNodes[1]:o.childNodes[0],s=i?o.childNodes[0]:o.childNodes[1],l=a.childNodes[0].childNodes[0],u=n.toArray(l.childNodes[0].childNodes),p=[];let h=1;for(const t of u)h%2&&p.push(t.childNodes[0]),h++;const d=r(p),f=r(n.toArray(s.childNodes[0].childNodes))[0],m=c.getInstance().factory_.makeBranchNode("premises",d,[]);m.mathmlTree=l;const y=c.getInstance().factory_.makeBranchNode("conclusion",[f],[]);return y.mathmlTree=s.childNodes[0].childNodes[0],{conclusion:y,premises:m}}findNestedRow(t,e,r){return c.getInstance().findNestedRow_(t,e,0,r)}cleanInference(t){return n.toArray(t).filter((function(t){return"MSPACE"!==n.tagName(t)}))}operatorNode(t){return"unknown"===t.type&&(t.type="operator"),i.run("multioperator",t)}implicitNode_(t){const e=c.getInstance().factory_.makeMultipleContentNodes(t.length-1,o.invisibleTimes());c.matchSpaces_(t,e);const r=c.getInstance().infixNode_(t,e[0]);return r.role="implicit",e.forEach((function(t){t.parent=r})),r.contentNodes=e,r}infixNode_(t,e){const r=c.getInstance().factory_.makeBranchNode("infixop",t,[e],l.getEmbellishedInner(e).textContent);return r.role=e.role,i.run("propagateSimpleFunction",r)}explicitMixed_(t){const e=l.partitionNodes(t,(function(t){return t.textContent===o.invisiblePlus()}));if(!e.rel.length)return t;let r=[];for(let t,n=0;t=e.rel[n];n++){const o=e.comp[n],i=e.comp[n+1],a=o.length-1;if(o[a]&&i[0]&&s.isType(o[a],"number")&&!s.isRole(o[a],"mixed")&&s.isType(i[0],"fraction")){const t=c.getInstance().factory_.makeBranchNode("number",[o[a],i[0]],[]);t.role="mixed",r=r.concat(o.slice(0,a)),r.push(t),i.shift()}else r=r.concat(o),r.push(t)}return r.concat(e.comp[e.comp.length-1])}concatNode_(t,e,r){if(0===e.length)return t;const n=e.map((function(t){return l.getEmbellishedInner(t).textContent})).join(" "),o=c.getInstance().factory_.makeBranchNode(r,[t],e,n);return e.length>1&&(o.role="multiop"),o}prefixNode_(t,e){const r=l.partitionNodes(e,(t=>s.isRole(t,"subtraction")));let n=c.getInstance().concatNode_(t,r.comp.pop(),"prefixop");for(1===n.contentNodes.length&&"addition"===n.contentNodes[0].role&&"+"===n.contentNodes[0].textContent&&(n.role="positive");r.rel.length>0;)n=c.getInstance().concatNode_(n,[r.rel.pop()],"prefixop"),n.role="negative",n=c.getInstance().concatNode_(n,r.comp.pop(),"prefixop");return n}postfixNode_(t,e){return e.length?c.getInstance().concatNode_(t,e,"postfixop"):t}combineUnits_(t){const e=l.partitionNodes(t,(function(t){return!s.isRole(t,"unit")}));if(t.length===e.rel.length)return e.rel;const r=[];let n,o;do{const t=e.comp.shift();n=e.rel.shift();let i=null;o=r.pop(),o&&(t.length&&s.isUnitCounter(o)?t.unshift(o):r.push(o)),1===t.length&&(i=t.pop()),t.length>1&&(i=c.getInstance().implicitNode_(t),i.role="unit"),i&&r.push(i),n&&r.push(n)}while(n);return r}getMixedNumbers_(t){const e=l.partitionNodes(t,(function(t){return s.isType(t,"fraction")&&s.isRole(t,"vulgar")}));if(!e.rel.length)return t;let r=[];for(let t,n=0;t=e.rel[n];n++){const o=e.comp[n],i=o.length-1;if(o[i]&&s.isType(o[i],"number")&&(s.isRole(o[i],"integer")||s.isRole(o[i],"float"))){const e=c.getInstance().factory_.makeBranchNode("number",[o[i],t],[]);e.role="mixed",r=r.concat(o.slice(0,i)),r.push(e)}else r=r.concat(o),r.push(t)}return r.concat(e.comp[e.comp.length-1])}getTextInRow_(t){if(t.length<=1)return t;const e=l.partitionNodes(t,(t=>s.isType(t,"text")));if(0===e.rel.length)return t;const r=[];let n=e.comp[0];n.length>0&&r.push(c.getInstance().row(n));for(let t,o=0;t=e.rel[o];o++)r.push(t),n=e.comp[o+1],n.length>0&&r.push(c.getInstance().row(n));return[c.getInstance().dummyNode_(r)]}relationsInRow_(t){const e=l.partitionNodes(t,s.isRelation),r=e.rel[0];if(!r)return c.getInstance().operationsInRow_(t);if(1===t.length)return t[0];const n=e.comp.map(c.getInstance().operationsInRow_);let o;return e.rel.some((function(t){return!t.equals(r)}))?(o=c.getInstance().factory_.makeBranchNode("multirel",n,e.rel),e.rel.every((function(t){return t.role===r.role}))&&(o.role=r.role),o):(o=c.getInstance().factory_.makeBranchNode("relseq",n,e.rel,l.getEmbellishedInner(r).textContent),o.role=r.role,o)}operationsInRow_(t){if(0===t.length)return c.getInstance().factory_.makeEmptyNode();if(1===(t=c.getInstance().explicitMixed_(t)).length)return t[0];const e=[];for(;t.length>0&&s.isOperator(t[0]);)e.push(t.shift());if(0===t.length)return c.getInstance().prefixNode_(e.pop(),e);if(1===t.length)return c.getInstance().prefixNode_(t[0],e);t=i.run("convert_juxtaposition",t);const r=l.sliceNodes(t,s.isOperator),n=c.getInstance().prefixNode_(c.getInstance().implicitNode(r.head),e);return r.div?c.getInstance().operationsTree_(r.tail,n,r.div):n}operationsTree_(t,e,r,n){const o=n||[];if(0===t.length){if(o.unshift(r),"infixop"===e.type){const t=c.getInstance().postfixNode_(e.childNodes.pop(),o);return e.appendChild(t),e}return c.getInstance().postfixNode_(e,o)}const i=l.sliceNodes(t,s.isOperator);if(0===i.head.length)return o.push(i.div),c.getInstance().operationsTree_(i.tail,e,r,o);const a=c.getInstance().prefixNode_(c.getInstance().implicitNode(i.head),o),u=c.getInstance().appendOperand_(e,r,a);return i.div?c.getInstance().operationsTree_(i.tail,u,i.div,[]):u}appendOperand_(t,e,r){if("infixop"!==t.type)return c.getInstance().infixNode_([t,r],e);const n=c.getInstance().appendDivisionOp_(t,e,r);return n||(c.getInstance().appendExistingOperator_(t,e,r)?t:"multiplication"===e.role?c.getInstance().appendMultiplicativeOp_(t,e,r):c.getInstance().appendAdditiveOp_(t,e,r))}appendDivisionOp_(t,e,r){return"division"===e.role?s.isImplicit(t)?c.getInstance().infixNode_([t,r],e):c.getInstance().appendLastOperand_(t,e,r):"division"===t.role?c.getInstance().infixNode_([t,r],e):null}appendLastOperand_(t,e,r){let n=t,o=t.childNodes[t.childNodes.length-1];for(;o&&"infixop"===o.type&&!s.isImplicit(o);)n=o,o=n.childNodes[t.childNodes.length-1];const i=c.getInstance().infixNode_([n.childNodes.pop(),r],e);return n.appendChild(i),t}appendMultiplicativeOp_(t,e,r){if(s.isImplicit(t))return c.getInstance().infixNode_([t,r],e);let n=t,o=t.childNodes[t.childNodes.length-1];for(;o&&"infixop"===o.type&&!s.isImplicit(o);)n=o,o=n.childNodes[t.childNodes.length-1];const i=c.getInstance().infixNode_([n.childNodes.pop(),r],e);return n.appendChild(i),t}appendAdditiveOp_(t,e,r){return c.getInstance().infixNode_([t,r],e)}appendExistingOperator_(t,e,r){return!(!t||"infixop"!==t.type||s.isImplicit(t))&&(t.contentNodes[0].equals(e)?(t.appendContentNode(e),t.appendChild(r),!0):c.getInstance().appendExistingOperator_(t.childNodes[t.childNodes.length-1],e,r))}getFencesInRow_(t){let e=l.partitionNodes(t,s.isFence);e=c.purgeFences_(e);const r=e.comp.shift();return c.getInstance().fences_(e.rel,e.comp,[],[r])}fences_(t,e,r,n){if(0===t.length&&0===r.length)return n[0];const o=t=>s.isRole(t,"open");if(0===t.length){const t=n.shift();for(;r.length>0;){if(o(r[0])){const e=r.shift();c.fenceToPunct_(e),t.push(e)}else{const e=l.sliceNodes(r,o),i=e.head.length-1,a=c.getInstance().neutralFences_(e.head,n.slice(0,i));n=n.slice(i),t.push(...a),e.div&&e.tail.unshift(e.div),r=e.tail}t.push(...n.shift())}return t}const i=r[r.length-1],a=t[0].role;if("open"===a||s.isNeutralFence(t[0])&&(!i||!s.compareNeutralFences(t[0],i))){r.push(t.shift());const o=e.shift();return o&&n.push(o),c.getInstance().fences_(t,e,r,n)}if(i&&"close"===a&&"open"===i.role){const o=c.getInstance().horizontalFencedNode_(r.pop(),t.shift(),n.pop());return n.push(n.pop().concat([o],e.shift())),c.getInstance().fences_(t,e,r,n)}if(i&&s.compareNeutralFences(t[0],i)){if(!s.elligibleLeftNeutral(i)||!s.elligibleRightNeutral(t[0])){r.push(t.shift());const o=e.shift();return o&&n.push(o),c.getInstance().fences_(t,e,r,n)}const o=c.getInstance().horizontalFencedNode_(r.pop(),t.shift(),n.pop());return n.push(n.pop().concat([o],e.shift())),c.getInstance().fences_(t,e,r,n)}if(i&&"close"===a&&s.isNeutralFence(i)&&r.some(o)){const i=l.sliceNodes(r,o,!0),a=n.pop(),s=n.length-i.tail.length+1,u=c.getInstance().neutralFences_(i.tail,n.slice(s));n=n.slice(0,s);const p=c.getInstance().horizontalFencedNode_(i.div,t.shift(),n.pop().concat(u,a));return n.push(n.pop().concat([p],e.shift())),c.getInstance().fences_(t,e,i.head,n)}const u=t.shift();return c.fenceToPunct_(u),n.push(n.pop().concat([u],e.shift())),c.getInstance().fences_(t,e,r,n)}neutralFences_(t,e){if(0===t.length)return t;if(1===t.length)return c.fenceToPunct_(t[0]),t;const r=t.shift();if(!s.elligibleLeftNeutral(r)){c.fenceToPunct_(r);const n=e.shift();return n.unshift(r),n.concat(c.getInstance().neutralFences_(t,e))}const n=l.sliceNodes(t,(function(t){return s.compareNeutralFences(t,r)}));if(!n.div){c.fenceToPunct_(r);const n=e.shift();return n.unshift(r),n.concat(c.getInstance().neutralFences_(t,e))}if(!s.elligibleRightNeutral(n.div))return c.fenceToPunct_(n.div),t.unshift(r),c.getInstance().neutralFences_(t,e);const o=c.getInstance().combineFencedContent_(r,n.div,n.head,e);if(n.tail.length>0){const t=o.shift(),e=c.getInstance().neutralFences_(n.tail,o);return t.concat(e)}return o[0]}combineFencedContent_(t,e,r,n){if(0===r.length){const r=c.getInstance().horizontalFencedNode_(t,e,n.shift());return n.length>0?n[0].unshift(r):n=[[r]],n}const o=n.shift(),i=r.length-1,a=n.slice(0,i),s=(n=n.slice(i)).shift(),l=c.getInstance().neutralFences_(r,a);o.push(...l),o.push(...s);const u=c.getInstance().horizontalFencedNode_(t,e,o);return n.length>0?n[0].unshift(u):n=[[u]],n}horizontalFencedNode_(t,e,r){const n=c.getInstance().row(r);let o=c.getInstance().factory_.makeBranchNode("fenced",[n],[t,e]);return"open"===t.role?(c.getInstance().classifyHorizontalFence_(o),o=i.run("propagateComposedFunction",o)):o.role=t.role,o=i.run("detect_cycle",o),c.rewriteFencedNode_(o)}classifyHorizontalFence_(t){t.role="leftright";const e=t.childNodes;if(!s.isSetNode(t)||e.length>1)return;if(0===e.length||"empty"===e[0].type)return void(t.role="set empty");const r=e[0].type;if(1===e.length&&s.isSingletonSetContent(e[0]))return void(t.role="set singleton");const n=e[0].role;if("punctuated"===r&&"sequence"===n){if("comma"!==e[0].contentNodes[0].role)return 1!==e[0].contentNodes.length||"vbar"!==e[0].contentNodes[0].role&&"colon"!==e[0].contentNodes[0].role?void 0:(t.role="set extended",void c.getInstance().setExtension_(t));t.role="set collection"}}setExtension_(t){const e=t.childNodes[0].childNodes[0];e&&"infixop"===e.type&&1===e.contentNodes.length&&s.isMembership(e.contentNodes[0])&&(e.addAnnotation("set","intensional"),e.contentNodes[0].addAnnotation("set","intensional"))}getPunctuationInRow_(t){if(t.length<=1)return t;const e=t=>{const e=t.type;return"punctuation"===e||"text"===e||"operator"===e||"relation"===e},r=l.partitionNodes(t,(function(r){if(!s.isPunctuation(r))return!1;if(s.isPunctuation(r)&&!s.isRole(r,"ellipsis"))return!0;const n=t.indexOf(r);if(0===n)return!t[1]||!e(t[1]);const o=t[n-1];if(n===t.length-1)return!e(o);const i=t[n+1];return!e(o)||!e(i)}));if(0===r.rel.length)return t;const n=[];let o=r.comp.shift();o.length>0&&n.push(c.getInstance().row(o));let i=0;for(;r.comp.length>0;)n.push(r.rel[i++]),o=r.comp.shift(),o.length>0&&n.push(c.getInstance().row(o));return[c.getInstance().punctuatedNode_(n,r.rel)]}punctuatedNode_(t,e){const r=c.getInstance().factory_.makeBranchNode("punctuated",t,e);if(e.length===t.length){const t=e[0].role;if("unknown"!==t&&e.every((function(e){return e.role===t})))return r.role=t,r}return s.singlePunctAtPosition(t,e,0)?r.role="startpunct":s.singlePunctAtPosition(t,e,t.length-1)?r.role="endpunct":e.every((t=>s.isRole(t,"dummy")))?r.role="text":e.every((t=>s.isRole(t,"space")))?r.role="space":r.role="sequence",r}dummyNode_(t){const e=c.getInstance().factory_.makeMultipleContentNodes(t.length-1,o.invisibleComma());return e.forEach((function(t){t.role="dummy"})),c.getInstance().punctuatedNode_(t,e)}accentRole_(t,e){if(!s.isAccent(t))return!1;const r=t.textContent,n=o.lookupSecondary("bar",r)||o.lookupSecondary("tilde",r)||t.role;return t.role="underscore"===e?"underaccent":"overaccent",t.addAnnotation("accent",n),!0}accentNode_(t,e,r,n,o){const i=(e=e.slice(0,n+1))[1],a=e[2];let s;if(!o&&a&&(s=c.getInstance().factory_.makeBranchNode("subscript",[t,i],[]),s.role="subsup",e=[s,a],r="superscript"),o){const n=c.getInstance().accentRole_(i,r);if(a){c.getInstance().accentRole_(a,"overscore")&&!n?(s=c.getInstance().factory_.makeBranchNode("overscore",[t,a],[]),e=[s,i],r="underscore"):(s=c.getInstance().factory_.makeBranchNode("underscore",[t,i],[]),e=[s,a],r="overscore"),s.role="underover"}}return c.getInstance().makeLimitNode_(t,e,s,r)}makeLimitNode_(t,e,r,n){if("limupper"===n&&"limlower"===t.type)return t.childNodes.push(e[1]),e[1].parent=t,t.type="limboth",t;if("limlower"===n&&"limupper"===t.type)return t.childNodes.splice(1,-1,e[1]),e[1].parent=t,t.type="limboth",t;const o=c.getInstance().factory_.makeBranchNode(n,e,[]),i=s.isEmbellished(t);return r&&(r.embellished=i),o.embellished=i,o.role=t.role,o}getFunctionsInRow_(t,e){const r=e||[];if(0===t.length)return r;const n=t.shift(),o=c.classifyFunction_(n,t);if(!o)return r.push(n),c.getInstance().getFunctionsInRow_(t,r);const i=c.getInstance().getFunctionsInRow_(t,[]),a=c.getInstance().getFunctionArgs_(n,i,o);return r.concat(a)}getFunctionArgs_(t,e,r){let n,o,i;switch(r){case"integral":{const r=c.getInstance().getIntegralArgs_(e);if(!r.intvar&&!r.integrand.length)return r.rest.unshift(t),r.rest;const n=c.getInstance().row(r.integrand);return i=c.getInstance().integralNode_(t,n,r.intvar),r.rest.unshift(i),r.rest}case"prefix":if(e[0]&&"fenced"===e[0].type){const r=e.shift();return s.isNeutralFence(r)||(r.role="leftright"),i=c.getInstance().functionNode_(t,r),e.unshift(i),e}if(n=l.sliceNodes(e,s.isPrefixFunctionBoundary),n.head.length)o=c.getInstance().row(n.head),n.div&&n.tail.unshift(n.div);else{if(!n.div||!s.isType(n.div,"appl"))return e.unshift(t),e;o=n.div}return i=c.getInstance().functionNode_(t,o),n.tail.unshift(i),n.tail;case"bigop":return n=l.sliceNodes(e,s.isBigOpBoundary),n.head.length?(o=c.getInstance().row(n.head),i=c.getInstance().bigOpNode_(t,o),n.div&&n.tail.unshift(n.div),n.tail.unshift(i),n.tail):(e.unshift(t),e);default:{if(0===e.length)return[t];const r=e[0];return"fenced"===r.type&&!s.isNeutralFence(r)&&s.isSimpleFunctionScope(r)?(r.role="leftright",c.propagateFunctionRole_(t,"simple function"),i=c.getInstance().functionNode_(t,e.shift()),e.unshift(i),e):(e.unshift(t),e)}}}getIntegralArgs_(t,e=[]){if(0===t.length)return{integrand:e,intvar:null,rest:t};const r=t[0];if(s.isGeneralFunctionBoundary(r))return{integrand:e,intvar:null,rest:t};if(s.isIntegralDxBoundarySingle(r))return r.role="integral",{integrand:e,intvar:r,rest:t.slice(1)};if(t[1]&&s.isIntegralDxBoundary(r,t[1])){const n=c.getInstance().prefixNode_(t[1],[r]);return n.role="integral",{integrand:e,intvar:n,rest:t.slice(2)}}return e.push(t.shift()),c.getInstance().getIntegralArgs_(t,e)}functionNode_(t,e){const r=c.getInstance().factory_.makeContentNode(o.functionApplication()),n=c.getInstance().funcAppls[t.id];n&&(r.mathmlTree=n.mathmlTree,r.mathml=n.mathml,r.annotation=n.annotation,r.attributes=n.attributes,delete c.getInstance().funcAppls[t.id]),r.type="punctuation",r.role="application";const i=c.getFunctionOp_(t,(function(t){return s.isType(t,"function")||s.isType(t,"identifier")&&s.isRole(t,"simple function")}));return c.getInstance().functionalNode_("appl",[t,e],i,[r])}bigOpNode_(t,e){const r=c.getFunctionOp_(t,(t=>s.isType(t,"largeop")));return c.getInstance().functionalNode_("bigop",[t,e],r,[])}integralNode_(t,e,r){e=e||c.getInstance().factory_.makeEmptyNode(),r=r||c.getInstance().factory_.makeEmptyNode();const n=c.getFunctionOp_(t,(t=>s.isType(t,"largeop")));return c.getInstance().functionalNode_("integral",[t,e,r],n,[])}functionalNode_(t,e,r,n){const o=e[0];let i;r&&(i=r.parent,n.push(r));const a=c.getInstance().factory_.makeBranchNode(t,e,n);return a.role=o.role,i&&(r.parent=i),a}fractionNode_(t,e){const r=c.getInstance().factory_.makeBranchNode("fraction",[t,e],[]);return r.role=r.childNodes.every((function(t){return s.isType(t,"number")&&s.isRole(t,"integer")}))?"vulgar":r.childNodes.every(s.isPureUnit)?"unit":"division",i.run("propagateSimpleFunction",r)}scriptNode_(t,e,r){let n;switch(t.length){case 0:n=c.getInstance().factory_.makeEmptyNode();break;case 1:if(n=t[0],r)return n;break;default:n=c.getInstance().dummyNode_(t)}return n.role=e,n}findNestedRow_(t,e,r,o){if(r>3)return null;for(let i,a=0;i=t[a];a++){const t=n.tagName(i);if("MSPACE"!==t){if("MROW"===t)return c.getInstance().findNestedRow_(n.toArray(i.childNodes),e,r+1,o);if(c.findSemantics(i,e,o))return i}}return null}}e.default=c,c.FENCE_TO_PUNCT_={metric:"metric",neutral:"vbar",open:"openfence",close:"closefence"},c.MML_TO_LIMIT_={MSUB:{type:"limlower",length:1},MUNDER:{type:"limlower",length:1},MSUP:{type:"limupper",length:1},MOVER:{type:"limupper",length:1},MSUBSUP:{type:"limboth",length:2},MUNDEROVER:{type:"limboth",length:2}},c.MML_TO_BOUNDS_={MSUB:{type:"subscript",length:1,accent:!1},MSUP:{type:"superscript",length:1,accent:!1},MSUBSUP:{type:"subscript",length:2,accent:!1},MUNDER:{type:"underscore",length:1,accent:!0},MOVER:{type:"overscore",length:1,accent:!0},MUNDEROVER:{type:"underscore",length:2,accent:!0}},c.CLASSIFY_FUNCTION_={integral:"integral",sum:"bigop","prefix function":"prefix","limit function":"prefix","simple function":"prefix","composed function":"prefix"},c.MATHJAX_FONTS={"-tex-caligraphic":"caligraphic","-tex-caligraphic-bold":"caligraphic-bold","-tex-calligraphic":"caligraphic","-tex-calligraphic-bold":"caligraphic-bold","-tex-oldstyle":"oldstyle","-tex-oldstyle-bold":"oldstyle-bold","-tex-mathit":"italic"}},5656:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticSkeleton=void 0;const n=r(707),o=r(5274),i=r(2298);class a{constructor(t){this.parents=null,this.levelsMap=null,t=0===t?t:t||[],this.array=t}static fromTree(t){return a.fromNode(t.root)}static fromNode(t){return new a(a.fromNode_(t))}static fromString(t){return new a(a.fromString_(t))}static simpleCollapseStructure(t){return"number"==typeof t}static contentCollapseStructure(t){return!!t&&!a.simpleCollapseStructure(t)&&"c"===t[0]}static interleaveIds(t,e){return n.interleaveLists(a.collapsedLeafs(t),a.collapsedLeafs(e))}static collapsedLeafs(...t){return t.reduce(((t,e)=>{return t.concat((r=e,a.simpleCollapseStructure(r)?[r]:(r=r,a.contentCollapseStructure(r[1])?r.slice(2):r.slice(1))));var r}),[])}static fromStructure(t,e){return new a(a.tree_(t,e.root))}static combineContentChildren(t,e,r){switch(t.type){case"relseq":case"infixop":case"multirel":return n.interleaveLists(r,e);case"prefixop":return e.concat(r);case"postfixop":return r.concat(e);case"fenced":return r.unshift(e[0]),r.push(e[1]),r;case"appl":return[r[0],e[0],r[1]];case"root":return[r[1],r[0]];case"row":case"line":return e.length&&r.unshift(e[0]),r;default:return r}}static makeSexp_(t){return a.simpleCollapseStructure(t)?t.toString():a.contentCollapseStructure(t)?"(c "+t.slice(1).map(a.makeSexp_).join(" ")+")":"("+t.map(a.makeSexp_).join(" ")+")"}static fromString_(t){let e=t.replace(/\(/g,"[");return e=e.replace(/\)/g,"]"),e=e.replace(/ /g,","),e=e.replace(/c/g,'"c"'),JSON.parse(e)}static fromNode_(t){if(!t)return[];const e=t.contentNodes;let r;e.length&&(r=e.map(a.fromNode_),r.unshift("c"));const n=t.childNodes;if(!n.length)return e.length?[t.id,r]:t.id;const o=n.map(a.fromNode_);return e.length&&o.unshift(r),o.unshift(t.id),o}static tree_(t,e){if(!e)return[];if(!e.childNodes.length)return e.id;const r=e.id,n=[r],s=o.evalXPath(`.//self::*[@${i.Attribute.ID}=${r}]`,t)[0],l=a.combineContentChildren(e,e.contentNodes.map((function(t){return t})),e.childNodes.map((function(t){return t})));s&&a.addOwns_(s,l);for(let e,r=0;e=l[r];r++)n.push(a.tree_(t,e));return n}static addOwns_(t,e){const r=t.getAttribute(i.Attribute.COLLAPSED),n=r?a.realLeafs_(a.fromString(r).array):e.map((t=>t.id));t.setAttribute(i.Attribute.OWNS,n.join(" "))}static realLeafs_(t){if(a.simpleCollapseStructure(t))return[t];if(a.contentCollapseStructure(t))return[];t=t;let e=[];for(let r=1;r<t.length;r++)e=e.concat(a.realLeafs_(t[r]));return e}populate(){this.parents&&this.levelsMap||(this.parents={},this.levelsMap={},this.populate_(this.array,this.array,[]))}toString(){return a.makeSexp_(this.array)}populate_(t,e,r){if(a.simpleCollapseStructure(t))return t=t,this.levelsMap[t]=e,void(this.parents[t]=t===r[0]?r.slice(1):r);const n=a.contentCollapseStructure(t)?t.slice(1):t,o=[n[0]].concat(r);for(let e=0,r=n.length;e<r;e++){const r=n[e];this.populate_(r,t,o)}}isRoot(t){return t===this.levelsMap[t][0]}directChildren(t){if(!this.isRoot(t))return[];return this.levelsMap[t].slice(1).map((t=>a.simpleCollapseStructure(t)?t:a.contentCollapseStructure(t)?t[1]:t[0]))}subtreeNodes(t){if(!this.isRoot(t))return[];const e=(t,r)=>{a.simpleCollapseStructure(t)?r.push(t):(t=t,a.contentCollapseStructure(t)&&(t=t.slice(1)),t.forEach((t=>e(t,r))))},r=this.levelsMap[t],n=[];return e(r.slice(1),n),n}}e.SemanticSkeleton=a},7075:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticTree=void 0;const n=r(5740),o=r(7630),i=r(9265),a=r(7228),s=r(5952),l=r(5609);r(94);class c{constructor(t){this.mathml=t,this.parser=new a.SemanticMathml,this.root=this.parser.parse(t),this.collator=this.parser.getFactory().leafMap.collateMeaning();const e=this.collator.newDefault();e&&(this.parser=new a.SemanticMathml,this.parser.getFactory().defaultMap=e,this.root=this.parser.parse(t)),u.visit(this.root,{}),(0,o.annotate)(this.root)}static empty(){const t=n.parseInput("<math/>"),e=new c(t);return e.mathml=t,e}static fromNode(t,e){const r=c.empty();return r.root=t,e&&(r.mathml=e),r}static fromRoot(t,e){let r=t;for(;r.parent;)r=r.parent;const n=c.fromNode(r);return e&&(n.mathml=e),n}static fromXml(t){const e=c.empty();return t.childNodes[0]&&(e.root=s.SemanticNode.fromXml(t.childNodes[0])),e}xml(t){const e=n.parseInput("<stree></stree>"),r=this.root.xml(e.ownerDocument,t);return e.appendChild(r),e}toString(t){return n.serializeXml(this.xml(t))}formatXml(t){const e=this.toString(t);return n.formatXml(e)}displayTree(){this.root.displayTree()}replaceNode(t,e){const r=t.parent;r?r.replaceChild(t,e):this.root=e}toJson(){const t={};return t.stree=this.root.toJson(),t}}e.SemanticTree=c;const u=new i.SemanticVisitor("general","unit",((t,e)=>{if("infixop"===t.type&&("multiplication"===t.role||"implicit"===t.role)){const e=t.childNodes;e.length&&(l.isPureUnit(e[0])||l.isUnitCounter(e[0]))&&t.childNodes.slice(1).every(l.isPureUnit)&&(t.role="unit")}return!1}))},4795:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.partitionNodes=e.sliceNodes=e.getEmbellishedInner=e.addAttributes=e.isZeroLength=e.purgeNodes=e.isOrphanedGlyph=e.hasDisplayTag=e.hasEmptyTag=e.hasIgnoreTag=e.hasLeafTag=e.hasMathTag=e.directSpeechKeys=e.DISPLAYTAGS=e.EMPTYTAGS=e.IGNORETAGS=e.LEAFTAGS=void 0;const n=r(5740);function o(t){return!!t&&-1!==e.LEAFTAGS.indexOf(n.tagName(t))}function i(t,e,r){r&&t.reverse();const n=[];for(let o,i=0;o=t[i];i++){if(e(o))return r?{head:t.slice(i+1).reverse(),div:o,tail:n.reverse()}:{head:n,div:o,tail:t.slice(i+1)};n.push(o)}return r?{head:[],div:null,tail:n.reverse()}:{head:n,div:null,tail:[]}}e.LEAFTAGS=["MO","MI","MN","MTEXT","MS","MSPACE"],e.IGNORETAGS=["MERROR","MPHANTOM","MALIGNGROUP","MALIGNMARK","MPRESCRIPTS","ANNOTATION","ANNOTATION-XML"],e.EMPTYTAGS=["MATH","MROW","MPADDED","MACTION","NONE","MSTYLE","SEMANTICS"],e.DISPLAYTAGS=["MROOT","MSQRT"],e.directSpeechKeys=["aria-label","exact-speech","alt"],e.hasMathTag=function(t){return!!t&&"MATH"===n.tagName(t)},e.hasLeafTag=o,e.hasIgnoreTag=function(t){return!!t&&-1!==e.IGNORETAGS.indexOf(n.tagName(t))},e.hasEmptyTag=function(t){return!!t&&-1!==e.EMPTYTAGS.indexOf(n.tagName(t))},e.hasDisplayTag=function(t){return!!t&&-1!==e.DISPLAYTAGS.indexOf(n.tagName(t))},e.isOrphanedGlyph=function(t){return!!t&&"MGLYPH"===n.tagName(t)&&!o(t.parentNode)},e.purgeNodes=function(t){const r=[];for(let o,i=0;o=t[i];i++){if(o.nodeType!==n.NodeType.ELEMENT_NODE)continue;const t=n.tagName(o);-1===e.IGNORETAGS.indexOf(t)&&(-1!==e.EMPTYTAGS.indexOf(t)&&0===o.childNodes.length||r.push(o))}return r},e.isZeroLength=function(t){if(!t)return!1;if(-1!==["negativeveryverythinmathspace","negativeverythinmathspace","negativethinmathspace","negativemediummathspace","negativethickmathspace","negativeverythickmathspace","negativeveryverythickmathspace"].indexOf(t))return!0;const e=t.match(/[0-9.]+/);return!!e&&0===parseFloat(e[0])},e.addAttributes=function(t,r){if(r.hasAttributes()){const n=r.attributes;for(let r=n.length-1;r>=0;r--){const o=n[r].name;o.match(/^ext/)&&(t.attributes[o]=n[r].value,t.nobreaking=!0),-1!==e.directSpeechKeys.indexOf(o)&&(t.attributes["ext-speech"]=n[r].value,t.nobreaking=!0),o.match(/texclass$/)&&(t.attributes.texclass=n[r].value),"href"===o&&(t.attributes.href=n[r].value,t.nobreaking=!0)}}},e.getEmbellishedInner=function t(e){return e&&e.embellished&&e.childNodes.length>0?t(e.childNodes[0]):e},e.sliceNodes=i,e.partitionNodes=function(t,e){let r=t;const n=[],o=[];let a=null;do{a=i(r,e),o.push(a.head),n.push(a.div),r=a.tail}while(a.div);return n.pop(),{rel:n,comp:o}}},6278:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.AbstractSpeechGenerator=void 0;const n=r(6828),o=r(2298),i=r(1214),a=r(9543);e.AbstractSpeechGenerator=class{constructor(){this.modality=o.addPrefix("speech"),this.rebuilt_=null,this.options_={}}getRebuilt(){return this.rebuilt_}setRebuilt(t){this.rebuilt_=t}setOptions(t){this.options_=t||{},this.modality=o.addPrefix(this.options_.modality||"speech")}getOptions(){return this.options_}start(){}end(){}generateSpeech(t,e){return this.rebuilt_||(this.rebuilt_=new i.RebuildStree(e)),(0,n.setup)(this.options_),a.computeMarkup(this.getRebuilt().xml)}}},1452:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.AdhocSpeechGenerator=void 0;const n=r(6278);class o extends n.AbstractSpeechGenerator{getSpeech(t,e){const r=this.generateSpeech(t,e);return t.setAttribute(this.modality,r),r}}e.AdhocSpeechGenerator=o},5152:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.ColorGenerator=void 0;const n=r(2298),o=r(8396),i=r(1214),a=r(1204),s=r(6278);class l extends s.AbstractSpeechGenerator{constructor(){super(...arguments),this.modality=(0,n.addPrefix)("foreground"),this.contrast=new o.ContrastPicker}static visitStree_(t,e,r){if(t.childNodes.length){if(t.contentNodes.length&&("punctuated"===t.type&&t.contentNodes.forEach((t=>r[t.id]=!0)),"implicit"!==t.role&&e.push(t.contentNodes.map((t=>t.id)))),t.childNodes.length){if("implicit"===t.role){const n=[];let o=[];for(const e of t.childNodes){const t=[];l.visitStree_(e,t,r),t.length<=2&&n.push(t.shift()),o=o.concat(t)}return e.push(n),void o.forEach((t=>e.push(t)))}t.childNodes.forEach((t=>l.visitStree_(t,e,r)))}}else r[t.id]||e.push(t.id)}getSpeech(t,e){return a.getAttribute(t,this.modality)}generateSpeech(t,e){return this.getRebuilt()||this.setRebuilt(new i.RebuildStree(t)),this.colorLeaves_(t),a.getAttribute(t,this.modality)}colorLeaves_(t){const e=[];l.visitStree_(this.getRebuilt().streeRoot,e,{});for(const r of e){const e=this.contrast.generate();let n=!1;n=Array.isArray(r)?r.map((r=>this.colorLeave_(t,r,e))).reduce(((t,e)=>t||e),!1):this.colorLeave_(t,r.toString(),e),n&&this.contrast.increment()}}colorLeave_(t,e,r){const n=a.getBySemanticId(t,e);return!!n&&(n.setAttribute(this.modality,r),!0)}}e.ColorGenerator=l},6604:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.DirectSpeechGenerator=void 0;const n=r(1204),o=r(6278);class i extends o.AbstractSpeechGenerator{getSpeech(t,e){return n.getAttribute(t,this.modality)}}e.DirectSpeechGenerator=i},3123:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.DummySpeechGenerator=void 0;const n=r(6278);class o extends n.AbstractSpeechGenerator{getSpeech(t,e){return""}}e.DummySpeechGenerator=o},5858:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.NodeSpeechGenerator=void 0;const n=r(1204),o=r(4598);class i extends o.TreeSpeechGenerator{getSpeech(t,e){return super.getSpeech(t,e),n.getAttribute(t,this.modality)}}e.NodeSpeechGenerator=i},9552:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.generatorMapping_=e.generator=void 0;const n=r(1452),o=r(5152),i=r(6604),a=r(3123),s=r(5858),l=r(597),c=r(4598);e.generator=function(t){return(e.generatorMapping_[t]||e.generatorMapping_.Direct)()},e.generatorMapping_={Adhoc:()=>new n.AdhocSpeechGenerator,Color:()=>new o.ColorGenerator,Direct:()=>new i.DirectSpeechGenerator,Dummy:()=>new a.DummySpeechGenerator,Node:()=>new s.NodeSpeechGenerator,Summary:()=>new l.SummarySpeechGenerator,Tree:()=>new c.TreeSpeechGenerator}},9543:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.computeSummary_=e.retrieveSummary=e.connectAllMactions=e.connectMactions=e.nodeAtPosition_=e.computePrefix_=e.retrievePrefix=e.addPrefix=e.addModality=e.addSpeech=e.recomputeMarkup=e.computeMarkup=e.recomputeSpeech=e.computeSpeech=void 0;const n=r(8290),o=r(5740),i=r(5274),a=r(2298),s=r(2362),l=r(7075),c=r(1204);function u(t){return s.SpeechRuleEngine.getInstance().evaluateNode(t)}function p(t){return u(l.SemanticTree.fromNode(t).xml())}function h(t){const e=p(t);return n.markup(e)}function d(t){const e=f(t);return n.markup(e)}function f(t){const e=l.SemanticTree.fromRoot(t),r=i.evalXPath('.//*[@id="'+t.id+'"]',e.xml());let n=r[0];return r.length>1&&(n=m(t,r)||n),n?s.SpeechRuleEngine.getInstance().runInSetting({modality:"prefix",domain:"default",style:"default",strict:!0,speech:!0},(function(){return s.SpeechRuleEngine.getInstance().evaluateNode(n)})):[]}function m(t,e){const r=e[0];if(!t.parent)return r;const n=[];for(;t;)n.push(t.id),t=t.parent;const o=function(t,e){for(;e.length&&e.shift().toString()===t.getAttribute("id")&&t.parentNode&&t.parentNode.parentNode;)t=t.parentNode.parentNode;return!e.length};for(let t,r=0;t=e[r];r++)if(o(t,n.slice()))return t;return r}function y(t){return t?s.SpeechRuleEngine.getInstance().runInSetting({modality:"summary",strict:!1,speech:!0},(function(){return s.SpeechRuleEngine.getInstance().evaluateNode(t)})):[]}e.computeSpeech=u,e.recomputeSpeech=p,e.computeMarkup=function(t){const e=u(t);return n.markup(e)},e.recomputeMarkup=h,e.addSpeech=function(t,e,r){const i=o.querySelectorAllByAttrValue(r,"id",e.id.toString())[0],s=i?n.markup(u(i)):h(e);t.setAttribute(a.Attribute.SPEECH,s)},e.addModality=function(t,e,r){const n=h(e);t.setAttribute(r,n)},e.addPrefix=function(t,e){const r=d(e);r&&t.setAttribute(a.Attribute.PREFIX,r)},e.retrievePrefix=d,e.computePrefix_=f,e.nodeAtPosition_=m,e.connectMactions=function(t,e,r){const n=o.querySelectorAll(e,"maction");for(let e,i=0;e=n[i];i++){const n=e.getAttribute("id"),i=o.querySelectorAllByAttrValue(t,"id",n)[0];if(!i)continue;const s=e.childNodes[1],l=s.getAttribute(a.Attribute.ID);let u=c.getBySemanticId(t,l);if(u&&"dummy"!==u.getAttribute(a.Attribute.TYPE))continue;if(u=i.childNodes[0],u.getAttribute("sre-highlighter-added"))continue;const p=s.getAttribute(a.Attribute.PARENT);p&&u.setAttribute(a.Attribute.PARENT,p),u.setAttribute(a.Attribute.TYPE,"dummy"),u.setAttribute(a.Attribute.ID,l);o.querySelectorAllByAttrValue(r,"id",l)[0].setAttribute("alternative",l)}},e.connectAllMactions=function(t,e){const r=o.querySelectorAll(t,"maction");for(let t,n=0;t=r[n];n++){const r=t.childNodes[1].getAttribute(a.Attribute.ID);o.querySelectorAllByAttrValue(e,"id",r)[0].setAttribute("alternative",r)}},e.retrieveSummary=function(t){const e=y(t);return n.markup(e)},e.computeSummary_=y},597:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SummarySpeechGenerator=void 0;const n=r(6278),o=r(9543);class i extends n.AbstractSpeechGenerator{getSpeech(t,e){return o.connectAllMactions(e,this.getRebuilt().xml),this.generateSpeech(t,e)}}e.SummarySpeechGenerator=i},4598:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.TreeSpeechGenerator=void 0;const n=r(2298),o=r(1204),i=r(6278),a=r(9543);class s extends i.AbstractSpeechGenerator{getSpeech(t,e){const r=this.generateSpeech(t,e),i=this.getRebuilt().nodeDict;for(const r in i){const s=i[r],l=o.getBySemanticId(e,r),c=o.getBySemanticId(t,r);l&&c&&(this.modality&&this.modality!==n.Attribute.SPEECH?a.addModality(c,s,this.modality):a.addSpeech(c,s,this.getRebuilt().xml),this.modality===n.Attribute.SPEECH&&a.addPrefix(c,s))}return r}}e.TreeSpeechGenerator=s},313:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.INTERVALS=e.makeLetter=e.numberRules=e.alphabetRules=e.getFont=e.makeInterval=e.generate=e.makeDomains_=e.Domains_=e.Base=e.Embellish=e.Font=void 0;const n=r(5897),o=r(7491),i=r(4356),a=r(2536),s=r(2780);var l,c,u;function p(){const t=i.LOCALE.ALPHABETS,r=(t,e)=>{const r={};return Object.keys(t).forEach((t=>r[t]=!0)),Object.keys(e).forEach((t=>r[t]=!0)),Object.keys(r)};e.Domains_.small=r(t.smallPrefix,t.letterTrans),e.Domains_.capital=r(t.capPrefix,t.letterTrans),e.Domains_.digit=r(t.digitPrefix,t.digitTrans)}function h(t){const e=t.toString(16).toUpperCase();return e.length>3?e:("000"+e).slice(-4)}function d([t,e],r){const n=parseInt(t,16),o=parseInt(e,16),i=[];for(let t=n;t<=o;t++){let e=h(t);!1!==r[e]&&(e=r[e]||e,i.push(e))}return i}function f(t){const e="normal"===t||"fullwidth"===t?"":i.LOCALE.MESSAGES.font[t]||i.LOCALE.MESSAGES.embellish[t]||"";return(0,a.localeFontCombiner)(e)}function m(t,r,n,o,a,s){const l=f(o);for(let o,c,u,p=0;o=t[p],c=r[p],u=n[p];p++){const t=s?i.LOCALE.ALPHABETS.capPrefix:i.LOCALE.ALPHABETS.smallPrefix,r=s?e.Domains_.capital:e.Domains_.small;g(l.combiner,o,c,u,l.font,t,a,i.LOCALE.ALPHABETS.letterTrans,r)}}function y(t,r,n,o,a){const s=f(n);for(let n,l,c=0;n=t[c],l=r[c];c++){const t=i.LOCALE.ALPHABETS.digitPrefix,r=c+a;g(s.combiner,n,l,r,s.font,t,o,i.LOCALE.ALPHABETS.digitTrans,e.Domains_.digit)}}function g(t,e,r,n,o,i,a,l,c){for(let u,p=0;u=c[p];p++){const c=u in l?l[u]:l.default,p=u in 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smartPreferences(t,e){const r=c.getLocalePreferences()[e];if(!r)return[];const n=t.explorers.speech,i=c.relevantPreferences(n.walker.getFocus().getSemanticPrimary()),a=o.DOMAIN_TO_STYLES.clearspeak;return[{type:"radio",content:"No Preferences",id:"clearspeak-default",variable:"speechRules"},{type:"radio",content:"Current Preferences",id:"clearspeak-"+a,variable:"speechRules"},{type:"rule"},{type:"label",content:"Preferences for "+i},{type:"rule"}].concat(r[i].map((function(t){const e=t.split("_");return{type:"radio",content:e[1],id:"clearspeak-"+c.addPreference(a,e[0],e[1]),variable:"speechRules"}})))}static relevantPreferences(t){const e=f[t.type];return e&&(e[t.role]||e[""])||"ImpliedTimes"}static findPreference(t,e){if("default"===t)return c.AUTO;return c.fromPreference(t)[e]||c.AUTO}static addPreference(t,e,r){if("default"===t)return e+"_"+r;const n=c.fromPreference(t);return n[e]=r,c.toPreference(n)}static getLocalePreferences_(t){const e={};for(const r in 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p extends a.DefaultComparator{constructor(t,e){super(t,e),this.preference=t instanceof c?t.preference:{}}match(t){if(!(t instanceof c))return super.match(t);if("default"===t.getComponents()[a.Axis.STYLE])return!0;const e=Object.keys(t.preference);for(let r,n=0;r=e[n];n++)if(this.preference[r]!==t.preference[r])return!1;return!0}compare(t,e){const r=super.compare(t,e);if(0!==r)return r;const n=t instanceof c,o=e instanceof c;if(!n&&o)return 1;if(n&&!o)return-1;if(!n&&!o)return 0;const i=Object.keys(t.preference).length,a=Object.keys(e.preference).length;return i>a?-1:i<a?1:0}}e.Comparator=p;class h extends a.DynamicCstrParser{constructor(){super([a.Axis.LOCALE,a.Axis.MODALITY,a.Axis.DOMAIN,a.Axis.STYLE])}parse(t){const e=super.parse(t);let r=e.getValue(a.Axis.STYLE);const n=e.getValue(a.Axis.LOCALE),o=e.getValue(a.Axis.MODALITY);let s={};return r!==i.DynamicCstr.DEFAULT_VALUE&&(s=this.fromPreference(r),r=this.toPreference(s)),new c({locale:n,modality:o,domain:"clearspeak",style:r},s)}fromPreference(t){return c.fromPreference(t)}toPreference(t){return c.toPreference(t)}}e.Parser=h;const d=[["AbsoluteValue","fenced","neutral","metric"],["Bar","overscore","overaccent"],["Caps","identifier","latinletter"],["CombinationPermutation","appl","unknown"],["Ellipses","punctuation","ellipsis"],["Exponent","superscript",""],["Fraction","fraction",""],["Functions","appl","simple function"],["ImpliedTimes","operator","implicit"],["Log","appl","prefix 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n=r(1676),o=r(365),i=r(9912),a=r(1378),s=r(8437),l=r(7283);e.ClearspeakRules=function(){l.addStore(n.DynamicCstr.BASE_LOCALE+".speech.clearspeak","",{CTFpauseSeparator:o.pauseSeparator,CTFnodeCounter:i.nodeCounter,CTFcontentIterator:o.contentIterator,CSFvulgarFraction:s.vulgarFraction,CQFvulgarFractionSmall:i.isSmallVulgarFraction,CQFcellsSimple:i.allCellsSimple,CSFordinalExponent:i.ordinalExponent,CSFwordOrdinal:i.wordOrdinal,CQFmatchingFences:i.matchingFences,CSFnestingDepth:i.nestingDepth,CQFfencedArguments:i.fencedArguments,CQFsimpleArguments:i.simpleArguments,CQFspaceoutNumber:a.spaceoutNumber})}},9912:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.wordOrdinal=e.layoutFactor_=e.fencedFactor_=e.simpleFactor_=e.simpleArguments=e.fencedArguments=e.insertNesting=e.matchingFences=e.nestingDepth=e.NESTING_DEPTH=e.ordinalExponent=e.allTextLastContent_=e.isUnitExpression=e.isSmallVulgarFraction=e.allCellsSimple=e.allIndices_=e.isInteger_=e.simpleCell_=e.simpleNode=e.hasPreference=e.isSimpleFraction_=e.isSimpleNumber_=e.isNumber_=e.isLetter_=e.isSimple_=e.isSimpleLetters_=e.isSimpleDegree_=e.isSimpleNegative_=e.isSimpleFunction_=e.isSimpleExpression=e.nodeCounter=void 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n>1&&n<=e?[t]:[]}},6922:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.implicitIterator=e.relationIterator=e.propagateNumber=e.checkParent_=e.NUMBER_INHIBITORS_=e.NUMBER_PROPAGATORS_=e.enlargeFence=e.indexRadical=e.closingRadical=e.openingRadical=e.radicalNestingDepth=e.nestedRadical=e.overBevelledFraction=e.overFraction=e.closingFraction=e.openingFraction=void 0;const n=r(7052),o=r(5740),i=r(5274),a=r(2105),s=r(5897),l=r(7630),c=r(9265),u=r(4356),p=r(1378);function h(t,e){const r=d(t);return 1===r?e:new Array(r).join(u.LOCALE.MESSAGES.MS.NESTED)+e}function d(t,e){const r=e||0;return t.parentNode?d(t.parentNode,"root"===t.tagName||"sqrt"===t.tagName?r+1:r):r}function f(t){const e="\u2820";if(1===t.length)return e+t;const r=t.split("");return r.every((function(t){return"\u2833"===t}))?e+r.join(e):t.slice(0,-1)+e+t.slice(-1)}function m(t,r){const n=t.parent;if(!n)return!1;const o=n.type;return-1!==e.NUMBER_PROPAGATORS_.indexOf(o)||"prefixop"===o&&"negative"===n.role&&!r.script||"prefixop"===o&&"geometry"===n.role||!("punctuated"!==o||r.enclosed&&"text"!==n.role)}function y(t,r){return t.childNodes.length?(-1!==e.NUMBER_INHIBITORS_.indexOf(t.type)&&(r.script=!0),"fenced"===t.type?(r.number=!1,r.enclosed=!0,["",r]):(m(t,r)&&(r.number=!0,r.enclosed=!1),["",r])):(m(t,r)&&(r.number=!0,r.script=!1,r.enclosed=!1),[r.number?"number":"",{number:!1,enclosed:r.enclosed,script:r.script}])}e.openingFraction=function(t){const e=p.fractionNestingDepth(t);return new Array(e).join(u.LOCALE.MESSAGES.MS.FRACTION_REPEAT)+u.LOCALE.MESSAGES.MS.FRACTION_START},e.closingFraction=function(t){const e=p.fractionNestingDepth(t);return new Array(e).join(u.LOCALE.MESSAGES.MS.FRACTION_REPEAT)+u.LOCALE.MESSAGES.MS.FRACTION_END},e.overFraction=function(t){const e=p.fractionNestingDepth(t);return new 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l.concat(c&&u&&"space"===s.getAttribute("role")?[n.AuditoryDescription.create({text:"\u2800"},{})]:[])}}},8437:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.ordinalPosition=e.vulgarFraction=e.wordCounter=e.ordinalCounter=void 0;const n=r(9536),o=r(5740),i=r(4356),a=r(4977);e.ordinalCounter=function(t,e){let r=0;return function(){return i.LOCALE.NUMBERS.numericOrdinal(++r)+" "+e}},e.wordCounter=function(t,e){let r=0;return function(){return i.LOCALE.NUMBERS.numberToOrdinal(++r,!1)+" "+e}},e.vulgarFraction=function(t){const e=(0,a.convertVulgarFraction)(t,i.LOCALE.MESSAGES.MS.FRAC_OVER);return e.convertible&&e.enumerator&&e.denominator?[new n.Span(i.LOCALE.NUMBERS.numberToWords(e.enumerator),{extid:t.childNodes[0].childNodes[0].getAttribute("extid"),separator:""}),new n.Span(i.LOCALE.NUMBERS.vulgarSep,{separator:""}),new n.Span(i.LOCALE.NUMBERS.numberToOrdinal(e.denominator,1!==e.enumerator),{extid:t.childNodes[0].childNodes[1].getAttribute("extid")})]:[new n.Span(e.content||"",{extid:t.getAttribute("extid")})]},e.ordinalPosition=function(t){const e=o.toArray(t.parentNode.childNodes);return i.LOCALE.NUMBERS.numericOrdinal(e.indexOf(t)+1).toString()}},9284:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.BrailleRules=e.OtherRules=e.PrefixRules=void 0;const 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n=r(7052),o=r(8290),i=r(5740),a=r(4440),s=r(6828),l=r(8496),c=r(2298),u=r(4356),p=r(2105),h=r(5656),d=r(9552),f=r(9543),m=r(8504),y=r(7730),g=r(1214),b=r(179),v=r(1204),_=r(5274);class S{constructor(t,e,r,n){this.node=t,this.generator=e,this.highlighter=r,this.modifier=!1,this.keyMapping=new Map([[l.KeyCode.UP,this.up.bind(this)],[l.KeyCode.DOWN,this.down.bind(this)],[l.KeyCode.RIGHT,this.right.bind(this)],[l.KeyCode.LEFT,this.left.bind(this)],[l.KeyCode.TAB,this.repeat.bind(this)],[l.KeyCode.DASH,this.expand.bind(this)],[l.KeyCode.SPACE,this.depth.bind(this)],[l.KeyCode.HOME,this.home.bind(this)],[l.KeyCode.X,this.summary.bind(this)],[l.KeyCode.Z,this.detail.bind(this)],[l.KeyCode.V,this.virtualize.bind(this)],[l.KeyCode.P,this.previous.bind(this)],[l.KeyCode.U,this.undo.bind(this)],[l.KeyCode.LESS,this.previousRules.bind(this)],[l.KeyCode.GREATER,this.nextRules.bind(this)]]),this.cursors=[],this.xml_=null,this.rebuilt_=null,this.focus_=null,this.active_=!1,this.node.id?this.id=this.node.id:this.node.hasAttribute(S.SRE_ID_ATTR)?this.id=this.node.getAttribute(S.SRE_ID_ATTR):(this.node.setAttribute(S.SRE_ID_ATTR,S.ID_COUNTER.toString()),this.id=S.ID_COUNTER++),this.rootNode=v.getSemanticRoot(t),this.rootId=this.rootNode.getAttribute(c.Attribute.ID),this.xmlString_=n,this.moved=b.WalkerMoves.ENTER}getXml(){return 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g.RebuildStree(this.getXml()),this.rootId=this.rebuilt_.stree.root.id.toString(),this.generator.setRebuilt(this.rebuilt_),this.skeleton=h.SemanticSkeleton.fromTree(this.rebuilt_.stree),this.skeleton.populate(),this.focus_=this.singletonFocus(this.rootId),this.levels=this.initLevels(),f.connectMactions(this.node,this.getXml(),this.rebuilt_.xml)}previousLevel(){const t=this.getFocus().getDomPrimary();return t?v.getAttribute(t,c.Attribute.PARENT):this.getFocus().getSemanticPrimary().parent.id.toString()}nextLevel(){const t=this.getFocus().getDomPrimary();let e,r;if(t){e=v.splitAttribute(v.getAttribute(t,c.Attribute.CHILDREN)),r=v.splitAttribute(v.getAttribute(t,c.Attribute.CONTENT));const n=v.getAttribute(t,c.Attribute.TYPE),o=v.getAttribute(t,c.Attribute.ROLE);return this.combineContentChildren(n,o,r,e)}const n=t=>t.id.toString(),o=this.getRebuilt().nodeDict[this.primaryId()];return e=o.childNodes.map(n),r=o.contentNodes.map(n),0===e.length?[]:this.combineContentChildren(o.type,o.role,r,e)}singletonFocus(t){this.getRebuilt();const e=this.retrieveVisuals(t);return this.focusFromId(t,e)}retrieveVisuals(t){if(!this.skeleton)return[t];const e=parseInt(t,10),r=this.skeleton.subtreeNodes(e);if(!r.length)return[t];r.unshift(e);const n={},o=[];_.updateEvaluator(this.getXml());for(const t of r)n[t]||(o.push(t.toString()),n[t]=!0,this.subtreeIds(t,n));return o}subtreeIds(t,e){const r=_.evalXPath(`//*[@data-semantic-id="${t}"]`,this.getXml());_.evalXPath("*//@data-semantic-id",r[0]).forEach((t=>e[parseInt(t.textContent,10)]=!0))}focusFromId(t,e){return y.Focus.factory(t,e,this.getRebuilt(),this.node)}summary(){return this.moved=this.isSpeech()?b.WalkerMoves.SUMMARY:b.WalkerMoves.REPEAT,this.getFocus().clone()}detail(){return this.moved=this.isSpeech()?b.WalkerMoves.DETAIL:b.WalkerMoves.REPEAT,this.getFocus().clone()}specialMove(){return null}virtualize(t){return this.cursors.push({focus:this.getFocus(),levels:this.levels,undo:t||!this.cursors.length}),this.levels=this.levels.clone(),this.getFocus().clone()}previous(){const t=this.cursors.pop();return t?(this.levels=t.levels,t.focus):this.getFocus()}undo(){let t;do{t=this.cursors.pop()}while(t&&!t.undo);return t?(this.levels=t.levels,t.focus):this.getFocus()}update(t){this.generator.setOptions(t),(0,s.setup)(t).then((()=>d.generator("Tree").getSpeech(this.node,this.getXml())))}nextRules(){const t=this.generator.getOptions();return"speech"!==t.modality?this.getFocus():(a.DOMAIN_TO_STYLES[t.domain]=t.style,t.domain="mathspeak"===t.domain?"clearspeak":"mathspeak",t.style=a.DOMAIN_TO_STYLES[t.domain],this.update(t),this.moved=b.WalkerMoves.REPEAT,this.getFocus().clone())}nextStyle(t,e){if("mathspeak"===t){const t=["default","brief","sbrief"],r=t.indexOf(e);return-1===r?e:r>=t.length-1?t[0]:t[r+1]}if("clearspeak"===t){const t=m.ClearspeakPreferences.getLocalePreferences().en;if(!t)return"default";const r=m.ClearspeakPreferences.relevantPreferences(this.getFocus().getSemanticPrimary()),n=m.ClearspeakPreferences.findPreference(e,r),o=t[r].map((function(t){return t.split("_")[1]})),i=o.indexOf(n);if(-1===i)return e;const a=i>=o.length-1?o[0]:o[i+1];return m.ClearspeakPreferences.addPreference(e,r,a)}return e}previousRules(){const t=this.generator.getOptions();return"speech"!==t.modality?this.getFocus():(t.style=this.nextStyle(t.domain,t.style),this.update(t),this.moved=b.WalkerMoves.REPEAT,this.getFocus().clone())}refocus(){let t,e=this.getFocus();for(;!e.getNodes().length;){t=this.levels.peek();const r=this.up();if(!r)break;this.setFocus(r),e=this.getFocus(!0)}this.levels.push(t),this.setFocus(e)}toggleActive_(){this.active_=!this.active_}mergePrefix_(t,e=[]){const r=this.isSpeech()?this.prefix_():"";r&&t.unshift(r);const n=this.isSpeech()?this.postfix_():"";return n&&t.push(n),o.finalize(o.merge(e.concat(t)))}prefix_(){const t=this.getFocus().getDomNodes(),e=this.getFocus().getSemanticNodes();return t[0]?v.getAttribute(t[0],c.Attribute.PREFIX):f.retrievePrefix(e[0])}postfix_(){const t=this.getFocus().getDomNodes();return t[0]?v.getAttribute(t[0],c.Attribute.POSTFIX):""}depth_(){const t=p.Grammar.getInstance().getParameter("depth");p.Grammar.getInstance().setParameter("depth",!0);const e=this.getFocus().getDomPrimary(),r=this.expandable(e)?u.LOCALE.MESSAGES.navigate.EXPANDABLE:this.collapsible(e)?u.LOCALE.MESSAGES.navigate.COLLAPSIBLE:"",i=u.LOCALE.MESSAGES.navigate.LEVEL+" "+this.getDepth(),a=this.getFocus().getSemanticNodes(),s=f.retrievePrefix(a[0]),l=[new n.AuditoryDescription({text:i,personality:{}}),new n.AuditoryDescription({text:s,personality:{}}),new n.AuditoryDescription({text:r,personality:{}})];return p.Grammar.getInstance().setParameter("depth",t),o.finalize(o.markup(l))}actionable_(t){const e=null==t?void 0:t.parentNode;return e&&this.highlighter.isMactionNode(e)?e:null}summary_(){const t=this.getFocus().getSemanticPrimary().id.toString(),e=this.getRebuilt().xml.getAttribute("id")===t?this.getRebuilt().xml:i.querySelectorAllByAttrValue(this.getRebuilt().xml,"id",t)[0],r=f.retrieveSummary(e);return this.mergePrefix_([r])}detail_(){const t=this.getFocus().getSemanticPrimary().id.toString(),e=this.getRebuilt().xml.getAttribute("id")===t?this.getRebuilt().xml:i.querySelectorAllByAttrValue(this.getRebuilt().xml,"id",t)[0],r=e.getAttribute("alternative");e.removeAttribute("alternative");const n=f.computeMarkup(e),o=this.mergePrefix_([n]);return e.setAttribute("alternative",r),o}}e.AbstractWalker=S,S.ID_COUNTER=0,S.SRE_ID_ATTR="sre-explorer-id"},162:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.DummyWalker=void 0;const n=r(3284);class o extends n.AbstractWalker{up(){return null}down(){return null}left(){return null}right(){return null}repeat(){return null}depth(){return null}home(){return null}getDepth(){return 0}initLevels(){return null}combineContentChildren(t,e,r,n){return[]}findFocusOnLevel(t){return null}}e.DummyWalker=o},7730:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.Focus=void 0;const n=r(1204);class o{constructor(t,e){this.nodes=t,this.primary=e,this.domNodes=[],this.domPrimary_=null,this.allNodes=[]}static factory(t,e,r,i){const a=t=>n.getBySemanticId(i,t),s=r.nodeDict,l=a(t),c=e.map(a),u=e.map((function(t){return s[t]})),p=new o(u,s[t]);return p.domNodes=c,p.domPrimary_=l,p.allNodes=o.generateAllVisibleNodes_(e,c,s,i),p}static generateAllVisibleNodes_(t,e,r,i){const a=t=>n.getBySemanticId(i,t);let s=[];for(let n=0,l=t.length;n<l;n++){if(e[n]){s.push(e[n]);continue}const l=r[t[n]];if(!l)continue;const c=l.childNodes.map((function(t){return t.id.toString()})),u=c.map(a);s=s.concat(o.generateAllVisibleNodes_(c,u,r,i))}return s}getSemanticPrimary(){return this.primary}getSemanticNodes(){return this.nodes}getNodes(){return this.allNodes}getDomNodes(){return this.domNodes}getDomPrimary(){return this.domPrimary_}toString(){return"Primary:"+this.domPrimary_+" Nodes:"+this.domNodes}clone(){const t=new o(this.nodes,this.primary);return t.domNodes=this.domNodes,t.domPrimary_=this.domPrimary_,t.allNodes=this.allNodes,t}}e.Focus=o},9797:function(t,e){Object.defineProperty(e,"__esModule",{value:!0}),e.Levels=void 0;class r{constructor(){this.level_=[]}push(t){this.level_.push(t)}pop(){return this.level_.pop()}peek(){return this.level_[this.level_.length-1]||null}indexOf(t){const e=this.peek();return e?e.indexOf(t):null}find(t){const e=this.peek();if(!e)return null;for(let r=0,n=e.length;r<n;r++)if(t(e[r]))return e[r];return null}get(t){const e=this.peek();return!e||t<0||t>=e.length?null:e[t]}depth(){return this.level_.length}clone(){const t=new r;return t.level_=this.level_.slice(0),t}toString(){let t="";for(let e,r=0;e=this.level_[r];r++)t+="\n"+e.map((function(t){return t.toString()}));return t}}e.Levels=r},1214:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.RebuildStree=void 0;const n=r(5740),o=r(2298),i=r(3588),a=r(6537),s=r(3308),l=r(5656),c=r(7075),u=r(4795),p=r(1204);class h{constructor(t){this.mathml=t,this.factory=new a.SemanticNodeFactory,this.nodeDict={},this.mmlRoot=p.getSemanticRoot(t),this.streeRoot=this.assembleTree(this.mmlRoot),this.stree=c.SemanticTree.fromNode(this.streeRoot,this.mathml),this.xml=this.stree.xml(),s.default.getInstance().setNodeFactory(this.factory)}static addAttributes(t,e,r){r&&1===e.childNodes.length&&e.childNodes[0].nodeType!==n.NodeType.TEXT_NODE&&u.addAttributes(t,e.childNodes[0]),u.addAttributes(t,e)}static textContent(t,e,r){if(!r&&e.textContent)return void(t.textContent=e.textContent);const n=p.splitAttribute(p.getAttribute(e,o.Attribute.OPERATOR));n.length>1&&(t.textContent=n[1])}static isPunctuated(t){return!l.SemanticSkeleton.simpleCollapseStructure(t)&&t[1]&&l.SemanticSkeleton.contentCollapseStructure(t[1])}getTree(){return this.stree}assembleTree(t){const e=this.makeNode(t),r=p.splitAttribute(p.getAttribute(t,o.Attribute.CHILDREN)),n=p.splitAttribute(p.getAttribute(t,o.Attribute.CONTENT));if(h.addAttributes(e,t,!(r.length||n.length)),0===n.length&&0===r.length)return h.textContent(e,t),e;if(n.length>0){const t=p.getBySemanticId(this.mathml,n[0]);t&&h.textContent(e,t,!0)}e.contentNodes=n.map((t=>this.setParent(t,e))),e.childNodes=r.map((t=>this.setParent(t,e)));const i=p.getAttribute(t,o.Attribute.COLLAPSED);return i?this.postProcess(e,i):e}makeNode(t){const e=p.getAttribute(t,o.Attribute.TYPE),r=p.getAttribute(t,o.Attribute.ROLE),n=p.getAttribute(t,o.Attribute.FONT),i=p.getAttribute(t,o.Attribute.ANNOTATION)||"",a=p.getAttribute(t,o.Attribute.ID),s=p.getAttribute(t,o.Attribute.EMBELLISHED),l=p.getAttribute(t,o.Attribute.FENCEPOINTER),c=this.createNode(parseInt(a,10));return c.type=e,c.role=r,c.font=n||"unknown",c.parseAnnotation(i),l&&(c.fencePointer=l),s&&(c.embellished=s),c}makePunctuation(t){const e=this.createNode(t);return e.updateContent((0,i.invisibleComma)()),e.role="dummy",e}makePunctuated(t,e,r){const n=this.createNode(e[0]);n.type="punctuated",n.embellished=t.embellished,n.fencePointer=t.fencePointer,n.role=r;const o=e.splice(1,1)[0].slice(1);n.contentNodes=o.map(this.makePunctuation.bind(this)),this.collapsedChildren_(e)}makeEmpty(t,e,r){const n=this.createNode(e);n.type="empty",n.embellished=t.embellished,n.fencePointer=t.fencePointer,n.role=r}makeIndex(t,e,r){if(h.isPunctuated(e))return this.makePunctuated(t,e,r),void(e=e[0]);l.SemanticSkeleton.simpleCollapseStructure(e)&&!this.nodeDict[e.toString()]&&this.makeEmpty(t,e,r)}postProcess(t,e){const r=l.SemanticSkeleton.fromString(e).array;if("subsup"===t.type){const e=this.createNode(r[1][0]);return e.type="subscript",e.role="subsup",t.type="superscript",e.embellished=t.embellished,e.fencePointer=t.fencePointer,this.makeIndex(t,r[1][2],"rightsub"),this.makeIndex(t,r[2],"rightsuper"),this.collapsedChildren_(r),t}if("subscript"===t.type)return this.makeIndex(t,r[2],"rightsub"),this.collapsedChildren_(r),t;if("superscript"===t.type)return this.makeIndex(t,r[2],"rightsuper"),this.collapsedChildren_(r),t;if("tensor"===t.type)return this.makeIndex(t,r[2],"leftsub"),this.makeIndex(t,r[3],"leftsuper"),this.makeIndex(t,r[4],"rightsub"),this.makeIndex(t,r[5],"rightsuper"),this.collapsedChildren_(r),t;if("punctuated"===t.type){if(h.isPunctuated(r)){const e=r.splice(1,1)[0].slice(1);t.contentNodes=e.map(this.makePunctuation.bind(this))}return t}if("underover"===t.type){const e=this.createNode(r[1][0]);return"overaccent"===t.childNodes[1].role?(e.type="overscore",t.type="underscore"):(e.type="underscore",t.type="overscore"),e.role="underover",e.embellished=t.embellished,e.fencePointer=t.fencePointer,this.collapsedChildren_(r),t}return t}createNode(t){const e=this.factory.makeNode(t);return this.nodeDict[t.toString()]=e,e}collapsedChildren_(t){const e=t=>{const r=this.nodeDict[t[0]];r.childNodes=[];for(let n=1,o=t.length;n<o;n++){const o=t[n];r.childNodes.push(l.SemanticSkeleton.simpleCollapseStructure(o)?this.nodeDict[o]:e(o))}return r};e(t)}setParent(t,e){const r=p.getBySemanticId(this.mathml,t),n=this.assembleTree(r);return n.parent=e,n}}e.RebuildStree=h},6295:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SemanticWalker=void 0;const n=r(3284),o=r(9797);class i extends n.AbstractWalker{constructor(t,e,r,n){super(t,e,r,n),this.node=t,this.generator=e,this.highlighter=r,this.levels=null,this.restoreState()}initLevels(){const t=new o.Levels;return t.push([this.getFocus()]),t}up(){super.up();const t=this.previousLevel();if(!t)return null;this.levels.pop();return this.levels.find((function(e){return e.getSemanticNodes().some((function(e){return e.id.toString()===t}))}))}down(){super.down();const t=this.nextLevel();return 0===t.length?null:(this.levels.push(t),t[0])}combineContentChildren(t,e,r,n){switch(t){case"relseq":case"infixop":case"multirel":return this.makePairList(n,r);case"prefixop":return[this.focusFromId(n[0],r.concat(n))];case"postfixop":return[this.focusFromId(n[0],n.concat(r))];case"matrix":case"vector":case"fenced":return[this.focusFromId(n[0],[r[0],n[0],r[1]])];case"cases":return[this.focusFromId(n[0],[r[0],n[0]])];case"punctuated":return"text"===e?n.map(this.singletonFocus.bind(this)):n.length===r.length?r.map(this.singletonFocus.bind(this)):this.combinePunctuations(n,r,[],[]);case"appl":return[this.focusFromId(n[0],[n[0],r[0]]),this.singletonFocus(n[1])];case"root":return[this.singletonFocus(n[1]),this.singletonFocus(n[0])];default:return n.map(this.singletonFocus.bind(this))}}combinePunctuations(t,e,r,n){if(0===t.length)return n;const o=t.shift(),i=e.shift();return o===i?(r.push(i),this.combinePunctuations(t,e,r,n)):(e.unshift(i),r.push(o),t.length===e.length?(n.push(this.focusFromId(o,r.concat(e))),n):(n.push(this.focusFromId(o,r)),this.combinePunctuations(t,e,[],n)))}makePairList(t,e){if(0===t.length)return[];if(1===t.length)return[this.singletonFocus(t[0])];const r=[this.singletonFocus(t.shift())];for(let n=0,o=t.length;n<o;n++)r.push(this.focusFromId(t[n],[e[n],t[n]]));return r}left(){super.left();const t=this.levels.indexOf(this.getFocus());if(null===t)return null;const e=this.levels.get(t-1);return e||null}right(){super.right();const t=this.levels.indexOf(this.getFocus());if(null===t)return null;const e=this.levels.get(t+1);return e||null}findFocusOnLevel(t){return this.levels.find((e=>e.getSemanticPrimary().id===t))}}e.SemanticWalker=i},9806:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.SyntaxWalker=void 0;const n=r(707),o=r(3284),i=r(9797);class a extends o.AbstractWalker{constructor(t,e,r,n){super(t,e,r,n),this.node=t,this.generator=e,this.highlighter=r,this.levels=null,this.restoreState()}initLevels(){const t=new i.Levels;return t.push([this.primaryId()]),t}up(){super.up();const t=this.previousLevel();return t?(this.levels.pop(),this.singletonFocus(t)):null}down(){super.down();const t=this.nextLevel();if(0===t.length)return null;const e=this.singletonFocus(t[0]);return e&&this.levels.push(t),e}combineContentChildren(t,e,r,o){switch(t){case"relseq":case"infixop":case"multirel":return(0,n.interleaveLists)(o,r);case"prefixop":return r.concat(o);case"postfixop":return o.concat(r);case"matrix":case"vector":case"fenced":return o.unshift(r[0]),o.push(r[1]),o;case"cases":return o.unshift(r[0]),o;case"punctuated":return"text"===e?(0,n.interleaveLists)(o,r):o;case"appl":return[o[0],r[0],o[1]];case"root":return[o[1],o[0]];default:return o}}left(){super.left();const t=this.levels.indexOf(this.primaryId());if(null===t)return null;const e=this.levels.get(t-1);return e?this.singletonFocus(e):null}right(){super.right();const t=this.levels.indexOf(this.primaryId());if(null===t)return null;const e=this.levels.get(t+1);return e?this.singletonFocus(e):null}findFocusOnLevel(t){return this.singletonFocus(t.toString())}focusDomNodes(){return[this.getFocus().getDomPrimary()]}focusSemanticNodes(){return[this.getFocus().getSemanticPrimary()]}}e.SyntaxWalker=a},1799:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.TableWalker=void 0;const n=r(5740),o=r(8496),i=r(9806),a=r(179);class s extends i.SyntaxWalker{constructor(t,e,r,n){super(t,e,r,n),this.node=t,this.generator=e,this.highlighter=r,this.firstJump=null,this.key_=null,this.row_=0,this.currentTable_=null,this.keyMapping.set(o.KeyCode.ZERO,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.ONE,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.TWO,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.THREE,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.FOUR,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.FIVE,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.SIX,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.SEVEN,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.EIGHT,this.jumpCell.bind(this)),this.keyMapping.set(o.KeyCode.NINE,this.jumpCell.bind(this))}move(t){this.key_=t;const e=super.move(t);return this.modifier=!1,e}up(){return this.moved=a.WalkerMoves.UP,this.eligibleCell_()?this.verticalMove_(!1):super.up()}down(){return this.moved=a.WalkerMoves.DOWN,this.eligibleCell_()?this.verticalMove_(!0):super.down()}jumpCell(){if(!this.isInTable_()||null===this.key_)return this.getFocus();if(this.moved===a.WalkerMoves.ROW){this.moved=a.WalkerMoves.CELL;const t=this.key_-o.KeyCode.ZERO;return this.isLegalJump_(this.row_,t)?this.jumpCell_(this.row_,t):this.getFocus()}const t=this.key_-o.KeyCode.ZERO;return t>this.currentTable_.childNodes.length?this.getFocus():(this.row_=t,this.moved=a.WalkerMoves.ROW,this.getFocus().clone())}undo(){const t=super.undo();return t===this.firstJump&&(this.firstJump=null),t}eligibleCell_(){const t=this.getFocus().getSemanticPrimary();return this.modifier&&"cell"===t.type&&-1!==s.ELIGIBLE_CELL_ROLES.indexOf(t.role)}verticalMove_(t){const e=this.previousLevel();if(!e)return null;const r=this.getFocus(),n=this.levels.indexOf(this.primaryId()),o=this.levels.pop(),i=this.levels.indexOf(e),a=this.levels.get(t?i+1:i-1);if(!a)return this.levels.push(o),null;this.setFocus(this.singletonFocus(a));const s=this.nextLevel();return s[n]?(this.levels.push(s),this.singletonFocus(s[n])):(this.setFocus(r),this.levels.push(o),null)}jumpCell_(t,e){this.firstJump?this.virtualize(!1):(this.firstJump=this.getFocus(),this.virtualize(!0));const r=this.currentTable_.id.toString();let n;do{n=this.levels.pop()}while(-1===n.indexOf(r));this.levels.push(n),this.setFocus(this.singletonFocus(r)),this.levels.push(this.nextLevel());const o=this.currentTable_.childNodes[t-1];return this.setFocus(this.singletonFocus(o.id.toString())),this.levels.push(this.nextLevel()),this.singletonFocus(o.childNodes[e-1].id.toString())}isLegalJump_(t,e){const r=n.querySelectorAllByAttrValue(this.getRebuilt().xml,"id",this.currentTable_.id.toString())[0];if(!r||r.hasAttribute("alternative"))return!1;const o=this.currentTable_.childNodes[t-1];if(!o)return!1;const i=n.querySelectorAllByAttrValue(r,"id",o.id.toString())[0];return!(!i||i.hasAttribute("alternative"))&&!(!o||!o.childNodes[e-1])}isInTable_(){let t=this.getFocus().getSemanticPrimary();for(;t;){if(-1!==s.ELIGIBLE_TABLE_TYPES.indexOf(t.type))return this.currentTable_=t,!0;t=t.parent}return!1}}e.TableWalker=s,s.ELIGIBLE_CELL_ROLES=["determinant","rowvector","binomial","squarematrix","multiline","matrix","vector","cases","table"],s.ELIGIBLE_TABLE_TYPES=["multiline","matrix","vector","cases","table"]},179:function(t,e){Object.defineProperty(e,"__esModule",{value:!0}),e.WalkerState=e.WalkerMoves=void 0,function(t){t.UP="up",t.DOWN="down",t.LEFT="left",t.RIGHT="right",t.REPEAT="repeat",t.DEPTH="depth",t.ENTER="enter",t.EXPAND="expand",t.HOME="home",t.SUMMARY="summary",t.DETAIL="detail",t.ROW="row",t.CELL="cell"}(e.WalkerMoves||(e.WalkerMoves={}));class r{static resetState(t){delete r.STATE[t]}static setState(t,e){r.STATE[t]=e}static getState(t){return r.STATE[t]}}e.WalkerState=r,r.STATE={}},3362:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.walkerMapping_=e.walker=void 0;const n=r(162),o=r(6295),i=r(9806),a=r(1799);e.walker=function(t,r,n,o,i){return(e.walkerMapping_[t.toLowerCase()]||e.walkerMapping_.dummy)(r,n,o,i)},e.walkerMapping_={dummy:(t,e,r,o)=>new n.DummyWalker(t,e,r,o),semantic:(t,e,r,n)=>new o.SemanticWalker(t,e,r,n),syntax:(t,e,r,n)=>new i.SyntaxWalker(t,e,r,n),table:(t,e,r,n)=>new a.TableWalker(t,e,r,n)}},1204:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.getBySemanticId=e.getSemanticRoot=e.getAttribute=e.splitAttribute=void 0;const n=r(5740),o=r(2298);e.splitAttribute=function(t){return t?t.split(/,/):[]},e.getAttribute=function(t,e){return t.getAttribute(e)},e.getSemanticRoot=function(t){if(t.hasAttribute(o.Attribute.TYPE)&&!t.hasAttribute(o.Attribute.PARENT))return t;const e=n.querySelectorAllByAttr(t,o.Attribute.TYPE);for(let 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<h1 class="title" style="text-align:center"><code class="sourceCode default">std::quantity</code>
as a numeric type</h1>
<table style="border:none;float:right">
  <tr>
    <td>Document #:</td>
    <td>P2982R1</td>
  </tr>
  <tr>
    <td>Date:</td>
    <td>2023-11-08</td>
  </tr>
  <tr>
    <td style="vertical-align:top">Project:</td>
    <td>Programming Language C++</td>
  </tr>
  <tr>
    <td style="vertical-align:top">Audience:</td>
    <td>
      SG6 Numerics<br>
    </td>
  </tr>
  <tr>
    <td style="vertical-align:top">Reply-to:</td>
    <td>
      Mateusz Pusz (<a href="http://www.epam.com">Epam
Systems</a>)<br>&lt;<a href="mailto:mateusz.pusz@gmail.com" class="email">mateusz.pusz@gmail.com</a>&gt;<br>
      Chip Hogg (<a href="https://aurora.tech/">Aurora
Innovation</a>)<br>&lt;<a href="mailto:charles.r.hogg@gmail.com" class="email">charles.r.hogg@gmail.com</a>&gt;<br>
    </td>
  </tr>
</table>
</header>
<div style="clear:both">
<div id="TOC" role="doc-toc">
<h1 id="toctitle">Contents</h1>
<ul>
<li><a href="#revision-history" id="toc-revision-history"><span class="toc-section-number">1</span> Revision history<span></span></a>
<ul>
<li><a href="#changes-since-p2982r0" id="toc-changes-since-p2982r0"><span class="toc-section-number">1.1</span> Changes since <span class="citation" data-cites="P2982R0">[<span>P2982R0</span>]</span><span></span></a></li>
</ul></li>
<li><a href="#introduction" id="toc-introduction"><span class="toc-section-number">2</span> Introduction<span></span></a></li>
<li><a href="#terms-and-definitions" id="toc-terms-and-definitions"><span class="toc-section-number">3</span>
Terms and definitions<span></span></a></li>
<li><a href="#systems-of-quantities" id="toc-systems-of-quantities"><span class="toc-section-number">4</span>
Systems of quantities<span></span></a>
<ul>
<li><a href="#dimension-is-not-enough-to-describe-a-quantity" id="toc-dimension-is-not-enough-to-describe-a-quantity"><span class="toc-section-number">4.1</span> Dimension is not enough to
describe a quantity<span></span></a></li>
<li><a href="#quantities-of-the-same-kind" id="toc-quantities-of-the-same-kind"><span class="toc-section-number">4.2</span> Quantities of the same
kind<span></span></a></li>
<li><a href="#system-of-quantities-is-not-only-about-kinds" id="toc-system-of-quantities-is-not-only-about-kinds"><span class="toc-section-number">4.3</span> System of quantities is not only
about kinds<span></span></a></li>
<li><a href="#converting-between-quantities-of-the-same-kind" id="toc-converting-between-quantities-of-the-same-kind"><span class="toc-section-number">4.4</span> Converting between quantities of
the same kind<span></span></a>
<ul>
<li><a href="#comparing-adding-and-subtracting-quantities-of-the-same-kind" id="toc-comparing-adding-and-subtracting-quantities-of-the-same-kind"><span class="toc-section-number">4.4.1</span> Comparing, adding, and
subtracting quantities of the same kind<span></span></a></li>
</ul></li>
<li><a href="#hierarchies-of-derived-quantities" id="toc-hierarchies-of-derived-quantities"><span class="toc-section-number">4.5</span> Hierarchies of derived
quantities<span></span></a></li>
<li><a href="#modeling-a-quantity-kind" id="toc-modeling-a-quantity-kind"><span class="toc-section-number">4.6</span> Modeling a quantity
kind<span></span></a></li>
</ul></li>
<li><a href="#systems-of-units" id="toc-systems-of-units"><span class="toc-section-number">5</span> Systems of units<span></span></a>
<ul>
<li><a href="#systems-of-units-are-based-on-systems-of-quantities" id="toc-systems-of-units-are-based-on-systems-of-quantities"><span class="toc-section-number">5.1</span> Systems of units are based on
systems of quantities<span></span></a></li>
<li><a href="#units-compose" id="toc-units-compose"><span class="toc-section-number">5.2</span> Units
compose<span></span></a></li>
<li><a href="#many-shades-of-the-same-unit" id="toc-many-shades-of-the-same-unit"><span class="toc-section-number">5.3</span> Many shades of the same
unit<span></span></a></li>
<li><a href="#constraining-a-derived-unit-to-work-only-with-a-specific-derived-quantity" id="toc-constraining-a-derived-unit-to-work-only-with-a-specific-derived-quantity"><span class="toc-section-number">5.4</span> Constraining a derived unit to
work only with a specific derived quantity<span></span></a></li>
<li><a href="#prefixed-units" id="toc-prefixed-units"><span class="toc-section-number">5.5</span> Prefixed
units<span></span></a></li>
<li><a href="#scaled-units" id="toc-scaled-units"><span class="toc-section-number">5.6</span> Scaled units<span></span></a></li>
<li><a href="#unit-symbols" id="toc-unit-symbols"><span class="toc-section-number">5.7</span> Unit symbols<span></span></a></li>
</ul></li>
<li><a href="#operations-on-units-dimensions-and-quantity-types" id="toc-operations-on-units-dimensions-and-quantity-types"><span class="toc-section-number">6</span> Operations on units, dimensions, and
quantity types<span></span></a>
<ul>
<li><a href="#expression-templates" id="toc-expression-templates"><span class="toc-section-number">6.1</span> Expression
templates<span></span></a>
<ul>
<li><a href="#derived-entities" id="toc-derived-entities"><span class="toc-section-number">6.1.1</span> Derived
entities<span></span></a></li>
<li><a href="#identities" id="toc-identities"><span class="toc-section-number">6.1.2</span> Identities<span></span></a></li>
<li><a href="#supported-operations-and-their-results" id="toc-supported-operations-and-their-results"><span class="toc-section-number">6.1.3</span> Supported operations and their
results<span></span></a></li>
<li><a href="#simplifying-the-resulting-expression-templates" id="toc-simplifying-the-resulting-expression-templates"><span class="toc-section-number">6.1.4</span> Simplifying the resulting
expression templates<span></span></a></li>
<li><a href="#example" id="toc-example"><span class="toc-section-number">6.1.5</span> Example<span></span></a></li>
</ul></li>
<li><a href="#scaled-units-1" id="toc-scaled-units-1"><span class="toc-section-number">6.2</span> Scaled units<span></span></a>
<ul>
<li><a href="#unit-magnitudes" id="toc-unit-magnitudes"><span class="toc-section-number">6.2.1</span> Unit
magnitudes<span></span></a></li>
<li><a href="#requirements-and-representation" id="toc-requirements-and-representation"><span class="toc-section-number">6.2.2</span> Requirements and
representation<span></span></a></li>
<li><a href="#construction-helpers" id="toc-construction-helpers"><span class="toc-section-number">6.2.3</span> Construction
helpers<span></span></a></li>
<li><a href="#common-units" id="toc-common-units"><span class="toc-section-number">6.2.4</span> Common
units<span></span></a></li>
</ul></li>
<li><a href="#faster-than-lightspeed-constants" id="toc-faster-than-lightspeed-constants"><span class="toc-section-number">6.3</span> Faster than lightspeed
constants<span></span></a>
<ul>
<li><a href="#simplifying-constants-in-an-equation" id="toc-simplifying-constants-in-an-equation"><span class="toc-section-number">6.3.1</span> Simplifying constants in an
equation<span></span></a></li>
<li><a href="#physical-constants-as-units" id="toc-physical-constants-as-units"><span class="toc-section-number">6.3.2</span> Physical constants as
units<span></span></a></li>
<li><a href="#usage-examples" id="toc-usage-examples"><span class="toc-section-number">6.3.3</span> Usage
examples<span></span></a></li>
</ul></li>
<li><a href="#equivalence" id="toc-equivalence"><span class="toc-section-number">6.4</span> Equivalence<span></span></a>
<ul>
<li><a href="#dimensions" id="toc-dimensions"><span class="toc-section-number">6.4.1</span> Dimensions<span></span></a></li>
<li><a href="#quantity-types" id="toc-quantity-types"><span class="toc-section-number">6.4.2</span> Quantity
types<span></span></a></li>
<li><a href="#units" id="toc-units"><span class="toc-section-number">6.4.3</span> Units<span></span></a></li>
</ul></li>
<li><a href="#ordering" id="toc-ordering"><span class="toc-section-number">6.5</span> Ordering<span></span></a></li>
</ul></li>
<li><a href="#quantities" id="toc-quantities"><span class="toc-section-number">7</span> Quantities<span></span></a>
<ul>
<li><a href="#quantity-references" id="toc-quantity-references"><span class="toc-section-number">7.1</span> Quantity
references<span></span></a></li>
<li><a href="#quantity-class-template" id="toc-quantity-class-template"><span class="toc-section-number">7.2</span>
<code class="sourceCode default">quantity</code> class
template<span></span></a></li>
<li><a href="#quantity-construction-helpers" id="toc-quantity-construction-helpers"><span class="toc-section-number">7.3</span> Quantity construction
helpers<span></span></a></li>
<li><a href="#quantity-constructors-and-value-conversions" id="toc-quantity-constructors-and-value-conversions"><span class="toc-section-number">7.4</span> Quantity constructors and value
conversions<span></span></a>
<ul>
<li><a href="#value-preserving-conversions" id="toc-value-preserving-conversions"><span class="toc-section-number">7.4.1</span> Value-preserving
conversions<span></span></a></li>
<li><a href="#value-truncating-conversions" id="toc-value-truncating-conversions"><span class="toc-section-number">7.4.2</span> Value-truncating
conversions<span></span></a></li>
</ul></li>
<li><a href="#character-of-a-quantity" id="toc-character-of-a-quantity"><span class="toc-section-number">7.5</span> Character of a
quantity<span></span></a>
<ul>
<li><a href="#scalars-vectors-and-tensors" id="toc-scalars-vectors-and-tensors"><span class="toc-section-number">7.5.1</span> Scalars, vectors, and
tensors<span></span></a></li>
<li><a href="#isq-defines-quantities-of-all-characters" id="toc-isq-defines-quantities-of-all-characters"><span class="toc-section-number">7.5.2</span> ISQ defines quantities of all
characters<span></span></a></li>
<li><a href="#characters-dont-apply-to-dimensions-and-units" id="toc-characters-dont-apply-to-dimensions-and-units"><span class="toc-section-number">7.5.3</span> Characters don’t apply to
dimensions and units<span></span></a></li>
<li><a href="#defining-vector-and-tensor-quantities" id="toc-defining-vector-and-tensor-quantities"><span class="toc-section-number">7.5.4</span> Defining vector and tensor
quantities<span></span></a></li>
<li><a href="#representation-types-for-vector-and-tensor-quantities" id="toc-representation-types-for-vector-and-tensor-quantities"><span class="toc-section-number">7.5.5</span> Representation types for vector
and tensor quantities<span></span></a></li>
<li><a href="#hacking-the-character" id="toc-hacking-the-character"><span class="toc-section-number">7.5.6</span> Hacking the
character<span></span></a></li>
</ul></li>
<li><a href="#dimensionless-quantities" id="toc-dimensionless-quantities"><span class="toc-section-number">7.6</span> Dimensionless
quantities<span></span></a>
<ul>
<li><a href="#dividing-two-quantities-of-the-same-kind" id="toc-dividing-two-quantities-of-the-same-kind"><span class="toc-section-number">7.6.1</span> Dividing two quantities of the
same kind<span></span></a></li>
<li><a href="#counts-of-things" id="toc-counts-of-things"><span class="toc-section-number">7.6.2</span> Counts of
things<span></span></a></li>
<li><a href="#lack-of-convertibility-from-fundamental-types" id="toc-lack-of-convertibility-from-fundamental-types"><span class="toc-section-number">7.6.3</span> Lack of convertibility from
fundamental types<span></span></a></li>
<li><a href="#predefined-units-of-the-dimensionless-quantity" id="toc-predefined-units-of-the-dimensionless-quantity"><span class="toc-section-number">7.6.4</span> Predefined units of the
dimensionless quantity<span></span></a></li>
<li><a href="#angular-quantities" id="toc-angular-quantities"><span class="toc-section-number">7.6.5</span> Angular
quantities<span></span></a></li>
<li><a href="#nested-quantity-kinds" id="toc-nested-quantity-kinds"><span class="toc-section-number">7.6.6</span> Nested quantity
kinds<span></span></a></li>
</ul></li>
<li><a href="#quantity-arithmetics" id="toc-quantity-arithmetics"><span class="toc-section-number">7.7</span> Quantity
arithmetics<span></span></a>
<ul>
<li><a href="#quantity-is-a-numeric-wrapper" id="toc-quantity-is-a-numeric-wrapper"><span class="toc-section-number">7.7.1</span>
<code class="sourceCode default">quantity</code> is a numeric
wrapper<span></span></a></li>
<li><a href="#addition-and-subtraction" id="toc-addition-and-subtraction"><span class="toc-section-number">7.7.2</span> Addition and
subtraction<span></span></a></li>
<li><a href="#multiplication-and-division" id="toc-multiplication-and-division"><span class="toc-section-number">7.7.3</span> Multiplication and
division<span></span></a></li>
<li><a href="#modulo" id="toc-modulo"><span class="toc-section-number">7.7.4</span> Modulo<span></span></a></li>
<li><a href="#comparison-against-zero" id="toc-comparison-against-zero"><span class="toc-section-number">7.7.5</span> Comparison against
zero<span></span></a></li>
<li><a href="#other-maths" id="toc-other-maths"><span class="toc-section-number">7.7.6</span> Other
maths<span></span></a></li>
</ul></li>
</ul></li>
<li><a href="#generic-interfaces" id="toc-generic-interfaces"><span class="toc-section-number">8</span> Generic Interfaces<span></span></a>
<ul>
<li><a href="#the-issues-with-unit-specific-interfaces" id="toc-the-issues-with-unit-specific-interfaces"><span class="toc-section-number">8.1</span> The issues with unit-specific
interfaces<span></span></a></li>
<li><a href="#a-naive-solution" id="toc-a-naive-solution"><span class="toc-section-number">8.2</span> A naive
solution<span></span></a></li>
<li><a href="#constraining-function-parameters-with-concepts" id="toc-constraining-function-parameters-with-concepts"><span class="toc-section-number">8.3</span> Constraining function parameters
with concepts<span></span></a></li>
<li><a href="#constraining-the-function-return-type" id="toc-constraining-the-function-return-type"><span class="toc-section-number">8.4</span> Constraining the function return
type<span></span></a></li>
<li><a href="#constraining-a-variable-on-the-stack" id="toc-constraining-a-variable-on-the-stack"><span class="toc-section-number">8.5</span> Constraining a variable on the
stack<span></span></a></li>
<li><a href="#basic-concepts" id="toc-basic-concepts"><span class="toc-section-number">8.6</span> Basic
concepts<span></span></a></li>
</ul></li>
<li><a href="#custom-representation-types" id="toc-custom-representation-types"><span class="toc-section-number">9</span> Custom representation
types<span></span></a></li>
<li><a href="#the-affine-space" id="toc-the-affine-space"><span class="toc-section-number">10</span> The affine space<span></span></a>
<ul>
<li><a href="#operations-in-the-affine-space" id="toc-operations-in-the-affine-space"><span class="toc-section-number">10.1</span> Operations in the affine
space<span></span></a></li>
<li><a href="#vector-is-modeled-by-quantity" id="toc-vector-is-modeled-by-quantity"><span class="toc-section-number">10.2</span> <em>Vector</em> is modeled by
<code class="sourceCode default">quantity</code><span></span></a></li>
<li><a href="#point-is-modeled-by-pointorigin-and-quantity_point" id="toc-point-is-modeled-by-pointorigin-and-quantity_point"><span class="toc-section-number">10.3</span> <em>Point</em> is modeled by
<code class="sourceCode default">PointOrigin</code> and
<code class="sourceCode default">quantity_point</code><span></span></a>
<ul>
<li><a href="#absolute-point-origin" id="toc-absolute-point-origin"><span class="toc-section-number">10.3.1</span> Absolute <em>point</em>
origin<span></span></a></li>
<li><a href="#quantity_point" id="toc-quantity_point"><span class="toc-section-number">10.3.2</span>
<code class="sourceCode default">quantity_point</code><span></span></a></li>
<li><a href="#relative-point-origin" id="toc-relative-point-origin"><span class="toc-section-number">10.3.3</span> Relative <em>point</em>
origin<span></span></a></li>
<li><a href="#converting-between-different-representations-of-the-same-point" id="toc-converting-between-different-representations-of-the-same-point"><span class="toc-section-number">10.3.4</span> Converting between different
representations of the same <em>point</em><span></span></a></li>
<li><a href="#point-arithmetics" id="toc-point-arithmetics"><span class="toc-section-number">10.3.5</span> <em>Point</em>
arithmetics<span></span></a></li>
<li><a href="#temperature-support" id="toc-temperature-support"><span class="toc-section-number">10.3.6</span> Temperature
support<span></span></a></li>
<li><a href="#no-text-output-for-points" id="toc-no-text-output-for-points"><span class="toc-section-number">10.3.7</span> No text output for
<em>points</em><span></span></a></li>
</ul></li>
<li><a href="#the-affine-space-is-about-type-safety" id="toc-the-affine-space-is-about-type-safety"><span class="toc-section-number">10.4</span> The affine space is about
type-safety<span></span></a></li>
</ul></li>
<li><a href="#acknowledgements" id="toc-acknowledgements"><span class="toc-section-number">11</span>
Acknowledgements<span></span></a></li>
<li><a href="#bibliography" id="toc-bibliography"><span class="toc-section-number">12</span> References<span></span></a></li>
</ul>
</div>
<h1 data-number="1" id="revision-history"><span class="header-section-number">1</span> Revision history<a href="#revision-history" class="self-link"></a></h1>
<h2 data-number="1.1" id="changes-since-p2982r0"><span class="header-section-number">1.1</span> Changes since <span class="citation" data-cites="P2982R0">[<a href="#ref-P2982R0" role="doc-biblioref">P2982R0</a>]</span><a href="#changes-since-p2982r0" class="self-link"></a></h2>
<ul>
<li>Fuel consumption example extended in <a href="#converting-between-quantities-of-the-same-kind">Converting
between quantities of the same kind</a>.</li>
<li><a href="#dimension-is-not-enough-to-describe-a-quantity">Dimension
is not enough to describe a quantity</a> extended with <span class="citation" data-cites="MSRMT_DATA">[<a href="#ref-MSRMT_DATA" role="doc-biblioref">Measurement Data</a>]</span>.</li>
<li><a href="#lack-of-convertibility-from-fundamental-types">Lack of
convertibility from fundamental types</a> added.</li>
<li><a href="#terms-and-definitions">Terms and definitions</a> extended
with the “Glossary” of <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span>.</li>
<li>Code samples in <a href="#equivalence">Equivalence</a>
reworked.</li>
<li>Some small editorial fixes.</li>
</ul>
<h1 data-number="2" id="introduction"><span class="header-section-number">2</span> Introduction<a href="#introduction" class="self-link"></a></h1>
<p>This document discusses the numeric and arithmetic aspects of the
physical quantities and units library. This subject is broader than it
could be initially imagined. Arithmetic operations are not only defined
for user-facing types like
<code class="sourceCode default">quantity</code> and
<code class="sourceCode default">quantity_point</code>, but also for
units and their magnitudes, dimensions, quantity specifications, and
references. Each has its own requirements and constraints, and we will
describe them in the following chapters.</p>
<p><em>Note: The code examples presented in this paper may not exactly
reflect the final interface design that is going to be proposed in the
follow-up papers. We are still doing some small fine-tuning to improve
the library.</em></p>
<h1 data-number="3" id="terms-and-definitions"><span class="header-section-number">3</span> Terms and definitions<a href="#terms-and-definitions" class="self-link"></a></h1>
<p>This document consistently uses the official metrology vocabulary
defined in the <span class="citation" data-cites="ISO-GUIDE">[<a href="#ref-ISO-GUIDE" role="doc-biblioref">ISO/IEC Guide 99</a>]</span>
and <span class="citation" data-cites="BIPM-VIM">[<a href="#ref-BIPM-VIM" role="doc-biblioref">JCGM 200:2012</a>]</span>. The
most important of those and also <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> implementation-specific terms
can be found in <a href="https://mpusz.github.io/mp-units/latest/appendix/glossary/">the
“Glossary” chapter of the project’s documentation</a>.</p>
<h1 data-number="4" id="systems-of-quantities"><span class="header-section-number">4</span> Systems of quantities<a href="#systems-of-quantities" class="self-link"></a></h1>
<p>The physical units libraries on the market typically only focus on
modeling one or more systems of units. However, this is not the only
system kind to model. Another, and maybe even more important is a system
of quantities. The most important example here is the International
System of Quantities (ISQ) defined by the series of <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> documents.</p>
<h2 data-number="4.1" id="dimension-is-not-enough-to-describe-a-quantity"><span class="header-section-number">4.1</span> Dimension is not enough to
describe a quantity<a href="#dimension-is-not-enough-to-describe-a-quantity" class="self-link"></a></h2>
<p>Most of the products on the market are aware of physical dimensions.
However, a dimension is not enough to describe a quantity. This has been
known for a long time now. The <span class="citation" data-cites="MSRMT_DATA">[<a href="#ref-MSRMT_DATA" role="doc-biblioref">Measurement Data</a>]</span> report from 1996 says
explicitly, “Dimensional analysis does not adequately model the
semantics of measurement data”.</p>
<p>A typical problem that most similar libraries struggle with is
supporting quantities like work and torque as being independent, strong
types. The problem here arises from the fact that both of them have
exactly the same dimension
<code class="sourceCode default">L²MT⁻²</code>, but a totally different
physical meaning. As a result, it is possible to mathematically add or
compare them in a quantity equation even though such an operation has no
sense from the physical point of view.</p>
<p>A similar question that we could ask ourselves is what should be the
result of:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> res <span class="op">=</span> <span class="dv">1</span> <span class="op">*</span> Hz <span class="op">+</span> <span class="dv">1</span> <span class="op">*</span> Bq <span class="op">+</span> <span class="dv">1</span> <span class="op">*</span> Bd;</span></code></pre></div>
<p>where:</p>
<ul>
<li><code class="sourceCode default">Hz</code> (hertz) - a unit of
frequency</li>
<li><code class="sourceCode default">Bq</code> (becquerel) - a unit of
activity</li>
<li><code class="sourceCode default">Bd</code> (baud) - a unit of
modulation rate</li>
</ul>
<p>All of those quantities have the same dimension, namely
<code class="sourceCode default">T⁻¹</code>, but it is probably not wise
to allow adding, subtracting, or comparing them, as they describe vastly
different physical properties.</p>
<p>It also is really tricky to separate handling of the quantities of
dimension one. For example, angular measure expressed in radians and
solid angular measure expressed in steradians should be independent. Any
attempts to add or compare them should be detected at compile-time.</p>
<p>Last but not least, let’s see the following implementation:</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> Box <span class="op">{</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>square<span class="op">(</span>si<span class="op">::</span>metre<span class="op">)&gt;</span> base_;</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span> height_;</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="kw">public</span><span class="op">:</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a>  Box<span class="op">(</span>quantity<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span> l, quantity<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span> w, quantity<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span> h<span class="op">)</span> <span class="op">:</span> base_<span class="op">(</span>l <span class="op">*</span> w<span class="op">)</span>, height_<span class="op">(</span>h<span class="op">)</span> <span class="op">{}</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>;</span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a>Box my_box<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m, <span class="dv">3</span> <span class="op">*</span> m, <span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<p>The above interface is far from being ideal. It does not provide any
type-safety and enables potentially severe errors caused by accidental
reordering of the constructor’s arguments.</p>
<p>It turns out that the above issues can’t be solved correctly without
proper modeling of systems of quantities.</p>
<h2 data-number="4.2" id="quantities-of-the-same-kind"><span class="header-section-number">4.2</span> Quantities of the same kind<a href="#quantities-of-the-same-kind" class="self-link"></a></h2>
<p>The <span class="citation" data-cites="ISO-GUIDE">[<a href="#ref-ISO-GUIDE" role="doc-biblioref">ISO/IEC Guide 99</a>]</span>
says:</p>
<ul>
<li>Quantities may be grouped together into categories of quantities
that are <strong>mutually comparable</strong></li>
<li>Mutually comparable quantities are called <strong>quantities of the
same kind</strong></li>
<li>Two or more quantities <strong>cannot be added or subtracted unless
they belong to the same category of mutually comparable
quantities</strong></li>
<li>Quantities of the <strong>same kind</strong> within a given system
of quantities <strong>have the same quantity dimension</strong></li>
<li>Quantities of the <strong>same dimension are not necessarily of the
same kind</strong></li>
</ul>
<p><span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> also explicitly
notes:</p>
<blockquote>
<p><strong>Measurement units of quantities of the same quantity
dimension may be designated by the same name and symbol even when the
quantities are not of the same kind</strong>. For example, joule per
kelvin and J/K are respectively the name and symbol of both a
measurement unit of heat capacity and a measurement unit of entropy,
which are generally not considered to be quantities of the same kind.
<strong>However, in some cases special measurement unit names are
restricted to be used with quantities of specific kind only</strong>.
For example, the measurement unit ‘second to the power minus one’ (1/s)
is called hertz (Hz) when used for frequencies and becquerel (Bq) when
used for activities of radionuclides. As another example, the joule (J)
is used as a unit of energy, but never as a unit of moment of force,
i.e. the newton metre (N · m).</p>
</blockquote>
<p>Those provide answers to all the issues mentioned above. More than
one quantity may be defined for the same dimension:</p>
<ul>
<li>quantities of different kinds (e.g. frequency, modulation rate,
activity)</li>
<li>quantities of the same kind (e.g. length, width, altitude, distance,
radius, wavelength, position vector)</li>
</ul>
<p>Two quantities can’t be added, subtracted, or compared unless they
belong to the same quantity kind.</p>
<h2 data-number="4.3" id="system-of-quantities-is-not-only-about-kinds"><span class="header-section-number">4.3</span> System of quantities is not
only about kinds<a href="#system-of-quantities-is-not-only-about-kinds" class="self-link"></a></h2>
<p>The <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>
specifies hundreds of different quantities. Plenty of various kinds are
provided, and often, each kind contains more than one quantity. It turns
out that such quantities form a hierarchy of quantities of the same
kind.</p>
<p>For example, here are all quantities of the kind length provided in
<span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> (part 1):</p>
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<p>Each of the above quantities expresses some kind of length, and each
can be measured with meters, which is the unit defined by the <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span> for quantities of length. However,
each has different properties, usage, and sometimes even a different
character (position vector and displacement are vector quantities).</p>
<p>The below presents how such a hierarchy tree can be defined in the
<span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library:</p>
<div class="sourceCode" id="cb3"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> dim_length <span class="op">:</span> base_dimension<span class="op">&lt;</span><span class="st">&quot;L&quot;</span><span class="op">&gt;</span> <span class="op">{}</span> dim_length;</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> length <span class="op">:</span> quantity_spec<span class="op">&lt;</span>dim_length<span class="op">&gt;</span> <span class="op">{}</span> length;</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> width <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length<span class="op">&gt;</span> <span class="op">{}</span> width;</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">auto</span> breadth <span class="op">=</span> width;</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> height <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length<span class="op">&gt;</span> <span class="op">{}</span> height;</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">auto</span> depth <span class="op">=</span> height;</span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">auto</span> altitude <span class="op">=</span> height;</span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> thickness <span class="op">:</span> quantity_spec<span class="op">&lt;</span>width<span class="op">&gt;</span> <span class="op">{}</span> thickness;</span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> diameter <span class="op">:</span> quantity_spec<span class="op">&lt;</span>width<span class="op">&gt;</span> <span class="op">{}</span> diameter;</span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> radius <span class="op">:</span> quantity_spec<span class="op">&lt;</span>width<span class="op">&gt;</span> <span class="op">{}</span> radius;</span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> radius_of_curvature <span class="op">:</span> quantity_spec<span class="op">&lt;</span>radius<span class="op">&gt;</span> <span class="op">{}</span> radius_of_curvature;</span>
<span id="cb3-13"><a href="#cb3-13" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> path_length <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length<span class="op">&gt;</span> <span class="op">{}</span> path_length;</span>
<span id="cb3-14"><a href="#cb3-14" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">auto</span> arc_length <span class="op">=</span> path_length;</span>
<span id="cb3-15"><a href="#cb3-15" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> distance <span class="op">:</span> quantity_spec<span class="op">&lt;</span>path_length<span class="op">&gt;</span> <span class="op">{}</span> distance;</span>
<span id="cb3-16"><a href="#cb3-16" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> radial_distance <span class="op">:</span> quantity_spec<span class="op">&lt;</span>distance<span class="op">&gt;</span> <span class="op">{}</span> radial_distance;</span>
<span id="cb3-17"><a href="#cb3-17" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> wavelength <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length<span class="op">&gt;</span> <span class="op">{}</span> wavelength;</span>
<span id="cb3-18"><a href="#cb3-18" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> position_vector <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length, quantity_character<span class="op">::</span>vector<span class="op">&gt;</span> <span class="op">{}</span> position_vector;</span>
<span id="cb3-19"><a href="#cb3-19" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> displacement <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length, quantity_character<span class="op">::</span>vector<span class="op">&gt;</span> <span class="op">{}</span> displacement;</span></code></pre></div>
<p>In the above code:</p>
<ul>
<li><code class="sourceCode default">length</code> takes the base
dimension to indicate that we are creating a base quantity that will
serve as a root for a tree of quantities of the same kind,</li>
<li><code class="sourceCode default">width</code> and following
quantities are branches and leaves of this tree with the parent always
provided as the argument to
<code class="sourceCode default">quantity_spec</code> class
template,</li>
<li><code class="sourceCode default">breadth</code> is an alias name for
the same quantity as <code class="sourceCode default">width</code>.</li>
</ul>
<p>Please note that some quantities may be specified by <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> as vector or tensor
quantities
(e.g. <code class="sourceCode default">displacement</code>).</p>
<h2 data-number="4.4" id="converting-between-quantities-of-the-same-kind"><span class="header-section-number">4.4</span> Converting between quantities
of the same kind<a href="#converting-between-quantities-of-the-same-kind" class="self-link"></a></h2>
<p>Quantity conversion rules can be defined based on the same hierarchy
of quantities of kind length.</p>
<ol type="1">
<li><p><strong>Implicit conversions</strong></p>
<ul>
<li>Every <code class="sourceCode default">width</code> is a
<code class="sourceCode default">length</code>.</li>
<li>Every <code class="sourceCode default">radius</code> is a
<code class="sourceCode default">width</code>.</li>
</ul>
<div class="sourceCode" id="cb4"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>width, isq<span class="op">::</span>length<span class="op">))</span>;</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>radius, isq<span class="op">::</span>length<span class="op">))</span>;</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>radius, isq<span class="op">::</span>width<span class="op">))</span>;</span></code></pre></div>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library,
implicit conversions are allowed on copy-initialization:</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="dt">void</span> foo<span class="op">(</span>quantity<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&lt;</span>m<span class="op">&gt;&gt;</span> q<span class="op">)</span>;</span></code></pre></div>
<div class="sourceCode" id="cb6"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>width<span class="op">&lt;</span>m<span class="op">&gt;&gt;</span> q1 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m;</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&lt;</span>m<span class="op">&gt;&gt;</span> q2 <span class="op">=</span> q1;  <span class="co">// implicit quantity conversion</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a>foo<span class="op">(</span>q1<span class="op">)</span>;                           <span class="co">// implicit quantity conversion</span></span></code></pre></div></li>
<li><p><strong>Explicit conversions</strong></p>
<ul>
<li>Not every <code class="sourceCode default">length</code> is a
<code class="sourceCode default">width</code>.</li>
<li>Not every <code class="sourceCode default">width</code> is a
<code class="sourceCode default">radius</code>.</li>
</ul>
<div class="sourceCode" id="cb7"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length, isq<span class="op">::</span>width<span class="op">))</span>;</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length, isq<span class="op">::</span>radius<span class="op">))</span>;</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>width, isq<span class="op">::</span>radius<span class="op">))</span>;</span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length, isq<span class="op">::</span>width<span class="op">))</span>;</span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length, isq<span class="op">::</span>radius<span class="op">))</span>;</span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>width, isq<span class="op">::</span>radius<span class="op">))</span>;</span></code></pre></div>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library,
explicit conversions are forced by passing the quantity to a call
operator of a <code class="sourceCode default">quantity_spec</code>
type:</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&lt;</span>m<span class="op">&gt;&gt;</span> q1 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m;</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">&lt;</span>m<span class="op">&gt;&gt;</span> q2 <span class="op">=</span> isq<span class="op">::</span>height<span class="op">(</span>q1<span class="op">)</span>;  <span class="co">// explicit quantity conversion</span></span></code></pre></div></li>
<li><p><strong>Explicit casts</strong></p>
<ul>
<li><code class="sourceCode default">height</code> is never a
<code class="sourceCode default">width</code>, and vice versa.</li>
<li>Both <code class="sourceCode default">height</code> and
<code class="sourceCode default">width</code> are quantities of kind
<code class="sourceCode default">length</code>.</li>
</ul>
<div class="sourceCode" id="cb9"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>height, isq<span class="op">::</span>width<span class="op">))</span>;</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>height, isq<span class="op">::</span>width<span class="op">))</span>;</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>castable<span class="op">(</span>isq<span class="op">::</span>height, isq<span class="op">::</span>width<span class="op">))</span>;</span></code></pre></div>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library,
explicit casts are forced with a dedicated
<code class="sourceCode default">quantity_cast</code> function:</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>width<span class="op">&lt;</span>m<span class="op">&gt;&gt;</span> q1 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m;</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">&lt;</span>m<span class="op">&gt;&gt;</span> q2 <span class="op">=</span> quantity_cast<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">&gt;(</span>q1<span class="op">)</span>;  <span class="co">// explicit quantity cast</span></span></code></pre></div></li>
<li><p><strong>No conversion</strong></p>
<ul>
<li><code class="sourceCode default">time</code> has nothing in common
with <code class="sourceCode default">length</code>.</li>
</ul>
<div class="sourceCode" id="cb11"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>time, isq<span class="op">::</span>length<span class="op">))</span>;</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>time, isq<span class="op">::</span>length<span class="op">))</span>;</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>castable<span class="op">(</span>isq<span class="op">::</span>time, isq<span class="op">::</span>length<span class="op">))</span>;</span></code></pre></div>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library,
even the explicit casts will not force such a conversion:</p>
<div class="sourceCode" id="cb12"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="dt">void</span> foo<span class="op">(</span>quantity<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">[</span>m<span class="op">]&gt;)</span>;</span></code></pre></div>
<div class="sourceCode" id="cb13"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>foo<span class="op">(</span>quantity_cast<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;(</span><span class="dv">42</span> <span class="op">*</span> s<span class="op">))</span>; <span class="co">// Compile-time error</span></span></code></pre></div></li>
</ol>
<p>With the above rules, one can write the following short application
to calculate a fuel consumption:</p>
<div class="sourceCode" id="cb14"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> fuel_volume <span class="op">:</span> quantity_spec<span class="op">&lt;</span>isq<span class="op">::</span>volume<span class="op">&gt;</span> <span class="op">{}</span> fuel_volume;</span>
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> fuel_consumption <span class="op">:</span> quantity_spec<span class="op">&lt;</span>fuel_volume <span class="op">/</span> isq<span class="op">::</span>distance<span class="op">&gt;</span> <span class="op">{}</span> fuel_consumption;</span>
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity fuel <span class="op">=</span> fuel_volume<span class="op">(</span><span class="fl">40.</span> <span class="op">*</span> l<span class="op">)</span>;</span>
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity distance <span class="op">=</span> isq<span class="op">::</span>distance<span class="op">(</span><span class="fl">550.</span> <span class="op">*</span> km<span class="op">)</span>;</span>
<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity<span class="op">&lt;</span>fuel_consumption<span class="op">[</span>l <span class="op">/</span> <span class="op">(</span>mag<span class="op">&lt;</span><span class="dv">100</span><span class="op">&gt;</span> <span class="op">*</span> km<span class="op">)]&gt;</span> q <span class="op">=</span> fuel <span class="op">/</span> distance;</span>
<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;Fuel consumption: &quot;</span> <span class="op">&lt;&lt;</span> q <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span></code></pre></div>
<p>The above code prints:</p>
<pre class="text"><code>Fuel consumption: 7.27273 × 10⁻² l/km</code></pre>
<p>Please note that, despite the dimensions of
<code class="sourceCode default">fuel_consumption</code> and
<code class="sourceCode default">isq::area</code> being the same (L²),
the constructor of a quantity <code class="sourceCode default">q</code>
below will fail to compile when we pass an argument being the quantity
of area:</p>
<div class="sourceCode" id="cb15"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>fuel_consumption<span class="op">.</span>dimension <span class="op">==</span> isq<span class="op">::</span>area<span class="op">.</span>dimension<span class="op">)</span>;</span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb15-3"><a href="#cb15-3" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity<span class="op">&lt;</span>isq<span class="op">::</span>area<span class="op">[</span>m2<span class="op">]&gt;</span> football_field <span class="op">=</span> isq<span class="op">::</span>length<span class="op">(</span><span class="dv">105</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">*</span> isq<span class="op">::</span>width<span class="op">(</span><span class="dv">68</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb15-4"><a href="#cb15-4" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity<span class="op">&lt;</span>fuel_consumption<span class="op">[</span>l <span class="op">/</span> <span class="op">(</span>mag<span class="op">&lt;</span><span class="dv">100</span><span class="op">&gt;</span> <span class="op">*</span> km<span class="op">)]&gt;</span> q2 <span class="op">=</span> football_field;  <span class="co">// Compile-time error</span></span>
<span id="cb15-5"><a href="#cb15-5" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity q3 <span class="op">=</span> q <span class="op">+</span> football_field;                                     <span class="co">// Compile-time error</span></span>
<span id="cb15-6"><a href="#cb15-6" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="op">(</span>q <span class="op">==</span> football_field<span class="op">)</span> <span class="op">{</span>                                                  <span class="co">// Compile-time error</span></span>
<span id="cb15-7"><a href="#cb15-7" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
<span id="cb15-8"><a href="#cb15-8" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<h3 data-number="4.4.1" id="comparing-adding-and-subtracting-quantities-of-the-same-kind"><span class="header-section-number">4.4.1</span> Comparing, adding, and
subtracting quantities of the same kind<a href="#comparing-adding-and-subtracting-quantities-of-the-same-kind" class="self-link"></a></h3>
<p><span class="citation" data-cites="ISO-GUIDE">[<a href="#ref-ISO-GUIDE" role="doc-biblioref">ISO/IEC Guide 99</a>]</span>
explicitly states that <code class="sourceCode default">width</code> and
<code class="sourceCode default">height</code> are quantities of the
same kind and as such they</p>
<ul>
<li>are mutually comparable, and</li>
<li>can be added and subtracted.</li>
</ul>
<p>If we take the above for granted, the only reasonable result of
<code class="sourceCode default">1 * width + 1 * height</code> is
<code class="sourceCode default">2 * length</code>, where the result of
<code class="sourceCode default">length</code> is known as a common
quantity type. A result of such an equation is always the first common
node in a hierarchy tree of the same kind. For example:</p>
<div class="sourceCode" id="cb16"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>common_quantity_spec<span class="op">(</span>isq<span class="op">::</span>width, isq<span class="op">::</span>height<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>length<span class="op">)</span>;</span>
<span id="cb16-2"><a href="#cb16-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>common_quantity_spec<span class="op">(</span>isq<span class="op">::</span>thickness, isq<span class="op">::</span>radius<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>width<span class="op">)</span>;</span>
<span id="cb16-3"><a href="#cb16-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>common_quantity_spec<span class="op">(</span>isq<span class="op">::</span>distance, isq<span class="op">::</span>path_length<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>path_length<span class="op">)</span>;</span></code></pre></div>
<div class="sourceCode" id="cb17"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a>quantity q <span class="op">=</span> isq<span class="op">::</span>thickness<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">+</span> isq<span class="op">::</span>radius<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q<span class="op">.</span>quantity_spec <span class="op">==</span> isq<span class="op">::</span>width<span class="op">)</span>;</span></code></pre></div>
<p>One could argue that allowing to add or compare quantities of height
and width might be a safety issue, but we need to be consistent with the
requirements of <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>.
Moreover, from our experience, disallowing such operations and requiring
an explicit cast to a common quantity in every single place makes the
code so cluttered with casts that it nearly renders the library
unusable.</p>
<p>Fortunately, the above-mentioned conversion rules make the code safe
by construction anyway. Let’s analyze the following example:</p>
<div class="sourceCode" id="cb18"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb18-1"><a href="#cb18-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> horizontal_length <span class="op">:</span> quantity_spec<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="op">{}</span> horizontal_length;</span>
<span id="cb18-2"><a href="#cb18-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-3"><a href="#cb18-3" aria-hidden="true" tabindex="-1"></a><span class="kw">namespace</span> christmas <span class="op">{</span></span>
<span id="cb18-4"><a href="#cb18-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-5"><a href="#cb18-5" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> gift <span class="op">{</span></span>
<span id="cb18-6"><a href="#cb18-6" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>horizontal_length<span class="op">[</span>m<span class="op">]&gt;</span> length;</span>
<span id="cb18-7"><a href="#cb18-7" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>isq<span class="op">::</span>width<span class="op">[</span>m<span class="op">]&gt;</span> width;</span>
<span id="cb18-8"><a href="#cb18-8" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">[</span>m<span class="op">]&gt;</span> height;</span>
<span id="cb18-9"><a href="#cb18-9" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>;</span>
<span id="cb18-10"><a href="#cb18-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-11"><a href="#cb18-11" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>array<span class="op">&lt;</span>quantity<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">[</span>m<span class="op">]&gt;</span>, <span class="dv">2</span><span class="op">&gt;</span> gift_wrapping_paper_size<span class="op">(</span><span class="kw">const</span> gift<span class="op">&amp;</span> g<span class="op">)</span></span>
<span id="cb18-12"><a href="#cb18-12" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb18-13"><a href="#cb18-13" aria-hidden="true" tabindex="-1"></a>  <span class="kw">const</span> <span class="kw">auto</span> dim1 <span class="op">=</span> <span class="dv">2</span> <span class="op">*</span> g<span class="op">.</span>width <span class="op">+</span> <span class="dv">2</span> <span class="op">*</span> g<span class="op">.</span>height <span class="op">+</span> <span class="fl">0.5</span> <span class="op">*</span> g<span class="op">.</span>width;</span>
<span id="cb18-14"><a href="#cb18-14" aria-hidden="true" tabindex="-1"></a>  <span class="kw">const</span> <span class="kw">auto</span> dim2 <span class="op">=</span> g<span class="op">.</span>length <span class="op">+</span> <span class="dv">2</span> <span class="op">*</span> <span class="fl">0.75</span> <span class="op">*</span> g<span class="op">.</span>height;</span>
<span id="cb18-15"><a href="#cb18-15" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> <span class="op">{</span> dim1, dim2 <span class="op">}</span>;</span>
<span id="cb18-16"><a href="#cb18-16" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span>
<span id="cb18-17"><a href="#cb18-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-18"><a href="#cb18-18" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>  <span class="co">// namespace christmas</span></span>
<span id="cb18-19"><a href="#cb18-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-20"><a href="#cb18-20" aria-hidden="true" tabindex="-1"></a><span class="dt">int</span> main<span class="op">()</span></span>
<span id="cb18-21"><a href="#cb18-21" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb18-22"><a href="#cb18-22" aria-hidden="true" tabindex="-1"></a>  <span class="kw">const</span> christmas<span class="op">::</span>gift lego <span class="op">=</span> <span class="op">{</span> horizontal_length<span class="op">(</span><span class="dv">40</span> <span class="op">*</span> cm<span class="op">)</span>, isq<span class="op">::</span>width<span class="op">(</span><span class="dv">30</span> <span class="op">*</span> cm<span class="op">)</span>, isq<span class="op">::</span>height<span class="op">(</span><span class="dv">15</span> <span class="op">*</span> cm<span class="op">)</span> <span class="op">}</span>;</span>
<span id="cb18-23"><a href="#cb18-23" aria-hidden="true" tabindex="-1"></a>  <span class="kw">auto</span> paper <span class="op">=</span> christmas<span class="op">::</span>gift_wrapping_paper_size<span class="op">(</span>lego<span class="op">)</span>;</span>
<span id="cb18-24"><a href="#cb18-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb18-25"><a href="#cb18-25" aria-hidden="true" tabindex="-1"></a>  std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;Paper needed to pack a lego box:</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb18-26"><a href="#cb18-26" aria-hidden="true" tabindex="-1"></a>  std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;- &quot;</span> <span class="op">&lt;&lt;</span> paper<span class="op">[</span><span class="dv">0</span><span class="op">]</span> <span class="op">&lt;&lt;</span> <span class="st">&quot; X &quot;</span> <span class="op">&lt;&lt;</span> paper<span class="op">[</span><span class="dv">1</span><span class="op">]</span> <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;  <span class="co">// - 1.05 m X 0.625 m</span></span>
<span id="cb18-27"><a href="#cb18-27" aria-hidden="true" tabindex="-1"></a>  std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;- area = &quot;</span> <span class="op">&lt;&lt;</span> paper<span class="op">[</span><span class="dv">0</span><span class="op">]</span> <span class="op">*</span> paper<span class="op">[</span><span class="dv">1</span><span class="op">]</span> <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;     <span class="co">// - area = 0.65625 m²</span></span>
<span id="cb18-28"><a href="#cb18-28" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<p>In the beginning, we introduce a custom quantity
<code class="sourceCode default">horizontal_length</code> of a kind
length, which then, together with
<code class="sourceCode default">isq::width</code> and
<code class="sourceCode default">isq::height</code>, are used to define
the dimensions of a Christmas gift. Next, we provide a function that
calculates the dimensions of a gift wrapping paper with some wraparound.
The result of both those expressions is a quantity of
<code class="sourceCode default">isq::length</code>, as this is the
closest common quantity for the arguments used in this quantity
equation.</p>
<p>Regarding safety, it is important to mention here, that thanks to the
conversion rules provided above, it would be impossible to accidentally
do the following:</p>
<div class="sourceCode" id="cb19"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="dt">void</span> foo<span class="op">(</span>quantity<span class="op">&lt;</span>horizontal_length<span class="op">[</span>m<span class="op">]&gt;</span> q<span class="op">)</span>;</span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>width<span class="op">[</span>m<span class="op">]&gt;</span> q1 <span class="op">=</span> dim1;  <span class="co">// Compile-time error</span></span>
<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">[</span>m<span class="op">]&gt;</span> q2<span class="op">{</span>dim1<span class="op">}</span>;  <span class="co">// Compile-time error</span></span>
<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a>foo<span class="op">(</span>dim1<span class="op">)</span>;                          <span class="co">// Compile-time error</span></span></code></pre></div>
<p>The reason of compilation errors above is the fact that
<code class="sourceCode default">isq::length</code> is not implicitly
convertible to the quantities defined based on it. To make the above
code compile, an explicit conversion of a quantity type is needed:</p>
<div class="sourceCode" id="cb20"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="dt">void</span> foo<span class="op">(</span>quantity<span class="op">&lt;</span>horizontal_length<span class="op">[</span>m<span class="op">]&gt;</span> q<span class="op">)</span>;</span>
<span id="cb20-2"><a href="#cb20-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb20-3"><a href="#cb20-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>width<span class="op">[</span>m<span class="op">]&gt;</span> q1 <span class="op">=</span> isq<span class="op">::</span>width<span class="op">(</span>dim1<span class="op">)</span>;</span>
<span id="cb20-4"><a href="#cb20-4" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">[</span>m<span class="op">]&gt;</span> q2<span class="op">{</span>isq<span class="op">::</span>height<span class="op">(</span>dim1<span class="op">)}</span>;</span>
<span id="cb20-5"><a href="#cb20-5" aria-hidden="true" tabindex="-1"></a>foo<span class="op">(</span>horizontal_length<span class="op">(</span>dim1<span class="op">))</span>;</span></code></pre></div>
<p>To summarize, rules for addition, subtraction, and comparison of
quantities improve the library usability, while the conversion rules
enhance the safety of the library compared to the libraries that do not
model quantity kinds.</p>
<h2 data-number="4.5" id="hierarchies-of-derived-quantities"><span class="header-section-number">4.5</span> Hierarchies of derived
quantities<a href="#hierarchies-of-derived-quantities" class="self-link"></a></h2>
<p>The same rules propagate to derived quantities. For example, we can
define strongly typed horizontal length and area:</p>
<div class="sourceCode" id="cb21"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> horizontal_length <span class="op">:</span> quantity_spec<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="op">{}</span> horizontal_length;</span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> horizontal_area <span class="op">:</span> quantity_spec<span class="op">&lt;</span>isq<span class="op">::</span>area, horizontal_length <span class="op">*</span> isq<span class="op">::</span>width<span class="op">&gt;</span> <span class="op">{}</span> horizontal_area;</span></code></pre></div>
<p>The first definition says that a horizontal length is a more
specialized quantity than length and belongs to the same quantity kind.
The second line defines a horizontal area, which is a more specialized
quantity than area, so it has a more constrained recipe as well. Thanks
to that:</p>
<div class="sourceCode" id="cb22"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>horizontal_length, isq<span class="op">::</span>length<span class="op">))</span>;</span>
<span id="cb22-2"><a href="#cb22-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length, horizontal_length<span class="op">))</span>;</span>
<span id="cb22-3"><a href="#cb22-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length, horizontal_length<span class="op">))</span>;</span>
<span id="cb22-4"><a href="#cb22-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-5"><a href="#cb22-5" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>horizontal_area, isq<span class="op">::</span>area<span class="op">))</span>;</span>
<span id="cb22-6"><a href="#cb22-6" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>area, horizontal_area<span class="op">))</span>;</span>
<span id="cb22-7"><a href="#cb22-7" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>area, horizontal_area<span class="op">))</span>;</span>
<span id="cb22-8"><a href="#cb22-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-9"><a href="#cb22-9" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length <span class="op">*</span> isq<span class="op">::</span>length, isq<span class="op">::</span>area<span class="op">))</span>;</span>
<span id="cb22-10"><a href="#cb22-10" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length <span class="op">*</span> isq<span class="op">::</span>length, horizontal_area<span class="op">))</span>;</span>
<span id="cb22-11"><a href="#cb22-11" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length <span class="op">*</span> isq<span class="op">::</span>length, horizontal_area<span class="op">))</span>;</span>
<span id="cb22-12"><a href="#cb22-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb22-13"><a href="#cb22-13" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>horizontal_length <span class="op">*</span> isq<span class="op">::</span>width, isq<span class="op">::</span>area<span class="op">))</span>;</span>
<span id="cb22-14"><a href="#cb22-14" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>horizontal_length <span class="op">*</span> isq<span class="op">::</span>width, horizontal_area<span class="op">))</span>;</span></code></pre></div>
<p>Unfortunately, derived quantity equations often do not automatically
form a hierarchy tree. This is why sometimes it is not obvious what such
a tree should look like. Also, the <span class="citation" data-cites="ISO-GUIDE">[<a href="#ref-ISO-GUIDE" role="doc-biblioref">ISO/IEC Guide 99</a>]</span> explicitly states:</p>
<blockquote>
<p>The division of ‘quantity’ according to ‘kind of quantity’ is, to
some extent, arbitrary.</p>
</blockquote>
<p>The below presents some arbitrary hierarchy of derived quantities of
kind energy:</p>
<p><img src="data:image/svg+xml;base64,<svg aria-roledescription="flowchart-v2" role="graphics-document document" viewBox="-8 -8 1389.5546875 394" style="max-width: 100%;" xmlns="http://www.w3.org/2000/svg" width="100%" id="graph-div" height="100%" xmlns:xlink="http://www.w3.org/1999/xlink"><style>@import url("https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/all.min.css");'</style><style>#graph-div{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;fill:#333;}#graph-div .error-icon{fill:#552222;}#graph-div .error-text{fill:#552222;stroke:#552222;}#graph-div .edge-thickness-normal{stroke-width:2px;}#graph-div .edge-thickness-thick{stroke-width:3.5px;}#graph-div .edge-pattern-solid{stroke-dasharray:0;}#graph-div .edge-pattern-dashed{stroke-dasharray:3;}#graph-div .edge-pattern-dotted{stroke-dasharray:2;}#graph-div .marker{fill:#333333;stroke:#333333;}#graph-div .marker.cross{stroke:#333333;}#graph-div svg{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;}#graph-div .label{font-family:"trebuchet ms",verdana,arial,sans-serif;color:#333;}#graph-div .cluster-label text{fill:#333;}#graph-div .cluster-label span,#graph-div p{color:#333;}#graph-div .label text,#graph-div span,#graph-div p{fill:#333;color:#333;}#graph-div .node rect,#graph-div .node circle,#graph-div .node ellipse,#graph-div .node polygon,#graph-div .node path{fill:#ECECFF;stroke:#9370DB;stroke-width:1px;}#graph-div .flowchart-label text{text-anchor:middle;}#graph-div .node .label{text-align:center;}#graph-div .node.clickable{cursor:pointer;}#graph-div .arrowheadPath{fill:#333333;}#graph-div .edgePath .path{stroke:#333333;stroke-width:2.0px;}#graph-div .flowchart-link{stroke:#333333;fill:none;}#graph-div .edgeLabel{background-color:#e8e8e8;text-align:center;}#graph-div .edgeLabel rect{opacity:0.5;background-color:#e8e8e8;fill:#e8e8e8;}#graph-div .labelBkg{background-color:rgba(232, 232, 232, 0.5);}#graph-div .cluster rect{fill:#ffffde;stroke:#aaaa33;stroke-width:1px;}#graph-div .cluster text{fill:#333;}#graph-div .cluster span,#graph-div p{color:#333;}#graph-div div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:12px;background:hsl(80, 100%, 96.2745098039%);border:1px solid #aaaa33;border-radius:2px;pointer-events:none;z-index:100;}#graph-div .flowchartTitleText{text-anchor:middle;font-size:18px;fill:#333;}#graph-div :root{--mermaid-font-family:"trebuchet ms",verdana,arial,sans-serif;}</style><g><marker orient="auto" markerHeight="12" markerWidth="12" markerUnits="userSpaceOnUse" refY="5" refX="6" viewBox="0 0 10 10" class="marker flowchart" id="graph-div_flowchart-pointEnd"><path style="stroke-width: 1; 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<p>Notice, that even though all of those quantities have the same
dimension and can be expressed in the same units, they have different
quantity equations used to create them implicitly:</p>
<ul>
<li><p><code class="sourceCode default">energy</code> is the most
generic one and thus can be created from base quantities of
<code class="sourceCode default">mass</code>,
<code class="sourceCode default">length</code>, and
<code class="sourceCode default">time</code>. As those are also the
roots of quantities of their kinds and all other quantities are
implicitly convertible to them, it means that an
<code class="sourceCode default">energy</code> can be implicitly
constructed from any quantity having proper powers of mass, length, and
time.</p>
<div class="sourceCode" id="cb23"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>mass <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>length<span class="op">)</span> <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>time<span class="op">)</span>, isq<span class="op">::</span>energy<span class="op">))</span>;</span>
<span id="cb23-2"><a href="#cb23-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>mass <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>height<span class="op">)</span> <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>time<span class="op">)</span>, isq<span class="op">::</span>energy<span class="op">))</span>;</span></code></pre></div></li>
<li><p><code class="sourceCode default">mechanical_energy</code> is a
more “specialized” quantity than
<code class="sourceCode default">energy</code> (not every
<code class="sourceCode default">energy</code> is a
<code class="sourceCode default">mechanical_energy</code>). It is why an
explicit cast is needed to convert from either
<code class="sourceCode default">energy</code> or the results of its
quantity equation.</p>
<div class="sourceCode" id="cb24"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>energy, isq<span class="op">::</span>mechanical_energy<span class="op">))</span>;</span>
<span id="cb24-2"><a href="#cb24-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>energy, isq<span class="op">::</span>mechanical_energy<span class="op">))</span>;</span>
<span id="cb24-3"><a href="#cb24-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>mass <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>length<span class="op">)</span> <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>time<span class="op">)</span>, isq<span class="op">::</span>mechanical_energy<span class="op">))</span>;</span>
<span id="cb24-4"><a href="#cb24-4" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>mass <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>length<span class="op">)</span> <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>time<span class="op">)</span>, isq<span class="op">::</span>mechanical_energy<span class="op">))</span>;</span></code></pre></div></li>
<li><p><code class="sourceCode default">gravitational_potential_energy</code>
is not only even more specialized one but additionally, it is special in
a way that it provides its own “constrained” quantity equation. Maybe
not every <code class="sourceCode default">mass * pow&lt;2&gt;(length) / pow&lt;2&gt;(time)</code>
is a
<code class="sourceCode default">gravitational_potential_energy</code>,
but every <code class="sourceCode default">mass * acceleration_of_free_fall * height</code>
is.</p>
<div class="sourceCode" id="cb25"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>energy, gravitational_potential_energy<span class="op">))</span>;</span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>energy, gravitational_potential_energy<span class="op">))</span>;</span>
<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>mass <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>length<span class="op">)</span> <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>time<span class="op">)</span>, gravitational_potential_energy<span class="op">))</span>;</span>
<span id="cb25-4"><a href="#cb25-4" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>explicitly_convertible<span class="op">(</span>isq<span class="op">::</span>mass <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>length<span class="op">)</span> <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>isq<span class="op">::</span>time<span class="op">)</span>, gravitational_potential_energy<span class="op">))</span>;</span>
<span id="cb25-5"><a href="#cb25-5" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>mass <span class="op">*</span> isq<span class="op">::</span>acceleration_of_free_fall <span class="op">*</span> isq<span class="op">::</span>height, gravitational_potential_energy<span class="op">))</span>;</span></code></pre></div></li>
</ul>
<h2 data-number="4.6" id="modeling-a-quantity-kind"><span class="header-section-number">4.6</span> Modeling a quantity kind<a href="#modeling-a-quantity-kind" class="self-link"></a></h2>
<p>In the physical units library, we also need an abstraction describing
an entire family of quantities of the same kind. Such quantities have
not only the same dimension but also can be expressed in the same
units.</p>
<p>To annotate a quantity to represent its kind (and not just a
hierarchy tree’s root quantity), in <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> we introduced a
<code class="sourceCode default">kind_of&lt;&gt;</code> specifier. For
example, to express any quantity of length, we need to specify
<code class="sourceCode default">kind_of&lt;isq::length&gt;</code>. That
entity behaves as any quantity of its kind. This means that it is
implicitly convertible to any quantity in a tree:</p>
<div class="sourceCode" id="cb26"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>implicitly_convertible<span class="op">(</span>isq<span class="op">::</span>length, isq<span class="op">::</span>height<span class="op">))</span>;</span>
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span>, isq<span class="op">::</span>height<span class="op">))</span>;</span></code></pre></div>
<p>Additionally, the result of operations on quantity kinds is also a
quantity kind:</p>
<div class="sourceCode" id="cb27"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>same_type<span class="op">&lt;</span>kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="op">/</span> kind_of<span class="op">&lt;</span>isq<span class="op">::</span>time<span class="op">&gt;</span>, kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length <span class="op">/</span> isq<span class="op">::</span>time<span class="op">&gt;&gt;)</span>;</span></code></pre></div>
<p>However, if at least one equation’s operand is not a quantity kind,
the result becomes a “strong” quantity where all the kinds are converted
to the hierarchy tree’s root quantities:</p>
<div class="sourceCode" id="cb28"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(!</span>same_type<span class="op">&lt;</span>kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="op">/</span> isq<span class="op">::</span>time, kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length <span class="op">/</span> isq<span class="op">::</span>time<span class="op">&gt;&gt;)</span>;</span>
<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>same_type<span class="op">&lt;</span>kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="op">/</span> isq<span class="op">::</span>time, isq<span class="op">::</span>length <span class="op">/</span> isq<span class="op">::</span>time<span class="op">&gt;)</span>;</span></code></pre></div>
<p>Please note that only a root quantity from the hierarchy tree or the
one marked with <code class="sourceCode default">is_kind</code>
specifier in the <code class="sourceCode default">quantity_spec</code>
definition can be put as a template parameter to the
<code class="sourceCode default">kind_of</code> specifier. For example,
<code class="sourceCode default">kind_of&lt;isq::width&gt;</code> will
fail to compile.</p>
<h1 data-number="5" id="systems-of-units"><span class="header-section-number">5</span> Systems of units<a href="#systems-of-units" class="self-link"></a></h1>
<p>Modeling a system of units is the most important feature and a
selling point of every physical units library. Thanks to that, the
library can protect users from performing invalid operations on
quantities and provide automated conversion factors between various
compatible units.</p>
<p>Probably all the libraries in the wild model the <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span>, and many of them provide support
for additional units belonging to various other systems
(e.g. imperial).</p>
<h2 data-number="5.1" id="systems-of-units-are-based-on-systems-of-quantities"><span class="header-section-number">5.1</span> Systems of units are based on
systems of quantities<a href="#systems-of-units-are-based-on-systems-of-quantities" class="self-link"></a></h2>
<p>Systems of quantities specify a set of quantities and equations
relating to those quantities. Those equations do not take any unit or a
numerical representation into account at all. In order to create a
quantity, we need to add those missing pieces of information. This is
where a system of units kicks in.</p>
<p>The <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span> is explicitly stated to be based on
the ISQ. Among others, it defines seven base units, one for each base
quantity. In <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span>, this is
expressed by associating a quantity kind to a unit being defined:</p>
<div class="sourceCode" id="cb29"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> metre <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;m&quot;</span>, kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;&gt;</span> <span class="op">{}</span> metre;</span></code></pre></div>
<p>The
<code class="sourceCode default">kind_of&lt;isq::length&gt;</code> above
states explicitly that this unit has an associated quantity kind. In
other words, <code class="sourceCode default">si::metre</code> (and
scaled units based on it) can be used to express the amount of any
quantity of kind length.</p>
<p>Associated units are so useful and common in the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library that they got their
own concepts
<code class="sourceCode default">AssociatedUnit&lt;T&gt;</code> to
improve the interfaces.</p>
<p>Please note that for some systems of units (e.g., natural units), a
unit may not have an associated quantity type. For example, if we define
the speed of light constant as
<code class="sourceCode default">c = 1</code>, we can define a system
where both length and time will be measured in seconds, and speed will
be a quantity measured with the unit
<code class="sourceCode default">one</code>. In such case, the
definition will look as follows:</p>
<div class="sourceCode" id="cb30"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> second <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;s&quot;</span><span class="op">&gt;</span> <span class="op">{}</span> second;</span></code></pre></div>
<h2 data-number="5.2" id="units-compose"><span class="header-section-number">5.2</span> Units compose<a href="#units-compose" class="self-link"></a></h2>
<p>One of the strongest points of the <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span>
system is that its units compose. This allows providing thousands of
different units for hundreds of various quantities with a really small
set of predefined units and prefixes. For example, one can write:</p>
<div class="sourceCode" id="cb31"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>metre <span class="op">/</span> si<span class="op">::</span>second<span class="op">&gt;</span> q;</span></code></pre></div>
<p>to express a quantity of speed. The resulting quantity type is
implicitly inferred from the unit equation by repeating exactly the same
operations on the associated quantity kinds.</p>
<p>Also, as units are regular values, we can easily provide a helper
ad-hoc unit with:</p>
<div class="sourceCode" id="cb32"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> mps <span class="op">=</span> si<span class="op">::</span>metre <span class="op">/</span> si<span class="op">::</span>second;</span>
<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>mps<span class="op">&gt;</span> q;</span></code></pre></div>
<h2 data-number="5.3" id="many-shades-of-the-same-unit"><span class="header-section-number">5.3</span> Many shades of the same unit<a href="#many-shades-of-the-same-unit" class="self-link"></a></h2>
<p>The <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span> provides the names for 22 common
coherent units of 22 derived quantities.</p>
<p>Each such named derived unit is a result of a specific predefined
unit equation. For example, a unit of power quantity is defined as:</p>
<div class="sourceCode" id="cb33"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> watt <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;W&quot;</span>, joule <span class="op">/</span> second<span class="op">&gt;</span> <span class="op">{}</span> watt;</span></code></pre></div>
<p>However, a power quantity can be expressed in other units as well.
For example, the following:</p>
<div class="sourceCode" id="cb34"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> W;</span>
<span id="cb34-2"><a href="#cb34-2" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb34-3"><a href="#cb34-3" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1<span class="op">.</span>in<span class="op">(</span>J <span class="op">/</span> s<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb34-4"><a href="#cb34-4" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1<span class="op">.</span>in<span class="op">(</span>N <span class="op">*</span> m <span class="op">/</span> s<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb34-5"><a href="#cb34-5" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1<span class="op">.</span>in<span class="op">(</span>kg <span class="op">*</span> m2 <span class="op">/</span> s3<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span></code></pre></div>
<p>prints:</p>
<pre class="text"><code>42 W
42 J/s
42 N m/s
42 kg m²/s³</code></pre>
<p>All of the above quantities are equivalent and mean exactly the
same.</p>
<h2 data-number="5.4" id="constraining-a-derived-unit-to-work-only-with-a-specific-derived-quantity"><span class="header-section-number">5.4</span> Constraining a derived unit to
work only with a specific derived quantity<a href="#constraining-a-derived-unit-to-work-only-with-a-specific-derived-quantity" class="self-link"></a></h2>
<p>Some derived units are valid only for specific derived quantities.
For example, <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span> specifies both
<code class="sourceCode default">hertz</code> and
<code class="sourceCode default">becquerel</code> derived units with the
same unit equation <code class="sourceCode default">1/s</code>. However,
it also explicitly states:</p>
<blockquote>
<p>The hertz shall only be used for periodic phenomena and the becquerel
shall only be used for stochastic processes in activity referred to a
radionuclide.</p>
</blockquote>
<p>This is why it is important for the library to allow constraining
such units to be used only with a specific quantity kind:</p>
<div class="sourceCode" id="cb35"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> hertz <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;Hz&quot;</span>, one <span class="op">/</span> second, kind_of<span class="op">&lt;</span>isq<span class="op">::</span>frequency<span class="op">&gt;&gt;</span> <span class="op">{}</span> hertz;</span>
<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> becquerel <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;Bq&quot;</span>, one <span class="op">/</span> second, kind_of<span class="op">&lt;</span>isq<span class="op">::</span>activity<span class="op">&gt;&gt;</span> <span class="op">{}</span> becquerel;</span></code></pre></div>
<p>With the above, <code class="sourceCode default">hertz</code> can
only be used for frequencies, while
<code class="sourceCode default">becquerel</code> should only be used
for quantities of activity. This means that the following equation will
not compile, improving the type-safety of the library:</p>
<div class="sourceCode" id="cb36"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q <span class="op">=</span> <span class="dv">1</span> <span class="op">*</span> Hz <span class="op">+</span> <span class="dv">1</span> <span class="op">*</span> Bq;   <span class="co">// Fails to compile</span></span></code></pre></div>
<h2 data-number="5.5" id="prefixed-units"><span class="header-section-number">5.5</span> Prefixed units<a href="#prefixed-units" class="self-link"></a></h2>
<p>Besides named units, the <a href="../../appendix/glossary.md#si">SI</a> specifies also 24 prefixes
(all being a power of <code class="sourceCode default">10</code>) that
can be prepended to all named units to obtain various scaled versions of
them.</p>
<p>Implementation of <code class="sourceCode default">std::ratio</code>
provided by all major compilers is able to express only 16 of them. This
is why, in the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span>, we had
to find an alternative way to represent a unit’s magnitude in a more
flexible way.</p>
<p>Each prefix is implemented as:</p>
<div class="sourceCode" id="cb37"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>PrefixableUnit <span class="kw">auto</span> U<span class="op">&gt;</span> <span class="kw">struct</span> quecto_ <span class="op">:</span> prefixed_unit<span class="op">&lt;</span><span class="st">&quot;q&quot;</span>, mag_power<span class="op">&lt;</span><span class="dv">10</span>, <span class="op">-</span><span class="dv">30</span><span class="op">&gt;</span>, U<span class="op">&gt;</span> <span class="op">{}</span>;</span>
<span id="cb37-2"><a href="#cb37-2" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>PrefixableUnit <span class="kw">auto</span> U<span class="op">&gt;</span> <span class="kw">inline</span> <span class="kw">constexpr</span> quecto_<span class="op">&lt;</span>U<span class="op">&gt;</span> quecto;</span></code></pre></div>
<p>and then a unit can be prefixed in the following way:</p>
<div class="sourceCode" id="cb38"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">auto</span> qm <span class="op">=</span> quecto<span class="op">&lt;</span>metre<span class="op">&gt;</span>;</span></code></pre></div>
<p>The usage of <code class="sourceCode default">mag_power</code> not
only enables providing support for SI prefixes, but it can also
efficiently represent any rational magnitude. For example, <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> (part 13) prefixes used
in the IT industry can be implemented as:</p>
<div class="sourceCode" id="cb39"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb39-1"><a href="#cb39-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>PrefixableUnit <span class="kw">auto</span> U<span class="op">&gt;</span> <span class="kw">struct</span> yobi_ <span class="op">:</span> prefixed_unit<span class="op">&lt;</span><span class="st">&quot;Yi&quot;</span>, mag_power<span class="op">&lt;</span><span class="dv">2</span>, <span class="dv">80</span><span class="op">&gt;</span>, U<span class="op">&gt;</span> <span class="op">{}</span>;</span>
<span id="cb39-2"><a href="#cb39-2" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>PrefixableUnit <span class="kw">auto</span> U<span class="op">&gt;</span> <span class="kw">inline</span> <span class="kw">constexpr</span> yobi_<span class="op">&lt;</span>U<span class="op">&gt;</span> yobi;</span></code></pre></div>
<p><em>Please note that we need two lines with two definitions in the
case of a class template and an associated variable template. The C and
C++ standards permit providing the same identifier for a class and its
variable and appending the variable name after the class definition.
None of this is allowed for class templates.</em></p>
<h2 data-number="5.6" id="scaled-units"><span class="header-section-number">5.6</span> Scaled units<a href="#scaled-units" class="self-link"></a></h2>
<p>In the <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span>, all units are either base or
derived units or prefixed versions of those. However, those are not the
only options possible.</p>
<p>For example, there is a list of off-system units accepted for use
with <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span>. All of those are scaled versions of
the <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span> units with ratios that can’t be
explicitly expressed with predefined SI prefixes. Those include units
like minute, hour, or electronvolt:</p>
<div class="sourceCode" id="cb40"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> minute <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;min&quot;</span>, mag<span class="op">&lt;</span><span class="dv">60</span><span class="op">&gt;</span> <span class="op">*</span> si<span class="op">::</span>second<span class="op">&gt;</span> <span class="op">{}</span> minute;</span>
<span id="cb40-2"><a href="#cb40-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> hour <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;h&quot;</span>, mag<span class="op">&lt;</span><span class="dv">60</span><span class="op">&gt;</span> <span class="op">*</span> minute<span class="op">&gt;</span> <span class="op">{}</span> hour;</span>
<span id="cb40-3"><a href="#cb40-3" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> electronvolt <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;eV&quot;</span>,</span>
<span id="cb40-4"><a href="#cb40-4" aria-hidden="true" tabindex="-1"></a>    mag<span class="op">&lt;</span>ratio<span class="op">{</span><span class="dv">1&#39;602&#39;176&#39;634</span>, <span class="dv">1&#39;000&#39;000&#39;000</span><span class="op">}&gt;</span> <span class="op">*</span> mag_power<span class="op">&lt;</span><span class="dv">10</span>, <span class="op">-</span><span class="dv">19</span><span class="op">&gt;</span> <span class="op">*</span> si<span class="op">::</span>joule<span class="op">&gt;</span> <span class="op">{}</span> electronvolt;</span></code></pre></div>
<p>Also, units of other systems of units are often defined in terms of
scaled versions of other (often SI) units. For example, the
international yard is defined as:</p>
<div class="sourceCode" id="cb41"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> yard <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;yd&quot;</span>, mag<span class="op">&lt;</span>ratio<span class="op">{</span><span class="dv">9&#39;144</span>, <span class="dv">10&#39;000</span><span class="op">}&gt;</span> <span class="op">*</span> si<span class="op">::</span>metre<span class="op">&gt;</span> <span class="op">{}</span> yard;</span></code></pre></div>
<p>and then a <code class="sourceCode default">foot</code> can be
defined as:</p>
<div class="sourceCode" id="cb42"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> foot <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;ft&quot;</span>, mag<span class="op">&lt;</span>ratio<span class="op">{</span><span class="dv">1</span>, <span class="dv">3</span><span class="op">}&gt;</span> <span class="op">*</span> yard<span class="op">&gt;</span> <span class="op">{}</span> foot;</span></code></pre></div>
<p>For some units, a magnitude might also be irrational. The best
example here is a <code class="sourceCode default">degree</code> which
is defined using a floating-point magnitude having a factor of the
number π (Pi):</p>
<div class="sourceCode" id="cb43"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> mag_pi <span class="op">:</span> magnitude<span class="op">&lt;</span>std<span class="op">::</span>numbers<span class="op">::</span>pi_v<span class="op">&lt;</span><span class="dt">long</span> <span class="dt">double</span><span class="op">&gt;&gt;</span> <span class="op">{}</span> mag_pi;</span></code></pre></div>
<div class="sourceCode" id="cb44"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb44-1"><a href="#cb44-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> degree <span class="op">:</span> named_unit<span class="op">&lt;</span>basic_symbol_text<span class="op">{</span><span class="st">&quot;°&quot;</span>, <span class="st">&quot;deg&quot;</span><span class="op">}</span>, mag_pi <span class="op">/</span> mag<span class="op">&lt;</span><span class="dv">180</span><span class="op">&gt;</span> <span class="op">*</span> si<span class="op">::</span>radian<span class="op">&gt;</span> <span class="op">{}</span> degree;</span></code></pre></div>
<h2 data-number="5.7" id="unit-symbols"><span class="header-section-number">5.7</span> Unit symbols<a href="#unit-symbols" class="self-link"></a></h2>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library,
units are available via their full names or through their short symbols.
To use a long version it is enough to type:</p>
<div class="sourceCode" id="cb45"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a><span class="pp">#include </span><span class="im">&lt;mp-units/systems/si/si.h&gt;</span></span>
<span id="cb45-2"><a href="#cb45-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb45-3"><a href="#cb45-3" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> <span class="kw">namespace</span> mp_units;</span>
<span id="cb45-4"><a href="#cb45-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb45-5"><a href="#cb45-5" aria-hidden="true" tabindex="-1"></a>quantity q <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> si<span class="op">::</span>metre;</span></code></pre></div>
<p>The same can be obtained using an optional unit symbol:</p>
<div class="sourceCode" id="cb46"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a><span class="pp">#include </span><span class="im">&lt;mp-units/systems/si/si.h&gt;</span></span>
<span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb46-3"><a href="#cb46-3" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> <span class="kw">namespace</span> mp_units;</span>
<span id="cb46-4"><a href="#cb46-4" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> <span class="kw">namespace</span> mp_units<span class="op">::</span>si<span class="op">::</span>unit_symbols;</span>
<span id="cb46-5"><a href="#cb46-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb46-6"><a href="#cb46-6" aria-hidden="true" tabindex="-1"></a>quantity q <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m;</span></code></pre></div>
<p>Unit symbols introduce a lot of short identifiers into the current
namespace, and that is why they are opt-in. A user has to explicitly
“import” them from a dedicated
<code class="sourceCode default">unit_symbols</code> namespace.</p>
<h1 data-number="6" id="operations-on-units-dimensions-and-quantity-types"><span class="header-section-number">6</span> Operations on units, dimensions,
and quantity types<a href="#operations-on-units-dimensions-and-quantity-types" class="self-link"></a></h1>
<p>Modern C++ physical quantities and units library should expose
compile-time constants for units, dimensions, and quantity types. Each
of such constants should be of a different type. Said otherwise, every
unit, dimension, and quantity type has a unique type and a compile-time
instance. This allows us to do regular algebra on such identifiers and
get proper types as results of such operations.</p>
<p>The operations exposed by such a library should include at least:</p>
<ul>
<li>multiplication
(e.g. <code class="sourceCode default">newton * metre</code>),</li>
<li>division
(e.g. <code class="sourceCode default">metre / second</code>),</li>
<li>power
(e.g. <code class="sourceCode default">pow&lt;2&gt;(metre)</code> or
<code class="sourceCode default">pow&lt;1, 2&gt;(metre * metre)</code>).</li>
</ul>
<p>To improve the usability of the library, we also recommend
adding:</p>
<ul>
<li>square root
(e.g. <code class="sourceCode default">sqrt(metre * metre)</code> as
equivalent to
<code class="sourceCode default">pow&lt;1, 2&gt;(metre * metre)</code>),</li>
<li>cubic root
(e.g. <code class="sourceCode default">cbrt(metre * metre * metre)</code>
as equivalent to <code class="sourceCode default">pow&lt;1, 3&gt;(metre * metre * metre)</code>),</li>
<li>inversion
(e.g. <code class="sourceCode default">inverse(second)</code> as
equivalent to
<code class="sourceCode default">one / second</code>).</li>
</ul>
<p>Additionally, for units only, to improve the readability of the code,
it makes sense to expose the following:</p>
<ul>
<li>square power
(e.g. <code class="sourceCode default">square(metre)</code> is
equivalent to
<code class="sourceCode default">pow&lt;2&gt;(metre)</code>),</li>
<li>cubic power
(e.g. <code class="sourceCode default">cubic(metre)</code> is equivalent
to <code class="sourceCode default">pow&lt;3&gt;(metre)</code>).</li>
</ul>
<p>The above two functions could also be considered for dimensions and
quantity types. However,
<code class="sourceCode default">cubic(length)</code> does not seem to
make much sense, and probably
<code class="sourceCode default">pow&lt;3&gt;(length)</code> should be
preferred instead.</p>
<h2 data-number="6.1" id="expression-templates"><span class="header-section-number">6.1</span> Expression templates<a href="#expression-templates" class="self-link"></a></h2>
<p>Modern C++ physical quantities and units libraries use opaque types
to improve the user experience while analyzing compile-time errors or
inspecting types in a debugger. This is a huge usability improvement
over the older libraries that use aliases to refer to long
instantiations of class templates.</p>
<h3 data-number="6.1.1" id="derived-entities"><span class="header-section-number">6.1.1</span> Derived entities<a href="#derived-entities" class="self-link"></a></h3>
<p>Having such strong types for entities is not enough. While doing
arithmetics on them, we get derived entities, and they also should be
easy to understand and correlate with the code written by the user. This
is where expression templates come into play.</p>
<p>The library should use the same unified approach to represent the
results of arithmetics on all kinds of entities. It is worth mentioning
that a generic purpose expression templates library is not a good
solution for a physical quantities and units library.</p>
<p>Let’s assume that we want to represent the results of the following
two unit equations:</p>
<ul>
<li><code class="sourceCode default">metre / second * second</code></li>
<li><code class="sourceCode default">metre * metre / metre</code></li>
</ul>
<p>Both of them should result in a type equivalent to
<code class="sourceCode default">metre</code>. A general-purpose library
will probably result with the types similar to the below:</p>
<ul>
<li><code class="sourceCode default">mul&lt;div&lt;metre, second&gt;, second&gt;</code></li>
<li><code class="sourceCode default">div&lt;mul&lt;metre, metre&gt;, metre&gt;</code></li>
</ul>
<p>Comparing such types for equivalence would not only be very expensive
at compile-time but would also be really confusing to the users
observing them in the compilation logs. This is why we need a dedicated
solution here.</p>
<p>In a physical quantities and units library, we need expression
templates to express the results of</p>
<ul>
<li>dimension equations,</li>
<li>quantity type equations,</li>
<li>unit equations, and</li>
<li>unit magnitude equations.</li>
</ul>
<p>If the above equation results in a derived entity, we must create a
type that clearly describes what we are dealing with. We need to pack a
simplified expression template into some container for that. There are
various possibilities here. The table below presents the types generated
from unit expressions by two leading products on the market in this
subject:</p>
<!-- markdownlint-disable MD013 -->
<table style="width:100%;">
<colgroup>
<col style="width: 9%" />
<col style="width: 42%" />
<col style="width: 47%" />
</colgroup>
<thead>
<tr class="header">
<th><div style="text-align:center">
<strong>Unit</strong>
</div></th>
<th><div style="text-align:center">
<strong><span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span></strong>
</div></th>
<th><div style="text-align:center">
<strong><span class="citation" data-cites="AU">[<a href="#ref-AU" role="doc-biblioref">Au</a>]</span></strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><code class="sourceCode default">N⋅m</code></td>
<td><code class="sourceCode default">derived_unit&lt;metre, newton&gt;</code></td>
<td><code class="sourceCode default">UnitProduct&lt;Meters, Newtons&gt;</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">1/s</code></td>
<td><code class="sourceCode default">derived_unit&lt;one, per&lt;second&gt;&gt;</code></td>
<td><code class="sourceCode default">Pow&lt;Seconds, -1&gt;</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">km/h</code></td>
<td><code class="sourceCode default">derived_unit&lt;kilo_&lt;metre&gt;, per&lt;hour&gt;&gt;</code></td>
<td><code class="sourceCode default">UnitProduct&lt;Kilo&lt;Meters&gt;, Pow&lt;Hours, -1&gt;&gt;</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">kg⋅m²/(s³⋅K)</code></td>
<td><code class="sourceCode default">derived_unit&lt;kilogram, pow&lt;metre, 2&gt;, per&lt;kelvin, power&lt;second, 3&gt;&gt;&gt;</code></td>
<td><code class="sourceCode default">UnitProduct&lt;Pow&lt;Meters, 2&gt;, Kilo&lt;Grams&gt;, Pow&lt;Seconds, -3&gt;, Pow&lt;Kelvins, -1&gt;&gt;</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">m²/m</code></td>
<td><code class="sourceCode default">metre</code></td>
<td><code class="sourceCode default">Meters</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">km/m</code></td>
<td><code class="sourceCode default">derived_unit&lt;kilo_&lt;metre&gt;, per&lt;metre&gt;&gt;</code></td>
<td><code class="sourceCode default">UnitProduct&lt;Pow&lt;Meters, -1&gt;, Kilo&lt;Meters&gt;&gt;</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">m/m</code></td>
<td><code class="sourceCode default">one</code></td>
<td><code class="sourceCode default">UnitProduct&lt;&gt;</code></td>
</tr>
</tbody>
</table>
<!-- markdownlint-enable MD013 -->
<p>It is a matter of taste which solution is better. While discussing
the pros and cons here, we should remember that our users often do not
have a scientific background. This is why the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library decided to use syntax
that is as similar to the correct English language as possible. It
consistently uses the <code class="sourceCode default">derived_</code>
prefix for types representing derived units, dimensions, and quantity
specifications. Those are instantiated first with the contents of the
numerator followed by the entities of the denominator (if present)
enclosed in the <code class="sourceCode default">per&lt;...&gt;</code>
expression template.</p>
<h3 data-number="6.1.2" id="identities"><span class="header-section-number">6.1.2</span> Identities<a href="#identities" class="self-link"></a></h3>
<p>The arithmetics on units, dimensions, and quantity types require a
special identity value. Such value can be returned as a result of the
division of the same entities, or using it should not modify the
expression template on multiplication.</p>
<p>The <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library
chose the following names here:</p>
<ul>
<li><code class="sourceCode default">one</code> in the domain of
units,</li>
<li><code class="sourceCode default">dimension_one</code> in the domain
of dimensions,</li>
<li><code class="sourceCode default">dimensionless</code> in the domain
of quantity types.</li>
</ul>
<p>The above names were selected based on the following quote from the
<span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>:</p>
<blockquote>
<p>A quantity whose dimensional exponents are all equal to zero has the
dimensional product denoted A<sup>0</sup>B<sup>0</sup>C<sup>0</sup>… =
1, where the symbol 1 denotes the corresponding dimension. There is no
agreement on how to refer to such quantities. They have been called
<strong>dimensionless</strong> quantities (although this term should now
be avoided), quantities with <strong>dimension one</strong>, quantities
with dimension number, or quantities with the <strong>unit one</strong>.
Such quantities are dimensionally simply numbers. To avoid confusion, it
is helpful to use explicit units with these quantities where possible,
e.g., m/m, nmol/mol, rad, as specified in the SI Brochure.</p>
</blockquote>
<h3 data-number="6.1.3" id="supported-operations-and-their-results"><span class="header-section-number">6.1.3</span> Supported operations and
their results<a href="#supported-operations-and-their-results" class="self-link"></a></h3>
<p>The table below presents all the operations that can be done on
units, dimensions, and quantity types in a physical quantities and units
library and corresponding expression templates chosen by the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> project as their results:</p>
<table>
<colgroup>
<col style="width: 53%" />
<col style="width: 46%" />
</colgroup>
<thead>
<tr class="header">
<th style="text-align: center;"><div style="text-align:center">
<strong>Operation</strong>
</div></th>
<th style="text-align: center;"><div style="text-align:center">
<strong>Resulting template expression arguments</strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">A * B</code></td>
<td style="text-align: center;"><code class="sourceCode default">A, B</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">B * A</code></td>
<td style="text-align: center;"><code class="sourceCode default">A, B</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">A * A</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 2&gt;</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">{identity} * A</code></td>
<td style="text-align: center;"><code class="sourceCode default">A</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">A * {identity}</code></td>
<td style="text-align: center;"><code class="sourceCode default">A</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">A / B</code></td>
<td style="text-align: center;"><code class="sourceCode default">A, per&lt;B&gt;</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">A / A</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">A / {identity}</code></td>
<td style="text-align: center;"><code class="sourceCode default">A</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">{identity} / A</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}, per&lt;A&gt;</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">pow&lt;2&gt;(A)</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 2&gt;</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">pow&lt;2&gt;({identity})</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">sqrt(A)</code>
or <code class="sourceCode default">pow&lt;1, 2&gt;(A)</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 1, 2&gt;</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">sqrt({identity})</code>
or
<code class="sourceCode default">pow&lt;1, 2&gt;({identity})</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}</code></td>
</tr>
</tbody>
</table>
<h3 data-number="6.1.4" id="simplifying-the-resulting-expression-templates"><span class="header-section-number">6.1.4</span> Simplifying the resulting
expression templates<a href="#simplifying-the-resulting-expression-templates" class="self-link"></a></h3>
<p>To limit the length and improve the readability of generated types,
there are many rules to simplify the resulting expression template.</p>
<ol type="1">
<li><p><strong>Ordering</strong></p>
<p>The resulting comma-separated arguments of multiplication are always
sorted according to a specific predicate. This is why:</p>
<div class="sourceCode" id="cb47"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>A <span class="op">*</span> B <span class="op">==</span> B <span class="op">*</span> A<span class="op">)</span>;</span>
<span id="cb47-2"><a href="#cb47-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>std<span class="op">::</span>is_same_v<span class="op">&lt;</span><span class="kw">decltype</span><span class="op">(</span>A <span class="op">*</span> B<span class="op">)</span>, <span class="kw">decltype</span><span class="op">(</span>B <span class="op">*</span> A<span class="op">)&gt;)</span>;</span></code></pre></div>
<p>This is probably the most important of all the steps, as it allows
comparing types and enables the rest of the simplification rules.</p>
<p>Units and dimensions have unique symbols, but ordering quantity types
might not be that trivial. Although the ISQ defined in <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> provides symbols for each
quantity, there is little use for them in the C++ code. This is caused
by the fact that such symbols use a lot of characters that are not
available with the Unicode encoding. Most of the limitations correspond
to Unicode providing only a minimal set of characters available as
subscripts, which are often used to differentiate various quantities of
the same kind. For example, it is impossible to encode the symbols of
the following quantities:</p>
<ul>
<li><em>c</em><sub>sat</sub> - specific heat capacity at saturated
vapour pressure,</li>
<li><em>μ</em><sub>JT</sub> - Joule-Thomson coefficient,</li>
<li><em>w</em><sub>H<sub>2</sub>O</sub> - mass fraction of water,</li>
<li><em>σ</em><sub>Ω,E</sub> - direction and energy distribution of
cross section,</li>
<li><em>d</em><sub>1/2</sub> - half-value thickness,</li>
<li><em>Φ</em><sub>e,λ</sub> - spectral radiant flux.</li>
</ul>
<p>This is why the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library
chose to use type name identifiers in such cases.</p></li>
<li><p><strong>Aggregation</strong></p>
<p>In case two of the same type identifiers are found next to each other
on the argument list, they will be aggregated in one entry:</p>
<table>
<thead>
<tr class="header">
<th style="text-align: center;"><div style="text-align:center">
<strong>Before</strong>
</div></th>
<th style="text-align: center;"><div style="text-align:center">
<strong>After</strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">A, A</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 2&gt;</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">A, power&lt;A, 2&gt;</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 3&gt;</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 1, 2&gt;, power&lt;A, 2&gt;</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 5, 2&gt;</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 1, 2&gt;, power&lt;A, 1, 2&gt;</code></td>
<td style="text-align: center;"><code class="sourceCode default">A</code></td>
</tr>
</tbody>
</table></li>
<li><p><strong>Simplification</strong></p>
<p>In case two of the same type identifiers are found in the numerator
and denominator argument lists, they are being simplified into one
entry:</p>
<table>
<thead>
<tr class="header">
<th style="text-align: center;"><div style="text-align:center">
<strong>Before</strong>
</div></th>
<th style="text-align: center;"><div style="text-align:center">
<strong>After</strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">A, per&lt;A&gt;</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 2&gt;, per&lt;A&gt;</code></td>
<td style="text-align: center;"><code class="sourceCode default">A</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 3&gt;, per&lt;A&gt;</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 2&gt;</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">A, per&lt;power&lt;A, 2&gt;&gt;</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}, per&lt;A&gt;</code></td>
</tr>
</tbody>
</table>
<p>It is important to notice here that only the elements with exactly
the same type are being simplified. This means that, for example,
<code class="sourceCode default">m/m</code> results in
<code class="sourceCode default">one</code>, but
<code class="sourceCode default">km/m</code> will not be simplified. The
resulting derived unit will preserve both symbols and their relative
magnitude. This allows us to properly print symbols of some units or
constants that require such behavior. For example, the Hubble constant
is expressed in <code class="sourceCode default">km⋅s⁻¹⋅Mpc⁻¹</code>,
where both <code class="sourceCode default">km</code> and
<code class="sourceCode default">Mpc</code> are units of
length.</p></li>
<li><p><strong>Repacking</strong></p>
<p>In case an expression uses two results of some other operations, the
components of its arguments are repacked into one resulting type and
simplified there.</p>
<p>For example, assuming:</p>
<div class="sourceCode" id="cb48"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb48-1"><a href="#cb48-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> X <span class="op">=</span> A <span class="op">/</span> B;</span></code></pre></div>
<p>then:</p>
<table>
<thead>
<tr class="header">
<th style="text-align: center;"><div style="text-align:center">
<strong>Operation</strong>
</div></th>
<th style="text-align: center;"><div style="text-align:center">
<strong>Resulting template expression arguments</strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">X * B</code></td>
<td style="text-align: center;"><code class="sourceCode default">A</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">X * A</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 2&gt;, per&lt;B&gt;</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">X * X</code></td>
<td style="text-align: center;"><code class="sourceCode default">power&lt;A, 2&gt;, per&lt;power&lt;B, 2&gt;&gt;</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">X / X</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}</code></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><code class="sourceCode default">X / A</code></td>
<td style="text-align: center;"><code class="sourceCode default">{identity}, per&lt;B&gt;</code></td>
</tr>
<tr class="even">
<td style="text-align: center;"><code class="sourceCode default">X / B</code></td>
<td style="text-align: center;"><code class="sourceCode default">A, per&lt;power&lt;B, 2&gt;&gt;</code></td>
</tr>
</tbody>
</table></li>
</ol>
<p>Please note that for as long as for the ordering step in some cases,
we use user-provided symbols, the aggregation, and the next steps do not
benefit from those. They always use type identifiers to determine
whether the operation should be performed.</p>
<p>Unit symbols are not guaranteed to be unique in the project. For
example, someone may use
<code class="sourceCode default">&quot;s&quot;</code> as a symbol for a
count of samples, which, when used in a unit expression with seconds,
would cause fatal consequences
(e.g. <code class="sourceCode default">sample * second</code> would
yield <code class="sourceCode default">s²</code>, or
<code class="sourceCode default">sample / second</code> would result in
<code class="sourceCode default">one</code>).</p>
<p>Some units would provide worse text output if the ordering step used
type identifiers rather than unit symbols. For example, <code class="sourceCode default">si::metre * si::second * cgs::second</code>
would result in <code class="sourceCode default">s m s</code>, or
<code class="sourceCode default">newton * metre</code> would result in
<code class="sourceCode default">m N</code>, which is not how we
typically spell this unit. However, for the sake of consistency, we may
also consider changing the algorithm used for ordering to be based on
type identifiers.</p>
<h3 data-number="6.1.5" id="example"><span class="header-section-number">6.1.5</span> Example<a href="#example" class="self-link"></a></h3>
<p>Thanks to all of the steps described above, a user may write the code
like this one:</p>
<div class="sourceCode" id="cb49"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> <span class="kw">namespace</span> mp_units<span class="op">::</span>si<span class="op">::</span>unit_symbols;</span>
<span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a>quantity speed <span class="op">=</span> isq<span class="op">::</span>speed<span class="op">(</span><span class="fl">60.</span> <span class="op">*</span> km <span class="op">/</span> h<span class="op">)</span>;</span>
<span id="cb49-3"><a href="#cb49-3" aria-hidden="true" tabindex="-1"></a>quantity duration <span class="op">=</span> <span class="dv">8</span> <span class="op">*</span> s;</span>
<span id="cb49-4"><a href="#cb49-4" aria-hidden="true" tabindex="-1"></a>quantity acceleration1 <span class="op">=</span> speed <span class="op">/</span> duration;</span>
<span id="cb49-5"><a href="#cb49-5" aria-hidden="true" tabindex="-1"></a>quantity acceleration2 <span class="op">=</span> isq<span class="op">::</span>acceleration<span class="op">(</span>acceleration1<span class="op">.</span>in<span class="op">(</span>m <span class="op">/</span> s2<span class="op">))</span>;</span>
<span id="cb49-6"><a href="#cb49-6" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;acceleration: &quot;</span> <span class="op">&lt;&lt;</span> acceleration1 <span class="op">&lt;&lt;</span> <span class="st">&quot; (&quot;</span> <span class="op">&lt;&lt;</span> acceleration2 <span class="op">&lt;&lt;</span> <span class="st">&quot;)</span><span class="sc">\n</span><span class="st">&quot;</span>;</span></code></pre></div>
<p>the text output provides:</p>
<pre class="text"><code>acceleration: 7.5 km h⁻¹ s⁻¹ (2.08333 m/s²)</code></pre>
<p>The above program will produce the following types for acceleration
quantities (after stripping the
<code class="sourceCode default">mp_units</code> namespace for
brevity):</p>
<ul>
<li><p><code class="sourceCode default">acceleration1</code></p>
<pre class="text"><code>quantity&lt;reference&lt;derived_quantity_spec&lt;isq::speed, per&lt;isq::time&gt;&gt;{},
                  derived_unit&lt;si::kilo_&lt;si::metre{}&gt;, per&lt;non_si::hour, si::second&gt;&gt;{}&gt;{},
         double&gt;</code></pre></li>
<li><p><code class="sourceCode default">acceleration2</code></p>
<pre class="text"><code>quantity&lt;reference&lt;isq::acceleration,
                   derived_unit&lt;si::metre, per&lt;power&lt;si::second, 2&gt;&gt;&gt;{}&gt;{},
         double&gt;&gt;</code></pre></li>
</ul>
<h2 data-number="6.2" id="scaled-units-1"><span class="header-section-number">6.2</span> Scaled units<a href="#scaled-units-1" class="self-link"></a></h2>
<p>Another very common operation is to multiply an existing unit by a
factor, creating a new, scaled unit. For example, the unit <em>foot</em>
can be multiplied by 3, producing the unit <em>yard</em>.</p>
<p>The process also works in reverse; the ratio between any two units of
the same dimension is a well-defined number. For example, the ratio
between one foot and one inch is 12.</p>
<h3 data-number="6.2.1" id="unit-magnitudes"><span class="header-section-number">6.2.1</span> Unit magnitudes<a href="#unit-magnitudes" class="self-link"></a></h3>
<p>In principle, this scaling factor can be any positive real number. In
<span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> and <span class="citation" data-cites="AU">[<a href="#ref-AU" role="doc-biblioref">Au</a>]</span>,
we have used the term “magnitude” to refer to this scaling factor. (This
should not be confused with other uses of the term, such as the
logarithmic “magnitude” unit commonly used in astronomy).</p>
<p>In the library implementation, each unit is associated with a
magnitude. However, for most units, the magnitude is a fully
encapsulated implementation detail, not a user-facing value.</p>
<p>This is because the notion of “the” magnitude of a unit is not
generally meaningful: it has no physically observable consequence. What
<em>is</em> meaningful is the <em>ratio</em> of magnitudes between two
units of the same quantity kind. We could associate the <em>foot</em>,
say, with any magnitude <span class="math inline">\(m_f\)</span> that we
like — but once we make that choice, we must assign <span class="math inline">\(3m_f\)</span> to the <em>yard</em>, and <span class="math inline">\(m_f/12\)</span> to the <em>inch</em>. Separately
and independently, we can assign any magnitude <span class="math inline">\(m_s\)</span> to the second, because it’s an
independent dimension — but once we make that choice, it fixes the
magnitude for derived units, and we must assign, say, <span class="math inline">\((5280 m_f) / (3600 m_s)\)</span> to the <em>mile
per hour</em>.</p>
<h3 data-number="6.2.2" id="requirements-and-representation"><span class="header-section-number">6.2.2</span> Requirements and
representation<a href="#requirements-and-representation" class="self-link"></a></h3>
<p>A magnitude is a positive real number. The best way to
<em>represent</em> it depends on how we will <em>use</em> it. To derive
our requirements, note that magnitudes must support every operation
which units do.</p>
<p>Units are closed under <em>products</em> and <em>rational
powers</em>. Therefore, our magnitude representation must support these
operations natively and robustly; this is the most basic requirement. We
must also support certain <em>irrational</em> “ratios”, such as the
factor of <span class="math inline">\(\frac{\pi}{180}\)</span> between
<em>degrees</em> and <em>radians</em>.</p>
<p>The usual approach,
<code class="sourceCode default">std::ratio</code>, fails to satisfy
these requirements in multiple ways.</p>
<ul>
<li>It is not closed under rational powers (rather infamously, in the
case of 2<sup>1/2</sup>).</li>
<li>It cannot represent irrational factors such as <span class="math inline">\(\pi\)</span>.</li>
<li>It is too vulnerable to overflow when raised to powers.</li>
</ul>
<p>One alternative is the <em>vector space magnitude</em>
representation. Here, we represent each magnitude as a <em>product of
powers of “basis” numbers</em>. This is the same representation we use
for dimensions, so it will naturally support all the same operations —
as long as we can find a suitable basis.</p>
<p>Each magnitude must have a unique representation. This requirement
constrains our choice of “basis” vectors; they must not be able to
represent any magnitude in more than one way. Prime numbers have this
property. Take any arbitrarily large (but finite) collection of primes,
raise each prime to some chosen exponent, and compute the product: the
result can’t be expressed by any other collection of exponents.</p>
<p>Already, this lets us represent every positive real number which
<code class="sourceCode default">std::ratio</code> can represent, by
breaking the numerator and denominator into their prime factorizations.
But we can go further, and handle irrational factors such as <span class="math inline">\(\pi\)</span> by introducing them as new basis
vectors. <span class="math inline">\(\pi\)</span> cannot be represented
by the product of powers of <em>any</em> finite collection of primes,
which means that it is “linearly independent” in the sense of our vector
space representation.</p>
<p>On the C++ implementation side, we use variadic templates to define
our magnitude. Each element is a basis number raised to some rational
power (which may be omitted or abbreviated as appropriate).</p>
<p>Here are some examples, using Astronomical Units (au), meters (m),
degrees (deg), and radians (rad).</p>
<!-- markdownlint-disable MD013 -->
<table>
<colgroup>
<col style="width: 23%" />
<col style="width: 15%" />
<col style="width: 61%" />
</colgroup>
<thead>
<tr class="header">
<th><div style="text-align:center">
<strong>Unit ratio</strong>
</div></th>
<th><div style="text-align:center">
<strong><code class="sourceCode default">std::ratio</code>
representation</strong>
</div></th>
<th><div style="text-align:center">
<strong>vector space representation</strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><span class="math inline">\(\left(\frac{\text{au}}{\text{m}}\right)\)</span></td>
<td><code class="sourceCode default">std::ratio&lt;149&#39;597&#39;870&#39;700&gt;</code></td>
<td><code class="sourceCode default">magnitude&lt;power_v&lt;2, 2&gt;(), 3, power_v&lt;5, 2&gt;(), 73, 877, 7789&gt;</code></td>
</tr>
<tr class="even">
<td><span class="math inline">\(\left(\frac{\text{au}}{\text{m}}\right)^2\)</span></td>
<td>Unrepresentable (overflow)</td>
<td><code class="sourceCode default">magnitude&lt;power_v&lt;2, 4&gt;(), power_v&lt;3, 2&gt;(), power_v&lt;5, 4&gt;(), power_v&lt;73, 2&gt;(), power_v&lt;877, 2&gt;(), power_v&lt;7789, 2&gt;()&gt;</code></td>
</tr>
<tr class="odd">
<td><span class="math inline">\(\sqrt{\frac{\text{au}}{\text{m}}}\)</span></td>
<td>Unrepresentable</td>
<td><code class="sourceCode default">magnitude&lt;2, power_v&lt;3, 1, 2&gt;(), 5, power_v&lt;73, 1, 2&gt;(), power_v&lt;877, 1, 2&gt;(), power_v&lt;7789, 1, 2&gt;()&gt;</code></td>
</tr>
<tr class="even">
<td><span class="math inline">\(\left(\frac{\text{rad}}{\text{deg}}\right)\)</span></td>
<td>Unrepresentable</td>
<td><code class="sourceCode default">magnitude&lt;power_v&lt;2, 2&gt;(), power_v&lt;3, 2&gt;(), power_v&lt;3.14159265358979323851e+0l, -1&gt;(), 5&gt;</code></td>
</tr>
</tbody>
</table>
<!-- markdownlint-enable MD013 -->
<p>The variadic magnitude types have one disadvantage: they are more
verbose. We may be able to hide them via opaque types with nicer names,
using a similar strategy as we have for units. In any case, their major
advantage is that they fulfill the requirements stated above — indeed,
they are the only solution we have seen which does.</p>
<h3 data-number="6.2.3" id="construction-helpers"><span class="header-section-number">6.2.3</span> Construction helpers<a href="#construction-helpers" class="self-link"></a></h3>
<p>We do not want end users to specify the variadic
<code class="sourceCode default">magnitude</code> implementation
manually. That would be too labor intensive and error prone. Instead, we
provide <em>construction helpers</em> for end users.</p>
<p><code class="sourceCode default">mag&lt;N&gt;</code> creates the
vector space representation of the integer
<code class="sourceCode default">N</code>. This is a simple value, so we
can multiply and divide it with other magnitudes.</p>
<p>For example,
<code class="sourceCode default">mag&lt;180&gt; / mag&lt;240&gt;</code>
produces <code class="sourceCode default">magnitude&lt;power_v&lt;2, -2&gt;(), 3&gt;</code>,
which is <span class="math inline">\(3/4\)</span>. Notice how the result
is always automatically represented in lowest terms, because a fraction
<em>not</em> in lowest terms cannot be represented as a product of
powers of primes.</p>
<h3 data-number="6.2.4" id="common-units"><span class="header-section-number">6.2.4</span> Common units<a href="#common-units" class="self-link"></a></h3>
<p>Some operations, such as addition or inequality comparison, are
“common unit” operations. In order to execute them on quantities of the
same dimension, we must first convert them to their <em>common
unit</em>.</p>
<p>For units which are <em>commensurable</em> — that is, units whose
ratio is a rational number — the “common unit” is the largest unit that
evenly divides both. For example, the common unit of the <em>foot</em>
and <em>inch</em> is the inch.</p>
<p>This also includes instances where neither unit is an integer
multiple of the other. For example, the common unit of the meter and the
yard does not have a name, but is equivalent to 800 micrometers. If we
call this unit <code class="sourceCode default">U</code>, then there are
1250 <code class="sourceCode default">U</code> per meter, and 1143
<code class="sourceCode default">U</code> per yard.</p>
<p>The practical benefit of this definition is that this unit conversion
will simply multiply each participating quantity by an exact integer. If
we use integer types to begin with, we can continue to use them without
losing precision.</p>
<p>Unfortunately, not every unit conversion makes this possible. No
angular unit could evenly divide both degrees and radians, for example.
In these instances, there is no uniquely defined notion of a “common
unit”.</p>
<p>In our vector space representation, we can easily compute the
magnitude of the common unit by taking the smallest exponent, across all
participating magnitudes, for each individual basis vector — as long as
we remember to use the implicit “0” exponent for any basis vector that
is omitted.</p>
<p>The following example may help make this clear. If we use <span class="math inline">\(\text{COM}\left[U_1, \cdots, U_n\right]\)</span>
as notation to represent “the common unit of <span class="math inline">\(U_1, \cdots, U_n\)</span>”, and we show only the
magnitudes for simplicity, here are the steps we would follow to find
the magnitude of the common unit.</p>
<!-- markdownlint-disable MD013 -->
<p><span class="math display">\[
\begin{align}
\text{COM}\left[18, \frac{80}{3}\right] &amp;= \text{COM}\left[(2 \cdot
3^2), (2^4 \cdot 3^{-1} \cdot 5)\right] \\
&amp;= \text{COM}\left[(2^1 \cdot 3^2 \cdot 5^0), (2^4 \cdot 3^{-1}
\cdot 5^1)\right] \\
&amp;= 2^{\text{min}[1, 4]} \cdot 3^{\text{min}[2, -1]} \cdot
5^{\text{min}[0, 1]} \\
&amp;= 2^1 \cdot 3^{-1} \cdot 5^0 \\
&amp;= \frac{2}{3}
\end{align}
\]</span></p>
<!-- markdownlint-enable MD013 -->
<p>This procedure produces the unambiguous correct answer whenever it is
well defined. It also produces an answer for irrational “ratios”, where
there is no uniquely defined result. This provides the practical benefit
of making it easy to compare, say, an angle in degrees to one in
radians, as long as at least one of them is represented in a floating
point type.</p>
<h2 data-number="6.3" id="faster-than-lightspeed-constants"><span class="header-section-number">6.3</span> Faster than lightspeed
constants<a href="#faster-than-lightspeed-constants" class="self-link"></a></h2>
<p>In most libraries, physical constants are implemented as constant
(possibly <code class="sourceCode default">constexpr</code>) quantity
values. Such an approach has some disadvantages, often resulting in
longer compilation times and a loss of precision.</p>
<h3 data-number="6.3.1" id="simplifying-constants-in-an-equation"><span class="header-section-number">6.3.1</span> Simplifying constants in an
equation<a href="#simplifying-constants-in-an-equation" class="self-link"></a></h3>
<p>When dealing with equations involving physical constants, they often
occur more than once in an expression. Such a constant may appear both
in a numerator and denominator of a quantity equation. As we know from
fundamental physics, we can simplify such an expression by simply
striking a constant out of the equation. Supporting such behavior allows
a faster runtime performance and often a better precision of the
resulting value.</p>
<h3 data-number="6.3.2" id="physical-constants-as-units"><span class="header-section-number">6.3.2</span> Physical constants as units<a href="#physical-constants-as-units" class="self-link"></a></h3>
<p>The <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library
allows and encourages implementing physical constants as regular units.
With that, the constant’s value is handled at compile-time, and under
favorable circumstances, it can be simplified in the same way as all
other repeated units do. If it is not simplified, the value is stored in
a type, and the expensive multiplication or division operations can be
delayed in time until a user selects a specific unit to represent/print
the data.</p>
<p>Such a feature often also allows the use of simpler or faster
representation types in the equation. For example, instead of always
multiplying a small integral value with a big floating-point constant
number, we can just use the integral type all the way. Only in case a
constant will not simplify in the equation, and the user will require a
specific unit, such a multiplication will be lazily invoked, and the
representation type will need to be expanded to facilitate that. With
that, addition, subtractions, multiplications, and divisions will always
be the fastest - compiled away or done on the fast arithmetic types or
in out-of-order execution.</p>
<p>To benefit from all of the above, in the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library, constants (defined by
SI or otherwise) are implemented as units in the following way:</p>
<div class="sourceCode" id="cb50"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a><span class="kw">namespace</span> si <span class="op">{</span></span>
<span id="cb50-2"><a href="#cb50-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb50-3"><a href="#cb50-3" aria-hidden="true" tabindex="-1"></a><span class="kw">namespace</span> si2019 <span class="op">{</span></span>
<span id="cb50-4"><a href="#cb50-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb50-5"><a href="#cb50-5" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> speed_of_light_in_vacuum <span class="op">:</span></span>
<span id="cb50-6"><a href="#cb50-6" aria-hidden="true" tabindex="-1"></a>  named_unit<span class="op">&lt;</span><span class="st">&quot;c&quot;</span>, mag<span class="op">&lt;</span><span class="dv">299&#39;792&#39;458</span><span class="op">&gt;</span> <span class="op">*</span> metre <span class="op">/</span> second<span class="op">&gt;</span> <span class="op">{}</span> speed_of_light_in_vacuum;</span>
<span id="cb50-7"><a href="#cb50-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb50-8"><a href="#cb50-8" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>  <span class="co">// namespace si2019</span></span>
<span id="cb50-9"><a href="#cb50-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb50-10"><a href="#cb50-10" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> magnetic_constant <span class="op">:</span></span>
<span id="cb50-11"><a href="#cb50-11" aria-hidden="true" tabindex="-1"></a>  named_unit<span class="op">&lt;</span>basic_symbol_text<span class="op">{</span><span class="st">&quot;μ₀&quot;</span>, <span class="st">&quot;u_0&quot;</span><span class="op">}</span>, mag<span class="op">&lt;</span><span class="dv">4</span><span class="op">&gt;</span> <span class="op">*</span> mag_pi <span class="op">*</span> mag_power<span class="op">&lt;</span><span class="dv">10</span>, <span class="op">-</span><span class="dv">7</span><span class="op">&gt;</span> <span class="op">*</span> henry <span class="op">/</span> metre<span class="op">&gt;</span> <span class="op">{}</span> magnetic_constant;</span>
<span id="cb50-12"><a href="#cb50-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb50-13"><a href="#cb50-13" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>  <span class="co">// namespace mp_units::si</span></span></code></pre></div>
<h3 data-number="6.3.3" id="usage-examples"><span class="header-section-number">6.3.3</span> Usage examples<a href="#usage-examples" class="self-link"></a></h3>
<p>With the above definitions, we can calculate vacuum permittivity
as:</p>
<div class="sourceCode" id="cb51"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> permeability_of_vacuum <span class="op">=</span> <span class="fl">1.</span> <span class="op">*</span> si<span class="op">::</span>magnetic_constant;</span>
<span id="cb51-2"><a href="#cb51-2" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> speed_of_light_in_vacuum <span class="op">=</span> <span class="dv">1</span> <span class="op">*</span> si<span class="op">::</span>si2019<span class="op">::</span>speed_of_light_in_vacuum;</span>
<span id="cb51-3"><a href="#cb51-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb51-4"><a href="#cb51-4" aria-hidden="true" tabindex="-1"></a>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>permittivity_of_vacuum<span class="op">&gt;</span> <span class="kw">auto</span> q <span class="op">=</span> <span class="dv">1</span> <span class="op">/</span> <span class="op">(</span>permeability_of_vacuum <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>speed_of_light_in_vacuum<span class="op">))</span>;</span>
<span id="cb51-5"><a href="#cb51-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb51-6"><a href="#cb51-6" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;permittivity of vacuum = &quot;</span> <span class="op">&lt;&lt;</span> q <span class="op">&lt;&lt;</span> <span class="st">&quot; = &quot;</span> <span class="op">&lt;&lt;</span> q<span class="op">.</span>in<span class="op">(</span>F <span class="op">/</span> m<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span></code></pre></div>
<p>The above first prints the following:</p>
<pre class="text"><code>permittivity of vacuum = 1  μ₀⁻¹ c⁻² = 8.85419e-12 F/m</code></pre>
<p>As we can clearly see, all the calculations above were just about
multiplying and dividing the number
<code class="sourceCode default">1</code> with the rest of the
information provided as a compile-time type. Only when a user wants a
specific SI unit as a result, the unit ratios are lazily resolved.</p>
<p>Another similar example can be an equation for total energy:</p>
<div class="sourceCode" id="cb52"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>mechanical_energy<span class="op">&gt;</span> <span class="kw">auto</span> total_energy<span class="op">(</span>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>momentum<span class="op">&gt;</span> <span class="kw">auto</span> p,</span>
<span id="cb52-2"><a href="#cb52-2" aria-hidden="true" tabindex="-1"></a>                                                     QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>mass<span class="op">&gt;</span> <span class="kw">auto</span> m,</span>
<span id="cb52-3"><a href="#cb52-3" aria-hidden="true" tabindex="-1"></a>                                                     QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">&gt;</span> <span class="kw">auto</span> c<span class="op">)</span></span>
<span id="cb52-4"><a href="#cb52-4" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb52-5"><a href="#cb52-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> isq<span class="op">::</span>mechanical_energy<span class="op">(</span>sqrt<span class="op">(</span>pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>p <span class="op">*</span> c<span class="op">)</span> <span class="op">+</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>m <span class="op">*</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>c<span class="op">))))</span>;</span>
<span id="cb52-6"><a href="#cb52-6" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<div class="sourceCode" id="cb53"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> GeV <span class="op">=</span> si<span class="op">::</span>giga<span class="op">&lt;</span>si<span class="op">::</span>electronvolt<span class="op">&gt;</span>;</span>
<span id="cb53-2"><a href="#cb53-2" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">&gt;</span> <span class="kw">auto</span> c <span class="op">=</span> <span class="fl">1.</span> <span class="op">*</span> si<span class="op">::</span>si2019<span class="op">::</span>speed_of_light_in_vacuum;</span>
<span id="cb53-3"><a href="#cb53-3" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> c2 <span class="op">=</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>c<span class="op">)</span>;</span>
<span id="cb53-4"><a href="#cb53-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb53-5"><a href="#cb53-5" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> p1 <span class="op">=</span> isq<span class="op">::</span>momentum<span class="op">(</span><span class="fl">4.</span> <span class="op">*</span> GeV <span class="op">/</span> c<span class="op">)</span>;</span>
<span id="cb53-6"><a href="#cb53-6" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>mass<span class="op">&gt;</span> <span class="kw">auto</span> m1 <span class="op">=</span> <span class="fl">3.</span> <span class="op">*</span> GeV <span class="op">/</span> c2;</span>
<span id="cb53-7"><a href="#cb53-7" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> E <span class="op">=</span> total_energy<span class="op">(</span>p1, m1, c<span class="op">)</span>;</span>
<span id="cb53-8"><a href="#cb53-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb53-9"><a href="#cb53-9" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;in `GeV` and `c`:</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-10"><a href="#cb53-10" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;p = &quot;</span> <span class="op">&lt;&lt;</span> p1 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-11"><a href="#cb53-11" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;m = &quot;</span> <span class="op">&lt;&lt;</span> m1 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-12"><a href="#cb53-12" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;E = &quot;</span> <span class="op">&lt;&lt;</span> E <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb53-13"><a href="#cb53-13" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb53-14"><a href="#cb53-14" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> p2 <span class="op">=</span> p1<span class="op">.</span>in<span class="op">(</span>GeV <span class="op">/</span> <span class="op">(</span>m <span class="op">/</span> s<span class="op">))</span>;</span>
<span id="cb53-15"><a href="#cb53-15" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> m2 <span class="op">=</span> m1<span class="op">.</span>in<span class="op">(</span>GeV <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>m <span class="op">/</span> s<span class="op">))</span>;</span>
<span id="cb53-16"><a href="#cb53-16" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> E2 <span class="op">=</span> total_energy<span class="op">(</span>p2, m2, c<span class="op">).</span>in<span class="op">(</span>GeV<span class="op">)</span>;</span>
<span id="cb53-17"><a href="#cb53-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb53-18"><a href="#cb53-18" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">in `GeV`:</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-19"><a href="#cb53-19" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;p = &quot;</span> <span class="op">&lt;&lt;</span> p2 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-20"><a href="#cb53-20" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;m = &quot;</span> <span class="op">&lt;&lt;</span> m2 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-21"><a href="#cb53-21" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;E = &quot;</span> <span class="op">&lt;&lt;</span> E2 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb53-22"><a href="#cb53-22" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb53-23"><a href="#cb53-23" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> p3 <span class="op">=</span> p1<span class="op">.</span>in<span class="op">(</span>kg <span class="op">*</span> m <span class="op">/</span> s<span class="op">)</span>;</span>
<span id="cb53-24"><a href="#cb53-24" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> m3 <span class="op">=</span> m1<span class="op">.</span>in<span class="op">(</span>kg<span class="op">)</span>;</span>
<span id="cb53-25"><a href="#cb53-25" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> <span class="kw">auto</span> E3 <span class="op">=</span> total_energy<span class="op">(</span>p3, m3, c<span class="op">).</span>in<span class="op">(</span>J<span class="op">)</span>;</span>
<span id="cb53-26"><a href="#cb53-26" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb53-27"><a href="#cb53-27" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">in SI base units:</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-28"><a href="#cb53-28" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;p = &quot;</span> <span class="op">&lt;&lt;</span> p3 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-29"><a href="#cb53-29" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;m = &quot;</span> <span class="op">&lt;&lt;</span> m3 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span></span>
<span id="cb53-30"><a href="#cb53-30" aria-hidden="true" tabindex="-1"></a>          <span class="op">&lt;&lt;</span> <span class="st">&quot;E = &quot;</span> <span class="op">&lt;&lt;</span> E3 <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span></code></pre></div>
<p>The above prints the following:</p>
<pre class="text"><code>in `GeV` and `c`:
p = 4 GeV/c
m = 3 GeV/c²
E = 5 GeV

in `GeV`:
p = 1.33426e-08 GeV s/m
m = 3.33795e-17 GeV s²/m²
E = 5 GeV

in SI base units:
p = 2.13771e-18 kg m/s
m = 5.34799e-27 kg
E = 8.01088e-10 J</code></pre>
<h2 data-number="6.4" id="equivalence"><span class="header-section-number">6.4</span> Equivalence<a href="#equivalence" class="self-link"></a></h2>
<p>Units, dimensions, and quantity types can be checked for equivalence
with <code class="sourceCode default">operator==</code>. However, what
it means to be an equivalent entity means something different for each
case here.</p>
<h3 data-number="6.4.1" id="dimensions"><span class="header-section-number">6.4.1</span> Dimensions<a href="#dimensions" class="self-link"></a></h3>
<p>Equivalence is the simplest to reason about in the case of
dimensions. The only thing to account for here is the point when a user
would like to derive its own strong type from a library-provided
one.</p>
<p>Please note that the library never provides strong types for derived
dimensions besides the
<code class="sourceCode default">dimension_one</code>. For example, ISQ
defines length (<code class="sourceCode default">L</code>) and time
(<code class="sourceCode default">T</code>) dimensions, but there is no
such thing as a speed dimension. It does not have its own symbol as
well. There is only a derived dimension of speed described with
<code class="sourceCode default">LT⁻¹</code>. This is the reason why a
user should also not derive strong types from the derived
dimensions.</p>
<p>The equality operator for dimensions can be implemented as:</p>
<div class="sourceCode" id="cb54"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>Dimension Lhs, Dimension Rhs<span class="op">&gt;</span></span>
<span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a><span class="op">[[</span><span class="at">nodiscard</span><span class="op">]]</span> <span class="kw">consteval</span> <span class="dt">bool</span> <span class="kw">operator</span><span class="op">==(</span>Lhs lhs, Rhs rhs<span class="op">)</span></span>
<span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb54-4"><a href="#cb54-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> is_same_v<span class="op">&lt;</span>Lhs, Rhs<span class="op">&gt;</span> <span class="op">||</span> detail<span class="op">::</span>derived_from_the_same_base_dimension<span class="op">(</span>lhs, rhs<span class="op">)</span>;</span>
<span id="cb54-5"><a href="#cb54-5" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<h3 data-number="6.4.2" id="quantity-types"><span class="header-section-number">6.4.2</span> Quantity types<a href="#quantity-types" class="self-link"></a></h3>
<p>Equality for quantity types is similar to dimensions. Again, users
are allowed to derive their own types, but only from the named strong
types provided by the library.</p>
<h3 data-number="6.4.3" id="units"><span class="header-section-number">6.4.3</span> Units<a href="#units" class="self-link"></a></h3>
<p>Equality for units is a bit more complicated. Not only watt
(<code class="sourceCode default">W</code>) should be equivalent to
<code class="sourceCode default">J/s</code> or
<code class="sourceCode default">kg m²/s³</code> but also litre
(<code class="sourceCode default">l</code>) should be equivalent to
cubic decimetre (<code class="sourceCode default">dm³</code>). This is
why in this case we do not compare user-provided types but first convert
each unit to its canonical representation and then we compare if the
reference unit and the magnitude is the same:</p>
<div class="sourceCode" id="cb55"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb55-1"><a href="#cb55-1" aria-hidden="true" tabindex="-1"></a><span class="op">[[</span><span class="at">nodiscard</span><span class="op">]]</span> <span class="kw">consteval</span> <span class="dt">bool</span> <span class="kw">operator</span><span class="op">==(</span>Unit <span class="kw">auto</span> lhs, Unit <span class="kw">auto</span> rhs<span class="op">)</span></span>
<span id="cb55-2"><a href="#cb55-2" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb55-3"><a href="#cb55-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">auto</span> canonical_lhs <span class="op">=</span> detail<span class="op">::</span>get_canonical_unit<span class="op">(</span>lhs<span class="op">)</span>;</span>
<span id="cb55-4"><a href="#cb55-4" aria-hidden="true" tabindex="-1"></a>  <span class="kw">auto</span> canonical_rhs <span class="op">=</span> detail<span class="op">::</span>get_canonical_unit<span class="op">(</span>rhs<span class="op">)</span>;</span>
<span id="cb55-5"><a href="#cb55-5" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> detail<span class="op">::</span>have_same_canonical_reference_unit<span class="op">(</span>canonical_lhs<span class="op">.</span>reference_unit, canonical_rhs<span class="op">.</span>reference_unit<span class="op">)</span> <span class="op">&amp;&amp;</span></span>
<span id="cb55-6"><a href="#cb55-6" aria-hidden="true" tabindex="-1"></a>         canonical_lhs<span class="op">.</span>mag <span class="op">==</span> canonical_rhs<span class="op">.</span>mag;</span>
<span id="cb55-7"><a href="#cb55-7" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<h2 data-number="6.5" id="ordering"><span class="header-section-number">6.5</span> Ordering<a href="#ordering" class="self-link"></a></h2>
<p>Please note that the ordering for dimensions and quantity types has
no physical sense.</p>
<p>We could entertain adding ordering for units, but this would work
only for quantities having the same reference unit, which would be
inconsistent with how equivalence works.</p>
<p>Let’s see the following example:</p>
<div class="sourceCode" id="cb56"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> Unit <span class="kw">auto</span> my_unit <span class="op">=</span> si<span class="op">::</span>second;</span>
<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="kw">constexpr</span> <span class="op">(</span>my_unit <span class="op">==</span> si<span class="op">::</span>metre<span class="op">)</span> <span class="op">{</span></span>
<span id="cb56-3"><a href="#cb56-3" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
<span id="cb56-4"><a href="#cb56-4" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span>
<span id="cb56-5"><a href="#cb56-5" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="kw">constexpr</span> <span class="op">(</span>my_unit <span class="op">&gt;</span> si<span class="op">::</span>metre<span class="op">)</span> <span class="op">{</span></span>
<span id="cb56-6"><a href="#cb56-6" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
<span id="cb56-7"><a href="#cb56-7" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span>
<span id="cb56-8"><a href="#cb56-8" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="kw">constexpr</span> <span class="op">(</span>my_unit <span class="op">&gt;</span> si<span class="op">::</span>nano<span class="op">(</span>si<span class="op">::</span>second<span class="op">))</span> <span class="op">{</span></span>
<span id="cb56-9"><a href="#cb56-9" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
<span id="cb56-10"><a href="#cb56-10" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<p>In the above code, the first check could be useful for some use
cases. However, the second one is impossible to implement and should not
compile. The third one could be considered useful, but the current
version of <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> does not
expose such an interface to limit potential confusion. Also, it is
really hard to mathematically prove that unit magnitude representation
that we us in the library is greater or smaller than the other one in
some cases.</p>
<h1 data-number="7" id="quantities"><span class="header-section-number">7</span> Quantities<a href="#quantities" class="self-link"></a></h1>
<p>The <code class="sourceCode default">quantity</code> class template
is the workhorse of the library. It can be considered a generalization
of <code class="sourceCode default">std::chrono::duration</code>, but is
not directly compatible with it.</p>
<h2 data-number="7.1" id="quantity-references"><span class="header-section-number">7.1</span> Quantity references<a href="#quantity-references" class="self-link"></a></h2>
<p><em>Note: We know that probably the term “reference” will not survive
too long in the Committee, but we couldn’t find a better name for it in
the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library
(<a href="https://github.com/mpusz/mp-units/issues/486" class="uri">https://github.com/mpusz/mp-units/issues/486</a>).</em></p>
<p><span class="citation" data-cites="ISO-GUIDE">[<a href="#ref-ISO-GUIDE" role="doc-biblioref">ISO/IEC Guide 99</a>]</span>
says:</p>
<blockquote>
<p><strong>quantity</strong> - property of a phenomenon, body, or
substance, where the property has a magnitude that can be expressed as a
number and a reference. … A reference can be a measurement unit, a
measurement procedure, a reference material, or a combination of
such.</p>
</blockquote>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library a
quantity reference represents all the domain-specific meta-data about
the quantity besides its representation type and its value. A
<code class="sourceCode default">Reference</code> concept is satisfied
by either of:</p>
<ul>
<li>an associated unit
(e.g. <code class="sourceCode default">si::metre</code>),</li>
<li>an instantiation of the <code class="sourceCode default">reference&lt;QuantitySpec, Unit&gt;</code>
class template explicitly specifying the quantity type and its
unit.</li>
</ul>
<p>A reference type is implicitly created as a result of the following
expression:</p>
<div class="sourceCode" id="cb57"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb57-1"><a href="#cb57-1" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> <span class="kw">namespace</span> mp_units<span class="op">::</span>si<span class="op">::</span>unit_symbols;</span>
<span id="cb57-2"><a href="#cb57-2" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> ref <span class="op">=</span> isq<span class="op">::</span>height<span class="op">[</span>m<span class="op">]</span>;</span></code></pre></div>
<p>The above example defines a variable of type <code class="sourceCode default">reference&lt;isq::height, si::metre&gt;</code>.</p>
<p>The <code class="sourceCode default">reference</code> class template
also exposes an arithmetic interface similar to the one that we have
already discussed in case of units and quantity types. It simply
forwards the operation to its quantity type and unit members. For
example:</p>
<div class="sourceCode" id="cb58"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>QuantitySpec <span class="kw">auto</span> Q, Unit <span class="kw">auto</span> U<span class="op">&gt;</span></span>
<span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> reference <span class="op">{</span></span>
<span id="cb58-3"><a href="#cb58-3" aria-hidden="true" tabindex="-1"></a>  <span class="kw">template</span><span class="op">&lt;</span><span class="kw">auto</span> Q2, <span class="kw">auto</span> U2<span class="op">&gt;</span></span>
<span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a>  <span class="op">[[</span><span class="at">nodiscard</span><span class="op">]]</span> <span class="kw">friend</span> <span class="kw">consteval</span> <span class="dt">bool</span> <span class="kw">operator</span><span class="op">==(</span>reference, reference<span class="op">&lt;</span>Q2, U2<span class="op">&gt;)</span></span>
<span id="cb58-5"><a href="#cb58-5" aria-hidden="true" tabindex="-1"></a>  <span class="op">{</span></span>
<span id="cb58-6"><a href="#cb58-6" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> Q <span class="op">==</span> Q2 <span class="op">&amp;&amp;</span> U <span class="op">==</span> U2;</span>
<span id="cb58-7"><a href="#cb58-7" aria-hidden="true" tabindex="-1"></a>  <span class="op">}</span></span>
<span id="cb58-8"><a href="#cb58-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb58-9"><a href="#cb58-9" aria-hidden="true" tabindex="-1"></a>  <span class="kw">template</span><span class="op">&lt;</span>AssociatedUnit U2<span class="op">&gt;</span></span>
<span id="cb58-10"><a href="#cb58-10" aria-hidden="true" tabindex="-1"></a>  <span class="op">[[</span><span class="at">nodiscard</span><span class="op">]]</span> <span class="kw">friend</span> <span class="kw">consteval</span> <span class="dt">bool</span> <span class="kw">operator</span><span class="op">==(</span>reference, U2 u2<span class="op">)</span></span>
<span id="cb58-11"><a href="#cb58-11" aria-hidden="true" tabindex="-1"></a>  <span class="op">{</span></span>
<span id="cb58-12"><a href="#cb58-12" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> Q <span class="op">==</span> get_quantity_spec<span class="op">(</span>u2<span class="op">)</span> <span class="op">&amp;&amp;</span> U <span class="op">==</span> u2;</span>
<span id="cb58-13"><a href="#cb58-13" aria-hidden="true" tabindex="-1"></a>  <span class="op">}</span></span>
<span id="cb58-14"><a href="#cb58-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb58-15"><a href="#cb58-15" aria-hidden="true" tabindex="-1"></a>  <span class="kw">template</span><span class="op">&lt;</span><span class="kw">auto</span> Q2, <span class="kw">auto</span> U2<span class="op">&gt;</span></span>
<span id="cb58-16"><a href="#cb58-16" aria-hidden="true" tabindex="-1"></a>  <span class="op">[[</span><span class="at">nodiscard</span><span class="op">]]</span> <span class="kw">friend</span> <span class="kw">consteval</span> reference<span class="op">&lt;</span>Q <span class="op">*</span> Q2, U <span class="op">*</span> U2<span class="op">&gt;</span> <span class="kw">operator</span><span class="op">*(</span>reference, reference<span class="op">&lt;</span>Q2, U2<span class="op">&gt;)</span></span>
<span id="cb58-17"><a href="#cb58-17" aria-hidden="true" tabindex="-1"></a>  <span class="op">{</span></span>
<span id="cb58-18"><a href="#cb58-18" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> <span class="op">{}</span>;</span>
<span id="cb58-19"><a href="#cb58-19" aria-hidden="true" tabindex="-1"></a>  <span class="op">}</span></span>
<span id="cb58-20"><a href="#cb58-20" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb58-21"><a href="#cb58-21" aria-hidden="true" tabindex="-1"></a>  <span class="kw">template</span><span class="op">&lt;</span>AssociatedUnit U2<span class="op">&gt;</span></span>
<span id="cb58-22"><a href="#cb58-22" aria-hidden="true" tabindex="-1"></a>  <span class="op">[[</span><span class="at">nodiscard</span><span class="op">]]</span> <span class="kw">friend</span> <span class="kw">consteval</span> reference<span class="op">&lt;</span>Q <span class="op">*</span> get_quantity_spec<span class="op">(</span>U2<span class="op">{})</span>, U <span class="op">*</span> U2<span class="op">{}&gt;</span> <span class="kw">operator</span><span class="op">*(</span>reference, U2<span class="op">)</span></span>
<span id="cb58-23"><a href="#cb58-23" aria-hidden="true" tabindex="-1"></a>  <span class="op">{</span></span>
<span id="cb58-24"><a href="#cb58-24" aria-hidden="true" tabindex="-1"></a>    <span class="cf">return</span> <span class="op">{}</span>;</span>
<span id="cb58-25"><a href="#cb58-25" aria-hidden="true" tabindex="-1"></a>  <span class="op">}</span></span>
<span id="cb58-26"><a href="#cb58-26" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb58-27"><a href="#cb58-27" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
<span id="cb58-28"><a href="#cb58-28" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>;</span></code></pre></div>
<p>Please note that all of the operators work on two
<code class="sourceCode default">reference</code>s, or one
<code class="sourceCode default">reference</code> and a
<code class="sourceCode default">AssociatedUnit</code>.</p>
<h2 data-number="7.2" id="quantity-class-template"><span class="header-section-number">7.2</span>
<code class="sourceCode default">quantity</code> class template<a href="#quantity-class-template" class="self-link"></a></h2>
<p>Based on the ISO definition above, the
<code class="sourceCode default">quantity</code> class template has the
following signature:</p>
<div class="sourceCode" id="cb59"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb59-1"><a href="#cb59-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>Reference <span class="kw">auto</span> R, RepresentationOf<span class="op">&lt;</span>get_quantity_spec<span class="op">(</span>R<span class="op">).</span>character<span class="op">&gt;</span> Rep <span class="op">=</span> <span class="dt">double</span><span class="op">&gt;</span></span>
<span id="cb59-2"><a href="#cb59-2" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> quantity;</span></code></pre></div>
<p>It has only one data member of
<code class="sourceCode default">Rep</code> type. Unfortunately, this
data member is publicly exposed to satisfy the C++ language requirements
for structural types. Hopefully, the language rules for structural types
will improve with time before this library gets standardized.</p>
<h2 data-number="7.3" id="quantity-construction-helpers"><span class="header-section-number">7.3</span> Quantity construction helpers<a href="#quantity-construction-helpers" class="self-link"></a></h2>
<p>As we already noticed in many examples above, a numerical value
multiplied or divided by a
<code class="sourceCode default">Reference</code> creates the value of
<code class="sourceCode default">quantity</code> class template with the
representation type and reference deduced from the types used in the
expression.</p>
<p>We have a few options to choose from:</p>
<ul>
<li>simple quantities (quantities of a quantity kind)</li>
</ul>
<div class="sourceCode" id="cb60"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb60-1"><a href="#cb60-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>metre <span class="op">/</span> si<span class="op">::</span>second, <span class="dt">int</span><span class="op">&gt;</span> q1 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m <span class="op">/</span> s;</span>
<span id="cb60-2"><a href="#cb60-2" aria-hidden="true" tabindex="-1"></a>quantity q2 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m <span class="op">/</span> s;</span>
<span id="cb60-3"><a href="#cb60-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb60-4"><a href="#cb60-4" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>std<span class="op">::</span>is_same_v<span class="op">&lt;</span><span class="kw">decltype</span><span class="op">(</span>q1<span class="op">)</span>, <span class="kw">decltype</span><span class="op">(</span>q2<span class="op">)&gt;)</span>;</span>
<span id="cb60-5"><a href="#cb60-5" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q1<span class="op">.</span>quantity_spec <span class="op">==</span> kind_of<span class="op">&lt;</span>isq<span class="op">::</span>length <span class="op">/</span> isq<span class="op">::</span>time<span class="op">&gt;)</span>;</span></code></pre></div>
<ul>
<li>typed quantities (quantities of a specific type within a kind)</li>
</ul>
<div class="sourceCode" id="cb61"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb61-1"><a href="#cb61-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>si<span class="op">::</span>metre <span class="op">/</span> si<span class="op">::</span>second<span class="op">]</span>, <span class="dt">int</span><span class="op">&gt;</span> q3 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m <span class="op">/</span> s;</span>
<span id="cb61-2"><a href="#cb61-2" aria-hidden="true" tabindex="-1"></a>quantity q4 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> isq<span class="op">::</span>speed<span class="op">[</span>m <span class="op">/</span> s<span class="op">]</span>;</span>
<span id="cb61-3"><a href="#cb61-3" aria-hidden="true" tabindex="-1"></a>quantity q5 <span class="op">=</span> isq<span class="op">::</span>speed<span class="op">(</span><span class="dv">42</span> <span class="op">*</span> m <span class="op">/</span> s<span class="op">)</span>;</span>
<span id="cb61-4"><a href="#cb61-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb61-5"><a href="#cb61-5" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>std<span class="op">::</span>is_same_v<span class="op">&lt;</span><span class="kw">decltype</span><span class="op">(</span>q3<span class="op">)</span>, <span class="kw">decltype</span><span class="op">(</span>q4<span class="op">)&gt;)</span>;</span>
<span id="cb61-6"><a href="#cb61-6" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>std<span class="op">::</span>is_same_v<span class="op">&lt;</span><span class="kw">decltype</span><span class="op">(</span>q3<span class="op">)</span>, <span class="kw">decltype</span><span class="op">(</span>q5<span class="op">)&gt;)</span>;</span>
<span id="cb61-7"><a href="#cb61-7" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q3<span class="op">.</span>quantity_spec <span class="op">==</span> isq<span class="op">::</span>speed<span class="op">)</span>;</span></code></pre></div>
<p>In case someone doesn’t like the multiply syntax or there is an
ambiguity between the <code class="sourceCode default">operator*</code>
provided by this and other libraries, a quantity can also be created
with a dedicated factory function:</p>
<div class="sourceCode" id="cb62"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb62-1"><a href="#cb62-1" aria-hidden="true" tabindex="-1"></a>quantity q <span class="op">=</span> make_quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>m <span class="op">/</span> s<span class="op">]&gt;(</span><span class="dv">42</span><span class="op">)</span>;</span></code></pre></div>
<h2 data-number="7.4" id="quantity-constructors-and-value-conversions"><span class="header-section-number">7.4</span> Quantity constructors and value
conversions<a href="#quantity-constructors-and-value-conversions" class="self-link"></a></h2>
<p>The <code class="sourceCode default">quantity</code> class template
has a converting constructor that participates in the overload
resolution only when:</p>
<ul>
<li>The source quantity specification is implicitly convertible to the
destination one,</li>
<li>both units share the same reference unit, and</li>
<li>the resulting value conversion will be value-preserving.</li>
</ul>
<p>Additionally, this constructor becomes explicit if the source
representation type is not convertible to the destination one.</p>
<h3 data-number="7.4.1" id="value-preserving-conversions"><span class="header-section-number">7.4.1</span> Value-preserving
conversions<a href="#value-preserving-conversions" class="self-link"></a></h3>
<p>As of today, the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library
follows <a href="https://en.cppreference.com/w/cpp/chrono/duration/duration">the
rules of <code class="sourceCode default">std::chrono::duration</code>
for value-preserving conversions</a>. We realize that some time has
passed now and maybe we can improve in this domain. However, to our
knowledge, as of today, we do not have any tools we could use in the C++
Standard to improve that.</p>
<p>Below we describe the current approach.</p>
<div class="sourceCode" id="cb63"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb63-1"><a href="#cb63-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="dv">5</span> <span class="op">*</span> km;</span>
<span id="cb63-2"><a href="#cb63-2" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1<span class="op">.</span>in<span class="op">(</span>m<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="ch">&#39;</span><span class="sc">\n</span><span class="ch">&#39;</span>;</span>
<span id="cb63-3"><a href="#cb63-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>metre, <span class="dt">int</span><span class="op">&gt;</span> q2 <span class="op">=</span> q1;</span></code></pre></div>
<p>The second line above converts the input quantity to be expressed in
metres and prints its value. The third line converts the quantity
expressed in kilometers into one measured in metres.</p>
<p>In both cases, it is assumed that we can convert a quantity into
another one with a unit of a higher resolution. There is no protection
against overflow of the representation type built into the library.</p>
<p>If we try similar, but this time opposite, operations to the above,
both conversions should fail to compile:</p>
<div class="sourceCode" id="cb64"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb64-1"><a href="#cb64-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="dv">5</span> <span class="op">*</span> m;</span>
<span id="cb64-2"><a href="#cb64-2" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1<span class="op">.</span>in<span class="op">(</span>km<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="ch">&#39;</span><span class="sc">\n</span><span class="ch">&#39;</span>;              <span class="co">// Compile-time error</span></span>
<span id="cb64-3"><a href="#cb64-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>kilo<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span>, <span class="dt">int</span><span class="op">&gt;</span> q2 <span class="op">=</span> q1;  <span class="co">// Compile-time error</span></span></code></pre></div>
<p>We can’t preserve the value of a source quantity when we convert it
to one with a unit of a lower resolution while dealing with an integral
representation type for a quantity. In the example above, converting
<code class="sourceCode default">5</code> meters would result in
<code class="sourceCode default">0</code> kilometers if internal
conversion is performed using regular integer arithmetic.</p>
<p>While this could be a valid behavior, the problem arises when the
user expects to be able to convert the quantity back to the original
unit without loss of information. So the library should prevent such
conversions from happening implicitly; <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> offers the named cast
<code class="sourceCode default">value_cast</code> for these conversions
marked as unsafe.</p>
<p>To make the above conversions compile, we could use a floating-point
representation type:</p>
<div class="sourceCode" id="cb65"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb65-1"><a href="#cb65-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="fl">5.</span> <span class="op">*</span> m;    <span class="co">// source quantity uses `double` as a representation type</span></span>
<span id="cb65-2"><a href="#cb65-2" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1<span class="op">.</span>in<span class="op">(</span>km<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="ch">&#39;</span><span class="sc">\n</span><span class="ch">&#39;</span>;</span>
<span id="cb65-3"><a href="#cb65-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>kilo<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;&gt;</span> q2 <span class="op">=</span> q1;</span></code></pre></div>
<p>or:</p>
<div class="sourceCode" id="cb66"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb66-1"><a href="#cb66-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="dv">5</span> <span class="op">*</span> m;     <span class="co">// source quantity uses `int` as a representation type</span></span>
<span id="cb66-2"><a href="#cb66-2" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> value_cast<span class="op">&lt;</span><span class="dt">double</span><span class="op">&gt;(</span>q1<span class="op">).</span>in<span class="op">(</span>km<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="ch">&#39;</span><span class="sc">\n</span><span class="ch">&#39;</span>;</span>
<span id="cb66-3"><a href="#cb66-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>kilo<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;&gt;</span> q2 <span class="op">=</span> q1;  <span class="co">// double by default</span></span></code></pre></div>
<p><em>The <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library
follows <code class="sourceCode default">std::chrono::duration</code>
logic and treats floating-point types as value-preserving.</em></p>
<h3 data-number="7.4.2" id="value-truncating-conversions"><span class="header-section-number">7.4.2</span> Value-truncating
conversions<a href="#value-truncating-conversions" class="self-link"></a></h3>
<p>Another possibility would be to force such a truncating conversion
explicitly from the code:</p>
<div class="sourceCode" id="cb67"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb67-1"><a href="#cb67-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="dv">5</span> <span class="op">*</span> m;     <span class="co">// source quantity uses `int` as a representation type</span></span>
<span id="cb67-2"><a href="#cb67-2" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> q1<span class="op">.</span>force_in<span class="op">(</span>km<span class="op">)</span> <span class="op">&lt;&lt;</span> <span class="ch">&#39;</span><span class="sc">\n</span><span class="ch">&#39;</span>;</span>
<span id="cb67-3"><a href="#cb67-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>kilo<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span>, <span class="dt">int</span><span class="op">&gt;</span> q2 <span class="op">=</span> value_cast<span class="op">&lt;</span>km<span class="op">&gt;(</span>q1<span class="op">)</span>;</span></code></pre></div>
<p>The code above makes it clear that “something bad” may happen here if
we are not extra careful.</p>
<p>Another case for truncation happens when we assign a quantity with a
floating-point representation type to the one using an integral
representation type for its value:</p>
<div class="sourceCode" id="cb68"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb68-1"><a href="#cb68-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="fl">2.5</span> <span class="op">*</span> m;</span>
<span id="cb68-2"><a href="#cb68-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>metre, <span class="dt">int</span><span class="op">&gt;</span> q2 <span class="op">=</span> q1;</span></code></pre></div>
<p>Such an operation should fail to compile as well. Again, to force
such a truncation, we have to be explicit in the code:</p>
<div class="sourceCode" id="cb69"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb69-1"><a href="#cb69-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q1 <span class="op">=</span> <span class="fl">2.5</span> <span class="op">*</span> m;</span>
<span id="cb69-2"><a href="#cb69-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>si<span class="op">::</span>metre, <span class="dt">int</span><span class="op">&gt;</span> q2 <span class="op">=</span> value_cast<span class="op">&lt;</span><span class="dt">int</span><span class="op">&gt;(</span>q1<span class="op">)</span>;</span></code></pre></div>
<p>As we can see, it is essential not to allow such truncating
conversions to happen implicitly and a good physical quantities and
units library should fail at compile-time in case an user makes such a
mistake.</p>
<h2 data-number="7.5" id="character-of-a-quantity"><span class="header-section-number">7.5</span> Character of a quantity<a href="#character-of-a-quantity" class="self-link"></a></h2>
<p>This chapter is just a short preview to the feature that will get its
own paper in the future if the Committee is interested in exploring this
path.</p>
<h3 data-number="7.5.1" id="scalars-vectors-and-tensors"><span class="header-section-number">7.5.1</span> Scalars, vectors, and
tensors<a href="#scalars-vectors-and-tensors" class="self-link"></a></h3>
<p><span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> explicitly states:</p>
<blockquote>
<p>Scalars, vectors and tensors are mathematical objects that can be
used to denote certain physical quantities and their values. They are as
such independent of the particular choice of a coordinate system,
whereas each scalar component of a vector or a tensor and each component
vector and component tensor depend on that choice.</p>
</blockquote>
<p>Such distinction is important because each quantity character
represents different properties and allows different operations to be
done on its quantities.</p>
<p>For example, imagine a physical units library that allows the
creation of a <code class="sourceCode default">speed</code> quantity
from both <code class="sourceCode default">length / time</code> and
<code class="sourceCode default">length * time</code>. It wouldn’t be
too safe to use such a product, right?</p>
<p>Now we have to realize that both of the above operations
(multiplication and division) are not even mathematically defined for
linear algebra types such as vectors or tensors. On the other hand, two
vectors can be passed as arguments to dot and cross-product operations.
The result of the first one is a scalar. The second one results in a
vector that is perpendicular to both input vectors. Again, it wouldn’t
be safe to allow replacing those two operations with each other or
expect the same results from both cases. This simply can’t work.</p>
<h3 data-number="7.5.2" id="isq-defines-quantities-of-all-characters"><span class="header-section-number">7.5.2</span> ISQ defines quantities of all
characters<a href="#isq-defines-quantities-of-all-characters" class="self-link"></a></h3>
<p>While defining quantities, <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> explicitly specifies when
a quantity has a vector or tensor character. Here are some examples:</p>
<table>
<colgroup>
<col style="width: 26%" />
<col style="width: 15%" />
<col style="width: 57%" />
</colgroup>
<thead>
<tr class="header">
<th><div style="text-align:center">
<strong>Quantity</strong>
</div></th>
<th style="text-align: center;"><div style="text-align:center">
<strong>Character</strong>
</div></th>
<th style="text-align: center;"><div style="text-align:center">
<strong>Quantity Equation</strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><code class="sourceCode default">duration</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><em>{base quantity}</em></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">mass</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><em>{base quantity}</em></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">length</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><em>{base quantity}</em></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">path_length</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><em>{base quantity}</em></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">radius</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><em>{base quantity}</em></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">position_vector</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><em>{base quantity}</em></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">velocity</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><code class="sourceCode default">position_vector / duration</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">acceleration</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><code class="sourceCode default">velocity / duration</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">force</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><code class="sourceCode default">mass * acceleration</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">power</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><code class="sourceCode default">force ⋅ velocity</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">moment_of_force</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><code class="sourceCode default">position_vector × force</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">torque</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><code class="sourceCode default">moment_of_force ⋅ {unit-vector}</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">surface_tension</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><code class="sourceCode default">|force| / length</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">angular_displacement</code></td>
<td style="text-align: center;">scalar</td>
<td style="text-align: center;"><code class="sourceCode default">path_length / radius</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">angular_velocity</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><code class="sourceCode default">angular_displacement / duration * {unit-vector}</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">momentum</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><code class="sourceCode default">mass * velocity</code></td>
</tr>
<tr class="odd">
<td><code class="sourceCode default">angular_momentum</code></td>
<td style="text-align: center;"><strong>vector</strong></td>
<td style="text-align: center;"><code class="sourceCode default">position_vector × momentum</code></td>
</tr>
<tr class="even">
<td><code class="sourceCode default">moment_of_inertia</code></td>
<td style="text-align: center;"><strong><em>tensor</em></strong></td>
<td style="text-align: center;"><code class="sourceCode default">angular_momentum ⊗ angular_velocity</code></td>
</tr>
</tbody>
</table>
<p>In the above equations:</p>
<ul>
<li><code class="sourceCode default">a * b</code> - regular
multiplication where one of the arguments has to be scalar</li>
<li><code class="sourceCode default">a / b</code> - regular division
where the divisor has to be scalar</li>
<li><code class="sourceCode default">a ⋅ b</code> - dot product of two
vectors</li>
<li><code class="sourceCode default">a × b</code> - cross product of two
vectors</li>
<li><code class="sourceCode default">|a|</code> - magnitude of a
vector</li>
<li><code class="sourceCode default">{unit-vector}</code> - a special
vector with the magnitude of
<code class="sourceCode default">1</code></li>
<li><code class="sourceCode default">a ⊗ b</code> - tensor product of
two vectors or tensors</li>
</ul>
<p>As of now, all of the C++ physical units libraries on the market
besides <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> do not
support the operations mentioned above. They expose only multiplication
and division operators, which do not work for linear algebra-based
representation types. If a user of those libraries would like to create
the quantities provided in the above table properly, this would result
in a compile-time error stating that multiplication and division of two
linear algebra vectors is impossible.</p>
<p>Outside of C++, only <span class="citation" data-cites="PINT">[<a href="#ref-PINT" role="doc-biblioref">Pint</a>]</span> provides great
support in this domain.</p>
<h3 data-number="7.5.3" id="characters-dont-apply-to-dimensions-and-units"><span class="header-section-number">7.5.3</span> Characters don’t apply to
dimensions and units<a href="#characters-dont-apply-to-dimensions-and-units" class="self-link"></a></h3>
<p><span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> explicitly states that
dimensions are orthogonal to quantity characters:</p>
<blockquote>
<p>In deriving the dimension of a quantity, no account is taken of its
scalar, vector, or tensor character.</p>
</blockquote>
<p>Also, it explicitly states that:</p>
<blockquote>
<p>All units are scalars.</p>
</blockquote>
<h3 data-number="7.5.4" id="defining-vector-and-tensor-quantities"><span class="header-section-number">7.5.4</span> Defining vector and tensor
quantities<a href="#defining-vector-and-tensor-quantities" class="self-link"></a></h3>
<p>To specify a quantity as having a vector or tensor character, a value
of the <code class="sourceCode default">quantity_character</code>
enumeration can be appended to the
<code class="sourceCode default">quantity_spec</code> that describes
it:</p>
<div class="sourceCode" id="cb70"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb70-1"><a href="#cb70-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> position_vector <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length, quantity_character<span class="op">::</span>vector<span class="op">&gt;</span> <span class="op">{}</span> position_vector;</span>
<span id="cb70-2"><a href="#cb70-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> displacement <span class="op">:</span> quantity_spec<span class="op">&lt;</span>length, quantity_character<span class="op">::</span>vector<span class="op">&gt;</span> <span class="op">{}</span> displacement;</span></code></pre></div>
<p>With the above, all the quantities derived from
<code class="sourceCode default">position_vector</code> or
<code class="sourceCode default">displacement</code> will have a correct
character determined according to the kind of operations included in the
quantity equation defining a derived quantity.</p>
<p>For example, <code class="sourceCode default">velocity</code> in the
below definition will be inferred to be a vector quantity (no explicit
character override is needed):</p>
<div class="sourceCode" id="cb71"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb71-1"><a href="#cb71-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> velocity <span class="op">:</span> quantity_spec<span class="op">&lt;</span>speed, position_vector <span class="op">/</span> duration<span class="op">&gt;</span> <span class="op">{}</span> velocity;</span></code></pre></div>
<h3 data-number="7.5.5" id="representation-types-for-vector-and-tensor-quantities"><span class="header-section-number">7.5.5</span> Representation types for
vector and tensor quantities<a href="#representation-types-for-vector-and-tensor-quantities" class="self-link"></a></h3>
<p>As we specified before, the
<code class="sourceCode default">quantity</code> class template is
defined as follows:</p>
<div class="sourceCode" id="cb72"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb72-1"><a href="#cb72-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>Reference <span class="kw">auto</span> R,</span>
<span id="cb72-2"><a href="#cb72-2" aria-hidden="true" tabindex="-1"></a>         RepresentationOf<span class="op">&lt;</span>get_quantity_spec<span class="op">(</span>R<span class="op">).</span>character<span class="op">&gt;</span> Rep <span class="op">=</span> <span class="dt">double</span><span class="op">&gt;</span></span>
<span id="cb72-3"><a href="#cb72-3" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> quantity;</span></code></pre></div>
<p>The second template parameter is constrained with a
<code class="sourceCode default">RepresentationOf</code> concept that
checks if the provided representation type satisfies the requirements
for the character associated with this quantity type.</p>
<p>Unfortunately, the current version of the C++ Standard Library does
not provide any types that could be used as a representation type for
vector and tensor quantities. This is why users are on their own
here.</p>
<p>However, thanks to the provided customization points, any linear
algebra library type can be used as a vector or tensor quantity
representation type.</p>
<p>To enable the usage of a user-defined type as a representation type
for vector or tensor quantities, users need to provide a partial
specialization of the <code class="sourceCode default">is_vector</code>
or <code class="sourceCode default">is_tensor</code> customization
points.</p>
<p>For example, here is how it can be done for the types in <a href="https://wg21.link/p1385">P1385</a>:</p>
<div class="sourceCode" id="cb73"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb73-1"><a href="#cb73-1" aria-hidden="true" tabindex="-1"></a><span class="pp">#include </span><span class="im">&lt;matrix&gt;</span></span>
<span id="cb73-2"><a href="#cb73-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb73-3"><a href="#cb73-3" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> la_vector <span class="op">=</span> STD_LA<span class="op">::</span>fixed_size_column_vector<span class="op">&lt;</span><span class="dt">double</span>, <span class="dv">3</span><span class="op">&gt;</span>;</span>
<span id="cb73-4"><a href="#cb73-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb73-5"><a href="#cb73-5" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;&gt;</span></span>
<span id="cb73-6"><a href="#cb73-6" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="dt">bool</span> mp_units<span class="op">::</span>is_vector<span class="op">&lt;</span>la_vector<span class="op">&gt;</span> <span class="op">=</span> <span class="kw">true</span>;</span></code></pre></div>
<p>With the above, we can use
<code class="sourceCode default">la_vector</code> as a representation
type for our quantity:</p>
<div class="sourceCode" id="cb74"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb74-1"><a href="#cb74-1" aria-hidden="true" tabindex="-1"></a>Quantity <span class="kw">auto</span> q <span class="op">=</span> la_vector<span class="op">{</span><span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span><span class="op">}</span> <span class="op">*</span> isq<span class="op">::</span>velocity<span class="op">[</span>m <span class="op">/</span> s<span class="op">]</span>;</span></code></pre></div>
<p>Please, note that the following does not work (but see the next
section):</p>
<div class="sourceCode" id="cb75"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb75-1"><a href="#cb75-1" aria-hidden="true" tabindex="-1"></a>Quantity <span class="kw">auto</span> q1 <span class="op">=</span> la_vector<span class="op">{</span><span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span><span class="op">}</span> <span class="op">*</span> m <span class="op">/</span> s;</span>
<span id="cb75-2"><a href="#cb75-2" aria-hidden="true" tabindex="-1"></a>Quantity <span class="kw">auto</span> q2 <span class="op">=</span> isq<span class="op">::</span>velocity<span class="op">(</span>la_vector<span class="op">{</span><span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span><span class="op">}</span> <span class="op">*</span> m <span class="op">/</span> s<span class="op">)</span>;</span>
<span id="cb75-3"><a href="#cb75-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>velocity<span class="op">[</span>m<span class="op">/</span>s<span class="op">]&gt;</span> q3<span class="op">{</span>la_vector<span class="op">{</span><span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span><span class="op">}</span> <span class="op">*</span> m <span class="op">/</span> s<span class="op">}</span>;</span></code></pre></div>
<p>In all the cases above, the SI unit
<code class="sourceCode default">m / s</code> has an associated scalar
quantity of
<code class="sourceCode default">isq::length / isq::time</code>.
<code class="sourceCode default">la_vector</code> is not a correct
representation type for a scalar quantity so the construction fails.</p>
<h3 data-number="7.5.6" id="hacking-the-character"><span class="header-section-number">7.5.6</span> Hacking the character<a href="#hacking-the-character" class="self-link"></a></h3>
<p>Sometimes, we want to use a vector quantity, but we don’t care about
its direction. For example, the standard gravity acceleration constant
always points down, so we might not care about this in a particular
scenario. In such a case, we may want to “hack” the library to allow
scalar types to be used as a representation type for scalar
quantities.</p>
<p>For example, we can do the following:</p>
<div class="sourceCode" id="cb76"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb76-1"><a href="#cb76-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span><span class="kw">class</span> T<span class="op">&gt;</span></span>
<span id="cb76-2"><a href="#cb76-2" aria-hidden="true" tabindex="-1"></a>  <span class="kw">requires</span> mp_units<span class="op">::</span>is_scalar<span class="op">&lt;</span>T<span class="op">&gt;</span></span>
<span id="cb76-3"><a href="#cb76-3" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="dt">bool</span> mp_units<span class="op">::</span>is_vector<span class="op">&lt;</span>T<span class="op">&gt;</span> <span class="op">=</span> <span class="kw">true</span>;</span></code></pre></div>
<p>which says that every type that can be used as a scalar
representation is also allowed for vector quantities.</p>
<p>Doing the above is actually not such a big “hack” as the <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> explicitly allows it:</p>
<blockquote>
<p>A vector is a tensor of the first order and a scalar is a tensor of
order zero.</p>
</blockquote>
<p>Despite it being allowed by <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>, for type-safety reasons,
we do not allow such a behavior by default, and a user has to opt into
such scenarios explicitly.</p>
<h2 data-number="7.6" id="dimensionless-quantities"><span class="header-section-number">7.6</span> Dimensionless quantities<a href="#dimensionless-quantities" class="self-link"></a></h2>
<p>The quantities we discussed so far always had some specific type and
physical dimension. However, this is not always the case. While
performing various computations, we sometimes end up with so-called
“dimensionless” quantities, which <span class="citation" data-cites="ISO-GUIDE">[<a href="#ref-ISO-GUIDE" role="doc-biblioref">ISO/IEC Guide 99</a>]</span> correctly defines as
quantities of dimension one:</p>
<blockquote>
<ul>
<li>Quantity for which all the exponents of the factors corresponding to
the base quantities in its quantity dimension are zero.</li>
<li>The measurement units and values of quantities of dimension one are
numbers, but such quantities convey more information than a number.</li>
<li>Some quantities of dimension one are defined as the ratios of two
quantities of the same kind.</li>
<li>Numbers of entities are quantities of dimension one.</li>
</ul>
</blockquote>
<h3 data-number="7.6.1" id="dividing-two-quantities-of-the-same-kind"><span class="header-section-number">7.6.1</span> Dividing two quantities of
the same kind<a href="#dividing-two-quantities-of-the-same-kind" class="self-link"></a></h3>
<p>Dividing two quantities of the same kind always results in a quantity
of dimension one. However, depending on what type of quantities we
divide or what their units are, we may end up with slightly different
types.</p>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library,
dividing two quantities of the same dimension always results in a
quantity with the dimension being
<code class="sourceCode default">dimension_one</code>. This is often
different for other physical units libraries, which may return a raw
representation type for such cases. A raw value is also always returned
from the division of two
<code class="sourceCode default">std::chrono::duration</code>
values.</p>
<p>In the initial design of this library, the resulting type of division
of two quantities was their common representation type (just like
<code class="sourceCode default">std::chrono::duration</code>):</p>
<div class="sourceCode" id="cb77"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb77-1"><a href="#cb77-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>std<span class="op">::</span>is_same_v<span class="op">&lt;</span><span class="kw">decltype</span><span class="op">(</span><span class="dv">10</span> <span class="op">*</span> km <span class="op">/</span> <span class="op">(</span><span class="dv">5</span> <span class="op">*</span> km<span class="op">))</span>, <span class="dt">int</span><span class="op">&gt;)</span>;</span></code></pre></div>
<p>The reasoning behind it was not providing a false impression of a
strong <code class="sourceCode default">quantity</code> type for
something that looks and feels like a regular number. Also, all of the
mathematic and trigonometric functions were working fine out of the box
with such representation types, so we did not have to rewrite
<code class="sourceCode default">sin()</code>,
<code class="sourceCode default">cos()</code>,
<code class="sourceCode default">exp()</code>, and others.</p>
<p>However, the feedback we got from the production usage was that such
an approach is really bad for generic programming. It is hard to handle
the result of the two quantities’ division (or multiplication) as it
might be either a quantity or a fundamental type. If we want to raise
such a result to some power, we must use
<code class="sourceCode default">units::pow</code> or
<code class="sourceCode default">std::pow</code> depending on the
resulting type (<code class="sourceCode default">units::pow</code> takes
the power as template arguments). Those are only a few issues related to
such an approach.</p>
<p>Moreover, suppose we divide quantities of the same dimension, but
with units of significantly different magnitudes. In that case, we may
end up with a really small or a huge floating-point value, which may
result in losing lots of precision. Returning a dimensionless quantity
from such cases allows us to benefit from all the properties of scaled
units and is consistent with the rest of the library.</p>
<h4 data-number="7.6.1.1" id="dividing-quantities-of-the-same-type"><span class="header-section-number">7.6.1.1</span> Dividing quantities of the
same type<a href="#dividing-quantities-of-the-same-type" class="self-link"></a></h4>
<p>First, let’s analyze what happens if we divide two quantities of the
same type:</p>
<div class="sourceCode" id="cb78"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb78-1"><a href="#cb78-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> QuantityOf<span class="op">&lt;</span>dimensionless<span class="op">&gt;</span> <span class="kw">auto</span> q <span class="op">=</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">200</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">/</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">50</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<p>In such a case, we end up with a dimensionless quantity that has the
following properties:</p>
<div class="sourceCode" id="cb79"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb79-1"><a href="#cb79-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q<span class="op">.</span>quantity_spec <span class="op">==</span> dimensionless<span class="op">)</span>;</span>
<span id="cb79-2"><a href="#cb79-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q<span class="op">.</span>dimension <span class="op">==</span> dimension_one<span class="op">)</span>;</span>
<span id="cb79-3"><a href="#cb79-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q<span class="op">.</span>unit <span class="op">==</span> one<span class="op">)</span>;</span></code></pre></div>
<p>In case we would like to print its value, we would see a raw value of
<code class="sourceCode default">4</code> in the output with no unit
being printed.</p>
<h4 data-number="7.6.1.2" id="dividing-quantities-of-different-types"><span class="header-section-number">7.6.1.2</span> Dividing quantities of
different types<a href="#dividing-quantities-of-different-types" class="self-link"></a></h4>
<p>Now let’s see what happens if we divide quantities of the same
dimension and unit but which have different quantity types:</p>
<div class="sourceCode" id="cb80"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb80-1"><a href="#cb80-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> QuantityOf<span class="op">&lt;</span>dimensionless<span class="op">&gt;</span> <span class="kw">auto</span> q <span class="op">=</span> isq<span class="op">::</span>work<span class="op">(</span><span class="dv">200</span> <span class="op">*</span> J<span class="op">)</span> <span class="op">/</span> isq<span class="op">::</span>heat<span class="op">(</span><span class="dv">50</span> <span class="op">*</span> J<span class="op">)</span>;</span></code></pre></div>
<p>Again we end up with
<code class="sourceCode default">dimension_one</code> and
<code class="sourceCode default">one</code>, but this time:</p>
<div class="sourceCode" id="cb81"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb81-1"><a href="#cb81-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q<span class="op">.</span>quantity_spec <span class="op">==</span> isq<span class="op">::</span>work <span class="op">/</span> isq<span class="op">::</span>heat<span class="op">)</span>;</span></code></pre></div>
<p>As shown above, the result is not of a
<code class="sourceCode default">dimensionless</code> type anymore.
Instead, we get a quantity type derived from the performed quantity
equation. According to the <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>, work divided by heat is
the recipe for the thermodynamic efficiency quantity, thus:</p>
<div class="sourceCode" id="cb82"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb82-1"><a href="#cb82-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>implicitly_convertible<span class="op">(</span>q<span class="op">.</span>quantity_spec, isq<span class="op">::</span>efficiency_thermodynamics<span class="op">))</span>;</span></code></pre></div>
<p>Please note that the quantity of
<code class="sourceCode default">isq::efficiency_thermodynamics</code>
is of a kind <code class="sourceCode default">dimensionless</code>, so
it is implicitly convertible to
<code class="sourceCode default">dimensionless</code> and satisfies the
<code class="sourceCode default">QuantityOf&lt;dimensionless&gt;</code>
concept.</p>
<h4 data-number="7.6.1.3" id="dividing-quantities-of-different-units"><span class="header-section-number">7.6.1.3</span> Dividing quantities of
different units<a href="#dividing-quantities-of-different-units" class="self-link"></a></h4>
<p>Now, let’s see what happens when we divide two quantities of the same
type but different units:</p>
<div class="sourceCode" id="cb83"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb83-1"><a href="#cb83-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> QuantityOf<span class="op">&lt;</span>dimensionless<span class="op">&gt;</span> <span class="kw">auto</span> q <span class="op">=</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">4</span> <span class="op">*</span> km<span class="op">)</span> <span class="op">/</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<p>This time we still get a quantity of
<code class="sourceCode default">dimensionless</code> type with a
<code class="sourceCode default">dimension_one</code> as its dimension.
However, the resulting unit is not
<code class="sourceCode default">one</code> anymore:</p>
<div class="sourceCode" id="cb84"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb84-1"><a href="#cb84-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>q<span class="op">.</span>unit <span class="op">==</span> mag_power<span class="op">&lt;</span><span class="dv">10</span>, <span class="dv">3</span><span class="op">&gt;</span> <span class="op">*</span> one<span class="op">)</span>;</span></code></pre></div>
<p>In case we would print the text output of this quantity, we would not
see a raw value of <code class="sourceCode default">2000</code>, but
<code class="sourceCode default">2 km/m</code>.</p>
<p>First, it may look surprising, but this is actually consistent with
the division of quantities of different dimensions. For example, if we
divide <code class="sourceCode default">4 * km / (2 * s)</code>, we do
not expect <code class="sourceCode default">km</code> to be “expanded”
to <code class="sourceCode default">m</code> before the division, right?
We would expect the result of
<code class="sourceCode default">2 * (km / s)</code>, which is exactly
what we get when we divide quantities of the same kind.</p>
<p>This is a compelling feature that allows us to express huge or tiny
ratios without the need for big and expensive representation types. With
this, we can easily define things like a <a href="https://en.wikipedia.org/wiki/Hubble%27s_law#Dimensionless_Hubble_constant">Hubble’s
constant</a> that uses a unit that is proportional to the ratio of
kilometers per megaparsecs, which are both units of length:</p>
<div class="sourceCode" id="cb85"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb85-1"><a href="#cb85-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> hubble_constant <span class="op">:</span></span>
<span id="cb85-2"><a href="#cb85-2" aria-hidden="true" tabindex="-1"></a>    named_unit<span class="op">&lt;</span>basic_symbol_text<span class="op">{</span><span class="st">&quot;H₀&quot;</span>, <span class="st">&quot;H_0&quot;</span><span class="op">}</span>, mag<span class="op">&lt;</span>ratio<span class="op">{</span><span class="dv">701</span>, <span class="dv">10</span><span class="op">}&gt;</span> <span class="op">*</span> si<span class="op">::</span>kilo<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span> <span class="op">/</span> si<span class="op">::</span>second <span class="op">/</span> si<span class="op">::</span>mega<span class="op">&lt;</span>parsec<span class="op">&gt;&gt;</span> <span class="op">{</span></span>
<span id="cb85-3"><a href="#cb85-3" aria-hidden="true" tabindex="-1"></a><span class="op">}</span> hubble_constant;</span></code></pre></div>
<h3 data-number="7.6.2" id="counts-of-things"><span class="header-section-number">7.6.2</span> Counts of things<a href="#counts-of-things" class="self-link"></a></h3>
<p>Another important use case for dimensionless quantities is to provide
strong types for counts of things. For example:</p>
<ul>
<li><span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>
(part 3) provides a <code class="sourceCode default">rotation</code>
quantity defined as the number of revolutions,</li>
<li><span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>
(part 6) provides a
<code class="sourceCode default">number_of_turns_in_a_winding</code>
quantity,</li>
<li><span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>
(part 13) provides a
<code class="sourceCode default">Hamming_distance</code> quantity
defined as the number of digit positions in which the corresponding
digits of two words of the same length are different.</li>
</ul>
<p>Thanks to assigning strong names to such quantities, they can be used
in the quantity equation of other quantities. For example,
<code class="sourceCode default">rotational_frequency</code> is defined
by <code class="sourceCode default">rotation / duration</code>.</p>
<h3 data-number="7.6.3" id="lack-of-convertibility-from-fundamental-types"><span class="header-section-number">7.6.3</span> Lack of convertibility from
fundamental types<a href="#lack-of-convertibility-from-fundamental-types" class="self-link"></a></h3>
<p>As stated <a href="#dividing-two-quantities-of-the-same-kind">before</a>, the
division of two quantities of the same kind results in a quantity of
dimension one. Even though it does not have a specific physical
dimension it still uses units with various ratios such as
<code class="sourceCode default">one</code>,
<code class="sourceCode default">percent</code>,
<code class="sourceCode default">radian</code>,
<code class="sourceCode default">degree</code>, etc. It is essential to
be explicit about which unit we want to use for such a quantity.</p>
<p>However, in some cases, this might look like an overkill. The <span class="citation" data-cites="P2980R0">[<a href="#ref-P2980R0" role="doc-biblioref">P2980R0</a>]</span> provides the motivation for
standardizing this library and some examples to allow the reader to
understand how the code looks and feels when such a library is used. <a href="https://wg21.link/p2980r0#basic-quantity-equations">One of the
examples provides the following line:</a></p>
<div class="sourceCode" id="cb86"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb86-1"><a href="#cb86-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">10</span> <span class="op">*</span> km <span class="op">/</span> <span class="op">(</span><span class="dv">5</span> <span class="op">*</span> km<span class="op">)</span> <span class="op">==</span> <span class="dv">2</span> <span class="op">*</span> one<span class="op">)</span>;</span></code></pre></div>
<p>Another example could be subtracting a value
<code class="sourceCode default">1</code> from the dimensionless
quantity:</p>
<div class="sourceCode" id="cb87"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb87-1"><a href="#cb87-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>time<span class="op">&gt;</span> <span class="kw">auto</span> fill_time_left <span class="op">=</span> <span class="op">(</span>height <span class="op">/</span> fill_level <span class="op">-</span> <span class="dv">1</span> <span class="op">*</span> one<span class="op">)</span> <span class="op">*</span> fill_time;</span></code></pre></div>
<p>Some physical quantities and units libraries (e.g. <span class="citation" data-cites="BOOST-UNITS">[<a href="#ref-BOOST-UNITS" role="doc-biblioref">Boost.Units</a>]</span>) provide <a href="https://www.boost.org/doc/libs/1_83_0/doc/html/boost_units/Quantities.html#boost_units.Quantities.Quantity_Construction_and_Conversion">implicit
conversions from the values of representation types to such
quantities</a>.</p>
<p>With such support, the above examples would look in the following
way:</p>
<div class="sourceCode" id="cb88"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb88-1"><a href="#cb88-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">10</span> <span class="op">*</span> km <span class="op">/</span> <span class="op">(</span><span class="dv">5</span> <span class="op">*</span> km<span class="op">)</span> <span class="op">==</span> <span class="dv">2</span><span class="op">)</span>;</span></code></pre></div>
<div class="sourceCode" id="cb89"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb89-1"><a href="#cb89-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>time<span class="op">&gt;</span> <span class="kw">auto</span> fill_time_left <span class="op">=</span> <span class="op">(</span>height <span class="op">/</span> fill_level <span class="op">-</span> <span class="dv">1</span><span class="op">)</span> <span class="op">*</span> fill_time;</span></code></pre></div>
<p>Such simplification might look tempting, and the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> initially provided a special
support that allowed the above to compile. However, in the V2 version of
the library, it was removed. There are a few reasons for that:</p>
<ul>
<li><p>Such support has sense only for quantities of dimension one with
a unit <code class="sourceCode default">one</code>. In the case of all
the other units, the specific unit should still be provided.</p></li>
<li><p>If we provide implicit conversions from the representation type
to a quantity of dimension one with a unit
<code class="sourceCode default">one</code> and we start depending on
such a feature, we might end up with compile-time errors after
refactoring the unit in a type of such a quantity. For example:</p></li>
</ul>
<table>
<thead>
<tr class="header">
<th><div style="text-align:center">
<strong>Before</strong>
</div></th>
<th><div style="text-align:center">
<strong>After</strong>
</div></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><div>

<div class="sourceCode" id="cb90"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb90-1"><a href="#cb90-1" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> my_data <span class="op">{</span></span>
<span id="cb90-2"><a href="#cb90-2" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>dimensionless<span class="op">[</span>one<span class="op">]&gt;</span> value;</span>
<span id="cb90-3"><a href="#cb90-3" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>;</span>
<span id="cb90-4"><a href="#cb90-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb90-5"><a href="#cb90-5" aria-hidden="true" tabindex="-1"></a>my_data data1<span class="op">{.</span>value <span class="op">=</span> <span class="dv">42</span><span class="op">}</span>;</span>
<span id="cb90-6"><a href="#cb90-6" aria-hidden="true" tabindex="-1"></a>my_data data2<span class="op">{.</span>value <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> one<span class="op">}</span>;</span></code></pre></div>

</div></td>
<td><div>

<div class="sourceCode" id="cb91"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb91-1"><a href="#cb91-1" aria-hidden="true" tabindex="-1"></a><span class="kw">struct</span> my_data <span class="op">{</span></span>
<span id="cb91-2"><a href="#cb91-2" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>dimensionless<span class="op">[</span>percent<span class="op">]&gt;</span> value;</span>
<span id="cb91-3"><a href="#cb91-3" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>;</span>
<span id="cb91-4"><a href="#cb91-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb91-5"><a href="#cb91-5" aria-hidden="true" tabindex="-1"></a>my_data data1<span class="op">{.</span>value <span class="op">=</span> <span class="dv">42</span><span class="op">}</span>;        <span class="co">// Compile-time error</span></span>
<span id="cb91-6"><a href="#cb91-6" aria-hidden="true" tabindex="-1"></a>my_data data2<span class="op">{.</span>value <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> one<span class="op">}</span>;  <span class="co">// OK</span></span></code></pre></div>

</div></td>
</tr>
</tbody>
</table>
<ul>
<li>Such support is purely additive (it can be added later) and requires
a bunch of additional overloads not only for a constructor but for
arithmetic operators and comparisons as well.</li>
</ul>
<p>Please also note that we will still need to use explicit units to
create a quantity for some cases. For example, in the following code,
<code class="sourceCode default">asin(-1)</code> would use the overload
from the <code class="sourceCode default">&lt;math&gt;</code> header of
the C++ standard library rather than the one for quantities provided in
<code class="sourceCode default">&lt;mp-units/math.h&gt;</code> header
file:</p>
<div class="sourceCode" id="cb92"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb92-1"><a href="#cb92-1" aria-hidden="true" tabindex="-1"></a>REQUIRE_THAT<span class="op">(</span>asin<span class="op">(-</span><span class="dv">1</span> <span class="op">*</span> one<span class="op">)</span>, AlmostEquals<span class="op">(-</span><span class="fl">90.</span> <span class="op">*</span> deg<span class="op">))</span>;</span></code></pre></div>
<h3 data-number="7.6.4" id="predefined-units-of-the-dimensionless-quantity"><span class="header-section-number">7.6.4</span> Predefined units of the
dimensionless quantity<a href="#predefined-units-of-the-dimensionless-quantity" class="self-link"></a></h3>
<p>As we observed above, the most common unit for dimensionless
quantities is <code class="sourceCode default">one</code>. It has the
ratio of <code class="sourceCode default">1</code> and does not output
any textual symbol.</p>
<p>A unit <code class="sourceCode default">one</code> is special in the
entire type system of units as it is considered to be an identity
operand in the unit expression templates. This means that, for
example:</p>
<div class="sourceCode" id="cb93"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb93-1"><a href="#cb93-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>one <span class="op">*</span> one <span class="op">==</span> one<span class="op">)</span>;</span>
<span id="cb93-2"><a href="#cb93-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>one <span class="op">*</span> si<span class="op">::</span>metre <span class="op">==</span> si<span class="op">::</span>metre<span class="op">)</span>;</span>
<span id="cb93-3"><a href="#cb93-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>si<span class="op">::</span>metre <span class="op">/</span> si<span class="op">::</span>metre <span class="op">==</span> one<span class="op">)</span>;</span></code></pre></div>
<p>The same is also true for
<code class="sourceCode default">dimension_one</code> and
<code class="sourceCode default">dimensionless</code> in the domains of
dimensions and quantity specifications, respectively.</p>
<p>Besides the unit <code class="sourceCode default">one</code>, there
are a few other scaled units predefined in the library for usage with
dimensionless quantities:</p>
<div class="sourceCode" id="cb94"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb94-1"><a href="#cb94-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> percent <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;%&quot;</span>, mag<span class="op">&lt;</span>ratio<span class="op">{</span><span class="dv">1</span>, <span class="dv">100</span><span class="op">}&gt;</span> <span class="op">*</span> one<span class="op">&gt;</span> <span class="op">{}</span> percent;</span>
<span id="cb94-2"><a href="#cb94-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> per_mille <span class="op">:</span> named_unit<span class="op">&lt;</span>basic_symbol_text<span class="op">{</span><span class="st">&quot;‰&quot;</span>, <span class="st">&quot;</span><span class="sc">%o</span><span class="st">&quot;</span><span class="op">}</span>, mag<span class="op">&lt;</span>ratio<span class="op">(</span><span class="dv">1</span>, <span class="dv">1000</span><span class="op">)&gt;</span> <span class="op">*</span> one<span class="op">&gt;</span> <span class="op">{}</span> per_mille;</span></code></pre></div>
<h3 data-number="7.6.5" id="angular-quantities"><span class="header-section-number">7.6.5</span> Angular quantities<a href="#angular-quantities" class="self-link"></a></h3>
<p>Special, often controversial, examples of dimensionless quantities
are the angular measure and solid angular measure quantities that are
defined in <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>
(part 3) to be the result of a division of
<code class="sourceCode default">arc_length / radius</code> and
<code class="sourceCode default">area / pow&lt;2&gt;(radius)</code>
respectively. Moreover, <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> also
explicitly states that both can be expressed in the unit
<code class="sourceCode default">one</code>. This means that both
<code class="sourceCode default">isq::angular_measure</code> and
<code class="sourceCode default">isq::solid_angular_measure</code>
should be of a kind of
<code class="sourceCode default">dimensionless</code>.</p>
<p>On the other hand, <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span> also
specifies that the unit <code class="sourceCode default">radian</code>
can be used for
<code class="sourceCode default">isq::angular_measure</code>, and the
unit <code class="sourceCode default">steradian</code> can be used for
<code class="sourceCode default">isq::solid_angular_measure</code>.
Those should not be mixed or used to express other types of
dimensionless quantities. This means that both
<code class="sourceCode default">isq::angular_measure</code> and
<code class="sourceCode default">isq::solid_angular_measure</code>
should also be quantity kinds by themselves.</p>
<p><em>Many people claim that angle being a dimensionless quantity is a
bad idea. There are proposals submitted to make an angle a base quantity
and <code class="sourceCode default">rad</code> to become a base unit in
both <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span> and <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>.</em></p>
<h3 data-number="7.6.6" id="nested-quantity-kinds"><span class="header-section-number">7.6.6</span> Nested quantity kinds<a href="#nested-quantity-kinds" class="self-link"></a></h3>
<p>Angular quantities are not the only ones with such a “strange”
behavior. A similar case is the
<code class="sourceCode default">storage_capacity</code> quantity
specified in <span class="citation" data-cites="ISO80000">[<a href="#ref-ISO80000" role="doc-biblioref">ISO/IEC 80000</a>]</span>
(part 13) that again allows expressing it in both
<code class="sourceCode default">one</code> and
<code class="sourceCode default">bit</code> units.</p>
<p>Those cases make dimensionless quantities an exceptional tree in the
library. This is the only quantity hierarchy that contains more than one
quantity kind in its tree:</p>
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<p>To provide such support in the library, we provided an
<code class="sourceCode default">is_kind</code> specifier that can be
appended to the quantity specification:</p>
<div class="sourceCode" id="cb95"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb95-1"><a href="#cb95-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> angular_measure <span class="op">:</span> quantity_spec<span class="op">&lt;</span>dimensionless, arc_length <span class="op">/</span> radius, is_kind<span class="op">&gt;</span> <span class="op">{}</span> angular_measure;</span>
<span id="cb95-2"><a href="#cb95-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> solid_angular_measure <span class="op">:</span> quantity_spec<span class="op">&lt;</span>dimensionless, area <span class="op">/</span> pow<span class="op">&lt;</span><span class="dv">2</span><span class="op">&gt;(</span>radius<span class="op">)</span>, is_kind<span class="op">&gt;</span> <span class="op">{}</span> solid_angular_measure;</span>
<span id="cb95-3"><a href="#cb95-3" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> storage_capacity <span class="op">:</span> quantity_spec<span class="op">&lt;</span>dimensionless, is_kind<span class="op">&gt;</span> <span class="op">{}</span> storage_capacity;</span></code></pre></div>
<p>With the above, we can constrain
<code class="sourceCode default">radian</code>,
<code class="sourceCode default">steradian</code>, and
<code class="sourceCode default">bit</code> to be allowed for usage with
specific quantity kinds only:</p>
<div class="sourceCode" id="cb96"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb96-1"><a href="#cb96-1" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> radian <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;rad&quot;</span>, metre <span class="op">/</span> metre, kind_of<span class="op">&lt;</span>isq<span class="op">::</span>angular_measure<span class="op">&gt;&gt;</span> <span class="op">{}</span> radian;</span>
<span id="cb96-2"><a href="#cb96-2" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> steradian <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;sr&quot;</span>, square<span class="op">(</span>metre<span class="op">)</span> <span class="op">/</span> square<span class="op">(</span>metre<span class="op">)</span>, kind_of<span class="op">&lt;</span>isq<span class="op">::</span>solid_angular_measure<span class="op">&gt;&gt;</span> <span class="op">{}</span> steradian;</span>
<span id="cb96-3"><a href="#cb96-3" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> bit <span class="op">:</span> named_unit<span class="op">&lt;</span><span class="st">&quot;bit&quot;</span>, one, kind_of<span class="op">&lt;</span>storage_capacity<span class="op">&gt;&gt;</span> <span class="op">{}</span> bit;</span></code></pre></div>
<p>This still allows the usage of
<code class="sourceCode default">one</code> (possibly scaled) for such
quantities.</p>
<h2 data-number="7.7" id="quantity-arithmetics"><span class="header-section-number">7.7</span> Quantity arithmetics<a href="#quantity-arithmetics" class="self-link"></a></h2>
<h3 data-number="7.7.1" id="quantity-is-a-numeric-wrapper"><span class="header-section-number">7.7.1</span>
<code class="sourceCode default">quantity</code> is a numeric wrapper<a href="#quantity-is-a-numeric-wrapper" class="self-link"></a></h3>
<p>If we think about it, the
<code class="sourceCode default">quantity</code> class template is just
a “smart” numeric wrapper. It exposes properly constrained set of
arithmetic operations on one or two operands.</p>
<p>Every single arithmetic operator is exposed by the
<code class="sourceCode default">quantity</code> class template only if
the underlying representation type provides it as well and its
implementation has proper semantics (e.g. returns a reasonable
type).</p>
<p>For example, in the following code,
<code class="sourceCode default">-a</code> will compile only if
<code class="sourceCode default">MyInt</code> exposes such an operation
as well:</p>
<div class="sourceCode" id="cb97"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb97-1"><a href="#cb97-1" aria-hidden="true" tabindex="-1"></a>quantity a <span class="op">=</span> MyInt<span class="op">{</span><span class="dv">42</span><span class="op">}</span> <span class="op">*</span> m;</span>
<span id="cb97-2"><a href="#cb97-2" aria-hidden="true" tabindex="-1"></a>quantity b <span class="op">=</span> <span class="op">-</span>a;</span></code></pre></div>
<p>Assuming that:</p>
<ul>
<li><code class="sourceCode default">q</code> is our quantity,</li>
<li><code class="sourceCode default">qq</code> is a quantity implicitly
convertible to <code class="sourceCode default">q</code>,</li>
<li><code class="sourceCode default">q2</code> is any other
quantity,</li>
<li><code class="sourceCode default">kind</code> is a quantity of the
same kind as <code class="sourceCode default">q</code>,</li>
<li><code class="sourceCode default">one</code> is a quantity of
<code class="sourceCode default">dimension_one</code> with the unit
<code class="sourceCode default">one</code>,</li>
<li><code class="sourceCode default">number</code> is a value of a type
“compatible” with <code class="sourceCode default">q</code>’s
representation type,</li>
</ul>
<p>here is the list of all the supported operators:</p>
<ul>
<li>unary:
<ul>
<li><code class="sourceCode default">+q</code></li>
<li><code class="sourceCode default">-q</code></li>
<li><code class="sourceCode default">++q</code></li>
<li><code class="sourceCode default">q++</code></li>
<li><code class="sourceCode default">--q</code></li>
<li><code class="sourceCode default">q--</code></li>
</ul></li>
<li>compound assignment:
<ul>
<li><code class="sourceCode default">q += qq</code></li>
<li><code class="sourceCode default">q -= qq</code></li>
<li><code class="sourceCode default">q %= qq</code></li>
<li><code class="sourceCode default">q *= number</code></li>
<li><code class="sourceCode default">q *= one</code></li>
<li><code class="sourceCode default">q /= number</code></li>
<li><code class="sourceCode default">q /= one</code></li>
</ul></li>
<li>binary:
<ul>
<li><code class="sourceCode default">q + kind</code></li>
<li><code class="sourceCode default">q - kind</code></li>
<li><code class="sourceCode default">q % kind</code></li>
<li><code class="sourceCode default">q * q2</code></li>
<li><code class="sourceCode default">q * number</code></li>
<li><code class="sourceCode default">number * q</code></li>
<li><code class="sourceCode default">q / q2</code></li>
<li><code class="sourceCode default">q / number</code></li>
<li><code class="sourceCode default">number / q</code></li>
</ul></li>
<li>ordering and comparison:
<ul>
<li><code class="sourceCode default">q == kind</code></li>
<li><code class="sourceCode default">q &lt;=&gt; kind</code></li>
</ul></li>
</ul>
<p>As we can see, there are plenty of operations one can do on a value
of a <code class="sourceCode default">quantity</code> type. As most of
them are obvious, in the following chapters, we will discuss only the
most important or non-trivial aspects of quantity arithmetics.</p>
<h3 data-number="7.7.2" id="addition-and-subtraction"><span class="header-section-number">7.7.2</span> Addition and subtraction<a href="#addition-and-subtraction" class="self-link"></a></h3>
<p>Quantities can easily be added or subtracted from each other:</p>
<div class="sourceCode" id="cb98"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb98-1"><a href="#cb98-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m <span class="op">+</span> <span class="dv">1</span> <span class="op">*</span> m <span class="op">==</span> <span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb98-2"><a href="#cb98-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m <span class="op">-</span> <span class="dv">1</span> <span class="op">*</span> m <span class="op">==</span> <span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb98-3"><a href="#cb98-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">+</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">))</span>;</span>
<span id="cb98-4"><a href="#cb98-4" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">-</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>height<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">))</span>;</span></code></pre></div>
<p>The above uses the same types for LHS, RHS, and the result, but in
general, we can add, subtract, or compare the values of any quantity
type as long as both quantities are of the same kind. The result of
addition and subtraction will be the common type of the arguments:</p>
<div class="sourceCode" id="cb99"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb99-1"><a href="#cb99-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">1</span> <span class="op">*</span> km <span class="op">+</span> <span class="fl">1.5</span> <span class="op">*</span> m <span class="op">==</span> <span class="fl">1001.5</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb99-2"><a href="#cb99-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">+</span> isq<span class="op">::</span>width<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>length<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">))</span>;</span>
<span id="cb99-3"><a href="#cb99-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">-</span> isq<span class="op">::</span>distance<span class="op">(</span><span class="fl">0.5</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> <span class="fl">1.5</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb99-4"><a href="#cb99-4" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>radius<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">-</span> <span class="fl">0.5</span> <span class="op">*</span> m <span class="op">==</span> isq<span class="op">::</span>radius<span class="op">(</span><span class="fl">0.5</span> <span class="op">*</span> m<span class="op">))</span>;</span></code></pre></div>
<p>Please note that for the compound assignment operators, both
arguments have to either be of the same type or the RHS has to be
implicitly convertible to the LHS, as the type of LHS is always the
result of such an operation:</p>
<div class="sourceCode" id="cb100"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb100-1"><a href="#cb100-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">((</span><span class="dv">1</span> <span class="op">*</span> m <span class="op">+=</span> <span class="dv">1</span> <span class="op">*</span> km<span class="op">)</span> <span class="op">==</span> <span class="dv">1001</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb100-2"><a href="#cb100-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">((</span>isq<span class="op">::</span>height<span class="op">(</span><span class="fl">1.5</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">-=</span> <span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>height<span class="op">(</span><span class="fl">0.5</span> <span class="op">*</span> m<span class="op">))</span>;</span></code></pre></div>
<p>If we break those rules, the code will not compile:</p>
<div class="sourceCode" id="cb101"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb101-1"><a href="#cb101-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">((</span><span class="dv">1</span> <span class="op">*</span> m <span class="op">-=</span> <span class="fl">0.5</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> <span class="fl">0.5</span> <span class="op">*</span> m<span class="op">)</span>;                       <span class="co">// Compile-time error (1)</span></span>
<span id="cb101-2"><a href="#cb101-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">((</span><span class="dv">1</span> <span class="op">*</span> km <span class="op">+=</span> <span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> <span class="dv">1001</span> <span class="op">*</span> m<span class="op">)</span>;                       <span class="co">// Compile-time error (2)</span></span>
<span id="cb101-3"><a href="#cb101-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">((</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">+=</span> isq<span class="op">::</span>length<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> m<span class="op">))</span> <span class="op">==</span> <span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span>; <span class="co">// Compile-time error (3)</span></span></code></pre></div>
<ol type="1">
<li>Floating-point to integral representation type is considered
narrowing.</li>
<li>Conversion of quantity with integral representation type from a unit
of a higher resolution to the one with a lower resolution is considered
narrowing.</li>
<li>Conversion from a more generic quantity type to a more specific one
is considered unsafe.</li>
</ol>
<h3 data-number="7.7.3" id="multiplication-and-division"><span class="header-section-number">7.7.3</span> Multiplication and division<a href="#multiplication-and-division" class="self-link"></a></h3>
<p>Multiplying or dividing a quantity by a number does not change its
quantity type or unit. However, its representation type may change. For
example:</p>
<div class="sourceCode" id="cb102"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb102-1"><a href="#cb102-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">3</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">*</span> <span class="fl">0.5</span> <span class="op">==</span> isq<span class="op">::</span>height<span class="op">(</span><span class="fl">1.5</span> <span class="op">*</span> m<span class="op">))</span>;</span></code></pre></div>
<p>Unless we use a compound assignment operator, in which case
truncating operations are again not allowed:</p>
<div class="sourceCode" id="cb103"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb103-1"><a href="#cb103-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">((</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">3</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">*=</span> <span class="fl">0.5</span><span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>height<span class="op">(</span><span class="fl">1.5</span> <span class="op">*</span> m<span class="op">))</span>; <span class="co">// Compile-time error (1)</span></span></code></pre></div>
<ol type="1">
<li>Floating-point to integral representation type is considered
narrowing.</li>
</ol>
<p>However, suppose we multiply or divide quantities of the same or
different types, or we divide a raw number by a quantity. In that case,
we most probably will end up in a quantity of yet another type:</p>
<div class="sourceCode" id="cb104"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb104-1"><a href="#cb104-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">120</span> <span class="op">*</span> km <span class="op">/</span> <span class="op">(</span><span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span> <span class="op">==</span> <span class="dv">60</span> <span class="op">*</span> km <span class="op">/</span> h<span class="op">)</span>;</span>
<span id="cb104-2"><a href="#cb104-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>width<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">*</span> isq<span class="op">::</span>length<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>area<span class="op">(</span><span class="dv">4</span> <span class="op">*</span> m2<span class="op">))</span>;</span>
<span id="cb104-3"><a href="#cb104-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">50</span> <span class="op">/</span> isq<span class="op">::</span>time<span class="op">(</span><span class="dv">1</span> <span class="op">*</span> s<span class="op">)</span> <span class="op">==</span> isq<span class="op">::</span>frequency<span class="op">(</span><span class="dv">50</span> <span class="op">*</span> Hz<span class="op">))</span>;</span></code></pre></div>
<p>An exception from the above rule happens when one of the arguments is
a dimensionless quantity. If we multiply or divide by such a quantity,
the quantity type will not change. If such a quantity has a unit
<code class="sourceCode default">one</code>, also the unit of a quantity
will not change:</p>
<div class="sourceCode" id="cb105"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb105-1"><a href="#cb105-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">120</span> <span class="op">*</span> m <span class="op">/</span> <span class="op">(</span><span class="dv">2</span> <span class="op">*</span> one<span class="op">)</span> <span class="op">==</span> <span class="dv">60</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<p>An interesting special case happens when we divide the same quantity
kinds or multiply a quantity by its inverted type. In such a case, we
end up with a dimensionless quantity.</p>
<div class="sourceCode" id="cb106"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb106-1"><a href="#cb106-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>isq<span class="op">::</span>height<span class="op">(</span><span class="dv">4</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">/</span> isq<span class="op">::</span>width<span class="op">(</span><span class="dv">2</span> <span class="op">*</span> m<span class="op">)</span> <span class="op">==</span> <span class="dv">2</span> <span class="op">*</span> one<span class="op">)</span>; <span class="co">// (1)!</span></span>
<span id="cb106-2"><a href="#cb106-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">5</span> <span class="op">*</span> h <span class="op">/</span> <span class="op">(</span><span class="dv">120</span> <span class="op">*</span> min<span class="op">)</span> <span class="op">==</span> <span class="dv">0</span> <span class="op">*</span> one<span class="op">)</span>;  <span class="co">// (2)!</span></span>
<span id="cb106-3"><a href="#cb106-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="fl">5.</span> <span class="op">*</span> h <span class="op">/</span> <span class="op">(</span><span class="dv">120</span> <span class="op">*</span> min<span class="op">)</span> <span class="op">==</span> <span class="fl">2.5</span> <span class="op">*</span> one<span class="op">)</span>;</span></code></pre></div>
<ol type="1">
<li>The resulting quantity type of the LHS is
<code class="sourceCode default">isq::height / isq::width</code>, which
is a quantity of the dimensionless kind.</li>
<li>The resulting quantity of the LHS is
<code class="sourceCode default">0 * dimensionless[h / min]</code>. To
be consistent with the division of different quantity types, we do not
convert quantity values to a common unit before the division.</li>
</ol>
<h4 data-number="7.7.3.1" id="beware-of-integral-division"><span class="header-section-number">7.7.3.1</span> Beware of integral
division<a href="#beware-of-integral-division" class="self-link"></a></h4>
<p>The physical units library can’t do any runtime branching logic for
the division operator. All logic has to be done at compile-time when the
actual values are not known, and the quantity types can’t change at
runtime.</p>
<p>If we expect
<code class="sourceCode default">120 * km / (2 * h)</code> to return
<code class="sourceCode default">60 km/h</code>, we have to agree with
the fact that <code class="sourceCode default">5 * km / (24 * h)</code>
returns <code class="sourceCode default">0 km/h</code>. We can’t do a
range check at runtime to dynamically adjust scales and types based on
the values of provided function arguments.</p>
<p><strong>This is why we often prefer floating-point representation
types when dealing with units.</strong> Some popular physical units
libraries even <a href="https://aurora-opensource.github.io/au/main/troubleshooting/#integer-division-forbidden">forbid
integer division at all</a>.</p>
<h3 data-number="7.7.4" id="modulo"><span class="header-section-number">7.7.4</span> Modulo<a href="#modulo" class="self-link"></a></h3>
<p>Now that we know how addition, subtraction, multiplication, and
division work, it is time to talk about modulo. What would we expect to
be returned from the following quantity equation?</p>
<div class="sourceCode" id="cb107"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb107-1"><a href="#cb107-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> q <span class="op">=</span> <span class="dv">5</span> <span class="op">*</span> h <span class="op">%</span> <span class="op">(</span><span class="dv">120</span> <span class="op">*</span> min<span class="op">)</span>;</span></code></pre></div>
<p>Most of us would probably expect to see
<code class="sourceCode default">1 h</code> or
<code class="sourceCode default">60 min</code> as a result. And this is
where the problems start.</p>
<p>The C++ language defines its
<code class="sourceCode default">/</code> and
<code class="sourceCode default">%</code> operators with the <a href="https://eel.is/c++draft/expr.mul#4">quotient-remainder
theorem</a>:</p>
<pre class="text"><code>q = a / b;
r = a % b;
q * b + r == a;</code></pre>
<p>The important property of the modulo operator is that it only works
for integral representation types (it is undefined what modulo for
floating-point types means). However, as we saw in the previous chapter,
integral types are tricky because they often truncate the value.</p>
<p>From the quotient-remainder theorem, the result of modulo operation
is <code class="sourceCode default">r = a - q * b</code>. Let’s see what
we get from such a quantity equation on integral representation
types:</p>
<div class="sourceCode" id="cb108"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb108-1"><a href="#cb108-1" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity a <span class="op">=</span> <span class="dv">5</span> <span class="op">*</span> h;</span>
<span id="cb108-2"><a href="#cb108-2" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity b <span class="op">=</span> <span class="dv">120</span> <span class="op">*</span> min;</span>
<span id="cb108-3"><a href="#cb108-3" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity q <span class="op">=</span> a <span class="op">/</span> b;</span>
<span id="cb108-4"><a href="#cb108-4" aria-hidden="true" tabindex="-1"></a><span class="kw">const</span> quantity r <span class="op">=</span> a <span class="op">-</span> q <span class="op">*</span> b;</span>
<span id="cb108-5"><a href="#cb108-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb108-6"><a href="#cb108-6" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;reminder: &quot;</span> <span class="op">&lt;&lt;</span> r <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span></code></pre></div>
<p>The above code outputs:</p>
<pre class="text"><code>reminder: 5 h</code></pre>
<p>And now, a tough question needs an answer. Do we really want modulo
operator on physical units to be consistent with the quotient-remainder
theorem and return <code class="sourceCode default">5 h</code> for
<code class="sourceCode default">5 * h % (120 * min)</code>?</p>
<p>This is exactly why we decided not to follow this hugely surprising
path in the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library.
The selected approach was also consistent with the feedback from C++
experts. For example, this is what Richard Smith said about this
issue:</p>
<blockquote>
<p>I think the quotient-remainder property is a less important
motivation here than other factors – the constraints on
<code class="sourceCode default">%</code> and
<code class="sourceCode default">/</code> are quite different, so they
lack the inherent connection they have for integers. In particular, I
would expect that <code class="sourceCode default">A / B</code> works
for all quantities <code class="sourceCode default">A</code> and
<code class="sourceCode default">B</code>, whereas
<code class="sourceCode default">A % B</code> is only meaningful when
<code class="sourceCode default">A</code> and
<code class="sourceCode default">B</code> have the same dimension. It
seems like a nice-to-have for the property to apply in the case where
both <code class="sourceCode default">/</code> and
<code class="sourceCode default">%</code> are defined, but internal
consistency of <code class="sourceCode default">/</code> across all
cases seems much more important to me.</p>
<p>I would expect <code class="sourceCode default">61 min % 1 h</code>
to be <code class="sourceCode default">1 min</code>, and
<code class="sourceCode default">1 h % 59 min</code> to also be
<code class="sourceCode default">1 min</code>, so my intuition tells me
that the result type of <code class="sourceCode default">A % B</code>,
where <code class="sourceCode default">A</code> and
<code class="sourceCode default">B</code> have the same dimension,
should have the smaller unit of
<code class="sourceCode default">A</code> and
<code class="sourceCode default">B</code> (and if the smaller one
doesn’t divide the larger one, we should either use the
<code class="sourceCode default">gcd / std::common_type</code> of the
units of <code class="sourceCode default">A</code> and
<code class="sourceCode default">B</code> or perhaps just produce an
error). I think any other behavior for
<code class="sourceCode default">%</code> is hard to defend.</p>
<p>On the other hand, for division it seems to me that the choice of
unit should probably not affect the result, and so if we want that
<code class="sourceCode default">5 mm / 120 min = 0 mm/min</code>, then
<code class="sourceCode default">5 h / 120 min == 0 hc</code> (where
<code class="sourceCode default">hc</code> is a dimensionless
“hexaconta”, or <code class="sourceCode default">60x</code>, unit). I
don’t like the idea of taking SI base units into account; that seems
arbitrary and like it would do the wrong thing as often as it does the
right thing, especially when the units have a multiplier that is very
large or small. We could special-case the situation of a dimensionless
quantity, but that could lead to problematic overflow pretty easily: a
calculation such as
<code class="sourceCode default">10 s * 5 GHz * 2 uW</code> would
overflow an <code class="sourceCode default">int</code> if it produces a
dimensionless quantity for
<code class="sourceCode default">10 s * 5 GHz</code>, but it could
equally produce
<code class="sourceCode default">50 G * 2 uW = 100 kW</code> without any
overflow, and presumably would if the terms were merely reordered.</p>
<p>If people want to use integer-valued quantities, I think it’s
fundamental that you need to know what the units of the result of an
operation will be, and take that into account in how you express
computations; the simplest rule for heterogeneous operators like
<code class="sourceCode default">*</code> or
<code class="sourceCode default">/</code> seems to be that the units of
the result are determined by applying the operator to the units of the
operands – and for homogeneous operators like
<code class="sourceCode default">+</code> or
<code class="sourceCode default">%</code>, it seems like the only
reasonable option is that you get the
<code class="sourceCode default">std::common_type</code> of the units of
the operands.</p>
</blockquote>
<p>To summarize, the modulo operator on physical units has more in
common with addition and division operators than with the
quotient-remainder theorem. To avoid surprising results, the operation
uses a common unit to do the calculation and provide its result:</p>
<div class="sourceCode" id="cb109"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb109-1"><a href="#cb109-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">5</span> <span class="op">*</span> h <span class="op">/</span> <span class="op">(</span><span class="dv">120</span> <span class="op">*</span> min<span class="op">)</span> <span class="op">==</span> <span class="dv">0</span> <span class="op">*</span> one<span class="op">)</span>;</span>
<span id="cb109-2"><a href="#cb109-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">5</span> <span class="op">*</span> h <span class="op">%</span> <span class="op">(</span><span class="dv">120</span> <span class="op">*</span> min<span class="op">)</span> <span class="op">==</span> <span class="dv">60</span> <span class="op">*</span> min<span class="op">)</span>;</span>
<span id="cb109-3"><a href="#cb109-3" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">61</span> <span class="op">*</span> min <span class="op">%</span> <span class="op">(</span><span class="dv">1</span> <span class="op">*</span> h<span class="op">)</span> <span class="op">==</span> <span class="dv">1</span> <span class="op">*</span> min<span class="op">)</span>;</span>
<span id="cb109-4"><a href="#cb109-4" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span><span class="dv">1</span> <span class="op">*</span> h <span class="op">%</span> <span class="op">(</span><span class="dv">59</span> <span class="op">*</span> min<span class="op">)</span> <span class="op">==</span> <span class="dv">1</span> <span class="op">*</span> min<span class="op">)</span>;</span></code></pre></div>
<h3 data-number="7.7.5" id="comparison-against-zero"><span class="header-section-number">7.7.5</span> Comparison against zero<a href="#comparison-against-zero" class="self-link"></a></h3>
<p>Zero is special. It is the only number that unambiguously defines the
value of any kind of quantity, regardless of its units: zero inches and
zero meters and zero miles are all identical. For this reason, it’s very
common to compare the value of a quantity against zero — for example,
when checking the sign of a quantity, or when making sure that it’s
nonzero.</p>
<p>We could implement such checks in the following way:</p>
<div class="sourceCode" id="cb110"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb110-1"><a href="#cb110-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="op">(</span>q1 <span class="op">/</span> q2 <span class="op">!=</span> <span class="dv">0</span> <span class="op">*</span> m <span class="op">/</span> s<span class="op">)</span></span>
<span id="cb110-2"><a href="#cb110-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span></code></pre></div>
<p>The above would work (assuming we are dealing with the quantity of
speed), but it’s not ideal. If the result of
<code class="sourceCode default">q1 / q2</code> is not expressed in
<code class="sourceCode default">m / s</code>, we’ll incur an extra unit
conversion. Even if it is in
<code class="sourceCode default">m / s</code>, it’s cumbersome to repeat
the unit in a context where it makes no difference.</p>
<p>We could avoid repeating the unit, and guarantee there won’t be an
extra conversion, by writing:</p>
<div class="sourceCode" id="cb111"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb111-1"><a href="#cb111-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="op">(</span><span class="kw">auto</span> q <span class="op">=</span> q1 <span class="op">/</span> q2; q <span class="op">!=</span> q<span class="op">.</span>zero<span class="op">())</span></span>
<span id="cb111-2"><a href="#cb111-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span></code></pre></div>
<p>But that is a bit inconvenient, and inexperienced users could be
unaware of this technique and its reasons.</p>
<h4 data-number="7.7.5.1" id="named-comparison-functions"><span class="header-section-number">7.7.5.1</span> Named comparison
functions<a href="#named-comparison-functions" class="self-link"></a></h4>
<p>For the above reasons, the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library provides dedicated
interfaces to compare against zero that follow the naming convention of
<a href="https://en.cppreference.com/w/cpp/utility/compare/named_comparison_functions">named
comparison functions</a> in the C++ Standard Library:</p>
<ul>
<li><code class="sourceCode default">is_eq_zero</code></li>
<li><code class="sourceCode default">is_neq_zero</code></li>
<li><code class="sourceCode default">is_lt_zero</code></li>
<li><code class="sourceCode default">is_gt_zero</code></li>
<li><code class="sourceCode default">is_lteq_zero</code></li>
<li><code class="sourceCode default">is_gteq_zero</code></li>
</ul>
<p>Thanks to them, to save typing and not pay for unneeded conversions,
our check could be implemented as follows:</p>
<div class="sourceCode" id="cb112"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb112-1"><a href="#cb112-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="op">(</span>is_neq_zero<span class="op">(</span>q1 <span class="op">/</span> q2<span class="op">))</span></span>
<span id="cb112-2"><a href="#cb112-2" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span></code></pre></div>
<p>Those functions will work with any type
<code class="sourceCode default">T</code> that exposes a
<code class="sourceCode default">zero()</code> member function returning
something comparable to <code class="sourceCode default">T</code>.
Thanks to that, we can use them not only with quantities but also with
<code class="sourceCode default">std::chrono::duration</code> or any
other type that exposes such an interface.</p>
<p>This approach has a downside, though: it produces a set of new APIs
which users must learn. Nor are these six the only such functions that
will need to exist: for example,
<code class="sourceCode default">max</code> and
<code class="sourceCode default">min</code> are perfectly reasonable to
use with <code class="sourceCode default">0</code> regardless of the
units, but supporting them under this strategy would require adding a
new utility function for each — and coming up with a name for those
functions.</p>
<p>It also introduces small opportunities for error and diffs that are
harder to review, because we’re replacing a pattern that uses an
operator (say, <code class="sourceCode default">a &gt; 0</code>) with a
named function call (say,
<code class="sourceCode default">is_gt_zero(a)</code>).</p>
<p>These pitfalls motivate us to consider other approaches as well.</p>
<h4 data-number="7.7.5.2" id="zero-type"><span class="header-section-number">7.7.5.2</span>
<code class="sourceCode default">Zero</code> type<a href="#zero-type" class="self-link"></a></h4>
<p>The <span class="citation" data-cites="AU">[<a href="#ref-AU" role="doc-biblioref">Au</a>]</span> library takes a different approach
to this problem. It provides an empty type,
<code class="sourceCode default">Zero</code>, which represents a value
of exactly <code class="sourceCode default">0</code> (in any units). It
also provides an instance <code class="sourceCode default">ZERO</code>
of this type. Every quantity is implicitly constructible from
<code class="sourceCode default">Zero</code>.</p>
<p>Consider this example legacy (i.e., pre-units-library) code:</p>
<div class="sourceCode" id="cb113"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb113-1"><a href="#cb113-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="op">(</span>speed_squared_m2ps2 <span class="op">&gt;</span> <span class="dv">0</span><span class="op">)</span> <span class="op">{</span> <span class="co">/* ... */</span> <span class="op">}</span></span></code></pre></div>
<p>When users upgrade to a units library, they will replace the raw
number <code class="sourceCode default">speed_squared_m2ps2</code> with
a strongly typed quantity
<code class="sourceCode default">speed_squared</code>. Unfortunately,
this replacement won’t compile, because quantities can’t be constructed
from raw numeric values such as
<code class="sourceCode default">0</code>. They can fix this problem by
using the instance, <code class="sourceCode default">ZERO</code>, which
encodes its value <em>in the type</em>:</p>
<div class="sourceCode" id="cb114"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb114-1"><a href="#cb114-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> <span class="op">(</span>speed_squared <span class="op">&gt;</span> ZERO<span class="op">)</span> <span class="op">{</span> <span class="co">/* ... */</span> <span class="op">}</span></span></code></pre></div>
<p>This has significant advantages. It preserves the <em>form</em> of
the code, making the transition less error prone than replacement with a
function such as <code class="sourceCode default">is_gt_zero</code>. It
also reduces the number of new comparison APIs a user must learn:
<code class="sourceCode default">Zero</code> handles them all.</p>
<p><code class="sourceCode default">Zero</code> has one downside: it
will not work when passed across <em>generic quantity</em> interfaces.
<code class="sourceCode default">Zero</code>’s value comes in situations
where the surrounding context makes it unambiguous which quantity type
it should construct. While it converts to any <em>specific</em> quantity
type, it is not itself a quantity. This could confuse users.</p>
<p>This downside manifests in several different ways. Here are some
examples:</p>
<ol type="1">
<li><p>While refactoring the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> code to try out this approach,
we found out a perfectly reasonable place where we could not replace the
numerical value <code class="sourceCode default">0</code> with
<code class="sourceCode default">ZERO</code>:</p>
<div class="sourceCode" id="cb115"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb115-1"><a href="#cb115-1" aria-hidden="true" tabindex="-1"></a>msl_altitude alt <span class="op">=</span> mean_sea_level <span class="op">+</span> <span class="dv">0</span> <span class="op">*</span> si<span class="op">::</span>metre;  <span class="co">// OK</span></span>
<span id="cb115-2"><a href="#cb115-2" aria-hidden="true" tabindex="-1"></a>msl_altitude alt <span class="op">=</span> mean_sea_level <span class="op">+</span> ZERO;           <span class="co">// Compile-time error</span></span></code></pre></div>
<p>This would not work because the
<code class="sourceCode default">mean_sea_level</code> is an absolute
point origin that stores the information about the quantity type but not
its value or unit, which
<code class="sourceCode default">msl_altitude</code> needs, but none of
which <code class="sourceCode default">ZERO</code> has.</p></li>
<li><p>Callsites passing <code class="sourceCode default">ZERO</code>
can add friction when refactoring a concrete interface to be more
generic.</p>
<div class="sourceCode" id="cb116"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb116-1"><a href="#cb116-1" aria-hidden="true" tabindex="-1"></a><span class="kw">namespace</span> v1 <span class="op">{</span> <span class="dt">void</span> foo<span class="op">(</span>quantity<span class="op">&lt;</span>si<span class="op">::</span>metre<span class="op">&gt;</span> q<span class="op">)</span>; <span class="op">}</span></span>
<span id="cb116-2"><a href="#cb116-2" aria-hidden="true" tabindex="-1"></a><span class="kw">namespace</span> v2 <span class="op">{</span> <span class="dt">void</span> foo<span class="op">(</span>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="kw">auto</span> q<span class="op">)</span>; <span class="op">}</span></span>
<span id="cb116-3"><a href="#cb116-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb116-4"><a href="#cb116-4" aria-hidden="true" tabindex="-1"></a>v1<span class="op">::</span>foo<span class="op">(</span>ZERO<span class="op">)</span>;   <span class="co">// OK</span></span>
<span id="cb116-5"><a href="#cb116-5" aria-hidden="true" tabindex="-1"></a>v2<span class="op">::</span>foo<span class="op">(</span>ZERO<span class="op">)</span>;   <span class="co">// Compile-time error</span></span></code></pre></div>
<p>In practice, this issue will be discovered at the point of
refactoring, so it mainly affects library authors, not their clients.
They can handle this by adding an overload for
<code class="sourceCode default">Zero</code>, if appropriate. However,
this wouldn’t scale well for APIs with <em>multiple</em> parameters
where users would want to pass
<code class="sourceCode default">Zero</code>.</p></li>
<li><p>For completeness, we mention that
<code class="sourceCode default">Zero</code> works for addition but not
multiplication. When multiplying, we do not know what units (or even
what dimension!) is desired for the result. However, this is not a
problem in practice because users would not be motivated to write this
in the first place, as simple multiplication with
<code class="sourceCode default">0</code> (including any necessary
units, if the result has a different dimension) would work.</p>
<div class="sourceCode" id="cb117"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb117-1"><a href="#cb117-1" aria-hidden="true" tabindex="-1"></a>quantity q1 <span class="op">=</span> <span class="dv">1</span> <span class="op">*</span> m <span class="op">/</span> s;</span>
<span id="cb117-2"><a href="#cb117-2" aria-hidden="true" tabindex="-1"></a>quantity q2 <span class="op">=</span> q1 <span class="op">+</span> <span class="dv">0</span> <span class="op">*</span> m <span class="op">/</span> s;   <span class="co">// OK</span></span>
<span id="cb117-3"><a href="#cb117-3" aria-hidden="true" tabindex="-1"></a>quantity q3 <span class="op">=</span> q1 <span class="op">*</span> <span class="op">(</span><span class="dv">0</span> <span class="op">*</span> s<span class="op">)</span>;     <span class="co">// OK</span></span></code></pre></div>
<div class="sourceCode" id="cb118"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb118-1"><a href="#cb118-1" aria-hidden="true" tabindex="-1"></a>quantity q1 <span class="op">=</span> <span class="dv">1</span> <span class="op">*</span> m <span class="op">/</span> s;</span>
<span id="cb118-2"><a href="#cb118-2" aria-hidden="true" tabindex="-1"></a>quantity q2 <span class="op">=</span> q1 <span class="op">+</span> ZERO;        <span class="co">// OK</span></span>
<span id="cb118-3"><a href="#cb118-3" aria-hidden="true" tabindex="-1"></a>quantity q3 <span class="op">=</span> q1 <span class="op">*</span> ZERO;        <span class="co">// Compile-time error</span></span></code></pre></div></li>
</ol>
<p>The main concern with the
<code class="sourceCode default">Zero</code> feature is that novices
might be tempted to replace every numeric value
<code class="sourceCode default">0</code> with the instance
<code class="sourceCode default">ZERO</code>, becoming confused when it
doesn’t work. We could address this with easy-to-read documentation that
clarifies its use cases and mental models.</p>
<h4 data-number="7.7.5.3" id="summary-comparison-against-zero"><span class="header-section-number">7.7.5.3</span> Summary (comparison against
zero)<a href="#summary-comparison-against-zero" class="self-link"></a></h4>
<p>Overall, these two approaches — special functions, and a
<code class="sourceCode default">Zero</code> type — represent two local
optima in design space. Each has its strengths and weaknesses; each
makes different tradeoffs. It’s currently an open question as to which
approach would be best suited for a quantity type in the standard
library.</p>
<h3 data-number="7.7.6" id="other-maths"><span class="header-section-number">7.7.6</span> Other maths<a href="#other-maths" class="self-link"></a></h3>
<p>This chapter scoped only on the
<code class="sourceCode default">quantity</code> type’s operators.
However, there are many named math functions provided in the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library. Among others, we can
find there the following:</p>
<ul>
<li><code class="sourceCode default">pow()</code>,
<code class="sourceCode default">sqrt()</code>, and
<code class="sourceCode default">cbrt()</code>,</li>
<li><code class="sourceCode default">exp()</code>,</li>
<li><code class="sourceCode default">abs()</code>,</li>
<li><code class="sourceCode default">epsilon()</code>,</li>
<li><code class="sourceCode default">floor()</code>,
<code class="sourceCode default">ceil()</code>,
<code class="sourceCode default">round()</code>,</li>
<li><code class="sourceCode default">inverse()</code>,</li>
<li><code class="sourceCode default">hypot()</code>,</li>
<li><code class="sourceCode default">sin()</code>,
<code class="sourceCode default">cos()</code>,
<code class="sourceCode default">tan()</code>,</li>
<li><code class="sourceCode default">asin()</code>,
<code class="sourceCode default">acos()</code>,
<code class="sourceCode default">atan()</code>.</li>
</ul>
<p>In the library, we can also find the
<code class="sourceCode default">&lt;mp-units/random.h&gt;</code> header
file with all the pseudo-random number generators.</p>
<p>We plan to provide a separate paper on those in the future.</p>
<h1 data-number="8" id="generic-interfaces"><span class="header-section-number">8</span> Generic Interfaces<a href="#generic-interfaces" class="self-link"></a></h1>
<p>Using a concrete unit in the interface often has a lot of sense. It
is especially useful if we store the data internally in the object. In
such a case, we have to select a specific unit anyway.</p>
<p>For example, let’s consider a simple storage tank:</p>
<div class="sourceCode" id="cb119"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb119-1"><a href="#cb119-1" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> StorageTank <span class="op">{</span></span>
<span id="cb119-2"><a href="#cb119-2" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>horizontal_area<span class="op">[</span>m2<span class="op">]&gt;</span> base_;</span>
<span id="cb119-3"><a href="#cb119-3" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">[</span>m<span class="op">]&gt;</span> height_;</span>
<span id="cb119-4"><a href="#cb119-4" aria-hidden="true" tabindex="-1"></a>  quantity<span class="op">&lt;</span>isq<span class="op">::</span>mass_density<span class="op">[</span>kg <span class="op">/</span> m3<span class="op">]&gt;</span> density_ <span class="op">=</span> air_density;</span>
<span id="cb119-5"><a href="#cb119-5" aria-hidden="true" tabindex="-1"></a><span class="kw">public</span><span class="op">:</span></span>
<span id="cb119-6"><a href="#cb119-6" aria-hidden="true" tabindex="-1"></a>  <span class="kw">constexpr</span> StorageTank<span class="op">(</span><span class="kw">const</span> quantity<span class="op">&lt;</span>horizontal_area<span class="op">[</span>m2<span class="op">]&gt;&amp;</span> base, <span class="kw">const</span> quantity<span class="op">&lt;</span>isq<span class="op">::</span>height<span class="op">[</span>m<span class="op">]&gt;&amp;</span> height<span class="op">)</span> <span class="op">:</span></span>
<span id="cb119-7"><a href="#cb119-7" aria-hidden="true" tabindex="-1"></a>      base_<span class="op">(</span>base<span class="op">)</span>, height_<span class="op">(</span>height<span class="op">)</span></span>
<span id="cb119-8"><a href="#cb119-8" aria-hidden="true" tabindex="-1"></a>  <span class="op">{</span></span>
<span id="cb119-9"><a href="#cb119-9" aria-hidden="true" tabindex="-1"></a>  <span class="op">}</span></span>
<span id="cb119-10"><a href="#cb119-10" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb119-11"><a href="#cb119-11" aria-hidden="true" tabindex="-1"></a>  <span class="co">// ...</span></span>
<span id="cb119-12"><a href="#cb119-12" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>;</span></code></pre></div>
<p>As the quantities provided in the function’s interface are then
stored in the class, there is probably no sense in using generic
interfaces here.</p>
<h2 data-number="8.1" id="the-issues-with-unit-specific-interfaces"><span class="header-section-number">8.1</span> The issues with unit-specific
interfaces<a href="#the-issues-with-unit-specific-interfaces" class="self-link"></a></h2>
<p>However, in many cases, using a specific unit in the interface is
counterproductive. Let’s consider the following function:</p>
<div class="sourceCode" id="cb120"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb120-1"><a href="#cb120-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>km <span class="op">/</span> h<span class="op">]&gt;</span> avg_speed<span class="op">(</span>quantity<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">[</span>km<span class="op">]&gt;</span> distance,</span>
<span id="cb120-2"><a href="#cb120-2" aria-hidden="true" tabindex="-1"></a>                                       quantity<span class="op">&lt;</span>isq<span class="op">::</span>time<span class="op">[</span>h<span class="op">]&gt;</span> duration<span class="op">)</span></span>
<span id="cb120-3"><a href="#cb120-3" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb120-4"><a href="#cb120-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> distance <span class="op">/</span> duration;</span>
<span id="cb120-5"><a href="#cb120-5" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<p>Everything seems fine for now. It also works great if we call it
with:</p>
<div class="sourceCode" id="cb121"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb121-1"><a href="#cb121-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>km <span class="op">/</span> h<span class="op">]&gt;</span> s1 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">220</span> <span class="op">*</span> km, <span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span>;</span></code></pre></div>
<p>However, if the user starts doing the following:</p>
<div class="sourceCode" id="cb122"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb122-1"><a href="#cb122-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>mi <span class="op">/</span> h<span class="op">]&gt;</span> s2 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">140</span> <span class="op">*</span> mi, <span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span>;</span>
<span id="cb122-2"><a href="#cb122-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>m <span class="op">/</span> s<span class="op">]&gt;</span> s3 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">20</span> <span class="op">*</span> m, <span class="dv">2</span> <span class="op">*</span> s<span class="op">)</span>;</span></code></pre></div>
<p>some issues start to be clearly visible:</p>
<ol type="1">
<li><p>The arguments must be converted to units mandated by the
function’s parameters at each call. This involves potentially expensive
multiplication/division operations at runtime.</p></li>
<li><p>After the function returns the speed in a unit of
<code class="sourceCode default">km/h</code>, another potentially
expensive multiplication/division operations have to be performed to
convert the resulting quantity into a unit being the derived unit of the
initial function’s arguments.</p></li>
<li><p>Besides the obvious runtime cost, some unit conversions may
result in a data truncation, which means that the result will not be
exactly equal to a direct division of the function’s arguments.</p></li>
<li><p>We have to use a floating-point representation type (the
<code class="sourceCode default">quantity</code> class template by
default uses <code class="sourceCode default">double</code> as a
representation type) which is considered value preserving. Trying to use
an integral type in this scenario will work only for
<code class="sourceCode default">s1</code>, while
<code class="sourceCode default">s2</code> and
<code class="sourceCode default">s3</code> will fail to compile. Failing
to compile is a good thing here as the library tries to prevent the user
from doing a clearly wrong thing. To make the code compile, the user
needs to use dedicated
<code class="sourceCode default">value_cast</code> or
<code class="sourceCode default">force_in</code> like this:</p>
<div class="sourceCode" id="cb123"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb123-1"><a href="#cb123-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>mi <span class="op">/</span> h<span class="op">]&gt;</span> s2 <span class="op">=</span> avg_speed<span class="op">(</span>value_cast<span class="op">&lt;</span>km<span class="op">&gt;(</span><span class="dv">140</span> <span class="op">*</span> mi<span class="op">)</span>, <span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span>;</span>
<span id="cb123-2"><a href="#cb123-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">[</span>m <span class="op">/</span> s<span class="op">]&gt;</span> s3 <span class="op">=</span> avg_speed<span class="op">((</span><span class="dv">20</span> <span class="op">*</span> m<span class="op">).</span>force_in<span class="op">(</span>km<span class="op">)</span>, <span class="op">(</span><span class="dv">2</span> <span class="op">*</span> s<span class="op">).</span>force_in<span class="op">(</span>h<span class="op">))</span>;</span></code></pre></div>
<p>But the above will obviously provide an incorrect behavior
(e.g. division by <code class="sourceCode default">0</code> in the
evaluation of <code class="sourceCode default">s3</code>).</p></li>
</ol>
<h2 data-number="8.2" id="a-naive-solution"><span class="header-section-number">8.2</span> A naive solution<a href="#a-naive-solution" class="self-link"></a></h2>
<p>A naive solution here would be to implement the function as an
unconstrained function template:</p>
<div class="sourceCode" id="cb124"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb124-1"><a href="#cb124-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> avg_speed<span class="op">(</span><span class="kw">auto</span> distance, <span class="kw">auto</span> duration<span class="op">)</span></span>
<span id="cb124-2"><a href="#cb124-2" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb124-3"><a href="#cb124-3" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> distance <span class="op">/</span> duration;</span>
<span id="cb124-4"><a href="#cb124-4" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<p>Beware that there are better solutions than this. The above code is
too generic. Such a function template accepts everything:</p>
<ul>
<li>quantities of other types
<ul>
<li>the compiler will not prevent accidental reordering of the
function’s arguments</li>
<li>quantities of different types can be passed as well</li>
</ul></li>
<li>plain <code class="sourceCode default">double</code> arguments</li>
<li><code class="sourceCode default">std::vector</code> and
<code class="sourceCode default">std::lock_guard</code> will be accepted
as well (of course, this will fail in the function’s body later in the
compilation process)</li>
</ul>
<h2 data-number="8.3" id="constraining-function-parameters-with-concepts"><span class="header-section-number">8.3</span> Constraining function
parameters with concepts<a href="#constraining-function-parameters-with-concepts" class="self-link"></a></h2>
<p>Much better generic code can be implemented using basic concepts
provided with the library:</p>
<div class="sourceCode" id="cb125"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb125-1"><a href="#cb125-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> avg_speed<span class="op">(</span>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="kw">auto</span> distance,</span>
<span id="cb125-2"><a href="#cb125-2" aria-hidden="true" tabindex="-1"></a>               QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>time<span class="op">&gt;</span> <span class="kw">auto</span> duration<span class="op">)</span></span>
<span id="cb125-3"><a href="#cb125-3" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb125-4"><a href="#cb125-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> isq<span class="op">::</span>speed<span class="op">(</span>distance <span class="op">/</span> duration<span class="op">)</span>;</span>
<span id="cb125-5"><a href="#cb125-5" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<p>This explicitly states that the arguments passed by the user must not
only satisfy a <code class="sourceCode default">Quantity</code> concept,
but also that their quantity specification must be implicitly
convertible to <code class="sourceCode default">isq::length</code> and
<code class="sourceCode default">isq::time</code>, respectively. This no
longer leaves room for error while still allowing the compiler to
generate the most efficient code.</p>
<p>Please, note that now it is safe just to use integral types all the
way, which again improves the runtime performance as the
multiplication/division operations are often faster on integral rather
than floating-point types.</p>
<h2 data-number="8.4" id="constraining-the-function-return-type"><span class="header-section-number">8.4</span> Constraining the function
return type<a href="#constraining-the-function-return-type" class="self-link"></a></h2>
<p>The above function template resolves all of the issues described
before. However, we can do even better here by additionally constraining
the return type:</p>
<div class="sourceCode" id="cb126"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb126-1"><a href="#cb126-1" aria-hidden="true" tabindex="-1"></a>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">&gt;</span> <span class="kw">auto</span> avg_speed<span class="op">(</span>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>length<span class="op">&gt;</span> <span class="kw">auto</span> distance,</span>
<span id="cb126-2"><a href="#cb126-2" aria-hidden="true" tabindex="-1"></a>                                      QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>time<span class="op">&gt;</span> <span class="kw">auto</span> duration<span class="op">)</span></span>
<span id="cb126-3"><a href="#cb126-3" aria-hidden="true" tabindex="-1"></a><span class="op">{</span></span>
<span id="cb126-4"><a href="#cb126-4" aria-hidden="true" tabindex="-1"></a>  <span class="cf">return</span> isq<span class="op">::</span>speed<span class="op">(</span>distance <span class="op">/</span> duration<span class="op">)</span>;</span>
<span id="cb126-5"><a href="#cb126-5" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<p>Doing so has two important benefits:</p>
<ol type="1">
<li>It informs the users of our interface about what to expect to be the
result of a function invocation. It is superior to just returning
<code class="sourceCode default">auto</code>, which does not provide any
hint about the thing being returned there.</li>
<li>Such a concept constrains the type returned from the function. This
means that it works as a unit test to verify if our function actually
performs what it is supposed to do. If there is an error in the quantity
equation, we will learn about it right away.</li>
</ol>
<h2 data-number="8.5" id="constraining-a-variable-on-the-stack"><span class="header-section-number">8.5</span> Constraining a variable on the
stack<a href="#constraining-a-variable-on-the-stack" class="self-link"></a></h2>
<p>If we know exactly what the function does in its internals and if we
know the exact argument types passed to such a function, we often know
the exact type that will be returned from its invocation.</p>
<p>However, if we care about performance, we should often use the
generic interfaces described in this chapter. A side effect is that we
sometimes are unsure about the return type. Even if we know it today, it
might change a week from now due to some code refactoring.</p>
<p>In such cases, we can again use
<code class="sourceCode default">auto</code> to denote the type:</p>
<div class="sourceCode" id="cb127"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb127-1"><a href="#cb127-1" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> s1 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">220</span> <span class="op">*</span> km, <span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span>;</span>
<span id="cb127-2"><a href="#cb127-2" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> s2 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">140</span> <span class="op">*</span> mi, <span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span>;</span>
<span id="cb127-3"><a href="#cb127-3" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> s3 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">20</span> <span class="op">*</span> m, <span class="dv">2</span> <span class="op">*</span> s<span class="op">)</span>;</span></code></pre></div>
<p>In this case, it is probably OK to do so as the
<code class="sourceCode default">avg_speed</code> function name
explicitly provides the information on what to expect as a result.</p>
<p>In other scenarios where the returned quantity type is not so
obvious, it is again helpful to constrain the type with a concept like
so:</p>
<div class="sourceCode" id="cb128"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb128-1"><a href="#cb128-1" aria-hidden="true" tabindex="-1"></a>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">&gt;</span> <span class="kw">auto</span> s1 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">220</span> <span class="op">*</span> km, <span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span>;</span>
<span id="cb128-2"><a href="#cb128-2" aria-hidden="true" tabindex="-1"></a>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">&gt;</span> <span class="kw">auto</span> s2 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">140</span> <span class="op">*</span> mi, <span class="dv">2</span> <span class="op">*</span> h<span class="op">)</span>;</span>
<span id="cb128-3"><a href="#cb128-3" aria-hidden="true" tabindex="-1"></a>QuantityOf<span class="op">&lt;</span>isq<span class="op">::</span>speed<span class="op">&gt;</span> <span class="kw">auto</span> s3 <span class="op">=</span> avg_speed<span class="op">(</span><span class="dv">20</span> <span class="op">*</span> m, <span class="dv">2</span> <span class="op">*</span> s<span class="op">)</span>;</span></code></pre></div>
<p>Again, this explicitly provides additional information about the
quantity we are dealing with in the code, and it serves as a unit test
checking if the “thing” returned from a function is actually what we
expected here.</p>
<h2 data-number="8.6" id="basic-concepts"><span class="header-section-number">8.6</span> Basic concepts<a href="#basic-concepts" class="self-link"></a></h2>
<p>Here are some more concepts exposed in the library and dependencies
between them:</p>
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<h1 data-number="9" id="custom-representation-types"><span class="header-section-number">9</span> Custom representation types<a href="#custom-representation-types" class="self-link"></a></h1>
<p>The library of physical quantities and units library should work with
any custom representation type. Those can be used to:</p>
<ul>
<li>Improve safety (e.g., prevent overflows, restrict the range of
accepted values, etc.),</li>
<li>provide additional information (e.g., not only a quantity value but
also the uncertainty of the measurement), and</li>
<li>enable linear algebra usage.</li>
</ul>
<p>As of right now, we have two other concurrent proposals to SG6 in
this subject on the fly (<span class="citation" data-cites="P2993_PRE">[<a href="#ref-P2993_PRE" role="doc-biblioref">P2993</a>]</span> and <span class="citation" data-cites="P3003R0">[<a href="#ref-P3003R0" role="doc-biblioref">P3003R0</a>]</span>), so we do not provide any
concrete requirements or recommendations here. Based on the results of
discussions on the mentioned proposals, we will provide correct
guidelines in the next revisions of this paper.</p>
<h1 data-number="10" id="the-affine-space"><span class="header-section-number">10</span> The affine space<a href="#the-affine-space" class="self-link"></a></h1>
<p>The affine space has two types of entities:</p>
<ul>
<li><strong><em>point</em></strong> - a position specified with
coordinate values (e.g. location, address, etc.)</li>
<li><strong><em>vector</em></strong> - the difference between two points
(e.g. shift, offset, displacement, duration, etc.)</li>
</ul>
<p>The <em>vector</em> described here is specific to the affine space
theory and is not the same thing as the quantity of a vector character
that we discussed in the before (although, in some cases, those terms
may overlap).</p>
<h2 data-number="10.1" id="operations-in-the-affine-space"><span class="header-section-number">10.1</span> Operations in the affine
space<a href="#operations-in-the-affine-space" class="self-link"></a></h2>
<p>Here are the primary operations one can do in the affine space:</p>
<ul>
<li><em>vector</em> + <em>vector</em> -&gt; <em>vector</em></li>
<li><em>vector</em> - <em>vector</em> -&gt; <em>vector</em></li>
<li>-<em>vector</em> -&gt; <em>vector</em></li>
<li><em>vector</em> * scalar -&gt; <em>vector</em></li>
<li>scalar * <em>vector</em> -&gt; <em>vector</em></li>
<li><em>vector</em> / scalar -&gt; <em>vector</em></li>
<li><em>point</em> - <em>point</em> -&gt; <em>vector</em></li>
<li><em>point</em> + <em>vector</em> -&gt; <em>point</em></li>
<li><em>vector</em> + <em>point</em> -&gt; <em>point</em></li>
<li><em>point</em> - <em>vector</em> -&gt; <em>point</em></li>
</ul>
<p>It is not possible to:</p>
<ul>
<li>add two <em>points</em>,</li>
<li>subtract a <em>point</em> from a <em>vector</em>,</li>
<li>multiply nor divide <em>points</em> with anything else.</li>
</ul>
<h2 data-number="10.2" id="vector-is-modeled-by-quantity"><span class="header-section-number">10.2</span> <em>Vector</em> is modeled by
<code class="sourceCode default">quantity</code><a href="#vector-is-modeled-by-quantity" class="self-link"></a></h2>
<p>Up until now, each time when we used a
<code class="sourceCode default">quantity</code> in our code, we were
modeling some kind of a difference between two things:</p>
<ul>
<li>the distance between two points</li>
<li>duration between two time points</li>
<li>the difference in speed (even if relative to
<code class="sourceCode default">0</code>)</li>
</ul>
<p>As we already know, a
<code class="sourceCode default">quantity</code> type provides all
operations required for the <em>vector</em> type in an affine space.</p>
<h2 data-number="10.3" id="point-is-modeled-by-pointorigin-and-quantity_point"><span class="header-section-number">10.3</span> <em>Point</em> is modeled by
<code class="sourceCode default">PointOrigin</code> and
<code class="sourceCode default">quantity_point</code><a href="#point-is-modeled-by-pointorigin-and-quantity_point" class="self-link"></a></h2>
<p>In the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library,
the <em>point</em> abstraction is modeled by:</p>
<ul>
<li>The <code class="sourceCode default">PointOrigin</code> concept that
specifies a measurement’s origin, and</li>
<li>the <code class="sourceCode default">quantity_point</code> class
template that specifies a <em>point</em> relative to a specified
origin.</li>
</ul>
<h3 data-number="10.3.1" id="absolute-point-origin"><span class="header-section-number">10.3.1</span> Absolute <em>point</em>
origin<a href="#absolute-point-origin" class="self-link"></a></h3>
<p>The <strong>absolute point origin</strong> specifies where the “zero”
of our measurement’s scale is. Users can specify such an origin by
deriving from the
<code class="sourceCode default">absolute_point_origin</code> class
template:</p>
<div class="sourceCode" id="cb129"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb129-1"><a href="#cb129-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">struct</span> mean_sea_level <span class="op">:</span> absolute_point_origin<span class="op">&lt;</span>isq<span class="op">::</span>altitude<span class="op">&gt;</span> <span class="op">{}</span> mean_sea_level;</span></code></pre></div>
<h3 data-number="10.3.2" id="quantity_point"><span class="header-section-number">10.3.2</span>
<code class="sourceCode default">quantity_point</code><a href="#quantity_point" class="self-link"></a></h3>
<p>The <code class="sourceCode default">quantity_point</code> class
template specifies an absolute quantity with respect to an origin:</p>
<div class="sourceCode" id="cb130"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb130-1"><a href="#cb130-1" aria-hidden="true" tabindex="-1"></a><span class="kw">template</span><span class="op">&lt;</span>Reference <span class="kw">auto</span> R,</span>
<span id="cb130-2"><a href="#cb130-2" aria-hidden="true" tabindex="-1"></a>         PointOriginFor<span class="op">&lt;</span>get_quantity_spec<span class="op">(</span>R<span class="op">)&gt;</span> <span class="kw">auto</span> PO,</span>
<span id="cb130-3"><a href="#cb130-3" aria-hidden="true" tabindex="-1"></a>         RepresentationOf<span class="op">&lt;</span>get_quantity_spec<span class="op">(</span>R<span class="op">).</span>character<span class="op">&gt;</span> Rep <span class="op">=</span> <span class="dt">double</span><span class="op">&gt;</span></span>
<span id="cb130-4"><a href="#cb130-4" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> quantity_point;</span></code></pre></div>
<p>As we can see above, the
<code class="sourceCode default">quantity_point</code> class template
exposes one additional parameter compared to
<code class="sourceCode default">quantity</code>. The
<code class="sourceCode default">PO</code> parameter satisfies a
<code class="sourceCode default">PointOriginFor</code> concept and
specifies the origin of our measurement scale.</p>
<p>As a <em>point</em> can be represented with a <em>vector</em> from
the origin, a <code class="sourceCode default">quantity_point</code>
class template can be created with the following operations:</p>
<div class="sourceCode" id="cb131"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb131-1"><a href="#cb131-1" aria-hidden="true" tabindex="-1"></a>quantity_point qp1 <span class="op">=</span> mean_sea_level <span class="op">+</span> <span class="dv">42</span> <span class="op">*</span> m;</span>
<span id="cb131-2"><a href="#cb131-2" aria-hidden="true" tabindex="-1"></a>quantity_point qp2 <span class="op">=</span> <span class="dv">42</span> <span class="op">*</span> m <span class="op">+</span> mean_sea_level;</span>
<span id="cb131-3"><a href="#cb131-3" aria-hidden="true" tabindex="-1"></a>quantity_point qp3 <span class="op">=</span> mean_sea_level <span class="op">-</span> <span class="dv">42</span> <span class="op">*</span> m;</span></code></pre></div>
<p>It is not allowed to subtract a <em>point</em> from a
<em>vector</em>, thus
<code class="sourceCode default">42 * m - mean_sea_level</code> is an
invalid operation.</p>
<p>Similarly to the creation of a quantity, if someone does not like the
operator-based syntax to create a
<code class="sourceCode default">quantity_point</code>, the same results
can be achieved with the
<code class="sourceCode default">make_quantity_point</code> factory
function:</p>
<div class="sourceCode" id="cb132"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb132-1"><a href="#cb132-1" aria-hidden="true" tabindex="-1"></a>quantity_point qp4 <span class="op">=</span> make_quantity_point<span class="op">&lt;</span>mean_sea_level<span class="op">&gt;(</span><span class="dv">42</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb132-2"><a href="#cb132-2" aria-hidden="true" tabindex="-1"></a>quantity_point qp5 <span class="op">=</span> make_quantity_point<span class="op">&lt;</span>mean_sea_level<span class="op">&gt;(-</span><span class="dv">42</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<p>The provided <code class="sourceCode default">quantity</code>
representing an offset from the origin is stored inside the
<code class="sourceCode default">quantity_point</code> class template
and can be obtained with the
<code class="sourceCode default">quantity_from_origin()</code> member
function:</p>
<div class="sourceCode" id="cb133"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb133-1"><a href="#cb133-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> quantity_point everest_base_camp_alt <span class="op">=</span> mean_sea_level <span class="op">+</span> isq<span class="op">::</span>altitude<span class="op">(</span><span class="dv">5364</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb133-2"><a href="#cb133-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>everest_base_camp_alt<span class="op">.</span>quantity_from_origin<span class="op">()</span> <span class="op">==</span> <span class="dv">5364</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<h3 data-number="10.3.3" id="relative-point-origin"><span class="header-section-number">10.3.3</span> Relative <em>point</em>
origin<a href="#relative-point-origin" class="self-link"></a></h3>
<p>We often do not have only one ultimate “zero” point when we measure
things.</p>
<p>Continuing the Mount Everest trip example above, measuring all daily
hikes from the <code class="sourceCode default">mean_sea_level</code>
might not be efficient. Maybe we know that we are not good climbers, so
all our climbs can be represented with an 8-bit integer type allowing us
to save memory in our database of climbs? Why not use
<code class="sourceCode default">everest_base_camp_alt</code> as our
reference point?</p>
<p>For this purpose, we can define a
<code class="sourceCode default">relative_point_origin</code> in the
following way:</p>
<div class="sourceCode" id="cb134"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb134-1"><a href="#cb134-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">struct</span> everest_base_camp <span class="op">:</span> relative_point_origin<span class="op">&lt;</span>everest_base_camp_alt<span class="op">&gt;</span> <span class="op">{}</span> everest_base_camp;</span></code></pre></div>
<p>The above can be used as an origin for subsequent
<em>points</em>:</p>
<div class="sourceCode" id="cb135"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb135-1"><a href="#cb135-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> quantity_point first_climb_alt <span class="op">=</span> everest_base_camp <span class="op">+</span> isq<span class="op">::</span>altitude<span class="op">(</span>std<span class="op">::</span><span class="dt">uint8_t</span><span class="op">{</span><span class="dv">42</span><span class="op">}</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb135-2"><a href="#cb135-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>first_climb_alt<span class="op">.</span>quantity_from_origin<span class="op">()</span> <span class="op">==</span> <span class="dv">42</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<p>As we can see above, the
<code class="sourceCode default">quantity_from_origin()</code> member
function returns a relative distance from the current point origin. In
case we would like to know the absolute altitude that we reached on this
climb, we can subtract the absolute point origin from the current
<em>point</em>:</p>
<div class="sourceCode" id="cb136"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb136-1"><a href="#cb136-1" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>first_climb_alt <span class="op">-</span> mean_sea_level <span class="op">==</span> <span class="dv">5406</span> <span class="op">*</span> m<span class="op">)</span>;</span>
<span id="cb136-2"><a href="#cb136-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>first_climb_alt <span class="op">-</span> first_climb_alt<span class="op">.</span>absolute_point_origin <span class="op">==</span> <span class="dv">5406</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div>
<h3 data-number="10.3.4" id="converting-between-different-representations-of-the-same-point"><span class="header-section-number">10.3.4</span> Converting between different
representations of the same <em>point</em><a href="#converting-between-different-representations-of-the-same-point" class="self-link"></a></h3>
<p>As we might represent the same <em>point</em> with <em>vectors</em>
from various origins, the <strong>mp-units</strong> library provides
facilities to convert the <em>point</em> to the
<code class="sourceCode default">quantity_point</code> class templates
expressed in terms of different origins.</p>
<p>For this purpose, we can either use:</p>
<ul>
<li><p>A converting constructor:</p>
<div class="sourceCode" id="cb137"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb137-1"><a href="#cb137-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> quantity_point<span class="op">&lt;</span>isq<span class="op">::</span>altitude<span class="op">[</span>m<span class="op">]</span>, mean_sea_level, <span class="dt">int</span><span class="op">&gt;</span> qp <span class="op">=</span> first_climb_alt;</span>
<span id="cb137-2"><a href="#cb137-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>qp<span class="op">.</span>quantity_from_origin<span class="op">()</span> <span class="op">==</span> <span class="dv">5406</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div></li>
<li><p>A dedicated conversion interface:</p>
<div class="sourceCode" id="cb138"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb138-1"><a href="#cb138-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> quantity_point qp <span class="op">=</span> first_climb_alt<span class="op">.</span>point_for<span class="op">(</span>mean_sea_level<span class="op">)</span>;</span>
<span id="cb138-2"><a href="#cb138-2" aria-hidden="true" tabindex="-1"></a><span class="kw">static_assert</span><span class="op">(</span>qp<span class="op">.</span>quantity_from_origin<span class="op">()</span> <span class="op">==</span> <span class="dv">5406</span> <span class="op">*</span> m<span class="op">)</span>;</span></code></pre></div></li>
</ul>
<p>It is only allowed to convert between various origins defined in
terms of the same
<code class="sourceCode default">absolute_point_origin</code>. Even if
it is possible to express the same <em>point</em> as a <em>vector</em>
from another
<code class="sourceCode default">absolute_point_origin</code>, the
library will not provide such a conversion. A custom user-defined
conversion function will be needed to add this functionality.</p>
<p>Said another way, in the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library, there is no way to
spell how two distinct
<code class="sourceCode default">absolute_point_origin</code> types
relate to each other.</p>
<h3 data-number="10.3.5" id="point-arithmetics"><span class="header-section-number">10.3.5</span> <em>Point</em> arithmetics<a href="#point-arithmetics" class="self-link"></a></h3>
<p>Let’s assume we will attend the CppCon conference hosted in Aurora,
CO, and we want to estimate the distance we will travel. We have to take
a taxi to a local airport, fly to DEN airport with a stopover in FRA,
and, in the end, get a cab to the Gaylord Rockies Resort &amp;
Convention Center:</p>
<div class="sourceCode" id="cb139"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb139-1"><a href="#cb139-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">struct</span> home <span class="op">:</span> absolute_point_origin<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">&gt;</span> <span class="op">{}</span> home;</span>
<span id="cb139-2"><a href="#cb139-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb139-3"><a href="#cb139-3" aria-hidden="true" tabindex="-1"></a>quantity_point<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]</span>, home<span class="op">&gt;</span> home_airport <span class="op">=</span> home <span class="op">+</span> <span class="dv">15</span> <span class="op">*</span> km;</span>
<span id="cb139-4"><a href="#cb139-4" aria-hidden="true" tabindex="-1"></a>quantity_point<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]</span>, home<span class="op">&gt;</span> fra_airport <span class="op">=</span> home_airport <span class="op">+</span> <span class="dv">829</span> <span class="op">*</span> km;</span>
<span id="cb139-5"><a href="#cb139-5" aria-hidden="true" tabindex="-1"></a>quantity_point<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]</span>, home<span class="op">&gt;</span> den_airport <span class="op">=</span> fra_airport <span class="op">+</span> <span class="dv">8115</span> <span class="op">*</span> km;</span>
<span id="cb139-6"><a href="#cb139-6" aria-hidden="true" tabindex="-1"></a>quantity_point<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]</span>, home<span class="op">&gt;</span> cppcon_venue <span class="op">=</span> den_airport <span class="op">+</span> <span class="fl">10.1</span> <span class="op">*</span> mi;</span></code></pre></div>
<p>As we can see above, we can easily get a new point by adding a
quantity to an origin or another quantity point.</p>
<p>If we want to find out the distance traveled between two points, we
simply subtract them:</p>
<div class="sourceCode" id="cb140"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb140-1"><a href="#cb140-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]&gt;</span> total <span class="op">=</span> cppcon_venue <span class="op">-</span> home;</span>
<span id="cb140-2"><a href="#cb140-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]&gt;</span> flight <span class="op">=</span> den_airport <span class="op">-</span> home_airport;</span></code></pre></div>
<p>If we would like to find out the total distance traveled by taxi as
well, we have to do a bit more calculations:</p>
<div class="sourceCode" id="cb141"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb141-1"><a href="#cb141-1" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]&gt;</span> taxi1 <span class="op">=</span> home_airport <span class="op">-</span> home;</span>
<span id="cb141-2"><a href="#cb141-2" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]&gt;</span> taxi2 <span class="op">=</span> cppcon_venue <span class="op">-</span> den_airport;</span>
<span id="cb141-3"><a href="#cb141-3" aria-hidden="true" tabindex="-1"></a>quantity<span class="op">&lt;</span>isq<span class="op">::</span>distance<span class="op">[</span>km<span class="op">]&gt;</span> taxi <span class="op">=</span> taxi1 <span class="op">+</span> taxi2;</span></code></pre></div>
<p>Now, if we print the results:</p>
<div class="sourceCode" id="cb142"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb142-1"><a href="#cb142-1" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;Total distance:  &quot;</span> <span class="op">&lt;&lt;</span> total <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb142-2"><a href="#cb142-2" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;Flight distance: &quot;</span> <span class="op">&lt;&lt;</span> flight <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span>
<span id="cb142-3"><a href="#cb142-3" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>cout <span class="op">&lt;&lt;</span> <span class="st">&quot;Taxi distance:   &quot;</span> <span class="op">&lt;&lt;</span> taxi <span class="op">&lt;&lt;</span> <span class="st">&quot;</span><span class="sc">\n</span><span class="st">&quot;</span>;</span></code></pre></div>
<p>we will see the following output:</p>
<pre class="text"><code>Total distance:  8975.25 km
Flight distance: 8944 km
Taxi distance:   31.2544 km</code></pre>
<p>It is not allowed to subtract two point origins defined in terms of
<code class="sourceCode default">absolute_point_origin</code>
(e.g. <code class="sourceCode default">mean_sea_level - mean_sea_level</code>).
Those do not contain information about the unit, so we are not able to
determine the resulting <code class="sourceCode default">quantity</code>
type.</p>
<h3 data-number="10.3.6" id="temperature-support"><span class="header-section-number">10.3.6</span> Temperature support<a href="#temperature-support" class="self-link"></a></h3>
<p>Another important example of relative point origins is support of
temperature quantity points in units different than kelvin (symbol
K).</p>
<p>The <span class="citation" data-cites="SI">[<a href="#ref-SI" role="doc-biblioref">SI</a>]</span> definition in the <span class="citation" data-cites="MP-UNITS">[<a href="#ref-MP-UNITS" role="doc-biblioref">mp-units</a>]</span> library provides two
predefined point origins:</p>
<div class="sourceCode" id="cb143"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb143-1"><a href="#cb143-1" aria-hidden="true" tabindex="-1"></a><span class="kw">namespace</span> si <span class="op">{</span></span>
<span id="cb143-2"><a href="#cb143-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb143-3"><a href="#cb143-3" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> absolute_zero <span class="op">:</span> absolute_point_origin<span class="op">&lt;</span>isq<span class="op">::</span>thermodynamic_temperature<span class="op">&gt;</span> <span class="op">{}</span> absolute_zero;</span>
<span id="cb143-4"><a href="#cb143-4" aria-hidden="true" tabindex="-1"></a><span class="kw">inline</span> <span class="kw">constexpr</span> <span class="kw">struct</span> ice_point <span class="op">:</span> relative_point_origin<span class="op">&lt;</span>absolute_zero <span class="op">+</span> <span class="fl">273.15</span> <span class="op">*</span> kelvin<span class="op">&gt;</span> <span class="op">{}</span> ice_point;</span>
<span id="cb143-5"><a href="#cb143-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb143-6"><a href="#cb143-6" aria-hidden="true" tabindex="-1"></a><span class="op">}</span></span></code></pre></div>
<p>With the above, we can be explicit about the origin of our
temperature point. For example, if we want to implement the degrees
Celsius scale, we can do it as follows:</p>
<div class="sourceCode" id="cb144"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb144-1"><a href="#cb144-1" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> Celsius_point <span class="op">=</span> quantity_point<span class="op">&lt;</span>isq<span class="op">::</span>Celsius_temperature<span class="op">[</span>deg_C<span class="op">]</span>, si<span class="op">::</span>ice_point<span class="op">&gt;</span>;</span></code></pre></div>
<p>Notice that while stacking point origins, we can use not only
different representation types, but also different units for an origin
and a <em>point</em>. In the above example, the relative point origin is
defined in terms of <code class="sourceCode default">si::kelvin</code>,
while the quantity point uses
<code class="sourceCode default">si::degree_Celsius</code>.</p>
<p>To play a bit with temperatures, we can implement a simple room’s AC
temperature controller in the following way:</p>
<div class="sourceCode" id="cb145"><pre class="sourceCode cpp"><code class="sourceCode cpp"><span id="cb145-1"><a href="#cb145-1" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">struct</span> room_reference_temp <span class="op">:</span> relative_point_origin<span class="op">&lt;</span>si<span class="op">::</span>ice_point <span class="op">+</span> <span class="dv">21</span> <span class="op">*</span> deg_C<span class="op">&gt;</span> <span class="op">{}</span> room_reference_temp;</span>
<span id="cb145-2"><a href="#cb145-2" aria-hidden="true" tabindex="-1"></a><span class="kw">using</span> room_temp <span class="op">=</span> quantity_point<span class="op">&lt;</span>isq<span class="op">::</span>Celsius_temperature<span class="op">[</span>deg_C<span class="op">]</span>, room_reference_temp<span class="op">&gt;</span>;</span>
<span id="cb145-3"><a href="#cb145-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb145-4"><a href="#cb145-4" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="kw">auto</span> step_delta <span class="op">=</span> isq<span class="op">::</span>Celsius_temperature<span class="op">(</span><span class="fl">0.5</span> <span class="op">*</span> deg_C<span class="op">)</span>;</span>
<span id="cb145-5"><a href="#cb145-5" aria-hidden="true" tabindex="-1"></a><span class="kw">constexpr</span> <span class="dt">int</span> number_of_steps <span class="op">=</span> <span class="dv">6</span>;</span>
<span id="cb145-6"><a href="#cb145-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb145-7"><a href="#cb145-7" aria-hidden="true" tabindex="-1"></a>room_temp room_low <span class="op">=</span> room_reference_temp <span class="op">-</span> number_of_steps <span class="op">*</span> step_delta;</span>
<span id="cb145-8"><a href="#cb145-8" aria-hidden="true" tabindex="-1"></a>room_temp room_high <span class="op">=</span> room_reference_temp <span class="op">+</span> number_of_steps <span class="op">*</span> step_delta;</span>
<span id="cb145-9"><a href="#cb145-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb145-10"><a href="#cb145-10" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>println<span class="op">(</span><span class="st">&quot;| {:&lt;14} | {:^18} | {:^18} | {:^18} |&quot;</span>, <span class="st">&quot;Temperature&quot;</span>, <span class="st">&quot;Room reference&quot;</span>, <span class="st">&quot;Ice point&quot;</span>, <span class="st">&quot;Absolute zero&quot;</span><span class="op">)</span>;</span>
<span id="cb145-11"><a href="#cb145-11" aria-hidden="true" tabindex="-1"></a>std<span class="op">::</span>println<span class="op">(</span><span class="st">&quot;|{0:=^16}|{0:=^20}|{0:=^20}|{0:=^20}|&quot;</span>, <span class="st">&quot;&quot;</span><span class="op">)</span>;</span>
<span id="cb145-12"><a href="#cb145-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb145-13"><a href="#cb145-13" aria-hidden="true" tabindex="-1"></a><span class="kw">auto</span> print <span class="op">=</span> <span class="op">[&amp;](</span>std<span class="op">::</span>string_view label, <span class="kw">auto</span> v<span class="op">){</span></span>
<span id="cb145-14"><a href="#cb145-14" aria-hidden="true" tabindex="-1"></a>  std<span class="op">::</span>println<span class="op">(</span><span class="st">&quot;| {:&lt;14} | {:^18} | {:^18} | {:^18} |&quot;</span>,</span>
<span id="cb145-15"><a href="#cb145-15" aria-hidden="true" tabindex="-1"></a>               label, v <span class="op">-</span> room_reference_temp, v <span class="op">-</span> si<span class="op">::</span>ice_point, v <span class="op">-</span> si<span class="op">::</span>absolute_zero<span class="op">)</span>;</span>
<span id="cb145-16"><a href="#cb145-16" aria-hidden="true" tabindex="-1"></a><span class="op">}</span>;</span>
<span id="cb145-17"><a href="#cb145-17" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb145-18"><a href="#cb145-18" aria-hidden="true" tabindex="-1"></a>print<span class="op">(</span><span class="st">&quot;Lowest&quot;</span>, room_low<span class="op">)</span>;</span>
<span id="cb145-19"><a href="#cb145-19" aria-hidden="true" tabindex="-1"></a>print<span class="op">(</span><span class="st">&quot;Default&quot;</span>, room_reference_temp<span class="op">)</span>;</span>
<span id="cb145-20"><a href="#cb145-20" aria-hidden="true" tabindex="-1"></a>print<span class="op">(</span><span class="st">&quot;Highest&quot;</span>, room_high<span class="op">)</span>;</span></code></pre></div>
<p>The above prints:</p>
<pre class="text"><code>| Temperature    |   Room reference   |     Ice point      |   Absolute zero    |
|================|====================|====================|====================|
| Lowest         |       -3 °C        |       18 °C        |     291.15 °C      |
| Default        |        0 °C        |       21 °C        |     294.15 °C      |
| Highest        |        3 °C        |       24 °C        |     297.15 °C      |</code></pre>
<h3 data-number="10.3.7" id="no-text-output-for-points"><span class="header-section-number">10.3.7</span> No text output for
<em>points</em><a href="#no-text-output-for-points" class="self-link"></a></h3>
<p>The library does not provide text output for quantity points, as
printing just a number and a unit is not enough to adequately describe a
quantity point. Often, an additional postfix is required.</p>
<p>For example, the text output of
<code class="sourceCode default">42 m</code> may mean many things and
can also be confused with an output of a regular quantity. On the other
hand, printing <code class="sourceCode default">42 m AMSL</code> for
altitudes above mean sea level is a much better solution, but the
library does not have enough information to print it that way by
itself.</p>
<h2 data-number="10.4" id="the-affine-space-is-about-type-safety"><span class="header-section-number">10.4</span> The affine space is about
type-safety<a href="#the-affine-space-is-about-type-safety" class="self-link"></a></h2>
<p>The following operations are not allowed in the affine space:</p>
<ul>
<li><strong>adding</strong> two
<code class="sourceCode default">quantity_point</code> objects
<ul>
<li>It is physically impossible to add the positions of home and Denver
airport.</li>
</ul></li>
<li><strong>subtracting</strong> a
<code class="sourceCode default">quantity_point</code> from a
<code class="sourceCode default">quantity</code>
<ul>
<li>What would it mean to subtract the DEN airport location from a
distance?</li>
</ul></li>
<li><strong>multiplying/dividing</strong> a
<code class="sourceCode default">quantity_point</code> with a scalar
<ul>
<li>What is the position of <code class="sourceCode default">2 *</code>
DEN airport location?</li>
</ul></li>
<li><strong>multiplying/dividing</strong> a
<code class="sourceCode default">quantity_point</code> with a quantity
<ul>
<li>What would multiplying a distance with the DEN airport location
mean?</li>
</ul></li>
<li><strong>multiplying/dividing</strong> two
<code class="sourceCode default">quantity_point</code> objects
<ul>
<li>What would multiplying the home and DEN airport location mean?</li>
</ul></li>
<li><strong>mixing</strong>
<code class="sourceCode default">quantity_points</code> of different
quantity kinds
<ul>
<li>It is physically impossible to subtract ISQ time from ISQ
length.</li>
</ul></li>
<li><strong>mixing</strong>
<code class="sourceCode default">quantity_points</code> of inconvertible
quantities
<ul>
<li>What does it mean to subtract a distance point to DEN airport from
the Mount Everest base camp altitude?</li>
</ul></li>
<li><strong>mixing</strong>
<code class="sourceCode default">quantity_points</code> of convertible
quantities but with unrelated origins
<ul>
<li>How to subtract a point on our trip to CppCon measured relatively to
our home location from a point measured relative to the center of the
Solar System?</li>
</ul></li>
</ul>
<p>The usage of <code class="sourceCode default">quantity_point</code>,
and affine space types in general, improves expressiveness and
type-safety of the code we write.</p>
<h1 data-number="11" id="acknowledgements"><span class="header-section-number">11</span> Acknowledgements<a href="#acknowledgements" class="self-link"></a></h1>
<p>Special thanks and recognition goes to <a href="http://www.epam.com">Epam Systems</a> for supporting Mateusz’s
membership in the ISO C++ Committee and the production of this
proposal.</p>
<p>We would also like to thank Peter Sommerlad for providing valuable
feedback that helped us shape the final version of this document.</p>
<!-- markdownlint-disable -->
<h1 data-number="12" id="bibliography"><span class="header-section-number">12</span> References<a href="#bibliography" class="self-link"></a></h1>
<div id="refs" class="references csl-bib-body hanging-indent" role="doc-bibliography">
<div id="ref-AU" class="csl-entry" role="doc-biblioentry">
[Au] The Au Units library. <a href="https://aurora-opensource.github.io/au"><div class="csl-block">https://aurora-opensource.github.io/au</div></a>
</div>
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[Boost.Units] Matthias C. Schabel and Steven Watanabe. Boost.Units. <a href="https://www.boost.org/doc/libs/1_83_0/doc/html/boost_units.html"><div class="csl-block">https://www.boost.org/doc/libs/1_83_0/doc/html/boost_units.html</div></a>
</div>
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[ISO/IEC 80000] ISO/IEC 80000: Quantities and units. <a href="https://www.iso.org/committee/46202/x/catalogue/"><div class="csl-block">https://www.iso.org/committee/46202/x/catalogue/</div></a>
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[ISO/IEC Guide 99] ISO/IEC Guide 99: International vocabulary of
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</div>
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[JCGM 200:2012] International vocabulary of metrology - Basic and
general concepts and associated terms (VIM) (JCGM 200:2012, 3rd
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[Measurement Data] William Kent, Stephanie Leichner Janowski, Bruce
Hamilton, and Dan Hepner. Measurement Data (Archive Report). <a href="https://www.bkent.net/Doc/mdarchiv.pdf"><div class="csl-block">https://www.bkent.net/Doc/mdarchiv.pdf</div></a>
</div>
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[mp-units] mp-units - A Physical Quantities and Units library for C++.
<a href="https://mpusz.github.io/mp-units"><div class="csl-block">https://mpusz.github.io/mp-units</div></a>
</div>
<div id="ref-P2980R0" class="csl-entry" role="doc-biblioentry">
[P2980R0] Mateusz Pusz, Dominik Berner, Johel Ernesto Guerrero Peña,
Charles Hogg, Nicolas Holthaus, Roth Michaels, Vincent Reverdy.
2023-10-15. A motivation, scope, and plan for a physical quantities and
units library. <a href="https://wg21.link/p2980r0"><div class="csl-block">https://wg21.link/p2980r0</div></a>
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<div id="ref-P2982R0" class="csl-entry" role="doc-biblioentry">
[P2982R0] Mateusz Pusz, Chip Hogg. 2023-10-15. `std::quantity` as a
numeric type. <a href="https://wg21.link/p2982r0"><div class="csl-block">https://wg21.link/p2982r0</div></a>
</div>
<div id="ref-P2993_PRE" class="csl-entry" role="doc-biblioentry">
[P2993] Luke Valenty. Constrained Numbers. <a href="https://wg21.link/p2993"><div class="csl-block">https://wg21.link/p2993</div></a>
</div>
<div id="ref-P3003R0" class="csl-entry" role="doc-biblioentry">
[P3003R0] Johel Ernesto Guerrero Peña. 2023-10-14. The design of a
library of number concepts. <a href="https://wg21.link/p3003r0"><div class="csl-block">https://wg21.link/p3003r0</div></a>
</div>
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[Pint] Pint: makes units easy. <a href="https://pint.readthedocs.io/en/stable/index.html"><div class="csl-block">https://pint.readthedocs.io/en/stable/index.html</div></a>
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[SI] SI Brochure: The International System of Units (SI). <a href="https://www.bipm.org/en/publications/si-brochure"><div class="csl-block">https://www.bipm.org/en/publications/si-brochure</div></a>
</div>
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