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 <head>
  <meta content="text/html; charset=utf-8" http-equiv="Content-Type">
  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport">
  <title>P0571r0: Type Requirements for &lt;numeric> Algorithms</title>
<style data-fill-with="stylesheet">/******************************************************************************
 *                   Style sheet for the W3C specifications                   *
 *
 * Special classes handled by this style sheet include:
 *
 * Indices
 *   - .toc for the Table of Contents (<ol class="toc">)
 *     + <span class="secno"> for the section numbers
 *   - #toc for the Table of Contents (<nav id="toc">)
 *   - ul.index for Indices (<a href="#ref">term</a><span>, in §N.M</span>)
 *   - table.index for Index Tables (e.g. for properties or elements)
 *
 * Structural Markup
 *   - table.data for general data tables
 *     -> use 'scope' attribute, <colgroup>, <thead>, and <tbody> for best results !
 *     -> use <table class='complex data'> for extra-complex tables
 *     -> use <td class='long'> for paragraph-length cell content
 *     -> use <td class='pre'> when manual line breaks/indentation would help readability
 *   - dl.switch for switch statements
 *   - ol.algorithm for algorithms (helps to visualize nesting)
 *   - .figure and .caption (HTML4) and figure and figcaption (HTML5)
 *     -> .sidefigure for right-floated figures
 *   - ins/del
 *
 * Code
 *   - pre and code
 *
 * Special Sections
 *   - .note       for informative notes             (div, p, span, aside, details)
 *   - .example    for informative examples          (div, p, pre, span)
 *   - .issue      for issues                        (div, p, span)
 *   - .assertion  for assertions                    (div, p, span)
 *   - .advisement for loud normative statements     (div, p, strong)
 *   - .annoying-warning for spec obsoletion notices (div, aside, details)
 *
 * Definition Boxes
 *   - pre.def   for WebIDL definitions
 *   - table.def for tables that define other entities (e.g. CSS properties)
 *   - dl.def    for definition lists that define other entitles (e.g. HTML elements)
 *
 * Numbering
 *   - .secno for section numbers in .toc and headings (<span class='secno'>3.2</span>)
 *   - .marker for source-inserted example/figure/issue numbers (<span class='marker'>Issue 4</span>)
 *   - ::before styled for CSS-generated issue/example/figure numbers:
 *     -> Documents wishing to use this only need to add
 *        figcaption::before,
 *        .caption::before { content: "Figure "  counter(figure) " ";  }
 *        .example::before { content: "Example " counter(example) " "; }
 *        .issue::before   { content: "Issue "   counter(issue) " ";   }
 *
 * Header Stuff (ignore, just don't conflict with these classes)
 *   - .head for the header
 *   - .copyright for the copyright
 *
 * Miscellaneous
 *   - .overlarge for things that should be as wide as possible, even if
 *     that overflows the body text area. This can be used on an item or
 *     on its container, depending on the effect desired.
 *     Note that this styling basically doesn't help at all when printing,
 *     since A4 paper isn't much wider than the max-width here.
 *     It's better to design things to fit into a narrower measure if possible.
 *   - js-added ToC jump links (see fixup.js)
 *
 ******************************************************************************/

/******************************************************************************/
/*                                   Body                                     */
/******************************************************************************/

	body {
		counter-reset: example figure issue;

		/* Layout */
		max-width: 50em;               /* limit line length to 50em for readability   */
		margin: 0 auto;                /* center text within page                     */
		padding: 1.6em 1.5em 2em 50px; /* assume 16px font size for downlevel clients */
		padding: 1.6em 1.5em 2em calc(26px + 1.5em); /* leave space for status flag     */

		/* Typography */
		line-height: 1.5;
		font-family: sans-serif;
		widows: 2;
		orphans: 2;
		word-wrap: break-word;
		overflow-wrap: break-word;
		hyphens: auto;

		/* Colors */
		color: black;
		background: white top left fixed no-repeat;
		background-size: 25px auto;
	}


/******************************************************************************/
/*                         Front Matter & Navigation                          */
/******************************************************************************/

/** Header ********************************************************************/

	div.head { margin-bottom: 1em }
	div.head hr { border-style: solid; }

	div.head h1 {
		font-weight: bold;
		margin: 0 0 .1em;
		font-size: 220%;
	}

	div.head h2 { margin-bottom: 1.5em;}

/** W3C Logo ******************************************************************/

	.head .logo {
		float: right;
		margin: 0.4rem 0 0.2rem .4rem;
	}

	.head img[src*="logos/W3C"] {
		display: block;
		border: solid #1a5e9a;
		border-width: .65rem .7rem .6rem;
		border-radius: .4rem;
		background: #1a5e9a;
		color: white;
		font-weight: bold;
	}

	.head a:hover > img[src*="logos/W3C"],
	.head a:focus > img[src*="logos/W3C"] {
		opacity: .8;
	}

	.head a:active > img[src*="logos/W3C"] {
		background: #c00;
		border-color: #c00;
	}

	/* see also additional rules in Link Styling section */

/** Copyright *****************************************************************/

	p.copyright,
	p.copyright small { font-size: small }

/** Back to Top / ToC Toggle **************************************************/

	@media print {
		#toc-nav {
			display: none;
		}
	}
	@media not print {
		#toc-nav {
			position: fixed;
			z-index: 2;
			bottom: 0; left: 0;
			margin: 0;
			min-width: 1.33em;
			border-top-right-radius: 2rem;
			box-shadow: 0 0 2px;
			font-size: 1.5em;
			color: black;
		}
		#toc-nav > a {
			display: block;
			white-space: nowrap;

			height: 1.33em;
			padding: .1em 0.3em;
			margin: 0;

			background: white;
			box-shadow: 0 0 2px;
			border: none;
			border-top-right-radius: 1.33em;
			background: white;
		}
		#toc-nav > #toc-jump {
			padding-bottom: 2em;
			margin-bottom: -1.9em;
		}

		#toc-nav > a:hover,
		#toc-nav > a:focus {
			background: #f8f8f8;
		}
		#toc-nav > a:not(:hover):not(:focus) {
			color: #707070;
		}

		/* statusbar gets in the way on keyboard focus; remove once browsers fix */
		#toc-nav > a[href="#toc"]:not(:hover):focus:last-child {
			padding-bottom: 1.5rem;
		}

		#toc-nav:not(:hover) > a:not(:focus) > span + span {
			/* Ideally this uses :focus-within on #toc-nav */
			display: none;
		}
		#toc-nav > a > span + span {
			padding-right: 0.2em;
		}

		#toc-toggle-inline {
			vertical-align: 0.05em;
			font-size: 80%;
			color: gray;
			color: hsla(203,20%,40%,.7);
			border-style: none;
			background: transparent;
			position: relative;
		}
		#toc-toggle-inline:hover:not(:active),
		#toc-toggle-inline:focus:not(:active) {
			text-shadow: 1px 1px silver;
			top: -1px;
			left: -1px;
		}

		#toc-nav :active {
			color: #C00;
		}
	}

/** ToC Sidebar ***************************************************************/

	/* Floating sidebar */
	@media screen {
		body.toc-sidebar #toc {
			position: fixed;
			top: 0; bottom: 0;
			left: 0;
			width: 23.5em;
			max-width: 80%;
			max-width: calc(100% - 2em - 26px);
			overflow: auto;
			padding: 0 1em;
			padding-left: 42px;
			padding-left: calc(1em + 26px);
			background: inherit;
			background-color: #f7f8f9;
			z-index: 1;
			box-shadow: -.1em 0 .25em rgba(0,0,0,.1) inset;
		}
		body.toc-sidebar #toc h2 {
			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
			font-weight: bold;
			color: gray;
			color: hsla(203,20%,40%,.7);
		}
		body.toc-sidebar #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
		}
	}
	/* Hide main scroller when only the ToC is visible anyway */
	@media screen and (max-width: 28em) {
		body.toc-sidebar {
			overflow: hidden;
		}
	}

	/* Sidebar with its own space */
	@media screen and (min-width: 78em) {
		body:not(.toc-inline) #toc {
			position: fixed;
			top: 0; bottom: 0;
			left: 0;
			width: 23.5em;
			overflow: auto;
			padding: 0 1em;
			padding-left: 42px;
			padding-left: calc(1em + 26px);
			background: inherit;
			background-color: #f7f8f9;
			z-index: 1;
			box-shadow: -.1em 0 .25em rgba(0,0,0,.1) inset;
		}
		body:not(.toc-inline) #toc h2 {
			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
			font-weight: bold;
			color: gray;
			color: hsla(203,20%,40%,.7);
		}

		body:not(.toc-inline) {
			padding-left: 29em;
		}
		/* See also Overflow section at the bottom */

		body:not(.toc-inline) #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
		}
	}
	@media screen and (min-width: 90em) {
		body:not(.toc-inline) {
			margin: 0 4em;
		}
	}

/******************************************************************************/
/*                                Sectioning                                  */
/******************************************************************************/

/** Headings ******************************************************************/

	h1, h2, h3, h4, h5, h6, dt {
		page-break-after: avoid;
		page-break-inside: avoid;
		font: 100% sans-serif;   /* Reset all font styling to clear out UA styles */
		font-family: inherit;    /* Inherit the font family. */
		line-height: 1.2;        /* Keep wrapped headings compact */
		hyphens: manual;         /* Hyphenated headings look weird */
	}

	h2, h3, h4, h5, h6 {
		margin-top: 3rem;
	}

	h1, h2, h3 {
		color: #005A9C;
		background: transparent;
	}

	h1 { font-size: 170%; }
	h2 { font-size: 140%; }
	h3 { font-size: 120%; }
	h4 { font-weight: bold; }
	h5 { font-style: italic; }
	h6 { font-variant: small-caps; }
	dt { font-weight: bold; }

/** Subheadings ***************************************************************/

	h1 + h2,
	#subtitle {
		/* #subtitle is a subtitle in an H2 under the H1 */
		margin-top: 0;
	}
	h2 + h3,
	h3 + h4,
	h4 + h5,
	h5 + h6 {
		margin-top: 1.2em; /* = 1 x line-height */
	}

/** Section divider ***********************************************************/

	:not(.head) > hr {
		font-size: 1.5em;
		text-align: center;
		margin: 1em auto;
		height: auto;
		border: transparent solid 0;
		background: transparent;
	}
	:not(.head) > hr::before {
		content: "\2727\2003\2003\2727\2003\2003\2727";
	}

/******************************************************************************/
/*                            Paragraphs and Lists                            */
/******************************************************************************/

	p {
		margin: 1em 0;
	}

	dd > p:first-child,
	li > p:first-child {
		margin-top: 0;
	}

	ul, ol {
		margin-left: 0;
		padding-left: 2em;
	}

	li {
		margin: 0.25em 0 0.5em;
		padding: 0;
	}

	dl dd {
		margin: 0 0 .5em 2em;
	}

	.head dd + dd { /* compact for header */
		margin-top: -.5em;
	}

	/* Style for algorithms */
	ol.algorithm ol:not(.algorithm),
	.algorithm > ol ol:not(.algorithm) {
	 border-left: 0.5em solid #DEF;
	}

	/* Put nice boxes around each algorithm. */
	[data-algorithm]:not(.heading) {
	  padding: .5em;
	  border: thin solid #ddd; border-radius: .5em;
	  margin: .5em 0;
	}
	[data-algorithm]:not(.heading) > :first-child {
	  margin-top: 0;
	}
	[data-algorithm]:not(.heading) > :last-child {
	  margin-bottom: 0;
	}

	/* Style for switch/case <dl>s */
	dl.switch > dd > ol.only,
	dl.switch > dd > .only > ol {
	 margin-left: 0;
	}
	dl.switch > dd > ol.algorithm,
	dl.switch > dd > .algorithm > ol {
	 margin-left: -2em;
	}
	dl.switch {
	 padding-left: 2em;
	}
	dl.switch > dt {
	 text-indent: -1.5em;
	 margin-top: 1em;
	}
	dl.switch > dt + dt {
	 margin-top: 0;
	}
	dl.switch > dt::before {
	 content: '\21AA';
	 padding: 0 0.5em 0 0;
	 display: inline-block;
	 width: 1em;
	 text-align: right;
	 line-height: 0.5em;
	}

/** Terminology Markup ********************************************************/


/******************************************************************************/
/*                                 Inline Markup                              */
/******************************************************************************/

/** Terminology Markup ********************************************************/
	dfn   { /* Defining instance */
		font-weight: bolder;
	}
	a > i { /* Instance of term */
		font-style: normal;
	}
	dt dfn code, code.idl {
		font-size: normal;
	}
	dfn var {
		font-style: normal;
	}

/** Change Marking ************************************************************/

	del { color: red !important;  background-color: #FFB0B0; text-decoration: line-through; }
	ins { color: #080 !important; background-color: #B0FFB0; text-decoration: none;         }

	del code, del pre, .del code, .del pre, code del, pre del, code .del, pre .del { background-color: #FFB0B0 !important; text-decoration: line-through; }
	ins code, ins pre, .ins code, .ins pre, code ins, pre ins, code .ins, pre .ins { background-color: #B0FFB0 !important; text-decoration: none;         }

/** Miscellaneous improvements to inline formatting ***************************/

	sup {
		vertical-align: super;
		font-size: 80%
	}

/******************************************************************************/
/*                                    Code                                    */
/******************************************************************************/

/** General monospace/pre rules ***********************************************/

	pre, code, samp {
		font-family: Menlo, Consolas, "DejaVu Sans Mono", Monaco, monospace;
		font-size: .9em;
		page-break-inside: avoid;
		hyphens: none;
		text-transform: none;
	}
	pre code,
	code code {
		font-size: 100%;
	}

	pre {
		margin-top: 1em;
		margin-bottom: 1em;
		overflow: auto;
	}

/** Inline Code fragments *****************************************************/

  /* Do something nice. */

/******************************************************************************/
/*                                    Links                                   */
/******************************************************************************/

/** General Hyperlinks ********************************************************/

	/* We hyperlink a lot, so make it less intrusive */
	a[href] {
		color: #034575;
		text-decoration: none;
		border-bottom: 1px solid #707070;
		/* Need a bit of extending for it to look okay */
		padding: 0 1px 0;
		margin: 0 -1px 0;
	}
	a:visited {
		border-bottom-color: #BBB;
	}

	/* Use distinguishing colors when user is interacting with the link */
	a[href]:focus,
	a[href]:hover {
		background: #f8f8f8;
		background: rgba(75%, 75%, 75%, .25);
		border-bottom-width: 3px;
		margin-bottom: -2px;
	}
	a[href]:active {
		color: #C00;
		border-color: #C00;
	}

	/* Backout above styling for W3C logo */
	.head .logo,
	.head .logo a {
		border: none;
		text-decoration: none;
		background: transparent;
	}

/******************************************************************************/
/*                                    Images                                  */
/******************************************************************************/

	img {
		border-style: none;
	}

	/* For autogen numbers, add
	   .caption::before, figcaption::before { content: "Figure " counter(figure) ". "; }
	*/

	figure, .figure, .sidefigure {
		page-break-inside: avoid;
		text-align: center;
		margin: 2.5em 0;
	}
	.figure img,    .sidefigure img,    figure img,
	.figure object, .sidefigure object, figure object {
		max-width: 100%;
		margin: auto;
	}
	.figure pre, .sidefigure pre, figure pre {
		text-align: left;
		display: table;
		margin: 1em auto;
	}
	.figure table, figure table {
		margin: auto;
	}
	@media screen and (min-width: 20em) {
		.sidefigure {
			float: right;
			width: 50%;
			margin: 0 0 0.5em 0.5em
		}
	}
	.caption, figcaption, caption {
		font-style: italic;
		font-size: 90%;
	}
	.caption::before, figcaption::before, figcaption > .marker {
		font-weight: bold;
	}
	.caption, figcaption {
		counter-increment: figure;
	}

	/* DL list is indented 2em, but figure inside it is not */
	dd > .figure, dd > figure { margin-left: -2em }

/******************************************************************************/
/*                             Colored Boxes                                  */
/******************************************************************************/

	.issue, .note, .example, .assertion, .advisement, blockquote {
		padding: .5em;
		border: .5em;
		border-left-style: solid;
		page-break-inside: avoid;
	}
	span.issue, span.note {
		padding: .1em .5em .15em;
		border-right-style: solid;
	}

	.issue,
	.note,
	.example,
	.advisement,
	.assertion,
	blockquote {
		margin: 1em auto;
	}
	.note  > p:first-child,
	.issue > p:first-child,
	blockquote > :first-child {
		margin-top: 0;
	}
	blockquote > :last-child {
		margin-bottom: 0;
	}

/** Blockquotes ***************************************************************/

	blockquote {
		border-color: silver;
	}

/** Open issue ****************************************************************/

	.issue {
		border-color: #E05252;
		background: #FBE9E9;
		counter-increment: issue;
		overflow: auto;
	}
	.issue::before, .issue > .marker {
		text-transform: uppercase;
		color: #AE1E1E;
		padding-right: 1em;
		text-transform: uppercase;
	}
	/* Add .issue::before { content: "Issue " counter(issue) " "; } for autogen numbers,
	   or use class="marker" to mark up the issue number in source. */

/** Example *******************************************************************/

	.example {
		border-color: #E0CB52;
		background: #FCFAEE;
		counter-increment: example;
		overflow: auto;
		clear: both;
	}
	.example::before, .example > .marker {
		text-transform: uppercase;
		color: #827017;
		min-width: 7.5em;
		display: block;
	}
	/* Add .example::before { content: "Example " counter(example) " "; } for autogen numbers,
	   or use class="marker" to mark up the example number in source. */

/** Non-normative Note ********************************************************/

	.note {
		border-color: #52E052;
		background: #E9FBE9;
		overflow: auto;
	}

	.note::before, .note > .marker,
	details.note > summary::before,
	details.note > summary > .marker {
		text-transform: uppercase;
		display: block;
		color: hsl(120, 70%, 30%);
	}
	/* Add .note::before { content: "Note"; } for autogen label,
	   or use class="marker" to mark up the label in source. */

	details.note > summary {
		display: block;
		color: hsl(120, 70%, 30%);
	}
	details.note[open] > summary {
		border-bottom: 1px silver solid;
	}

/** Assertion Box *************************************************************/
	/*  for assertions in algorithms */

	.assertion {
		border-color: #AAA;
		background: #EEE;
	}

/** Advisement Box ************************************************************/
	/*  for attention-grabbing normative statements */

	.advisement {
		border-color: orange;
		border-style: none solid;
		background: #FFEECC;
	}
	strong.advisement {
		display: block;
		text-align: center;
	}
	.advisement > .marker {
		color: #B35F00;
	}

/** Spec Obsoletion Notice ****************************************************/
	/* obnoxious obsoletion notice for older/abandoned specs. */

	details {
		display: block;
	}
	summary {
		font-weight: bolder;
	}

	.annoying-warning:not(details),
	details.annoying-warning:not([open]) > summary,
	details.annoying-warning[open] {
		background: #fdd;
		color: red;
		font-weight: bold;
		padding: .75em 1em;
		border: thick red;
		border-style: solid;
		border-radius: 1em;
	}
	.annoying-warning :last-child {
		margin-bottom: 0;
	}

@media not print {
	details.annoying-warning[open] {
		position: fixed;
		left: 1em;
		right: 1em;
		bottom: 1em;
		z-index: 1000;
	}
}

	details.annoying-warning:not([open]) > summary {
		text-align: center;
	}

/** Entity Definition Boxes ***************************************************/

	.def {
		padding: .5em 1em;
		background: #DEF;
		margin: 1.2em 0;
		border-left: 0.5em solid #8CCBF2;
	}

/******************************************************************************/
/*                                    Tables                                  */
/******************************************************************************/

	th, td {
		text-align: left;
		text-align: start;
	}

/** Property/Descriptor Definition Tables *************************************/

	table.def {
		/* inherits .def box styling, see above */
		width: 100%;
		border-spacing: 0;
	}

	table.def td,
	table.def th {
		padding: 0.5em;
		vertical-align: baseline;
		border-bottom: 1px solid #bbd7e9;
	}

	table.def > tbody > tr:last-child th,
	table.def > tbody > tr:last-child td {
		border-bottom: 0;
	}

	table.def th {
		font-style: italic;
		font-weight: normal;
		padding-left: 1em;
		width: 3em;
	}

	/* For when values are extra-complex and need formatting for readability */
	table td.pre {
		white-space: pre-wrap;
	}

	/* A footnote at the bottom of a def table */
	table.def           td.footnote {
		padding-top: 0.6em;
	}
	table.def           td.footnote::before {
		content: " ";
		display: block;
		height: 0.6em;
		width: 4em;
		border-top: thin solid;
	}

/** Data tables (and properly marked-up index tables) *************************/
	/*
		 <table class="data"> highlights structural relationships in a table
		 when correct markup is used (e.g. thead/tbody, th vs. td, scope attribute)

		 Use class="complex data" for particularly complicated tables --
		 (This will draw more lines: busier, but clearer.)

		 Use class="long" on table cells with paragraph-like contents
		 (This will adjust text alignment accordingly.)
		 Alternately use class="longlastcol" on tables, to have the last column assume "long".
	*/

	table {
		word-wrap: normal;
		overflow-wrap: normal;
		hyphens: manual;
	}

	table.data,
	table.index {
		margin: 1em auto;
		border-collapse: collapse;
		border: hidden;
		width: 100%;
	}
	table.data caption,
	table.index caption {
		max-width: 50em;
		margin: 0 auto 1em;
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   <h1 class="p-name no-ref" id="title">P0571r0<br>Type Requirements for &lt;numeric> Algorithms</h1>
   <h2 class="no-num no-toc no-ref heading settled" id="subtitle"><span class="content">Published Proposal, <time class="dt-updated" datetime="2017-02-02">2 February 2017</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt>This version:
     <dd><a class="u-url" href="http://wg21.link/P0571r0">http://wg21.link/P0571r0</a>
     <dt>Author:
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:balelbach@lbl.gov">Bryce Adelstein Lelbach</a> (<span class="p-org org">Lawrence Berkeley National Laboratory</span>)
     <dt>Audience:
     <dd>SG1, LEWG, LWG
     <dt>Toggle Diffs:
     <dd>
      <label for="hidedel" id="hidedel-label">Hide deleted text</label>
     <dt>Project:
     <dd><span>ISO JTC1/SC22/WG21: Programming Language C++</span>
    </dl>
   </div>
   <div data-fill-with="warning"></div>
   <hr title="Separator for header">
  </div>
  <div class="p-summary" data-fill-with="abstract"></div>
  <nav data-fill-with="table-of-contents" id="toc">
   <h2 class="no-num no-toc no-ref" id="contents">Table of Contents</h2>
   <ol class="toc" role="directory">
    <li><a href="#overview"><span class="secno">1</span> <span class="content">Overview</span></a>
    <li><a href="#oldalgos"><span class="secno">2</span> <span class="content">Existing <code class="highlight"><span class="o">&lt;</span><span class="n">numeric</span><span class="o">></span></code> Algorithms</span></a>
    <li><a href="#paraalgos"><span class="secno">3</span> <span class="content">New Parallel <code class="highlight"><span class="o">&lt;</span><span class="n">numeric</span><span class="o">></span></code> Algorithms</span></a>
    <li>
     <a href="#references"><span class="secno"></span> <span class="content">References</span></a>
     <ol class="toc">
      <li><a href="#informative"><span class="secno"></span> <span class="content">Informative References</span></a>
     </ol>
   </ol>
  </nav>
  <main>
   <h2 class="heading settled" data-level="1" id="overview"><span class="secno">1. </span><span class="content">Overview</span><a class="self-link" href="#overview"></a></h2>
   <p>At the Issaquah 2016 meeting, during the LWG review of <a data-link-type="biblio" href="#biblio-p0452r0">[P0452r0]</a>, it was
realized that some of the numeric algorithms (both old algorithms and new ones
from the Parallelism TS) had insufficient or unclear type requirements.</p>
   <p>Consider:</p>
<pre class="highlight"><span class="n">vector</span><span class="o">&lt;</span><span class="kt">double</span><span class="o">></span> <span class="n">v</span><span class="p">;</span>
<span class="kt">double</span> <span class="n">d</span> <span class="o">=</span> <span class="n">accumulate</span><span class="p">(</span><span class="n">v</span><span class="p">.</span><span class="n">begin</span><span class="p">(</span><span class="p">)</span><span class="p">,</span> <span class="n">v</span><span class="p">.</span><span class="n">end</span><span class="p">(</span><span class="p">)</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span><span class="p">;</span>
</pre>
   <p>What type is used to store the intermediary result during accumulation in the
above example? Is it <code class="highlight"><span class="n">T</span></code>, the type of the initial value (which is <code class="highlight"><span class="kt">int</span></code> here)? Or
is it <code class="highlight"><span class="kt">double</span></code>, the value type of the <code class="highlight"><span class="n">InputIterator</span></code>s?</p>
   <p>The answer: it is <code class="highlight"><span class="n">T</span></code>, or <code class="highlight"><span class="kt">int</span></code> in the above example. For algorithms
which do not take an initial value (such as <code class="highlight"><span class="n">partial_sum</span></code> and <code class="highlight"><span class="n">adjacent_difference</span></code>) and instead write to an output sequence via an <code class="highlight"><span class="n">OutputIterator</span></code>, the value type of the <code class="highlight"><span class="n">InputIterator</span></code> is used. This may not
be the best or most consistent behavior, but it appears to have be the status
quo in major implementations.</p>
   <p>However, the specification lacks clarity about which types are used for
accumulation and the constraints on the template parameters to the algorithms.
This paper identifies some of the issues and proposes potential solutions.</p>
   <h2 class="heading settled" data-level="2" id="oldalgos"><span class="secno">2. </span><span class="content">Existing <code class="highlight"><span class="o">&lt;</span><span class="n">numeric</span><span class="o">></span></code> Algorithms</span><a class="self-link" href="#oldalgos"></a></h2>
   <p><code class="highlight"><span class="n">accumulate</span></code>, <code class="highlight"><span class="n">inner_product</span></code>, <code class="highlight"><span class="n">partial_sum</span></code> and <code class="highlight"><span class="n">adjacent_difference</span></code> are all
specified using similar "accumulator-style" wording:</p>
   <blockquote>
     <code class="highlight"><span class="n">accumulate</span></code> (26.8.2 [<strong>accumulate</strong>] paragraph 1): 
    <blockquote> <em>Effects:</em> Computes its result by initializing the accumulator <code class="highlight"><span class="n">acc</span></code> with the
initial value <code class="highlight"><span class="n">init</span></code> and then modifies it with <code class="highlight"><span class="n">acc</span> <span class="o">=</span> <span class="n">acc</span> <span class="o">+</span> <span class="o">*</span><span class="n">i</span></code> or <code class="highlight"><span class="n">acc</span> <span class="o">=</span> <span class="n">binary_op</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span> <span class="o">*</span><span class="n">i</span><span class="p">)</span></code> for every iterator <code class="highlight"><span class="n">i</span></code> in the range <code class="highlight"><span class="p">[</span><span class="n">first</span><span class="p">,</span> <span class="n">last</span><span class="p">)</span></code> in
order. </blockquote>
    <blockquote> <em>Requires:</em> <code class="highlight"><span class="n">T</span></code> shall meet the requirements of <code class="highlight"><span class="n">CopyConstructible</span></code> (Table 22) and <code class="highlight"><span class="n">CopyAssignable</span></code> (Table 24) types. In the range <code class="highlight"><span class="p">[</span><span class="n">first</span><span class="p">,</span> <span class="n">last</span><span class="p">]</span></code>, <code class="highlight"><span class="n">binary_op</span></code> shall
neither modify elements nor invalidate iterators or subranges. </blockquote>
   </blockquote>
   <blockquote>
     <code class="highlight"><span class="n">inner_product</span></code> (26.8.5 [<strong>inner.product</strong>] paragraph 1): 
    <blockquote> <em>Effects:</em> Computes its result by initializing the accumulator <code class="highlight"><span class="n">acc</span></code> with the
initial value <code class="highlight"><span class="n">init</span></code> and then modifying it with <code class="highlight"><span class="n">acc</span> <span class="o">=</span> <span class="n">acc</span> <span class="o">+</span> <span class="p">(</span><span class="o">*</span><span class="n">i1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="o">*</span><span class="n">i2</span><span class="p">)</span></code> or <code class="highlight"><span class="n">acc</span> <span class="o">=</span> <span class="n">binary_op1</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span> <span class="n">binary_op2</span><span class="p">(</span><span class="o">*</span><span class="n">i1</span><span class="p">,</span> <span class="o">*</span><span class="n">i2</span><span class="p">)</span><span class="p">)</span></code> for every iterator <code class="highlight"><span class="n">i1</span></code> in the
range <code class="highlight"><span class="p">[</span><span class="n">first1</span><span class="p">,</span> <span class="n">last1</span><span class="p">)</span></code> and iterator <code class="highlight"><span class="n">i2</span></code> in the range <code class="highlight"><span class="p">[</span><span class="n">first2</span><span class="p">,</span> <span class="n">first2</span> <span class="o">+</span> <span class="p">(</span><span class="n">last1</span> <span class="o">-</span> <span class="n">first1</span><span class="p">)</span><span class="p">)</span></code> in order. </blockquote>
    <blockquote> <em>Requires:</em> <code class="highlight"><span class="n">T</span></code> shall meet the requirements of <code class="highlight"><span class="n">CopyConstructible</span></code> (Table 22) and <code class="highlight"><span class="n">CopyAssignable</span></code> (Table 24) types. In the ranges <code class="highlight"><span class="p">[</span><span class="n">first1</span><span class="p">,</span> <span class="n">last1</span><span class="p">]</span></code> and <code class="highlight"><span class="p">[</span><span class="n">first2</span><span class="p">,</span> <span class="n">first2</span> <span class="o">+</span> <span class="p">(</span><span class="n">last1</span> <span class="o">-</span> <span class="n">first1</span><span class="p">)</span><span class="p">]</span></code> <code class="highlight"><span class="n">binary_op1</span></code> and <code class="highlight"><span class="n">binary_op2</span></code> shall
neither modify elements nor invalidate iterators or subranges. </blockquote>
   </blockquote>
   <blockquote>
     <code class="highlight"><span class="n">partial_sum</span></code> (26.8.6 [<strong>partial.sum</strong>] paragraph 1 and 4) 
    <blockquote> <em>Effects:</em> For a non-empty range, the function creates an accumulator <code class="highlight"><span class="n">acc</span></code> whose type is <code class="highlight"><span class="n">InputIterator</span></code>'s value type, initializes it with <code class="highlight"><span class="o">*</span><span class="n">first</span></code>, and
assigns the result to <code class="highlight"><span class="o">*</span><span class="n">result</span></code>. For every iterator <code class="highlight"><span class="n">i</span></code> in <code class="highlight"><span class="p">[</span><span class="n">first</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">last</span><span class="p">)</span></code> in order, <code class="highlight"><span class="n">acc</span></code> is then modified by <code class="highlight"><span class="n">acc</span> <span class="o">=</span> <span class="n">acc</span> <span class="o">+</span> <span class="o">*</span><span class="n">i</span></code> or <code class="highlight"><span class="n">acc</span> <span class="o">=</span> <span class="n">binary_op</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span> <span class="o">*</span><span class="n">i</span><span class="p">)</span></code> and the result is assigned to <code class="highlight"><span class="o">*</span><span class="p">(</span><span class="n">result</span> <span class="o">+</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">first</span><span class="p">)</span><span class="p">)</span></code>. </blockquote>
    <blockquote> <em>Requires:</em> <code class="highlight"><span class="n">InputIterator</span></code>'s value type shall be constructible from the type of <code class="highlight"><span class="o">*</span><span class="n">first</span></code>. The result of the expression <code class="highlight"><span class="n">acc</span> <span class="o">+</span> <span class="o">*</span><span class="n">i</span></code> or <code class="highlight"><span class="n">binary_op</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span> <span class="o">*</span><span class="n">i</span><span class="p">)</span></code> shall be
implicitly convertible to <code class="highlight"><span class="n">InputIterator</span></code>'s value type. <code class="highlight"><span class="n">acc</span></code> shall be writable
(24.2.1) to the result output iterator. In the ranges <code class="highlight"><span class="p">[</span><span class="n">first</span><span class="p">,</span> <span class="n">last</span><span class="p">]</span></code> and <code class="highlight"><span class="p">[</span><span class="n">result</span><span class="p">,</span> <span class="n">result</span> <span class="o">+</span> <span class="p">(</span><span class="n">last</span> <span class="o">-</span> <span class="n">first</span><span class="p">)</span><span class="p">]</span></code> <code class="highlight"><span class="n">binary_op</span></code> shall neither modify elements nor
invalidate iterators or subranges. </blockquote>
   </blockquote>
   <blockquote>
     <code class="highlight"><span class="n">adjacent_difference</span></code> (26.8.11 [<strong>adjacent.difference</strong>] paragraph 1 and 2) 
    <blockquote> <em>Effects:</em> For a non-empty range, the function creates an accumulator <code class="highlight"><span class="n">acc</span></code> whose type is <code class="highlight"><span class="n">InputIterator</span></code>'s value type, initializes it with *first, and
assigns the result to <code class="highlight"><span class="o">*</span><span class="n">result</span></code>. For every iterator <code class="highlight"><span class="n">i</span></code> in <code class="highlight"><span class="p">[</span><span class="n">first</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">last</span><span class="p">)</span></code> in order, creates an object val whose type is InputIterator’s value type,
initializes it with <code class="highlight"><span class="o">*</span><span class="n">i</span></code>, computes <code class="highlight"><span class="n">val</span> <span class="o">-</span> <span class="n">acc</span></code> or <code class="highlight"><span class="n">binary_op</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">acc</span><span class="p">)</span></code>,
assigns the result to <code class="highlight"><span class="o">*</span><span class="p">(</span><span class="n">result</span> <span class="o">+</span> <span class="p">(</span><span class="n">i</span> <span class="o">-</span> <span class="n">first</span><span class="p">)</span><span class="p">)</span></code>, and move assigns from <code class="highlight"><span class="n">val</span></code> to <code class="highlight"><span class="n">acc</span></code>. </blockquote>
    <blockquote> <em>Requires:</em> <code class="highlight"><span class="n">InputIterator</span></code>'s value type shall be <code class="highlight"><span class="n">MoveAssignable</span></code> (Table 23) and
shall be constructible from the type of <code class="highlight"><span class="o">*</span><span class="n">first</span></code>. <code class="highlight"><span class="n">acc</span></code> shall be writable (24.2.1)
to the result output iterator. The result of the expression <code class="highlight"><span class="n">val</span> <span class="o">-</span> <span class="n">acc</span></code> or <code class="highlight"><span class="n">binary_op</span><span class="p">(</span><span class="n">val</span><span class="p">,</span> <span class="n">acc</span><span class="p">)</span></code> shall be writable to the result output iterator. In the
ranges <code class="highlight"><span class="p">[</span><span class="n">first</span><span class="p">,</span> <span class="n">last</span><span class="p">]</span></code> and <code class="highlight"><span class="p">[</span><span class="n">result</span><span class="p">,</span> <span class="n">result</span> <span class="o">+</span> <span class="p">(</span><span class="n">last</span> <span class="o">-</span> <span class="n">first</span><span class="p">)</span><span class="p">]</span></code>, <code class="highlight"><span class="n">binary_op</span></code> shall
neither modify elements nor invalidate iterators or subranges. </blockquote>
   </blockquote>
   <p>The description of these four algorithms shares a similar structure:</p>
   <ul>
    <li data-md="">
     <p>Create an accumulator <code class="highlight"><span class="n">acc</span></code> which is initialized with <code class="highlight"><span class="n">init</span></code> or <code class="highlight"><span class="o">*</span><span class="n">first</span></code>.</p>
    <li data-md="">
     <p>For (the) iterator(s) in the range(s) <strong>in order</strong>, modify <code class="highlight"><span class="n">acc</span></code> by applying
an update operation which takes <code class="highlight"><span class="n">acc</span></code> and the dereferenced value(s) of the
iterator(s) as arguments.</p>
   </ul>
   <p>For the numeric algorithms which do not take an initial value (<code class="highlight"><span class="n">partial_sum</span></code> and <code class="highlight"><span class="n">adjacent_difference</span></code>), the accumulator <code class="highlight"><span class="n">acc</span></code>'s type is the <code class="highlight"><span class="n">InputIterator</span></code>'s 
value type.</p>
   <p>For the numeric algorithms which do take an initial value (<code class="highlight"><span class="n">accumulate</span></code> and <code class="highlight"><span class="n">inner_product</span></code>), it is not entirely clear what the type of the accumulator
should be, although it is strongly implied that it should be the type of the
initial value (<code class="highlight"><span class="n">T</span></code>) not the <code class="highlight"><span class="n">InputIterator</span></code>'s value type. The return value of
these two algorithms is <code class="highlight"><span class="n">T</span></code>, and the <em>Requires:</em> clauses for both algorithms
specify that <code class="highlight"><span class="n">T</span></code> must be <code class="highlight"><span class="n">CopyConstructible</span></code> and <code class="highlight"><span class="n">CopyAssignable</span></code>, which would
only make sense if the accumulator is to be of type <code class="highlight"><span class="n">T</span></code>. In libstdc++ and
libc++, the <code class="highlight"><span class="n">init</span></code> parameter is used as the accumulator (e.g. the accumulator
type is <code class="highlight"><span class="n">T</span></code>).</p>
   <p>I suggest that we change <code class="highlight"><span class="n">accumulate</span></code> and <code class="highlight"><span class="n">inner_product</span></code> to</p>
   <ul>
    <li data-md="">
     <p>Specify that the accumulator type is <code class="highlight"><span class="n">T</span></code>.</p>
    <li data-md="">
     <p>Require the result of the update expression for these algorithms ( <code class="highlight"><span class="n">acc</span> <span class="o">+</span> <span class="o">*</span><span class="n">i</span></code> or <code class="highlight"><span class="n">binary_op</span><span class="p">(</span><span class="n">acc</span><span class="p">,</span> <span class="o">*</span><span class="n">i</span><span class="p">)</span></code> for <code class="highlight"><span class="n">accumulate</span></code>, for example) be implicitly
convertible to <code class="highlight"><span class="n">T</span></code>. <code class="highlight"><span class="n">partial_sum</span></code> and <code class="highlight"><span class="n">adjacent_difference</span></code> already have
similar language requiring implicit convertibilty to <code class="highlight"><span class="n">InputIterator</span></code>'s value
type.</p>
   </ul>
   <h2 class="heading settled" data-level="3" id="paraalgos"><span class="secno">3. </span><span class="content">New Parallel <code class="highlight"><span class="o">&lt;</span><span class="n">numeric</span><span class="o">></span></code> Algorithms</span><a class="self-link" href="#paraalgos"></a></h2>
   <p>For the parallel numeric algorithms (<code class="highlight"><span class="n">reduce</span></code>, <code class="highlight"><span class="n">transform_reduce</span></code>, <code class="highlight"><span class="n">inclusive_scan</span></code>, <code class="highlight"><span class="n">exclusive_scan</span></code>, <code class="highlight"><span class="n">transform_inclusive_scan</span></code> and <code class="highlight"><span class="n">transform_exclusive_scan</span></code>), things are a bit more complicated.  All of these
algorithms are specified in terms of GENERALIZED_SUM or
GENERALIZED_NONCOMMUTATIVE_SUM:</p>
   <blockquote>
     26.2 [<strong>numeric.defns</strong>] 
    <blockquote>
      Define <code class="highlight"><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <span class="n">a1</span><span class="p">,</span> <span class="p">.</span><span class="p">.</span><span class="p">.</span><span class="p">,</span> <span class="n">aN</span><span class="p">)</span></code> as follows: 
     <ul>
      <li data-md="">
       <p><code class="highlight"><span class="n">a1</span></code> when <code class="highlight"><span class="n">N</span></code> is <code class="highlight"><span class="mi">1</span></code>, otherwise</p>
      <li data-md="">
       <p><code class="highlight"><span class="n">op</span><span class="p">(</span><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <span class="n">a1</span><span class="p">,</span> <span class="p">.</span><span class="p">.</span><span class="p">.</span><span class="p">,</span> <span class="n">aK</span><span class="p">)</span><span class="p">,</span> <span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <span class="n">aM</span><span class="p">,</span> <span class="p">.</span><span class="p">.</span><span class="p">.</span><span class="p">,</span> <span class="n">aN</span><span class="p">)</span><span class="p">)</span></code> for any <code class="highlight"><span class="n">K</span></code> where <code class="highlight"><span class="mi">1</span> <span class="o">&lt;</span> <span class="n">K</span> <span class="o">+</span> <span class="mi">1</span> <span class="o">=</span> <span class="n">M</span> ≤ <span class="n">N</span></code>.</p>
     </ul>
    </blockquote>
    <blockquote> Define <code class="highlight"><span class="n">GENERALIZED_SUM</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <span class="n">a1</span><span class="p">,</span> <span class="p">.</span><span class="p">.</span><span class="p">.</span><span class="p">,</span> <span class="n">aN</span><span class="p">)</span></code> as <code class="highlight"><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <span class="n">b1</span><span class="p">,</span> <span class="p">.</span><span class="p">.</span><span class="p">.</span><span class="p">,</span> <span class="n">bN</span><span class="p">)</span></code> where <code class="highlight"><span class="n">b1</span><span class="p">,</span> <span class="p">.</span><span class="p">.</span><span class="p">.</span><span class="p">,</span> <span class="n">bN</span></code> may be any permutation of <code class="highlight"><span class="n">a1</span><span class="p">,</span> <span class="p">.</span><span class="p">.</span><span class="p">.</span><span class="p">,</span> <span class="n">aN</span></code>. </blockquote>
   </blockquote>
   <p>Arbitrary partition of the reduction calculations at the implementation’s
discretion is allowed for <code class="highlight"><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span></code> algorithms
(<code class="highlight"><span class="n">exclusive_scan</span></code>, <code class="highlight"><span class="n">inclusive_scan</span></code>, <code class="highlight"><span class="n">transform_exclusive_scan</span></code> and <code class="highlight"><span class="n">transform_inclusive_scan</span></code>). In addition to arbitrary partitioning, <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> algorithms (<code class="highlight"><span class="n">reduce</span></code> and <code class="highlight"><span class="n">transform_reduce</span></code>) can arbitrarily
re-order the reduction calculations. All of the algorithms have overloads that
take an initial value of type <code class="highlight"><span class="n">T</span></code>. For the <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> algorithms, all
overloads take an initial value of type <code class="highlight"><span class="n">T</span></code>.</p>
   <p>These algorithms present a conundrum for us. Should they:</p>
   <ul>
    <li data-md="">
     <p>break the precedence of their serial counterparts (<code class="highlight"><span class="n">accumulate</span></code>, <code class="highlight"><span class="n">inner_product</span></code> and <code class="highlight"><span class="n">adjacent_difference</span></code>) and use the value type of
InputIterator instead of <code class="highlight"><span class="n">T</span></code> for intermediate storage (the implied  status
quo of the working text, given the current definitions of <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> and <code class="highlight"><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span></code>).</p>
    <li data-md="">
     <p>OR should the definition of algorithms be changed to require conversions to <code class="highlight"><span class="n">T</span></code>, making the <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> algorithms consistent with their serial
counterparts?</p>
   </ul>
   <p>The second option seems very messy. We’d have to change the definition of <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> to explicitly take the initial value and initial value type
into account. The second option would potentially force conversions to <code class="highlight"><span class="n">T</span></code> that
would be unnecessary and undesirable. Additionally, what about the algorithms
which can take an initial value, but also have overloads which do not take one
(<code class="highlight"><span class="n">inclusive_scan</span></code> and <code class="highlight"><span class="n">transform_inclusive_scan</span></code>)? The first option seems best
and appears to be closest to the design intent.</p>
   <p>I suggest that we change all the parallel numeric algorithms to:</p>
   <ul>
    <li data-md="">
     <p>Require that the type of the initial value (<code class="highlight"><span class="n">T</span></code>) be <code class="highlight"><span class="n">CopyConstructible</span></code> (and <code class="highlight"><span class="n">CopyAssignable</span></code>?).</p>
    <li data-md="">
     <p>Require that InputIterator’s value type can be constructed from the type of
the initial value (T).</p>
    <li data-md="">
     <p>Replace uses of <code class="highlight"><span class="n">init</span></code> in the <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> or <code class="highlight"><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span></code> expression used to define the algorithms with a temporary of the
InputIterator’s value type constructed from the initial value.</p>
    <li data-md="">
     <p>Require that the result of the <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> or <code class="highlight"><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span></code> expression be implicitly convertible to the correct return type for the
algorithm (<code class="highlight"><span class="n">T</span></code>).</p>
   </ul>
   <p>Additionally, for the parallel numeric algorithms which take <code class="highlight"><span class="n">OutputIterator</span></code>s, I
suggest the following changes:</p>
   <ul>
    <li data-md="">
     <p>Require that the result of the <code class="highlight"><span class="n">GENERALIZED_SUM</span></code> or <code class="highlight"><span class="n">GENERALIZED_NONCOMMUTATIVE_SUM</span></code> expression be writable (24.2.1) to the result output iterator.</p>
   </ul>
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  <h2 class="no-num no-ref heading settled" id="references"><span class="content">References</span><a class="self-link" href="#references"></a></h2>
  <h3 class="no-num no-ref heading settled" id="informative"><span class="content">Informative References</span><a class="self-link" href="#informative"></a></h3>
  <dl>
   <dt id="biblio-p0452r0">[P0452r0]
   <dd>Bryce Adelstein Lelbach. <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2016/p0452r0.html">Binary transform_reduce(): The Missing Overload</a>. 14 October 2016. URL: <a href="http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2016/p0452r0.html">http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2016/p0452r0.html</a>
  </dl>