<!doctype html><html lang="en">
 <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>P1468R3: Fixed-layout floating-point type aliases</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 calc(-0.5em - 1px);
	}
	[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: medium;
	}
	dfn var {
		font-style: normal;
	}

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

	del { color: red;  text-decoration: line-through; }
	ins { color: #080; text-decoration: underline;    }

/** 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;
	}

	table.data td,  table.data th,
	table.index td, table.index th {
		padding: 0.5em 1em;
		border-width: 1px;
		border-color: silver;
		border-top-style: solid;
	}

	table.data thead td:empty {
		padding: 0;
		border: 0;
	}

	table.data  thead,
	table.index thead,
	table.data  tbody,
	table.index tbody {
		border-bottom: 2px solid;
	}

	table.data colgroup,
	table.index colgroup {
		border-left: 2px solid;
	}

	table.data  tbody th:first-child,
	table.index tbody th:first-child  {
		border-right: 2px solid;
		border-top: 1px solid silver;
		padding-right: 1em;
	}

	table.data th[colspan],
	table.data td[colspan] {
		text-align: center;
	}

	table.complex.data th,
	table.complex.data td {
		border: 1px solid silver;
		text-align: center;
	}

	table.data.longlastcol td:last-child,
	table.data td.long {
	 vertical-align: baseline;
	 text-align: left;
	}

	table.data img {
		vertical-align: middle;
	}


/*
Alternate table alignment rules

	table.data,
	table.index {
		text-align: center;
	}

	table.data  thead th[scope="row"],
	table.index thead th[scope="row"] {
		text-align: right;
	}

	table.data  tbody th:first-child,
	table.index tbody th:first-child  {
		text-align: right;
	}

Possible extra rowspan handling

	table.data  tbody th[rowspan]:not([rowspan='1']),
	table.index tbody th[rowspan]:not([rowspan='1']),
	table.data  tbody td[rowspan]:not([rowspan='1']),
	table.index tbody td[rowspan]:not([rowspan='1']) {
		border-left: 1px solid silver;
	}

	table.data  tbody th[rowspan]:first-child,
	table.index tbody th[rowspan]:first-child,
	table.data  tbody td[rowspan]:first-child,
	table.index tbody td[rowspan]:first-child{
		border-left: 0;
		border-right: 1px solid silver;
	}
*/

/******************************************************************************/
/*                                  Indices                                   */
/******************************************************************************/


/** Table of Contents *********************************************************/

	.toc a {
		/* More spacing; use padding to make it part of the click target. */
		padding-top: 0.1rem;
		/* Larger, more consistently-sized click target */
		display: block;
		/* Reverse color scheme */
		color: black;
		border-color: #3980B5;
		border-bottom-width: 3px !important;
		margin-bottom: 0px !important;
	}
	.toc a:visited {
		border-color: #054572;
	}
	.toc a:not(:focus):not(:hover) {
		/* Allow colors to cascade through from link styling */
		border-bottom-color: transparent;
	}

	.toc, .toc ol, .toc ul, .toc li {
		list-style: none; /* Numbers must be inlined into source */
		/* because generated content isn't search/selectable and markers can't do multilevel yet */
		margin:  0;
		padding: 0;
		line-height: 1.1rem; /* consistent spacing */
	}

	/* ToC not indented until third level, but font style & margins show hierarchy */
	.toc > li             { font-weight: bold;   }
	.toc > li li          { font-weight: normal; }
	.toc > li li li       { font-size:   95%;    }
	.toc > li li li li    { font-size:   90%;    }
	.toc > li li li li .secno { font-size: 85%; }
	.toc > li li li li li { font-size:   85%;    }
	.toc > li li li li li .secno { font-size: 100%; }

	/* @supports not (display:grid) { */
		.toc > li             { margin: 1.5rem 0;    }
		.toc > li li          { margin: 0.3rem 0;    }
		.toc > li li li       { margin-left: 2rem;   }

		/* Section numbers in a column of their own */
		.toc .secno {
			float: left;
			width: 4rem;
			white-space: nowrap;
		}

		.toc li {
			clear: both;
		}

		:not(li) > .toc              { margin-left:  5rem; }
		.toc .secno                  { margin-left: -5rem; }
		.toc > li li li .secno       { margin-left: -7rem; }
		.toc > li li li li .secno    { margin-left: -9rem; }
		.toc > li li li li li .secno { margin-left: -11rem; }

		/* Tighten up indentation in narrow ToCs */
		@media (max-width: 30em) {
			:not(li) > .toc              { margin-left:  4rem; }
			.toc .secno                  { margin-left: -4rem; }
			.toc > li li li              { margin-left:  1rem; }
			.toc > li li li .secno       { margin-left: -5rem; }
			.toc > li li li li .secno    { margin-left: -6rem; }
			.toc > li li li li li .secno { margin-left: -7rem; }
		}
	/* } */

	@supports (display:grid) and (display:contents) {
		/* Use #toc over .toc to override non-@supports rules. */
		#toc {
			display: grid;
			align-content: start;
			grid-template-columns: auto 1fr;
			grid-column-gap: 1rem;
			column-gap: 1rem;
			grid-row-gap: .6rem;
			row-gap: .6rem;
		}
		#toc h2 {
			grid-column: 1 / -1;
			margin-bottom: 0;
		}
		#toc ol,
		#toc li,
		#toc a {
			display: contents;
			/* Switch <a> to subgrid when supported */
		}
		#toc span {
			margin: 0;
		}
		#toc > .toc > li > a > span {
			/* The spans of the top-level list,
			   comprising the first items of each top-level section. */
			margin-top: 1.1rem;
		}
		#toc#toc .secno { /* Ugh, need more specificity to override base.css */
			grid-column: 1;
			width: auto;
			margin-left: 0;
		}
		#toc .content {
			grid-column: 2;
			width: auto;
			margin-right: 1rem;
		}
		#toc .content:hover {
			background: rgba(75%, 75%, 75%, .25);
			border-bottom: 3px solid #054572;
			margin-bottom: -3px;
		}
		#toc li li li .content {
			margin-left: 1rem;
		}
		#toc li li li li .content {
			margin-left: 2rem;
		}
	}


/** Index *********************************************************************/

	/* Index Lists: Layout */
	ul.index       { margin-left: 0; columns: 15em; text-indent: 1em hanging; }
	ul.index li    { margin-left: 0; list-style: none; break-inside: avoid; }
	ul.index li li { margin-left: 1em }
	ul.index dl    { margin-top: 0; }
	ul.index dt    { margin: .2em 0 .2em 20px;}
	ul.index dd    { margin: .2em 0 .2em 40px;}
	/* Index Lists: Typography */
	ul.index ul,
	ul.index dl { font-size: smaller; }
	@media not print {
		ul.index li span {
			white-space: nowrap;
			color: transparent; }
		ul.index li a:hover + span,
		ul.index li a:focus + span {
			color: #707070;
		}
	}

/** Index Tables *****************************************************/
	/* See also the data table styling section, which this effectively subclasses */

	table.index {
		font-size: small;
		border-collapse: collapse;
		border-spacing: 0;
		text-align: left;
		margin: 1em 0;
	}

	table.index td,
	table.index th {
		padding: 0.4em;
	}

	table.index tr:hover td:not([rowspan]),
	table.index tr:hover th:not([rowspan]) {
		background: #f7f8f9;
	}

	/* The link in the first column in the property table (formerly a TD) */
	table.index th:first-child a {
		font-weight: bold;
	}

/******************************************************************************/
/*                                    Print                                   */
/******************************************************************************/

	@media print {
		/* Pages have their own margins. */
		html {
			margin: 0;
		}
		/* Serif for print. */
		body {
			font-family: serif;
		}
	}
	@page {
		margin: 1.5cm 1.1cm;
	}

/******************************************************************************/
/*                                    Legacy                                  */
/******************************************************************************/

	/* This rule is inherited from past style sheets. No idea what it's for. */
	.hide { display: none }



/******************************************************************************/
/*                             Overflow Control                               */
/******************************************************************************/

	.figure .caption, .sidefigure .caption, figcaption {
		/* in case figure is overlarge, limit caption to 50em */
		max-width: 50rem;
		margin-left: auto;
		margin-right: auto;
	}
	.overlarge {
		/* Magic to create good table positioning:
		   "content column" is 50ems wide at max; less on smaller screens.
		   Extra space (after ToC + content) is empty on the right.

		   1. When table < content column, centers table in column.
		   2. When content < table < available, left-aligns.
		   3. When table > available, fills available + scroll bar.
		*/ 
		display: grid;
		grid-template-columns: minmax(0, 50em);
	}
	.overlarge > table {
		/* limit preferred width of table */
		max-width: 50em;
		margin-left: auto;
		margin-right: auto;
	}

	@media (min-width: 55em) {
		.overlarge {
			margin-right: calc(13px + 26.5rem - 50vw);
			max-width: none;
		}
	}
	@media screen and (min-width: 78em) {
		body:not(.toc-inline) .overlarge {
			/* 30.5em body padding 50em content area */
			margin-right: calc(40em - 50vw) !important;
		}
	}
	@media screen and (min-width: 90em) {
		body:not(.toc-inline) .overlarge {
			/* 4em html margin 30.5em body padding 50em content area */
			margin-right: calc(84.5em - 100vw) !important;
		}
	}

	@media not print {
		.overlarge {
			overflow-x: auto;
			/* See Lea Verou's explanation background-attachment:
			 * http://lea.verou.me/2012/04/background-attachment-local/
			 *
			background: top left  / 4em 100% linear-gradient(to right,  #ffffff, rgba(255, 255, 255, 0)) local,
			            top right / 4em 100% linear-gradient(to left, #ffffff, rgba(255, 255, 255, 0)) local,
			            top left  / 1em 100% linear-gradient(to right,  #c3c3c5, rgba(195, 195, 197, 0)) scroll,
			            top right / 1em 100% linear-gradient(to left, #c3c3c5, rgba(195, 195, 197, 0)) scroll,
			            white;
			background-repeat: no-repeat;
			*/
		}
	}
</style>
<style type="text/css">
    table, th, td {
      border: 1px solid black;
      border-collapse: collapse;
      vertical-align: top;
    }
    th, td {
      border-left: none;
      border-right: none;
      padding: 0px 10px;
    }
    th {
      text-align: center;
    }

    del { background: #fcc; color: #000; text-decoration: line-through; }
    ins { background: #cfc; color: #000; }
    blockquote .highlight:not(.idl) { background: initial; margin: initial; padding: 0.5em }
    blockquote ul { background: inherit; }
    blockquote code.highlight:not(.idl) { padding: initial; }
    blockquote c-[a] { color: inherit; } /* Keyword.Declaration */
    blockquote c-[b] { color: inherit; } /* Keyword.Type */
    blockquote c-[c] { color: inherit; } /* Comment */
    blockquote c-[d] { color: inherit; } /* Comment.Multiline */
    blockquote c-[e] { color: inherit; } /* Name.Attribute */
    blockquote c-[f] { color: inherit; } /* Name.Tag */
    blockquote c-[g] { color: inherit; } /* Name.Variable */
    blockquote c-[k] { color: inherit; } /* Keyword */
    blockquote c-[l] { color: inherit; } /* Literal */
    blockquote c-[m] { color: inherit; } /* Literal.Number */
    blockquote c-[n] { color: inherit; } /* Name */
    blockquote c-[o] { color: inherit; } /* Operator */
    blockquote c-[p] { color: inherit; } /* Punctuation */
    blockquote c-[s] { color: inherit; } /* Literal.String */
    blockquote c-[t] { color: inherit; } /* Literal.String.Single */
    blockquote c-[u] { color: inherit; } /* Literal.String.Double */
    blockquote c-[cp] { color: inherit; } /* Comment.Preproc */
    blockquote c-[c1] { color: inherit; } /* Comment.Single */
    blockquote c-[cs] { color: inherit; } /* Comment.Special */
    blockquote c-[kc] { color: inherit; } /* Keyword.Constant */
    blockquote c-[kn] { color: inherit; } /* Keyword.Namespace */
    blockquote c-[kp] { color: inherit; } /* Keyword.Pseudo */
    blockquote c-[kr] { color: inherit; } /* Keyword.Reserved */
    blockquote c-[ld] { color: inherit; } /* Literal.Date */
    blockquote c-[nc] { color: inherit; } /* Name.Class */
    blockquote c-[no] { color: inherit; } /* Name.Constant */
    blockquote c-[nd] { color: inherit; } /* Name.Decorator */
    blockquote c-[ni] { color: inherit; } /* Name.Entity */
    blockquote c-[ne] { color: inherit; } /* Name.Exception */
    blockquote c-[nf] { color: inherit; } /* Name.Function */
    blockquote c-[nl] { color: inherit; } /* Name.Label */
    blockquote c-[nn] { color: inherit; } /* Name.Namespace */
    blockquote c-[py] { color: inherit; } /* Name.Property */
    blockquote c-[ow] { color: inherit; } /* Operator.Word */
    blockquote c-[mb] { color: inherit; } /* Literal.Number.Bin */
    blockquote c-[mf] { color: inherit; } /* Literal.Number.Float */
    blockquote c-[mh] { color: inherit; } /* Literal.Number.Hex */
    blockquote c-[mi] { color: inherit; } /* Literal.Number.Integer */
    blockquote c-[mo] { color: inherit; } /* Literal.Number.Oct */
    blockquote c-[sb] { color: inherit; } /* Literal.String.Backtick */
    blockquote c-[sc] { color: inherit; } /* Literal.String.Char */
    blockquote c-[sd] { color: inherit; } /* Literal.String.Doc */
    blockquote c-[se] { color: inherit; } /* Literal.String.Escape */
    blockquote c-[sh] { color: inherit; } /* Literal.String.Heredoc */
    blockquote c-[si] { color: inherit; } /* Literal.String.Interpol */
    blockquote c-[sx] { color: inherit; } /* Literal.String.Other */
    blockquote c-[sr] { color: inherit; } /* Literal.String.Regex */
    blockquote c-[ss] { color: inherit; } /* Literal.String.Symbol */
    blockquote c-[vc] { color: inherit; } /* Name.Variable.Class */
    blockquote c-[vg] { color: inherit; } /* Name.Variable.Global */
    blockquote c-[vi] { color: inherit; } /* Name.Variable.Instance */
    blockquote c-[il] { color: inherit; } /* Literal.Number.Integer.Long */
  </style>
  <meta content="Bikeshed version 6a4c097ae8b731ee02752a258e8ac5ffbb752299" name="generator">
  <link href="https://wg21.link/P1468" rel="canonical">
  <link href="https://isocpp.org/favicon.ico" rel="icon">
<style>
ins             {background-color: #CCFFCC;             text-decoration: underline;}
del             {background-color: #FFCACA;             text-decoration: line-through;}
ins .highlight  {background-color: #CCFFCC !important;  text-decoration: underline;}
ins blockquote  {border-color:     #008000 !important;  text-decoration: underline;}
del .highlight  {background-color: #FFCACA !important;  text-decoration: line-through;}
del blockquote  {border-color:     #FF0000 !important;  text-decoration: line-through;}
blockquote {
  counter-reset: paragraph;
}
div.numbered, div.newnumbered {
  margin-left: 2em;
  margin-top: 1em;
  margin-bottom: 1em;
}
div.numbered:before, div.newnumbered:before {
  position: absolute;
  margin-left: -2em;
  display-style: block;
}
div.numbered:before {
  content: counter(paragraph);
  counter-increment: paragraph;
}
div.newnumbered:before {
  content: "�";
}
div.numbered ul, div.newnumbered ul {
  counter-reset: list_item;
}
div.numbered li, div.newnumbered li {
  margin-left: 3em;
}
div.numbered li:before, div.newnumbered li:before {
  position: absolute;
  margin-left: -4.8em;
  display-style: block;
}
div.numbered li:before {
  content: "(" counter(paragraph) "." counter(list_item) ")";
  counter-increment: list_item;
}
div.newnumbered li:before {
  content: "(�." counter(list_item) ")";
  counter-increment: list_item;
}
</style>
<style>/* style-md-lists */

/* This is a weird hack for me not yet following the commonmark spec
   regarding paragraph and lists. */
[data-md] > :first-child {
    margin-top: 0;
}
[data-md] > :last-child {
    margin-bottom: 0;
}</style>
<style>/* style-hidedel */

            #hidedel:checked ~ del, #hidedel:checked ~ * del { display:none; }
            #hidedel ~ #hidedel-label::before, #hidedel ~ * #hidedel-label::before { content: "☐ "; }
            #hidedel:checked ~ #hidedel-label::before, #hidedel:checked ~ * #hidedel-label::before { content: "☑ "; }
        </style>
<style>/* style-selflinks */

.heading, .issue, .note, .example, li, dt {
    position: relative;
}
a.self-link {
    position: absolute;
    top: 0;
    left: calc(-1 * (3.5rem - 26px));
    width: calc(3.5rem - 26px);
    height: 2em;
    text-align: center;
    border: none;
    transition: opacity .2s;
    opacity: .5;
}
a.self-link:hover {
    opacity: 1;
}
.heading > a.self-link {
    font-size: 83%;
}
li > a.self-link {
    left: calc(-1 * (3.5rem - 26px) - 2em);
}
dfn > a.self-link {
    top: auto;
    left: auto;
    opacity: 0;
    width: 1.5em;
    height: 1.5em;
    background: gray;
    color: white;
    font-style: normal;
    transition: opacity .2s, background-color .2s, color .2s;
}
dfn:hover > a.self-link {
    opacity: 1;
}
dfn > a.self-link:hover {
    color: black;
}

a.self-link::before            { content: "¶"; }
.heading > a.self-link::before { content: "§"; }
dfn > a.self-link::before      { content: "#"; }</style>
<style>/* style-counters */

body {
    counter-reset: example figure issue;
}
.issue {
    counter-increment: issue;
}
.issue:not(.no-marker)::before {
    content: "Issue " counter(issue);
}

.example {
    counter-increment: example;
}
.example:not(.no-marker)::before {
    content: "Example " counter(example);
}
.invalid.example:not(.no-marker)::before,
.illegal.example:not(.no-marker)::before {
    content: "Invalid Example" counter(example);
}

figcaption {
    counter-increment: figure;
}
figcaption:not(.no-marker)::before {
    content: "Figure " counter(figure) " ";
}</style>
<style>/* style-autolinks */

.css.css, .property.property, .descriptor.descriptor {
    color: #005a9c;
    font-size: inherit;
    font-family: inherit;
}
.css::before, .property::before, .descriptor::before {
    content: "‘";
}
.css::after, .property::after, .descriptor::after {
    content: "’";
}
.property, .descriptor {
    /* Don't wrap property and descriptor names */
    white-space: nowrap;
}
.type { /* CSS value <type> */
    font-style: italic;
}
pre .property::before, pre .property::after {
    content: "";
}
[data-link-type="property"]::before,
[data-link-type="propdesc"]::before,
[data-link-type="descriptor"]::before,
[data-link-type="value"]::before,
[data-link-type="function"]::before,
[data-link-type="at-rule"]::before,
[data-link-type="selector"]::before,
[data-link-type="maybe"]::before {
    content: "‘";
}
[data-link-type="property"]::after,
[data-link-type="propdesc"]::after,
[data-link-type="descriptor"]::after,
[data-link-type="value"]::after,
[data-link-type="function"]::after,
[data-link-type="at-rule"]::after,
[data-link-type="selector"]::after,
[data-link-type="maybe"]::after {
    content: "’";
}

[data-link-type].production::before,
[data-link-type].production::after,
.prod [data-link-type]::before,
.prod [data-link-type]::after {
    content: "";
}

[data-link-type=element],
[data-link-type=element-attr] {
    font-family: Menlo, Consolas, "DejaVu Sans Mono", monospace;
    font-size: .9em;
}
[data-link-type=element]::before { content: "<" }
[data-link-type=element]::after  { content: ">" }

[data-link-type=biblio] {
    white-space: pre;
}</style>
<style>/* style-syntax-highlighting */

.highlight:not(.idl) { background: hsl(24, 20%, 95%); }
code.highlight { padding: .1em; border-radius: .3em; }
pre.highlight, pre > code.highlight { display: block; padding: 1em; margin: .5em 0; overflow: auto; border-radius: 0; }
c-[a] { color: #990055 } /* Keyword.Declaration */
c-[b] { color: #990055 } /* Keyword.Type */
c-[c] { color: #708090 } /* Comment */
c-[d] { color: #708090 } /* Comment.Multiline */
c-[e] { color: #0077aa } /* Name.Attribute */
c-[f] { color: #669900 } /* Name.Tag */
c-[g] { color: #222222 } /* Name.Variable */
c-[k] { color: #990055 } /* Keyword */
c-[l] { color: #000000 } /* Literal */
c-[m] { color: #000000 } /* Literal.Number */
c-[n] { color: #0077aa } /* Name */
c-[o] { color: #999999 } /* Operator */
c-[p] { color: #999999 } /* Punctuation */
c-[s] { color: #a67f59 } /* Literal.String */
c-[t] { color: #a67f59 } /* Literal.String.Single */
c-[u] { color: #a67f59 } /* Literal.String.Double */
c-[cp] { color: #708090 } /* Comment.Preproc */
c-[c1] { color: #708090 } /* Comment.Single */
c-[cs] { color: #708090 } /* Comment.Special */
c-[kc] { color: #990055 } /* Keyword.Constant */
c-[kn] { color: #990055 } /* Keyword.Namespace */
c-[kp] { color: #990055 } /* Keyword.Pseudo */
c-[kr] { color: #990055 } /* Keyword.Reserved */
c-[ld] { color: #000000 } /* Literal.Date */
c-[nc] { color: #0077aa } /* Name.Class */
c-[no] { color: #0077aa } /* Name.Constant */
c-[nd] { color: #0077aa } /* Name.Decorator */
c-[ni] { color: #0077aa } /* Name.Entity */
c-[ne] { color: #0077aa } /* Name.Exception */
c-[nf] { color: #0077aa } /* Name.Function */
c-[nl] { color: #0077aa } /* Name.Label */
c-[nn] { color: #0077aa } /* Name.Namespace */
c-[py] { color: #0077aa } /* Name.Property */
c-[ow] { color: #999999 } /* Operator.Word */
c-[mb] { color: #000000 } /* Literal.Number.Bin */
c-[mf] { color: #000000 } /* Literal.Number.Float */
c-[mh] { color: #000000 } /* Literal.Number.Hex */
c-[mi] { color: #000000 } /* Literal.Number.Integer */
c-[mo] { color: #000000 } /* Literal.Number.Oct */
c-[sb] { color: #a67f59 } /* Literal.String.Backtick */
c-[sc] { color: #a67f59 } /* Literal.String.Char */
c-[sd] { color: #a67f59 } /* Literal.String.Doc */
c-[se] { color: #a67f59 } /* Literal.String.Escape */
c-[sh] { color: #a67f59 } /* Literal.String.Heredoc */
c-[si] { color: #a67f59 } /* Literal.String.Interpol */
c-[sx] { color: #a67f59 } /* Literal.String.Other */
c-[sr] { color: #a67f59 } /* Literal.String.Regex */
c-[ss] { color: #a67f59 } /* Literal.String.Symbol */
c-[vc] { color: #0077aa } /* Name.Variable.Class */
c-[vg] { color: #0077aa } /* Name.Variable.Global */
c-[vi] { color: #0077aa } /* Name.Variable.Instance */
c-[il] { color: #000000 } /* Literal.Number.Integer.Long */
</style>
 <body class="h-entry">
  <input id="hidedel" style="display:none" type="checkbox">
  <div class="head">
   <p data-fill-with="logo"></p>
   <h1 class="p-name no-ref" id="title">P1468R3<br>Fixed-layout floating-point type aliases</h1>
   <h2 class="no-num no-toc no-ref heading settled" id="subtitle"><span class="content">Published Proposal, <time class="dt-updated" datetime="2020-01-10">2020-01-10</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt>This version:
     <dd><a class="u-url" href="https://wg21.link/P1468">https://wg21.link/P1468</a>
     <dt>Issue Tracking:
     <dd><a href="#issues-index">Inline In Spec</a>
     <dt>Authors:
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:griwes@griwes.info">Michał Dominiak</a> (<span class="p-org org">NVIDIA</span>)
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:dolsen@nvidia.com">David Olsen</a> (<span class="p-org org">NVIDIA</span>)
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:boris@stlport.com">Boris Fomitchev</a> (<span class="p-org org">NVIDIA</span>)
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:me@cvmlib.com">Sergei Nikolaev</a> (<span class="p-org org">NVIDIA</span>)
     <dt>Audience:
     <dd>EWG, LEWG
     <dt>Toggle Diffs:
     <dd>
      <label for="hidedel" id="hidedel-label">Hide deleted text</label>
     <dt>Project:
     <dd>ISO/IEC JTC1/SC22/WG21 14882: Programming Language — C++
    </dl>
   </div>
   <div data-fill-with="warning"></div>
   <hr title="Separator for header">
  </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="#abstract"><span class="secno">1</span> <span class="content">Abstract</span></a>
    <li>
     <a href="#revisions"><span class="secno">2</span> <span class="content">Revision history</span></a>
     <ol class="toc">
      <li><a href="#r1"><span class="secno">2.1</span> <span class="content">R0 -> R1 (pre-Cologne)</span></a>
      <li><a href="#r2"><span class="secno">2.2</span> <span class="content">R1 -> R2 (pre-Belfast)</span></a>
      <li><a href="#r3"><span class="secno">2.3</span> <span class="content">R2 -> R3 (pre-Prague)</span></a>
     </ol>
    <li><a href="#motivation"><span class="secno">3</span> <span class="content">Motivation</span></a>
    <li><a href="#header"><span class="secno">4</span> <span class="content">Header name</span></a>
    <li>
     <a href="#alias"><span class="secno">5</span> <span class="content">Type aliases</span></a>
     <ol class="toc">
      <li><a href="#alias-formats"><span class="secno">5.1</span> <span class="content">Supported formats</span></a>
      <li><a href="#alias-alias"><span class="secno">5.2</span> <span class="content">Aliasing standard types</span></a>
      <li><a href="#layout-vs-behavior"><span class="secno">5.3</span> <span class="content">Layout vs. behavior</span></a>
      <li><a href="#test-macros"><span class="secno">5.4</span> <span class="content">Feature test macros</span></a>
      <li>
       <a href="#names"><span class="secno">5.5</span> <span class="content">Names</span></a>
       <ol class="toc">
        <li><a href="#floatX_t"><span class="secno">5.5.1</span> <span class="content"><code class="highlight"><c- n>floatX_t</c-></code></span></a>
        <li><a href="#fp_namespace"><span class="secno">5.5.2</span> <span class="content"><code class="highlight"><c- n>fp</c-><c- o>::</c-><c- n>binaryX_t</c-></code></span></a>
        <li><a href="#fp_binaryX_t"><span class="secno">5.5.3</span> <span class="content"><code class="highlight"><c- n>fp_binaryX_t</c-></code></span></a>
       </ol>
     </ol>
    <li><a href="#literals"><span class="secno">6</span> <span class="content">Literal suffixes</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>
    <li><a href="#issues-index"><span class="secno"></span> <span class="content">Issues Index</span></a>
   </ol>
  </nav>
  <main>
   <h2 class="heading settled" data-level="1" id="abstract"><span class="secno">1. </span><span class="content">Abstract</span><a class="self-link" href="#abstract"></a></h2>
   <p>This paper proposes a set of <code class="highlight"><c- o>&lt;</c-><c- n>cstdint</c-><c- o>></c-></code>-style type aliases for floating point types matching specific, well-know floating-point layouts.</p>
   <p>This is a companion paper to <a data-link-type="biblio" href="#biblio-p1467">[P1467]</a>, which allows implementations to define floating-point types beyond the three standard types.  This paper gives convenient names to some of those types.</p>
   <h2 class="heading settled" data-level="2" id="revisions"><span class="secno">2. </span><span class="content">Revision history</span><a class="self-link" href="#revisions"></a></h2>
   <h3 class="heading settled" data-level="2.1" id="r1"><span class="secno">2.1. </span><span class="content">R0 -> R1 (pre-Cologne)</span><a class="self-link" href="#r1"></a></h3>
   <ol>
    <li data-md>
     <p>Add the requirement that the types must not alias any of the standard floating-point types.</p>
    <li data-md>
     <p>Add a design question about feature-test macros.</p>
    <li data-md>
     <p>Add a section on QoI - should we strongly encourage that the aliases to have a hardware implementation?</p>
   </ol>
   <h3 class="heading settled" data-level="2.2" id="r2"><span class="secno">2.2. </span><span class="content">R1 -> R2 (pre-Belfast)</span><a class="self-link" href="#r2"></a></h3>
   <p>Changes based on feedback in Cologne from <a href="http://wiki.edg.com/bin/view/Wg21cologne2019/SG6P1468">SG6</a>, <a href="http://wiki.edg.com/bin/view/Wg21cologne2019/P1467">LEWGI</a>, and <a href="http://wiki.edg.com/bin/view/Wg21cologne2019/P1468">EWGI</a>.  Further changes came from further development of the paper by the authors.</p>
   <ul>
    <li data-md>
     <p>Expanded the <a href="#alias-alias">section</a> about whether or not the fixed-layout aliases are allowed to alias standard floating-point types.</p>
    <li data-md>
     <p>Added a <a href="#layout-vs-behavior">section</a> about whether the aliases only need to guarantee layout, or should also guarantee behavior.</p>
    <li data-md>
     <p>Added some text, still preliminary, about <a href="#literals">literal suffixes</a>.</p>
   </ul>
   <h3 class="heading settled" data-level="2.3" id="r3"><span class="secno">2.3. </span><span class="content">R2 -> R3 (pre-Prague)</span><a class="self-link" href="#r3"></a></h3>
   <p>Changes based on feedback in Belfast from <a href="http://wiki.edg.com/bin/view/Wg21belfast/P1468-EWG">EWG</a>.</p>
   <ul>
    <li data-md>
     <p>Added <code class="highlight"><c- o>&lt;</c-><c- n>stdfloat</c-><c- o>></c-></code> as a possible name for the header containing the proposed type aliases.</p>
    <li data-md>
     <p>Resolved the issue of <a href="#layout-vs-behavior">layout vs. behavior</a> by stating that <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>numeric_limits</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- n>is_iec559</c-></code> is true for the IEEE aliases.</p>
    <li data-md>
     <p>Reduced the proposed sets of names for the type alias from six different naming schemes down to three.</p>
   </ul>
   <h2 class="heading settled" data-level="3" id="motivation"><span class="secno">3. </span><span class="content">Motivation</span><a class="self-link" href="#motivation"></a></h2>
   <p>16-bit floating-point support is becoming more widely available in both hardware (ARM CPUs and NVIDIA GPUs) and software (OpenGL, CUDA, and LLVM IR).  Programmers wanting to take advantage of 16-bit floating-point support have been stymied by the lack of built-in compiler support for the type.  A common workaround is to define a class type with all of the conversion operators and overloaded arithmetic operators to make it behave as much as possible like a built-in type.  But that approach is cumbersome and incomplete, requiring inline assembly or other compiler-specific magic to generate efficient code.</p>
   <p>The problem of efficiently using newer floating-point types that haven’t traditionally been supported can’t be solved through user-defined libraries.  A possible solution of an implementation changing <code class="highlight"><c- b>float</c-></code> to be a 16-bit type would be unpopular because users want support for newer floating-point types in addition to the standard types, and because users have come to expect <code class="highlight"><c- b>float</c-></code> and <code class="highlight"><c- b>double</c-></code> to be 32- and 64-bit types and have lots of existing code written with that assumption.</p>
   <p>This problem is worth solving, and there is no viable solution under the current standard.  So changing the core language in an extensible and backward-compatible way is appropriate.  Providing a standard way for implementations to support 16-bit floating-point types will result in better code, more portable code, and wider use of those types.</p>
   <p><a data-link-type="biblio" href="#biblio-p1467">[P1467]</a> changes the language so that implementations can support 16-bit and other non-standard floating-point types.  This paper gives well-known names to 16-bit and other commonly used floating-point types.</p>
   <p>These two papers are the follow-up to <a data-link-type="biblio" href="#biblio-p0192">[P0192]</a>, the <code class="highlight"><c- b>short</c-> <c- b>float</c-></code> proposal, which was not approved by EWG.  This paper is also a revival, with modifications, of <a data-link-type="biblio" href="#biblio-n1703">[N1703]</a>, which in 2013 proposed adding typedefs for fixed-layout floating-point types to both C and C++, but was not adopted by either language.</p>
   <p>The language rules in this paper and in <a data-link-type="biblio" href="#biblio-p1467">[P1467]</a> are designed to work together to simplify the safe adoption of the new floating-point types into existing applications.  Programmers should be able to start using the 16-bit types in one part of the application without having to change other parts.  When <code class="highlight"><c- b>float</c-></code> and <code class="highlight"><c- b>double</c-></code> are IEEE-conformant types, it should be possible to mix the standard types with their fixed-layout aliases without problems.  This proposal would be a failure if code using the IEEE 64-bit type alias had to be kept mostly separate from code using <code class="highlight"><c- b>double</c-></code>.</p>
   <h2 class="heading settled" data-level="4" id="header"><span class="secno">4. </span><span class="content">Header name</span><a class="self-link" href="#header"></a></h2>
   <p>The type aliases proposed here do not fit neatly into any existing header.  So we are offering up two possibilities for new header names, neither of which we are thrilled with: <code class="highlight"><c- o>&lt;</c-><c- n>fixed_float</c-><c- o>></c-></code> and <code class="highlight"><c- o>&lt;</c-><c- n>stdfloat</c-><c- o>></c-></code>.  We are open to other names for the header and to arguments that the type aliases should be added to an existing header.</p>
   <p class="issue" id="issue-0ebb1eda"><a class="self-link" href="#issue-0ebb1eda"></a> What new or existing header should the type aliases go into?</p>
   <h2 class="heading settled" data-level="5" id="alias"><span class="secno">5. </span><span class="content">Type aliases</span><a class="self-link" href="#alias"></a></h2>
   <p>This paper introduces type aliases for several fixed-layout floating-point types.  Each alias will be defined only if a type with that layout is supported by the implementation, similar to the <code class="highlight"><c- n>intN_t</c-></code> and <code class="highlight"><c- n>uintN_t</c-></code> aliases.</p>
   <h3 class="heading settled" data-level="5.1" id="alias-formats"><span class="secno">5.1. </span><span class="content">Supported formats</span><a class="self-link" href="#alias-formats"></a></h3>
   <p>We propose aliases for the following layouts:</p>
   <ul>
    <li data-md>
     <p><a data-link-type="biblio" href="#biblio-ieee-754-2008">[IEEE-754-2008]</a> <code class="highlight"><c- n>binary16</c-></code> - IEEE 16-bit.</p>
    <li data-md>
     <p><a data-link-type="biblio" href="#biblio-ieee-754-2008">[IEEE-754-2008]</a> <code class="highlight"><c- n>binary32</c-></code> - IEEE 32-bit.</p>
    <li data-md>
     <p><a data-link-type="biblio" href="#biblio-ieee-754-2008">[IEEE-754-2008]</a> <code class="highlight"><c- n>binary64</c-></code> - IEEE 64-bit.</p>
    <li data-md>
     <p><a data-link-type="biblio" href="#biblio-ieee-754-2008">[IEEE-754-2008]</a> <code class="highlight"><c- n>binary128</c-></code> - IEEE 128-bit.</p>
    <li data-md>
     <p><code class="highlight"><c- n>bfloat16</c-></code>, which is <code class="highlight"><c- n>binary32</c-></code> with 16 bits of precision truncated; see <a data-link-type="biblio" href="#biblio-bfloat16">[bfloat16]</a>.</p>
   </ul>
   <p><code class="highlight"><c- n>binary32</c-></code> and <code class="highlight"><c- n>binary64</c-></code> are the most widely used floating-point types, and are the formats that <code class="highlight"><c- b>float</c-></code> and <code class="highlight"><c- b>double</c-></code> have in most implementations. <code class="highlight"><c- n>binary16</c-></code> is becoming more widely used; see this paper’s motivation for details. <code class="highlight"><c- n>binary128</c-></code> has hardware support in IBM POWER P9 chips. <code class="highlight"><c- n>bfloat16</c-></code> is used in Google’s TPUs and in TensorFlow.</p>
   <p>The most widely used format that is not in this list is X87 80-bit.  Even though there is hardware support for this format in all current x86 chips, it is used most often because it is the largest type available, not because users specifically want that format.</p>
   <h3 class="heading settled" data-level="5.2" id="alias-alias"><span class="secno">5.2. </span><span class="content">Aliasing standard types</span><a class="self-link" href="#alias-alias"></a></h3>
   <p>This has turned out to be the most contentious issue raised in this proposal with strong opinions on both sides.  In Cologne, SG6 and LEWGI voted in favor of allowing aliasing of standard types, while EWGI was strongly against the idea.  The authors are in favor of prohibiting aliasing of standard types, but realize that not everyone else is convinced of that yet.</p>
   <p>The header <code class="highlight"><c- o>&lt;</c-><c- n>cstdint</c-><c- o>></c-></code> defines integer type aliases for certain integer types, such as <code class="highlight"><c- n>std</c-><c- o>::</c-><c- b>int32_t</c-></code> and <code class="highlight"><c- n>std</c-><c- o>::</c-><c- b>int64_t</c-></code>.  These are similar in many ways to the aliases proposed here.  The types in <code class="highlight"><c- o>&lt;</c-><c- n>cstdint</c-><c- o>></c-></code> are allowed to alias standard integer types.  That has resulted in compilation errors when users try to create an overload set with both standard types and fixed-layout aliases, such as:</p>
<pre class="highlight"><code class="highlight"><c- b><c- b>int</c-></c-> <c- nf><c- nf>bit_count</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>int</c-></c-> <c- n><c- n>x</c-></c-><c- p><c- p>)</c-></c-> <c- p><c- p>{</c-></c-> <c- d><c- d>/* ... */</c-></c-> <c- p><c- p>}</c-></c->
<c- b><c- b>int</c-></c-> <c- nf><c- nf>bit_count</c-></c-><c- p><c- p>(</c-></c-><c- n><c- n>std</c-></c-><c- o><c- o>::</c-></c-><c- b><c- b>int32_t</c-></c-> <c- n><c- n>x</c-></c-><c- p><c- p>)</c-></c-> <c- p><c- p>{</c-></c-> <c- d><c- d>/* ... */</c-></c-> <c- p><c- p>}</c-></c->
</code></pre>
   <p>If aliasing of standard types is allowed for the floating-point type aliases, then similar compilation errors will likely result:</p>
<pre class="highlight"><code class="highlight"><c- b><c- b>int</c-></c-> <c- nf><c- nf>get_exponent</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>double</c-></c-> <c- n><c- n>x</c-></c-><c- p><c- p>)</c-></c-> <c- p><c- p>{</c-></c-> <c- d><c- d>/* ... */</c-></c-> <c- p><c- p>}</c-></c->
<c- b><c- b>int</c-></c-> <c- nf><c- nf>get_exponent</c-></c-><c- p><c- p>(</c-></c-><c- n><c- n>std</c-></c-><c- o><c- o>::</c-></c-><c- n><c- n>float64_t</c-></c-> <c- n><c- n>x</c-></c-><c- p><c- p>)</c-></c-> <c- p><c- p>{</c-></c-> <c- d><c- d>/* ... */</c-></c-> <c- p><c- p>}</c-></c->
</code></pre>
   <p>This is the strongest argument against allowing aliasing of standard types.  People who don’t find this argument persuasive point out that users should not create overload sets with both standard types and fixed-layout type aliases.  An overload set should contain just the standard floating-point types or just the fixed-layout types, but not both.  The example above that fails to compile is considered poor design and should not be encouraged.</p>
   <p>(The arguments about overload sets apply equally to explicit template specializations.)</p>
   <p>Not allowing the aliasing of standard types imposes an implementation burden.  If aliasing were allowed, then implementations that don’t define any extended floating-point types could define some of the aliases with a little bit of library code that boils down to something like:</p>
<pre class="highlight"><code class="highlight"><c- k><c- k>namespace</c-></c-> <c- n><c- n>std</c-></c-> <c- p><c- p>{</c-></c->
  <c- k><c- k>using</c-></c-> <c- n><c- n>float32_t</c-></c-> <c- o><c- o>=</c-></c-> <c- b><c- b>float</c-></c-><c- p><c- p>;</c-></c->
  <c- k><c- k>using</c-></c-> <c- n><c- n>float64_t</c-></c-> <c- o><c- o>=</c-></c-> <c- b><c- b>double</c-></c-><c- p><c- p>;</c-></c->
<c- p><c- p>}</c-></c->
</code></pre>
   <p>But when aliasing is not allowed, implementations have to support extended floating-point types in at least the compiler front end, which is not a trivial task.  There is also a burden on the name mangling ABI, which will have to define how to encode these extended floating-point types.</p>
   <p>The authors feel that the burden on users of allowing aliasing of standard types is greater than the burden on implementers of not allowing such aliasing.  Therefore, the authors recommend not allowing aliasing of standard types.</p>
   <p>(This argument is predicated on the changes to overload resolution proposed in <a data-link-type="biblio" href="#biblio-p1467">[P1467]</a>.  If those changes don’t go through, then having <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>float64_t</c-></code> be an alias of an extended floating-point type rather than an alias of <code class="highlight"><c- b>double</c-></code> will cause the following code to not compile:</p>
<pre class="highlight"><code class="highlight"><c- b><c- b>void</c-></c-> <c- nf><c- nf>f</c-></c-><c- p><c- p>(</c-></c-><c- n><c- n>std</c-></c-><c- o><c- o>::</c-></c-><c- n><c- n>float32_t</c-></c-><c- p><c- p>);</c-></c->
<c- b><c- b>void</c-></c-> <c- nf><c- nf>f</c-></c-><c- p><c- p>(</c-></c-><c- n><c- n>std</c-></c-><c- o><c- o>::</c-></c-><c- n><c- n>float64_t</c-></c-><c- p><c- p>);</c-></c->
<c- b><c- b>void</c-></c-> <c- nf><c- nf>g</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>double</c-></c-> <c- n><c- n>x</c-></c-><c- p><c- p>)</c-></c-> <c- p><c- p>{</c-></c->
  <c- n><c- n>f</c-></c-><c- p><c- p>(</c-></c-><c- n><c- n>x</c-></c-><c- p><c- p>);</c-></c-> <c- c1><c- c1>// error - ambiguous call without overload resolution changes</c-></c->
<c- p><c- p>}</c-></c->
</code></pre>
   <p>If that code doesn’t compile, that would be a bigger burden on users than not being able to overload on both <code class="highlight"><c- b>double</c-></code> and <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>float64_t</c-></code>.  That would change the authors' opinion on the best resolution for this issue.)</p>
   <h3 class="heading settled" data-level="5.3" id="layout-vs-behavior"><span class="secno">5.3. </span><span class="content">Layout vs. behavior</span><a class="self-link" href="#layout-vs-behavior"></a></h3>
   <p>The IEEE-conforming type aliases must have the specified IEEE layout and should have the required behavior.  For the four IEEE-conforming type aliases, <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>numeric_limits</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- n>is_iec559</c-></code> is true.</p>
   <h3 class="heading settled" data-level="5.4" id="test-macros"><span class="secno">5.4. </span><span class="content">Feature test macros</span><a class="self-link" href="#test-macros"></a></h3>
   <p>Since implementations may choose to support (or not) each of the fixed-layout aliases individually, there should be a separate test macro for detecting each of the type aliases.  The names of the test macros would be derived from whichever type alias names we settle on.  (The authors are not thrilled with introducing so many new test macros, but they have yet to come up with a better idea.)</p>
   <p class="issue" id="issue-50e4e46f"><a class="self-link" href="#issue-50e4e46f"></a> How should feature test macros be handled for this feature?</p>
   <h3 class="heading settled" data-level="5.5" id="names"><span class="secno">5.5. </span><span class="content">Names</span><a class="self-link" href="#names"></a></h3>
   <p>We are proposing several different naming schemes for fixed-layout type alias, and are open to other suggested naming schemes.  In committee discussions so far, no set of names has emerged as the favorites.  The authors have whittled proposed names down to what they feel are the three best choices, and are comfortable leaving it up to the committee to choose between those.</p>
   <h4 class="heading settled" data-level="5.5.1" id="floatX_t"><span class="secno">5.5.1. </span><span class="content"><code class="highlight"><c- n>floatX_t</c-></code></span><a class="self-link" href="#floatX_t"></a></h4>
   <ul>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>float16_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>float32_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>float64_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>float128_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>bfloat16_t</c-></code></p>
   </ul>
   <p>This is the simplest of all the options being presented.  It is the naming scheme used by Boost.Math’s fixed-layout floating-point types.</p>
   <p>Nothing in the names of the IEEE aliases implies that they are in fact IEEE binary formats.  Additionally, <code class="highlight"><c- n>float16_t</c-></code> and <code class="highlight"><c- n>bfloat16_t</c-></code> are similar enough that we aren’t fully comfortable using these names.</p>
   <h4 class="heading settled" data-level="5.5.2" id="fp_namespace"><span class="secno">5.5.2. </span><span class="content"><code class="highlight"><c- n>fp</c-><c- o>::</c-><c- n>binaryX_t</c-></code></span><a class="self-link" href="#fp_namespace"></a></h4>
   <ul>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp</c-><c- o>::</c-><c- n>binary16_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp</c-><c- o>::</c-><c- n>binary32_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp</c-><c- o>::</c-><c- n>binary64_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp</c-><c- o>::</c-><c- n>binary128_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp</c-><c- o>::</c-><c- n>bfloat16_t</c-></code></p>
   </ul>
   <p>The namespace <code class="highlight"><c- n>fp</c-></code> makes it more obvious that these types are floating-point types, assisting in the recognition of <code class="highlight"><c- n>binary16</c-></code> as an <a data-link-type="biblio" href="#biblio-ieee-754-2008">[IEEE-754-2008]</a> format.  A using namespace directive can be used to avoid repeating <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp</c-><c- o>::</c-></code> everywhere.</p>
   <p>The drawbacks of this approach are that it introduces a new namespace with a very small purpose, and that <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp</c-><c- o>::</c-><c- n>bloat16_t</c-></code> is somewhat redundant with two different floating-point indications (<code class="highlight"><c- n>fp</c-></code> and the <code class="highlight"><c- b>float</c-></code> in <code class="highlight"><c- n>bfloat16_t</c-></code>).</p>
   <h4 class="heading settled" data-level="5.5.3" id="fp_binaryX_t"><span class="secno">5.5.3. </span><span class="content"><code class="highlight"><c- n>fp_binaryX_t</c-></code></span><a class="self-link" href="#fp_binaryX_t"></a></h4>
   <ul>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp_binary16_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp_binary32_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp_binary64_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp_binary128_t</c-></code></p>
    <li data-md>
     <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>fp_bfloat16_t</c-></code></p>
   </ul>
   <p>This is a slight modification of the previous scheme, which trades the nested namespace for an <code class="highlight"><c- n>fp_</c-></code> prefix.  The advantages and disadvantages are similar.</p>
   <h2 class="heading settled" data-level="6" id="literals"><span class="secno">6. </span><span class="content">Literal suffixes</span><a class="self-link" href="#literals"></a></h2>
   <p>Once the names of the aliases have been decided on, a literal suffix for each of those types will be defined, similar to what is proposed in <a data-link-type="biblio" href="#biblio-p1280">[P1280]</a>.  Each type will have either two literal operators with <code class="highlight"><c- b>long</c-> <c- b>double</c-></code> and <code class="highlight"><c- b>unsigned</c-> <c- b>long</c-> <c- b>long</c-></code> parameters, or (for types whose conversion rank is not less than <code class="highlight"><c- b>long</c-> <c- b>double</c-></code>) one literal operator with a <code class="highlight"><c- k>const</c-> <c- b>char</c-> <c- o>*</c-></code> parameter.  The literal operators for an implementation might look like this (with all names subject to change):</p>
   <blockquote>
<pre class="highlight"><code class="highlight"><c- k><c- k>namespace</c-></c-> <c- n><c- n>std</c-></c-> <c- p><c- p>{</c-></c->
  <c- kr><c- kr>inline</c-></c-> <c- k><c- k>namespace</c-></c-> <c- n><c- n>literals</c-></c-> <c- p><c- p>{</c-></c->
  <c- kr><c- kr>inline</c-></c-> <c- k><c- k>namespace</c-></c-> <c- n><c- n>fixed_float_literals</c-></c-> <c- p><c- p>{</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>float16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>float16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>float32_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp32</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>float32_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp32</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>float64_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp64</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>float64_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp64</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>float128_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp128</c-></c-><c- p><c- p>(</c-></c-><c- k><c- k>const</c-></c-> <c- b><c- b>char</c-></c-> <c- o><c- o>*</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>bfloat16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>bf16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-><c- p><c- p>);</c-></c->
    <c- k><c- k>constexpr</c-></c-> <c- n><c- n>bfloat16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>bf16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-><c- p><c- p>);</c-></c->
  <c- p><c- p>}</c-></c->
  <c- p><c- p>}</c-></c->
<c- p><c- p>}</c-></c->
</code></pre>
   </blockquote>
   <blockquote>
<pre class="highlight"><code class="highlight"><c- k><c- k>constexpr</c-></c-> <c- n><c- n>float16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
<c- k><c- k>constexpr</c-></c-> <c- n><c- n>float16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
</code></pre>
    <div class="numbered"> <em>Returns</em>: <code class="highlight"><c- n>float16_t</c-><c- p>{</c-><c- n>d</c-><c- p>}</c-></code>. </div>
<pre class="highlight"><code class="highlight"><c- k><c- k>constexpr</c-></c-> <c- n><c- n>float32_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp32</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
<c- k><c- k>constexpr</c-></c-> <c- n><c- n>float32_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp32</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
</code></pre>
    <div class="numbered"> <em>Returns</em>: <code class="highlight"><c- n>float32_t</c-><c- p>{</c-><c- n>d</c-><c- p>}</c-></code>. </div>
<pre class="highlight"><code class="highlight"><c- k><c- k>constexpr</c-></c-> <c- n><c- n>float64_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp64</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
<c- k><c- k>constexpr</c-></c-> <c- n><c- n>float64_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp64</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
</code></pre>
    <div class="numbered"> <em>Returns</em>: <code class="highlight"><c- n>float64_t</c-><c- p>{</c-><c- n>d</c-><c- p>}</c-></code>. </div>
<pre class="highlight"><code class="highlight"><c- k><c- k>constexpr</c-></c-> <c- n><c- n>float128_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>fp128</c-></c-><c- p><c- p>(</c-></c-><c- k><c- k>const</c-></c-> <c- b><c- b>char</c-></c-> <c- o><c- o>*</c-></c-><c- n><c- n>s</c-></c-><c- p><c- p>);</c-></c->
</code></pre>
    <div class="numbered">
      <em>Effects</em>: Equivalent to: 
<pre class="highlight"><code class="highlight"><c- n><c- n>float128_t</c-></c-> <c- n><c- n>x</c-></c-><c- p><c- p>{</c-></c-><c- mi><c- mi>0</c-></c-><c- p><c- p>};</c-></c->
<c- n><c- n>from_chars</c-></c-><c- p><c- p>(</c-></c-><c- n><c- n>s</c-></c-><c- p><c- p>,</c-></c-> <c- n><c- n>s</c-></c-> <c- o><c- o>+</c-></c-> <c- n><c- n>strlen</c-></c-><c- p><c- p>(</c-></c-><c- n><c- n>s</c-></c-><c- p><c- p>),</c-></c-> <c- o><c- o>&amp;</c-></c-><c- n><c- n>x</c-></c-><c- p><c- p>);</c-></c->
<c- k><c- k>return</c-></c-> <c- n><c- n>x</c-></c-><c- p><c- p>;</c-></c->
</code></pre>
    </div>
<pre class="highlight"><code class="highlight"><c- k><c- k>constexpr</c-></c-> <c- n><c- n>bfloat16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>bf16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>long</c-></c-> <c- b><c- b>double</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
<c- k><c- k>constexpr</c-></c-> <c- n><c- n>bfloat16_t</c-></c-> <c- k><c- k>operator</c-></c-><c- s><c- s>""</c-></c-><c- n><c- n>bf16</c-></c-><c- p><c- p>(</c-></c-><c- b><c- b>unsigned</c-></c-> <c- b><c- b>long</c-></c-> <c- b><c- b>long</c-></c-> <c- n><c- n>d</c-></c-><c- p><c- p>);</c-></c->
</code></pre>
    <div class="numbered"> <em>Returns</em>: <code class="highlight"><c- n>bfloat16_t</c-><c- p>{</c-><c- n>d</c-><c- p>}</c-></code>. </div>
   </blockquote>
  </main>
<script>
(function() {
  "use strict";
  var collapseSidebarText = '<span aria-hidden="true">←</span> '
                          + '<span>Collapse Sidebar</span>';
  var expandSidebarText   = '<span aria-hidden="true">→</span> '
                          + '<span>Pop Out Sidebar</span>';
  var tocJumpText         = '<span aria-hidden="true">↑</span> '
                          + '<span>Jump to Table of Contents</span>';

  var sidebarMedia = window.matchMedia('screen and (min-width: 78em)');
  var autoToggle   = function(e){ toggleSidebar(e.matches) };
  if(sidebarMedia.addListener) {
    sidebarMedia.addListener(autoToggle);
  }

  function toggleSidebar(on) {
    if (on == undefined) {
      on = !document.body.classList.contains('toc-sidebar');
    }

    /* Don’t scroll to compensate for the ToC if we’re above it already. */
    var headY = 0;
    var head = document.querySelector('.head');
    if (head) {
      // terrible approx of "top of ToC"
      headY += head.offsetTop + head.offsetHeight;
    }
    var skipScroll = window.scrollY < headY;

    var toggle = document.getElementById('toc-toggle');
    var tocNav = document.getElementById('toc');
    if (on) {
      var tocHeight = tocNav.offsetHeight;
      document.body.classList.add('toc-sidebar');
      document.body.classList.remove('toc-inline');
      toggle.innerHTML = collapseSidebarText;
      if (!skipScroll) {
        window.scrollBy(0, 0 - tocHeight);
      }
      tocNav.focus();
      sidebarMedia.addListener(autoToggle); // auto-collapse when out of room
    }
    else {
      document.body.classList.add('toc-inline');
      document.body.classList.remove('toc-sidebar');
      toggle.innerHTML = expandSidebarText;
      if (!skipScroll) {
        window.scrollBy(0, tocNav.offsetHeight);
      }
      if (toggle.matches(':hover')) {
        /* Unfocus button when not using keyboard navigation,
           because I don’t know where else to send the focus. */
        toggle.blur();
      }
    }
  }

  function createSidebarToggle() {
    /* Create the sidebar toggle in JS; it shouldn’t exist when JS is off. */
    var toggle = document.createElement('a');
      /* This should probably be a button, but appearance isn’t standards-track.*/
    toggle.id = 'toc-toggle';
    toggle.class = 'toc-toggle';
    toggle.href = '#toc';
    toggle.innerHTML = collapseSidebarText;

    sidebarMedia.addListener(autoToggle);
    var toggler = function(e) {
      e.preventDefault();
      sidebarMedia.removeListener(autoToggle); // persist explicit off states
      toggleSidebar();
      return false;
    }
    toggle.addEventListener('click', toggler, false);


    /* Get <nav id=toc-nav>, or make it if we don’t have one. */
    var tocNav = document.getElementById('toc-nav');
    if (!tocNav) {
      tocNav = document.createElement('p');
      tocNav.id = 'toc-nav';
      /* Prepend for better keyboard navigation */
      document.body.insertBefore(tocNav, document.body.firstChild);
    }
    /* While we’re at it, make sure we have a Jump to Toc link. */
    var tocJump = document.getElementById('toc-jump');
    if (!tocJump) {
      tocJump = document.createElement('a');
      tocJump.id = 'toc-jump';
      tocJump.href = '#toc';
      tocJump.innerHTML = tocJumpText;
      tocNav.appendChild(tocJump);
    }

    tocNav.appendChild(toggle);
  }

  var toc = document.getElementById('toc');
  if (toc) {
    createSidebarToggle();
    toggleSidebar(sidebarMedia.matches);

    /* If the sidebar has been manually opened and is currently overlaying the text
       (window too small for the MQ to add the margin to body),
       then auto-close the sidebar once you click on something in there. */
    toc.addEventListener('click', function(e) {
      if(e.target.tagName.toLowerCase() == "a" && document.body.classList.contains('toc-sidebar') && !sidebarMedia.matches) {
        toggleSidebar(false);
      }
    }, false);
  }
  else {
    console.warn("Can’t find Table of Contents. Please use <nav id='toc'> around the ToC.");
  }

  /* Wrap tables in case they overflow */
  var tables = document.querySelectorAll(':not(.overlarge) > table.data, :not(.overlarge) > table.index');
  var numTables = tables.length;
  for (var i = 0; i < numTables; i++) {
    var table = tables[i];
    var wrapper = document.createElement('div');
    wrapper.className = 'overlarge';
    table.parentNode.insertBefore(wrapper, table);
    wrapper.appendChild(table);
  }

})();
</script>
  <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-bfloat16">[BFLOAT16]
   <dd><a href="https://en.wikipedia.org/wiki/Bfloat16_floating-point_format">bfloat16 floating-point format</a>. URL: <a href="https://en.wikipedia.org/wiki/Bfloat16_floating-point_format">https://en.wikipedia.org/wiki/Bfloat16_floating-point_format</a>
   <dt id="biblio-ieee-754-2008">[IEEE-754-2008]
   <dd><a href="http://ieeexplore.ieee.org/servlet/opac?punumber=4610933">IEEE Standard for Floating-Point Arithmetic</a>. 29 August 2008. URL: <a href="http://ieeexplore.ieee.org/servlet/opac?punumber=4610933">http://ieeexplore.ieee.org/servlet/opac?punumber=4610933</a>
   <dt id="biblio-n1703">[N1703]
   <dd>Paul A. Bristow; Christopher Kormanyos; John Maddock. <a href="http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1703.pdf">Floating-Point Typedefs Having Specified Widths</a>. URL: <a href="http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1703.pdf">http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1703.pdf</a>
   <dt id="biblio-p0192">[P0192]
   <dd>Michał Dominiak; et al. <a href="https://wg21.link/P0192">`short float` and fixed-size floating point types</a>. URL: <a href="https://wg21.link/P0192">https://wg21.link/P0192</a>
   <dt id="biblio-p1280">[P1280]
   <dd>Isabella Muerte. <a href="https://wg21.link/P1280">Integer Width Literals</a>. URL: <a href="https://wg21.link/P1280">https://wg21.link/P1280</a>
   <dt id="biblio-p1467">[P1467]
   <dd>Michał Dominiak; David Olsen. <a href="https://wg21.link/P1467">Extended floating-point types</a>. URL: <a href="https://wg21.link/P1467">https://wg21.link/P1467</a>
  </dl>
  <h2 class="no-num no-ref heading settled" id="issues-index"><span class="content">Issues Index</span><a class="self-link" href="#issues-index"></a></h2>
  <div style="counter-reset:issue">
   <div class="issue"> What new or existing header should the type aliases go into?<a href="#issue-0ebb1eda"> ↵ </a></div>
   <div class="issue"> How should feature test macros be handled for this feature?<a href="#issue-50e4e46f"> ↵ </a></div>
  </div>