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  <title>P1696R0: Refinement proposal for P0920 Precalculated hash values in lookup</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,
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 <body class="h-entry">
  <div class="head">
   <p data-fill-with="logo"></p>
   <h1 class="p-name no-ref" id="title">P1696R0<br>Refinement proposal for P0920 Precalculated hash values in lookup</h1>
   <h2 class="no-num no-toc no-ref heading settled" id="subtitle"><span class="content">Draft Proposal, <time class="dt-updated" datetime="2019-06-12">2019-06-12</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt>This version:
     <dd><a class="u-url" href="http://wg21.link/p1696r0">http://wg21.link/p1696r0</a>
     <dt>Authors:
     <dd>
      <dd class="editor p-author h-card vcard"><span class="p-name fn"></span>
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-url url" href="xshi.fb@gmail.com">Xiao Shi</a> (<a class="p-org org" href="https://www.facebook.com">Facebook</a>) <a class="u-email email" href="mailto:xshi.fb@gmail.com">xshi.fb@gmail.com</a>
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-url url" href="http://www.train-it.eu">Mateusz Pusz</a> (<a class="p-org org" href="http://www.epam.com">Epam Systems</a>) <a class="u-email email" href="mailto:mateusz.pusz@gmail.com">mateusz.pusz@gmail.com</a>
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-url url" href="gromer@google.com">Geoffrey Romer</a> (<a class="p-org org" href="https://www.google.com">Google</a>) <a class="u-email email" href="mailto:gromer@google.com">gromer@google.com</a>
     <dd>
      <dd class="editor p-author h-card vcard"><a class="p-name fn u-url url" href="yfeldblum@gmail.com">Jay Feldblum</a> (<a class="p-org org" href="https://www.facebook.com">Facebook</a>) <a class="u-email email" href="mailto:yfeldblum@gmail.com">yfeldblum@gmail.com</a>
     <dd>
      <dd class="editor p-author h-card vcard"><span class="p-name fn"></span>
     <dt>Audience:
     <dd>LEWG
     <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>
  <div class="p-summary" data-fill-with="abstract">
   <h2 class="no-num no-toc no-ref heading settled" id="abstract"><span class="content">Abstract</span></h2>
   <p><a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a> proposed an extension of the lookup API of unordered associative

  containers that allows the user to precalculate hash values and use them
  directly. We have found the proposed API error-prone from implementation
  experiences. This paper details the problems and proposes corresponding design
  alternatives for using precalculated hash values.</p>
  </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="#motivation-and-scope"><span class="secno">1</span> <span class="content">Motivation and Scope</span></a>
     <ol class="toc">
      <li><a href="#hasher-misuse"><span class="secno">1.1</span> <span class="content">Hasher misuse</span></a>
      <li><a href="#unclear-sharing-semantics"><span class="secno">1.2</span> <span class="content">Unclear semantics of sharing hash values across containers</span></a>
      <li><a href="#limit-opt"><span class="secno">1.3</span> <span class="content">Use of raw hash values limits optimization opportunities</span></a>
      <li><a href="#extensibility"><span class="secno">1.4</span> <span class="content">Extensibility considerations</span></a>
     </ol>
    <li>
     <a href="#design-alternatives"><span class="secno">2</span> <span class="content">Design alternatives and respective concerns</span></a>
     <ol class="toc">
      <li>
       <a href="#encapsulate-with-hash-token"><span class="secno">2.1</span> <span class="content">Encapsulate hash values with <code class="highlight"><c- n>hash_token</c-></code></span></a>
       <ol class="toc">
        <li><a href="#possible-mismatch"><span class="secno">2.1.1</span> <span class="content">Possible mismatch between <code class="highlight"><c- n>hash_token</c-></code> and key</span></a>
        <li><a href="#store-key"><span class="secno">2.1.2</span> <span class="content">Whether to store reference / copy of key in hash_token</span></a>
        <li><a href="#sharing-tokens"><span class="secno">2.1.3</span> <span class="content">Sharing <code class="highlight"><c- n>hash_token</c-></code>s across different containers</span></a>
       </ol>
      <li><a href="#caching-hash-values"><span class="secno">2.2</span> <span class="content">“Caching” hash values in keys</span></a>
     </ol>
    <li>
     <a href="#wording"><span class="secno">3</span> <span class="content">Proposed Wording</span></a>
     <ol class="toc">
      <li><a href="#wording-revert"><span class="secno">3.1</span> <span class="content">Option 1: Revert <span>[P0920R2]</span></span></a>
      <li><a href="#wording-opaque-token"><span class="secno">3.2</span> <span class="content">Option 2: Replace <code class="highlight"><c- b>size_t</c-></code> with an opaque token</span></a>
     </ol>
    <li><a href="#implementation-usage-experiences"><span class="secno">4</span> <span class="content">Implementation and Usage Experiences</span></a>
    <li><a href="#acknowledgements"><span class="secno">5</span> <span class="content">Acknowledgements</span></a>
    <li>
     <a href="#references"><span class="secno"></span> <span class="content">References</span></a>
     <ol class="toc">
      <li><a href="#normative"><span class="secno"></span> <span class="content">Normative References</span></a>
      <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="motivation-and-scope"><span class="secno">1. </span><span class="content">Motivation and Scope</span><a class="self-link" href="#motivation-and-scope"></a></h2>
    <a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a> extends the interface of unordered containers with member function
overloads that have one additional argument taking a precalculated hash value
for the value being queried. 
   <div class="example" id="example-orig">
    <a class="self-link" href="#example-orig"></a> The motivating example below (taken from <a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a> allows the code to look
for the same keyword in more than one container at a time. 
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>array</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>unordered_map</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>string</c-><c- p>,</c-> <c- b>int</c-><c- o>></c-><c- p>,</c-> <c- n>array_size</c-><c- o>></c-> <c- n>maps</c-><c- p>;</c->

<c- b>void</c-> <c- nf>update</c-><c- p>(</c-><c- k>const</c-> <c- n>std</c-><c- o>::</c-><c- n>string</c-><c- o>&amp;</c-> <c- n>user</c-><c- p>)</c->
<c- p>{</c->
  <c- k>const</c-> <c- k>auto</c-> <c- n>hash</c-> <c- o>=</c-> <c- n>maps</c-><c- p>.</c-><c- n>front</c-><c- p>().</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>user</c-><c- p>);</c->
  <c- k>for</c-><c- p>(</c-><c- k>auto</c-><c- o>&amp;</c-> <c- nl>m</c-> <c- p>:</c-> <c- n>maps</c-><c- p>)</c-> <c- p>{</c->
    <c- k>auto</c-> <c- n>it</c-> <c- o>=</c-> <c- n>m</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>user</c-><c- p>,</c-> <c- n>hash</c-><c- p>);</c->
    <c- c1>// ...</c->
  <c- p>}</c->
<c- p>}</c->
</pre>
   </div>
   <p>This approach has a number of problems, including the lack of safety around
hasher misuse and the lack of clarity around sharing hash values across multiple
containers.</p>
   <h3 class="heading settled" data-level="1.1" id="hasher-misuse"><span class="secno">1.1. </span><span class="content">Hasher misuse</span><a class="self-link" href="#hasher-misuse"></a></h3>
    Instead of obtaining the hasher instance via <code class="highlight"><c- n>map</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()</c-></code>, users often
precalculate the key using a different hasher type or the wrong hasher instance.
(See the <a href="#implementation-usage-experiences">§4 Implementation and Usage Experiences</a> section for more details.) 
   <p>Firstly, this is plain incorrect if the hasher for the container is stateful.
Secondly, even if the user used a different hasher instance of the correct
hasher type and the hasher is not stateful, it increases the chance of mismatch
between the hash value and the key, especially when the hasher of the container
gets changed. It would require auditing and potentially changing all call sites
of this API.</p>
   <h3 class="heading settled" data-level="1.2" id="unclear-sharing-semantics"><span class="secno">1.2. </span><span class="content">Unclear semantics of sharing hash values across containers</span><a class="self-link" href="#unclear-sharing-semantics"></a></h3>
    The motivating example in <a href="#motivation-and-scope">§1 Motivation and Scope</a> assumes the hasher type <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>hash</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>string</c-><c- o>></c-></code> is not stateful and the maps in the array all use the
same hasher type. This assumption may not be true for other use cases or
implementations, and is susceptible to future changes. For example, if the
different maps are in different part of a code base, there is no way for the
user to tell whether hash values from one <code class="highlight"><c- n>unordered_map</c-></code> can be reused in
another other than comparing the types of the hashers. 
   <h3 class="heading settled" data-level="1.3" id="limit-opt"><span class="secno">1.3. </span><span class="content">Use of raw hash values limits optimization opportunities</span><a class="self-link" href="#limit-opt"></a></h3>
    Some hash table implementations may choose to further process the hash values
obtained from <code class="highlight"><c- n>Hash</c-></code>. For example, <a data-link-type="biblio" href="#biblio-f14">[F14]</a> mixes hash values from hashers not
known to be avalanching (e.g., <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>hash</c-></code> for numeric types). The existing
approach does not allow hash table implementations to  preprocess. 
   <p>In addition, in a few use cases, sizable performance gains came mainly from
prefetching the pertinent cache lines, as opposed to merely precalculating the
hash value. This optimization is not possible with the API <a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a> proposed.</p>
   <h3 class="heading settled" data-level="1.4" id="extensibility"><span class="secno">1.4. </span><span class="content">Extensibility considerations</span><a class="self-link" href="#extensibility"></a></h3>
    In the future, we may wish to extend the mutation methods (<code class="highlight"><c- n>emplace</c-></code>, <code class="highlight"><c- n>try_emplace</c-></code>, <code class="highlight"><c- n>insert</c-></code>, etc.) with both heterogeneous operations and
precalculated hash API. Keeping the position of the precalculated hash argument
consistent and putting it as the first argument would be advantageous. 
   <div class="example" id="example-extensibility">
    <a class="self-link" href="#example-extensibility"></a> If the <code class="highlight"><c- n>mapped_type</c-></code> is constructible both from an N-tuple that starts with <code class="highlight"><c- b>size_t</c-></code> as well as an (N-1)-tuple, overload resolution will be unable to
resolve the following: 
<pre class="language-cpp highlight"><c- n>try_emplace</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-><c- p>,</c-> <c- n>Args</c-><c- o>&amp;&amp;</c-><c- p>...</c-> <c- n>args</c-><c- p>);</c->
<c- n>try_emplace</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>,</c-> <c- n>Args</c-><c- o>&amp;&amp;</c-><c- p>...</c-> <c- n>args</c-><c- p>);</c->
</pre>
   </div>
   <p>For <code class="highlight"><c- n>emplace</c-></code>, disambiguation is more difficult, which suggests an API akin to <code class="highlight"><c- n>emplace_hint</c-></code> where the <em>first</em> parameter is the precalculated hash / position
hint.</p>
   <h2 class="heading settled" data-level="2" id="design-alternatives"><span class="secno">2. </span><span class="content">Design alternatives and respective concerns</span><a class="self-link" href="#design-alternatives"></a></h2>
   <h3 class="heading settled" data-level="2.1" id="encapsulate-with-hash-token"><span class="secno">2.1. </span><span class="content">Encapsulate hash values with <code class="highlight"><c- n>hash_token</c-></code></span><a class="self-link" href="#encapsulate-with-hash-token"></a></h3>
    Some of the above issues can be addressed by encapsulating the raw hash value in
a <code class="highlight"><c- n>hash_token</c-></code> and introducing a API which allows the container to return an
opaque token to the user that can be used in lookups. 
   <div class="example" id="example-token">
    <a class="self-link" href="#example-token"></a> 
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>unordered_map</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>string</c-><c- p>,</c-> <c- b>int</c-><c- o>></c-> <c- n>m</c-><c- p>;</c->
<c- k>auto</c-> <c- n>ht</c-> <c- o>=</c-> <c- n>m</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>key</c-><c- p>);</c->
<c- n>m</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>ht</c-><c- p>,</c-> <c- n>key</c-><c- p>);</c->
</pre>
   </div>
    Since the token can only be meaningfully generated via the member function <code class="highlight"><c- n>token</c-><c- p>()</c-></code>, it prevents misuse of the hasher instance. However, it does <em>not</em> preclude possible mismatch between <code class="highlight"><c- n>hash_token</c-></code> and key. 
   <h4 class="heading settled" data-level="2.1.1" id="possible-mismatch"><span class="secno">2.1.1. </span><span class="content">Possible mismatch between <code class="highlight"><c- n>hash_token</c-></code> and key</span><a class="self-link" href="#possible-mismatch"></a></h4>
    The intention of lookup using precalculated hash is to improve performance in
specific use cases which need it. These use cases can presumably be trusted to
provide a matching <code class="highlight"><c- n>hash_token</c-></code> and key. The semantics of the look up API would
be "find the table entry identified by the key if the provided <code class="highlight"><c- n>hash_token</c-></code> matches the key, returning an iterator to the entry, otherwise returning <code class="highlight"><c- n>end</c-><c- p>()</c-></code>." 
   <p>Conceivably, the hash table implementation could check whether the supplied <code class="highlight"><c- n>hash_token</c-></code>s match the keys in debug builds. (Note that we could do that with
raw hash values as well and we acknowledge this is an unresolved problem with
this approach.)</p>
   <h4 class="heading settled" data-level="2.1.2" id="store-key"><span class="secno">2.1.2. </span><span class="content">Whether to store reference / copy of key in hash_token</span><a class="self-link" href="#store-key"></a></h4>
    Storing a reference or a copy of the key in the <code class="highlight"><c- n>hash_token</c-></code> would prevent any
mismatch between the <code class="highlight"><c- n>hash_token</c-></code> and the key, but could easily result in
dangling references or expensive key copies. 
   <h4 class="heading settled" data-level="2.1.3" id="sharing-tokens"><span class="secno">2.1.3. </span><span class="content">Sharing <code class="highlight"><c- n>hash_token</c-></code>s across different containers</span><a class="self-link" href="#sharing-tokens"></a></h4>
    This approach does not provide a satisfying solution to specifying sharing
semantics across different containers. 
   <h3 class="heading settled" data-level="2.2" id="caching-hash-values"><span class="secno">2.2. </span><span class="content">“Caching” hash values in keys</span><a class="self-link" href="#caching-hash-values"></a></h3>
    Another approach for using precalculating hash is to “cache” hash values
alongside <code class="highlight"><c- n>key_type</c-></code>s and rely on heterogeneous lookup (<a data-link-type="biblio" href="#biblio-p0919r3">[P0919R3]</a>, P1690R0)
to use the precalculated hash values. (P1661R0 has a detailed account of a
similar approach.) 
   <div class="example" id="example-key-with-hash">
    <a class="self-link" href="#example-key-with-hash"></a> 
<pre class="language-cpp highlight"><c- k>struct</c-> <c- n>KeyWithHash</c-> <c- p>{</c->
  <c- n>std</c-><c- o>::</c-><c- n>string</c-> <c- n>key_</c-><c- p>;</c->
  <c- b>size_t</c-> <c- n>hash_</c-><c- p>;</c->
<c- p>};</c->

<c- n>std</c-><c- o>::</c-><c- n>unordered_map</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>string</c-><c- p>,</c-> <c- b>int</c-><c- p>,</c-> <c- n>MyHash</c-><c- p>,</c-> <c- n>MyEqualTo</c-><c- o>></c-> <c- n>m</c-><c- p>;</c->
<c- n>m</c-><c- p>.</c-><c- n>emplace</c-><c- p>(</c->“<c- n>foo</c->”<c- p>,</c-> <c- mi>42</c-><c- p>);</c->

<c- n>KeyWithHash</c-> <c- n>kh</c-><c- p>;</c->
<c- n>kh</c-><c- p>.</c-><c- n>key_</c-> <c- o>=</c-> <c- s>"foo"</c-><c- p>;</c->
<c- n>kh</c-><c- p>.</c-><c- n>hash_</c-> <c- o>=</c-> <c- n>m</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>kh</c-><c- p>.</c-><c- n>key_</c-><c- p>);</c-> <c- c1>// precalculated</c->
<c- n>m</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>kh</c-><c- p>);</c-> <c- c1>// heterogeneous lookup</c->
</pre>
   </div>
    This approach allows the user to exploit the benefit of precalculating hash
without standard library API changes. It does, however, put more onus on the
user, who is required to implement a custom hasher and key equality comparison
operator. It does <em>not</em> address some of the problems stated above, as the user
is still required to use the right hasher instance and the raw hash value is
still exposed. 
   <p>Note that this approach does not need the committee need to adopt any changes to
the standard. This is an option available to users with or without <a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a>.
This could also be a mitigation that would buy us time to explore the design
space further.</p>
   <h2 class="heading settled" data-level="3" id="wording"><span class="secno">3. </span><span class="content">Proposed Wording</span><a class="self-link" href="#wording"></a></h2>
   <h3 class="heading settled" data-level="3.1" id="wording-revert"><span class="secno">3.1. </span><span class="content">Option 1: Revert <a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a></span><a class="self-link" href="#wording-revert"></a></h3>
    Given the issues detailed above, as well as the lack of an obviously clean
design or agreeable approach to use precalculated hash values for lookups, we
recommend reverting the changes in <a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a>, as the first option. This
allows the LEWG to come to a safer and better-explored design in the near
future. 
   <h3 class="heading settled" data-level="3.2" id="wording-opaque-token"><span class="secno">3.2. </span><span class="content">Option 2: Replace <code class="highlight"><c- b>size_t</c-></code> with an opaque token</span><a class="self-link" href="#wording-opaque-token"></a></h3>
    As a second option, we recommend using an opaque hash token to encapsulate the
generation and usage of precalculated hash, with the following wording changes.
This is a relatively small forward-fix to <a data-link-type="biblio" href="#biblio-p0920r2">[P0920R2]</a>. 
   <p>The proposed changes are relative to the working draft of the standard as of <a data-link-type="biblio" href="#biblio-n4810">[N4810]</a>.</p>
   <p>Modify <strong>22.2.7 [unord.req]</strong> paragraph 11.23 as follows:</p>
   <blockquote>
     <code class="highlight"><c- n>hk</c-></code> and <code class="highlight"><c- n>hke</c-></code> denote values of type 
    <del><code class="highlight"><c- b>size_t</c-></code></del>
    <ins><code class="highlight"><c- n>X</c-><c- o>::</c-><c- n>hash_token_type</c-></code></ins>
    . 
   </blockquote>
   <p>Add the following paragraph to <strong>22.2.7 [unord.req]</strong>:</p>
   <blockquote>
    <ins>Member function templates <code class="highlight"><c- n>find</c-></code>, <code class="highlight"><c- n>count</c-></code>, <code class="highlight"><c- n>equal_range</c-></code>, and <code class="highlight"><c- n>contains</c-></code> that
take a <code class="highlight"><c- n>hash_token_type</c-></code> shall not participate in overload resolution unless <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>is_empty_v</c-><c- o>&lt;</c-><c- n>Hash</c-><c- o>></c-></code> is true.</ins>
   </blockquote>
   <p>Modify Table 70 in <strong>22.2.7 [unord.req]</strong> as follows:</p>
   <blockquote>
    <table>
     <tbody>
      <tr>
       <th>Expression
       <th>Return type
       <th>Assertion/note pre-/post-condition
       <th>Complexity
      <tr>
       <td>
        <ins><code class="highlight"><c- n>X</c-><c- o>::</c-><c- n>hash_token_type</c-></code></ins>
       <td>
        <ins>An implementation-defined type that can be used to look up its associated key.</ins>
       <td>
        <ins><em>Expects</em>: <code class="highlight"><c- n>hash_token_type</c-></code> meets the <em>Cpp17DefaultConstructible</em> requirements.</ins>
       <td>
        <ins>compile time</ins>
      <tr>
       <td>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code></ins>
       <td>
        <ins><code class="highlight"><c- n>X</c-><c- o>::</c-><c- n>hash_token_type</c-></code></ins>
       <td>
        <ins><em>Returns</em>: returns a value <code class="highlight"><c- n>hk</c-></code> of type <code class="highlight"><c- n>X</c-><c- o>::</c-><c- n>hash_token_type</c-></code> such that <code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>hk</c-><c- p>,</c-> <c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code></ins>
       <td>
        <ins>constant</ins>
      <tr>
       <td>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code></ins>
       <td>
        <ins><code class="highlight"><c- n>X</c-><c- o>::</c-><c- n>hash_token_type</c-></code></ins>
       <td>
        <ins><em>Returns</em>: returns a value <code class="highlight"><c- n>hke</c-></code> of type <code class="highlight"><c- n>X</c-><c- o>::</c-><c- n>hash_token_type</c-></code> such that <code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>hke</c-><c- p>,</c-> <c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code></ins>
       <td>
        <ins>constant</ins>
      <tr>
       <td>
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>k</c-><c- p>,</c-> <c- n>hk</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>hk</c-><c- p>,</c-> <c- n>k</c-><c- p>)</c-></code></ins>
       <td><code class="highlight"><c- n>iterator</c-></code>; <code class="highlight"><c- n>const_iterator</c-></code> for const <code class="highlight"><c- n>b</c-></code>.
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</ins>
        <br> <i>Returns:</i> an iterator pointing to an element with key equivalent to <code class="highlight"><c- n>k</c-></code>, or <code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>end</c-><c- p>()</c-></code> if no such element exists.
       <td>Average case O(1), worst case O(<code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>).
      <tr>
       <td>
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>ke</c-><c- p>,</c-> <c- n>hke</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>hke</c-><c- p>,</c-> <c- n>ke</c-><c- p>)</c-></code></ins>
       <td><code class="highlight"><c- n>iterator</c-></code>; <code class="highlight"><c- n>const_iterator</c-></code> for const <code class="highlight"><c- n>a_tran</c-></code>.
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</ins>
        <br> <i>Returns:</i> an iterator pointing to an element with key equivalent to <code class="highlight"><c- n>ke</c-></code>, or <code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>end</c-><c- p>()</c-></code> if no such
        element exists.
       <td>
        <ins>Average case O(1), worst case O(<code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>).</ins>
      <tr>
       <td>
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>k</c-><c- p>,</c-> <c- n>hk</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>hk</c-><c- p>,</c-> <c- n>k</c-><c- p>)</c-></code></ins>
       <td><code class="highlight"><c- n>size_type</c-></code>
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</ins>
        <br> <i>Returns:</i> the number of elements with key equivalent to <code class="highlight"><c- n>k</c-></code>.
       <td>Average case O(<code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code>), worst case O(<code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>).
      <tr>
       <td>
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>ke</c-><c- p>,</c-> <c- n>hke</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>hke</c-><c- p>,</c-> <c- n>ke</c-><c- p>)</c-></code></ins>
       <td><code class="highlight"><c- n>size_type</c-></code>
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</ins>
        <br> <i>Returns:</i> the number of elements with key equivalent to <code class="highlight"><c- n>ke</c-></code>.
       <td>Average case O(<code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code>), worst case O(<code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>).
      <tr>
       <td>
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>contains</c-><c- p>(</c-><c- n>k</c-><c- p>,</c-> <c- n>hk</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>contains</c-><c- p>(</c-><c- n>hk</c-><c- p>,</c-> <c- n>k</c-><c- p>)</c-></code></ins>
       <td>bool
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</ins>
        <br> <i>Effects:</i> Equivalent to <code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>k</c-><c- p>,</c-> <c- n>hk</c-><c- p>)</c-> <c- o>!=</c-> <c- n>b</c-><c- p>.</c-><c- n>end</c-><c- p>()</c-></code>
       <td>Average case O(1), worst case O(<code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>)
      <tr>
       <td>
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>contains</c-><c- p>(</c-><c- n>ke</c-><c- p>,</c-> <c- n>hke</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>contains</c-><c- p>(</c-><c- n>hke</c-><c- p>,</c-> <c- n>ke</c-><c- p>)</c-></code></ins>
       <td>bool
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</ins>
        <br> <i>Effects:</i> Equivalent to <code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>find</c-><c- p>(</c-><c- n>ke</c-><c- p>,</c-> <c- n>hke</c-><c- p>)</c-> <c- o>!=</c-> <c- n>a_tran</c-><c- p>.</c-><c- n>end</c-><c- p>()</c-></code>
       <td>Average case O(1), worst case O(<code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>)
      <tr>
       <td>
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>equal_range</c-><c- p>(</c-><c- n>k</c-><c- p>,</c-> <c- n>hk</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>equal_range</c-><c- p>(</c-><c- n>hk</c-><c- p>,</c-> <c- n>k</c-><c- p>)</c-></code></ins>
       <td><code class="highlight"><c- n>pair</c-><c- o>&lt;</c-><c- n>iterator</c-><c- p>,</c-> <c- n>iterator</c-><c- o>></c-></code>; <code class="highlight"><c- n>pair</c-><c- o>&lt;</c-><c- n>const_iterator</c-><c- p>,</c-> <c- n>const_iterator</c-><c- o>></c-></code> for const <code class="highlight"><c- n>b</c-></code>.
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</del>
        <ins><code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hk</c-></code>,</ins>
        <br> <i>Returns:</i> a range containing all elements with keys equivalent to <code class="highlight"><c- n>k</c-></code>. Returns <code class="highlight"><c- n>make_pair</c-><c- p>(</c-><c- n>b</c-><c- p>.</c-><c- n>end</c-><c- p>(),</c-> <c- n>b</c-><c- p>.</c-><c- n>end</c-><c- p>())</c-></code> if no such elements exist.
       <td>Average case O(<code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>k</c-><c- p>)</c-></code>), worst case O(<code class="highlight"><c- n>b</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>).
      <tr>
       <td>
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>equal_range</c-><c- p>(</c-><c- n>ke</c-><c- p>,</c-> <c- n>hke</c-><c- p>)</c-></code></del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>equal_range</c-><c- p>(</c-><c- n>hke</c-><c- p>,</c-> <c- n>ke</c-><c- p>)</c-></code></ins>
       <td><code class="highlight"><c- n>pair</c-><c- o>&lt;</c-><c- n>iterator</c-><c- p>,</c-> <c- n>iterator</c-><c- o>></c-></code>; <code class="highlight"><c- n>pair</c-><c- o>&lt;</c-><c- n>const_iterator</c-><c- p>,</c-> <c- n>const_iterator</c-><c- o>></c-></code> for const <code class="highlight"><c- n>a_tran</c-></code>.
       <td>
        <i>Expects:</i> 
        <del><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>hash_function</c-><c- p>()(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</del>
        <ins><code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>token</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code> equals <code class="highlight"><c- n>hke</c-></code>,</ins>
        <br> <i>Returns:</i> a range containing all elements with keys equivalent to <code class="highlight"><c- n>ke</c-></code>. Returns <code class="highlight"><c- n>make_pair</c-><c- p>(</c-><c- n>a_tran</c-><c- p>.</c-><c- n>end</c-><c- p>(),</c-> <c- n>a_tran</c-><c- p>.</c-><c- n>end</c-><c- p>())</c-></code> if no such elements exist.
       <td>Average case O(<code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>count</c-><c- p>(</c-><c- n>ke</c-><c- p>)</c-></code>), worst case O(<code class="highlight"><c- n>a_tran</c-><c- p>.</c-><c- n>size</c-><c- p>()</c-></code>).
    </table>
   </blockquote>
   <p>Make the following changes to:</p>
   <ul>
    <li data-md>
     <p><strong>22.5.4.1 [unord.map.overview]</strong></p>
    <li data-md>
     <p><strong>22.5.5.1 [unord.multimap.overview]</strong></p>
    <li data-md>
     <p><strong>22.5.6.1 [unord.set.overview]</strong></p>
    <li data-md>
     <p><strong>22.5.7.1 [unord.multiset.overview]</strong></p>
   </ul>
   <blockquote>
<pre class="highlight"><c- k>using</c-> <c- n>const_local_iterator</c-> <c- o>=</c-> <i><c- n>implementation</c-><c- o>-</c-><c- n>defined</c-></i><c- p>;</c-> <c- c1>// see 22.2</c->
<c- k>using</c-> <c- n>node_type</c->            <c- o>=</c-> <i><c- n>unspecified</c-></i><c- p>;</c->
<c- k>using</c-> <c- n>insert_return_type</c->   <c- o>=</c-> <i><c- n>insert</c-><c- o>-</c-><c- k>return</c-><c- o>-</c-><c- n>type</c-></i><c- o>&lt;</c-><c- n>iterator</c-><c- p>,</c-> <c- n>node_type</c-><c- o>></c-><c- p>;</c->
<ins><c- k>using</c-> <c- n>hash_token_type</c->      <c- o>=</c-> <i><c- n>implementation</c-><c- o>-</c-><c- n>defined</c-></i><c- p>;</c-> <c- c1>// see 22.2</c-></ins>
<c- p>...</c->
<ins><c- n>hash_token_type</c->                    <c- n>token</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>);</c-></ins>
<ins><c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>hash_token_type</c-> <c- n>token</c-><c- p>(</c-><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>);</c-></ins>
<c- n>iterator</c->       <c- nf>find</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>);</c->
<c- n>const_iterator</c-> <c- nf>find</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- n>iterator</c->       <c- nf>find</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>);</c->
<c- n>const_iterator</c-> <c- nf>find</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>iterator</c->       <c- n>find</c-><c- p>(</c-><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>);</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>const_iterator</c-> <c- n>find</c-><c- p>(</c-><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>iterator</c->       <c- n>find</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>);</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>const_iterator</c-> <c- n>find</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- n>size_type</c-> <c- nf>count</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- n>size_type</c-> <c- nf>count</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>size_type</c-> <c- n>count</c-><c- p>(</c-><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>size_type</c-> <c- n>count</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- b>bool</c-> <c- nf>contains</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- b>bool</c-> <c- nf>contains</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- b>bool</c-> <c- n>contains</c-><c- p>(</c-><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- b>bool</c-> <c- n>contains</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- n>pair</c-><c- o>&lt;</c-><c- n>iterator</c-><c- p>,</c-> <c- n>iterator</c-><c- o>></c->             <c- n>equal_range</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>);</c->
<c- n>pair</c-><c- o>&lt;</c-><c- n>const_iterator</c-><c- p>,</c-> <c- n>const_iterator</c-><c- o>></c-> <c- n>equal_range</c-><c- p>(</c-><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- n>pair</c-><c- o>&lt;</c-><c- n>iterator</c-><c- p>,</c-> <c- n>iterator</c-><c- o>></c->             <c- n>equal_range</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>);</c->
<c- n>pair</c-><c- o>&lt;</c-><c- n>const_iterator</c-><c- p>,</c-> <c- n>const_iterator</c-><c- o>></c-> <c- n>equal_range</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>key_type</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>pair</c-><c- o>&lt;</c-><c- n>iterator</c-><c- p>,</c-> <c- n>iterator</c-><c- o>></c->             <c- n>equal_range</c-><c- p>(</c-><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>);</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>pair</c-><c- o>&lt;</c-><c- n>const_iterator</c-><c- p>,</c-> <c- n>const_iterator</c-><c- o>></c-> <c- n>equal_range</c-><c- p>(</c-><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><c- p>)</c-> <c- k>const</c-><c- p>;</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>pair</c-><c- o>&lt;</c-><c- n>iterator</c-><c- p>,</c-> <c- n>iterator</c-><c- o>></c->             <c- n>equal_range</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>);</c->
<c- k>template</c-> <c- o>&lt;</c-><c- k>class</c-> <c- nc>K</c-><c- o>></c-> <c- n>pair</c-><c- o>&lt;</c-><c- n>const_iterator</c-><c- p>,</c-> <c- n>const_iterator</c-><c- o>></c-> <c- n>equal_range</c-><c- p>(</c-><ins><c- k>const</c-> <c- n>hash_token_type</c-><c- o>&amp;</c-> <c- n>hash_token</c-><c- p>,</c-> </ins><c- k>const</c-> <c- n>K</c-><c- o>&amp;</c-> <c- n>k</c-><del><c- p>,</c-> <c- b>size_t</c-> <c- n>hash</c-></del><c- p>)</c-> <c- k>const</c-><c- p>;</c->
</pre>
   </blockquote>
   <h2 class="heading settled" data-level="4" id="implementation-usage-experiences"><span class="secno">4. </span><span class="content">Implementation and Usage Experiences</span><a class="self-link" href="#implementation-usage-experiences"></a></h2>
    Google’s widely used hash table implementation, <a data-link-type="biblio" href="#biblio-swisstables">[SwissTables]</a>, provides the
exact same API as <a data-link-type="biblio" href="#biblio-p0929r0">[P0929R0]</a>. There are fewer than a dozen or so uses of the
precalculated hash in <code class="highlight"><c- n>find</c-><c- p>()</c-></code>. None of them used the table’s hash instance (the
correct one). 
   <p>Facebook’s <a data-link-type="biblio" href="#biblio-f14">[F14]</a> hash tables used an opaque hash token approach. In our
experiences, such an approach allowed several use cases to precalculate hash and
prefetch the relevant cache lines, resulting in CPU and memory efficiency
improvement.</p>
   <h2 class="heading settled" data-level="5" id="acknowledgements"><span class="secno">5. </span><span class="content">Acknowledgements</span><a class="self-link" href="#acknowledgements"></a></h2>
    We would like to thank Samuel Benzaquen, Nathan Bronson, David Goldblatt, Chris
Kennelly, Matthew Fowles Kulukundis, and Titus Winters for various discussions,
insights, and implementation experiences. 
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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="normative"><span class="content">Normative References</span><a class="self-link" href="#normative"></a></h3>
  <dl>
   <dt id="biblio-n4810">[N4810]
   <dd>Richard Smith. <a href="https://wg21.link/n4810">Working Draft, Standard for Programming Language C++</a>. 15 March 2019. URL: <a href="https://wg21.link/n4810">https://wg21.link/n4810</a>
   <dt id="biblio-p0919r3">[P0919R3]
   <dd>Mateusz Pusz. <a href="https://wg21.link/p0919r3">Heterogeneous lookup for unordered containers</a>. 9 November 2018. URL: <a href="https://wg21.link/p0919r3">https://wg21.link/p0919r3</a>
   <dt id="biblio-p0920r2">[P0920R2]
   <dd>Mateusz Pusz. <a href="https://wg21.link/p0920r2">Precalculated hash values in lookup</a>. 22 February 2019. URL: <a href="https://wg21.link/p0920r2">https://wg21.link/p0920r2</a>
   <dt id="biblio-p0929r0">[P0929R0]
   <dd>Jens Maurer. <a href="https://wg21.link/p0929r0">Checking for abstract class types</a>. 8 February 2018. URL: <a href="https://wg21.link/p0929r0">https://wg21.link/p0929r0</a>
  </dl>
  <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-f14">[F14]
   <dd>Nathan Bronson; Xiao Shi. <a href="https://code.fb.com/developer-tools/f14/">Open-sourcing F14 for faster, more memory-efficient hash tables</a>. URL: <a href="https://code.fb.com/developer-tools/f14/">https://code.fb.com/developer-tools/f14/</a>
   <dt id="biblio-swisstables">[SwissTables]
   <dd>Sam Benzaquen; et al. <a href="https://abseil.io/blog/20180927-swisstables">Swiss Tables and absl::Hash</a>. URL: <a href="https://abseil.io/blog/20180927-swisstables">https://abseil.io/blog/20180927-swisstables</a>
  </dl>