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  <title>P2880R0: Algorithm-like vs std::simd based RNG API</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
 *     -> ins/del.c### for candidate and proposed changes (amendments)
 *
 * 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)
 *   - .advisement for loud normative statements     (div, p, strong)
 *   - .annoying-warning for spec obsoletion notices (div, aside, details)
 *   - .correction for "candidate corrections"       (div, aside, details, section)
 *   - .addition   for "candidate additions"         (div, aside, details, section)
 *   - .correction.proposed for "proposed corrections" (div, aside, details, section)
 *   - .addition.proposed   for "proposed additions"   (div, aside, details, section)
 *
 * 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
 *
 * Outdated warning for old specs
 *
 * 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)
 *
 ******************************************************************************/

/* color variables included separately for reliability */

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

	html {
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	body {
		counter-reset: example figure issue;

		/* Layout */
		max-width: 50em;			  /* limit line length to 50em for readability   */
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		background: white top left fixed no-repeat;
		background: var(--bg) top left fixed no-repeat;
		background-size: 25px auto;
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/******************************************************************************/
/*                         Front Matter & Navigation                          */
/******************************************************************************/

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

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

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

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

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

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

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

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

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	@media not print {
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			height: 1.33em;
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			box-shadow: 0 0 2px;
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			margin-bottom: -1.9em;
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		/* statusbar gets in the way on keyboard focus; remove once browsers fix */
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		#toc-nav:not(:hover) > a:not(:focus) > span + span {
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		#toc-nav > a > span + span {
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/** ToC Sidebar ***************************************************************/

	/* Floating sidebar */
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			width: 23.5em;
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			color: var(--tocsidebar-text);
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			font-variant: small-caps;
			font-variant: all-small-caps;
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			color: var(--tocsidebar-text);
			background: inherit;
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			box-shadow: -.1em 0 .25em var(--tocsidebar-shadow) inset;
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			font-variant: small-caps;
			font-variant: all-small-caps;
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			color: gray;
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		body:not(.toc-inline) {
			padding-left: 29em;
		}
		/* See also Overflow section at the bottom */

		body:not(.toc-inline) #toc-jump:not(:focus) {
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			height: 0;
			padding: 0;
			position: absolute;
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		body:not(.toc-inline) {
			margin: 0 4em;
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/******************************************************************************/
/*                                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 */
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	h2, h3, h4, h5, h6 {
		margin-top: 3rem;
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	h1, h2, h3 {
		color: #005A9C;
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	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,
	#profile-and-date {
		/* #profile-and-date is a subtitle in an H2 under the H1 */
		margin-top: 0;
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	h2 + h3,
	h3 + h4,
	h4 + h5,
	h5 + h6 {
		margin-top: 1.2em; /* = 1 x line-height */
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/** Section divider ***********************************************************/

	:not(.head) > :not(.head) + hr {
		font-size: 1.5em;
		text-align: center;
		margin: 1em auto;
		height: auto;
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		color: var(--hr-text);
		border: transparent solid 0;
		background: transparent;
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	:not(.head) > hr::before {
		content: "\2727\2003\2003\2727\2003\2003\2727";
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/******************************************************************************/
/*                            Paragraphs and Lists                            */
/******************************************************************************/

	p {
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	dd > p:first-child,
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	ul, ol {
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		padding-left: 2em;
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	li {
		margin: 0.25em 0 0.5em;
		padding: 0;
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	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;
	border-left: 0.5em solid var(--algo-border);
	}

	/* Put nice boxes around each algorithm. */
	[data-algorithm]:not(.heading) {
	 padding: .5em;
	 border: thin solid #ddd;
	 border: thin solid var(--algo-border);
	 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: inherit;
	}
	dfn var {
		font-style: normal;
	}

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

	del {
		color: #aa0000;
		color: var(--del-text);
		background: transparent;
		background: var(--del-bg);
		text-decoration: line-through;
	}
	ins {
		color: #006100;
		color: var(--ins-text);
		background: transparent;
		background: var(--ins-bg);
		text-decoration: underline;
	}

	/* for amendments (candidate/proposed changes) */

	.amendment ins, .correction ins, .addition ins,
	ins[class^=c] {
		text-decoration-style: dotted;
	}
	.amendment del, .correction del, .addition del,
	del[class^=c] {
		text-decoration-style: dotted;
	}
	.amendment.proposed ins, .correction.proposed ins, .addition.proposed ins,
	ins[class^=c].proposed {
		text-decoration-style: double;
	}
	.amendment.proposed del, .correction.proposed del, .addition.proposed del,
	del[class^=c].proposed {
		text-decoration-style: double;
	}

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

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

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

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

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		font-size: .9em;
		hyphens: none;
		text-transform: none;
		text-align: left;
		text-align: start;
		font-variant: normal;
		orphans: 3;
		widows: 3;
		page-break-before: avoid;
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	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;
		color: var(--a-normal-text);
		text-decoration: underline #707070;
		text-decoration: underline var(--a-normal-underline);
		text-decoration-skip-ink: none;
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	a:visited {
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		color: var(--a-visited-text);
		text-decoration-color: #bbb;
		text-decoration-color: var(--a-visited-underline);
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	/* Indicate interaction with the link */
	a[href]:focus,
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		text-decoration-thickness: 2px;
	}
	a[href]:active {
		color: #c00;
		color: var(--a-active-text);
		text-decoration-color: #c00;
		text-decoration-color: var(--a-active-underline);
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	/* Backout above styling for W3C logo */
	.head .logo,
	.head .logo a {
		border: none;
		text-decoration: none;
		background: transparent;
	}

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

	img {
		border-style: none;
	}

	img, svg {
		/* Intentionally not color-scheme aware. */
		background: white;
	}

	/* 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;
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	.figure img,	.sidefigure img,	figure img,
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		max-width: 100%;
		margin: auto;
		height: auto;
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	.figure pre, .sidefigure pre, figure pre {
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		display: table;
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	.figure table, figure table {
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		.sidefigure {
			float: right;
			width: 50%;
			margin: 0 0 0.5em 0.5em;
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	}
	.caption, figcaption, caption {
		font-style: italic;
		font-size: 90%;
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	.caption::before, figcaption::before, figcaption > .marker {
		font-weight: bold;
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	.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,
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		margin: 1em auto;
		padding: .5em;
		border: .5em;
		border-left-style: solid;
		page-break-inside: avoid;
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	span.issue, span.note {
		padding: .1em .5em .15em;
		border-right-style: solid;
	}

	blockquote > :first-child,
	.note  > p:first-child,
	.issue > p:first-child,
	.amendment > p:first-child,
	.correction > p:first-child,
	.addition > p:first-child {
		margin-top: 0;
	}
	blockquote > :last-child,
	.note  > p:last-child,
	.issue > p:last-child,
	.amendment > p:last-child,
	.correction > p:last-child,
	.addition > p:last-child {
		margin-bottom: 0;
	}


	.issue::before, .issue > .marker,
	.example::before, .example > .marker,
	.note::before, .note > .marker,
	details.note > summary > .marker,
	.amendment::before, .amendment > .marker,
	details.amendment > summary > .marker,
	.addition::before, .addition > .marker,
	addition.amendment > summary > .marker,
	.correction::before, .correction > .marker,
	correction.amendment > summary > .marker
	{
		text-transform: uppercase;
		padding-right: 1em;
	}

	.example::before, .example > .marker {
		display: block;
		padding-right: 0em;
	}

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

	blockquote {
		border-color: silver;
		border-color: var(--blockquote-border);
		background: transparent;
		background: var(--blockquote-bg);
		color: currentcolor;
		color: var(--blockquote-text);
	}

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

	.issue {
		border-color: #e05252;
		border-color: var(--issue-border);
		background: #fbe9e9;
		background: var(--issue-bg);
		color: black;
		color: var(--issue-text);
		counter-increment: issue;
		overflow: auto;
	}
	.issue::before, .issue > .marker {
		color: #831616;
		color: var(--issueheading-text);
	}
	/* 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;
		border-color: var(--example-border);
		background: #fcfaee;
		background: var(--example-bg);
		color: black;
		color: var(--example-text);
		counter-increment: example;
		overflow: auto;
		clear: both;
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	.example::before, .example > .marker {
		color: #574b0f;
		color: var(--exampleheading-text);
	}
	/* 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;
		border-color: var(--note-border);
		background: #e9fbe9;
		background: var(--note-bg);
		color: black;
		color: var(--note-text);
		overflow: auto;
	}

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

	details.note[open] > summary {
		border-bottom: 1px silver solid;
		border-bottom: 1px var(--notesummary-underline) solid;
	}

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

	.assertion {
		border-color: #AAA;
		border-color: var(--assertion-border);
		background: #EEE;
		background: var(--assertion-bg);
		color: black;
		color: var(--assertion-text);
	}

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

	.advisement {
		border-color: orange;
		border-color: var(--advisement-border);
		border-style: none solid;
		background: #fec;
		background: var(--advisement-bg);
		color: black;
		color: var(--advisement-text);
	}
	strong.advisement {
		display: block;
		text-align: center;
	}
	.advisement::before, .advisement > .marker {
		color: #b35f00;
		color: var(--advisementheading-text);
	}

/** Amendment Box *************************************************************/

	.amendment, .correction, .addition {
		border-color: #330099;
		border-color: var(--amendment-border);
		background: #F5F0FF;
		background: var(--amendment-bg);
		color: black;
		color: var(--amendment-text);
	}
	.amendment.proposed, .correction.proposed, .addition.proposed {
		border-style: solid;
		border-block-width: 0.25em;
	}
	.amendment::before, .amendment > .marker,
	details.amendment > summary::before, details.amendment > summary > .marker,
	.correction::before, .correction > .marker,
	details.correction > summary::before, details.correction > summary > .marker,
	.addition::before, .addition > .marker,
	details.addition > summary::before, details.addition > summary > .marker {
		color: #220066;
		color: var(--amendmentheading-text);
	}
	.amendment.proposed::before, .amendment.proposed > .marker,
	details.amendment.proposed > summary::before, details.amendment.proposed > summary > .marker,
	.correction.proposed::before, .correction.proposed > .marker,
	details.correction.proposed > summary::before, details.correction.proposed > summary > .marker,
	.addition.proposed::before, .addition.proposed > .marker,
	details.addition.proposed > summary::before, details.addition.proposed > summary > .marker {
		font-weight: bold;
	}

/** 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: hsla(40,100%,50%,0.95);
		background: var(--warning-bg);
		color: black;
		color: var(--warning-text);
		padding: .75em 1em;
		border: red;
		border: var(--warning-border);
		border-style: solid none;
		box-shadow: 0 2px 8px black;
		text-align: center;
	}
	.annoying-warning :last-child {
		margin-bottom: 0;
	}

@media not print {
	details.annoying-warning[open] {
		position: fixed;
		left: 0;
		right: 0;
		bottom: 2em;
		z-index: 1000;
	}
}

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

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

	.def {
		padding: .5em 1em;
		background: #def;
		background: var(--def-bg);
		margin: 1.2em 0;
		border-left: 0.5em solid #8ccbf2;
		border-left: 0.5em solid var(--def-border);
		color: black;
		color: var(--def-text);
	}

/******************************************************************************/
/*                                    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;
		border-bottom: 1px solid var(--defrow-border);
	}

	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-color: var(--datacell-border);
		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;
		border-top: 1px solid var(--datacell-border);
		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;
		border: 1px solid var(--datacell-border);
		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: 0.1rem 1px 0;
		/* Larger, more consistently-sized click target */
		display: block;
		/* Switch to using border-bottom for underlines */
		text-decoration: none;
		border-bottom: 1px solid;
		/* Reverse color scheme */
		color: black;
		color: var(--toclink-text);
		border-color: #3980b5;
		border-color: var(--toclink-underline);
	}
	.toc a:visited {
		color: black;
		color: var(--toclink-visited-text);
		border-color: #054572;
		border-color: var(--toclink-visited-underline);
	}
	.toc a:focus,
	.toc a:hover {
		background: rgba(75%, 75%, 75%, .25);
		background: var(--a-hover-bg);
		border-bottom-width: 3px;
		margin-bottom: -2px;
	}
	.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;
	}
	.toc {
		line-height: 1.1em;
	}

	/* 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 li { font-size:   85%;	}

	/* @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 li li li .secno { font-size: 85%; }
		.toc > li li li li li .secno { font-size: 100%; }

		.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; }
		}
		/* Loosen it on wide screens */
		@media screen and (min-width: 78em) {
			body:not(.toc-inline) :not(li) > .toc			 { margin-left:  4rem; }
			body:not(.toc-inline) .toc .secno				 { margin-left: -4rem; }
			body:not(.toc-inline) .toc > li li li			 { margin-left:  1rem; }
			body:not(.toc-inline) .toc > li li li .secno	  { margin-left: -5rem; }
			body:not(.toc-inline) .toc > li li li li .secno	{ margin-left: -6rem; }
			body:not(.toc-inline) .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,
		#toc .content:focus {
			background: rgba(75%, 75%, 75%, .25);
			background: var(--a-hover-bg);
			border-bottom: 3px solid #054572;
			border-bottom: 3px solid var(--toclink-underline);
			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 a + span {
			white-space: nowrap;
			color: transparent; }
		ul.index li a:hover + span,
		ul.index li a:focus + span {
			color: #707070;
			color: var(--indexinfo-text);
		}
	}

/** 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]) {
		color: black;
		color: var(--indextable-hover-text);
		background: #f7f8f9;
		background: var(--indextable-hover-bg);
	}

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

/** Outdated warning **********************************************************/

.outdated-spec {
	color: black;
	color: var(--outdatedspec-text);
	background-color: rgba(0,0,0,0.5);
	background-color: var(--outdatedspec-bg);
}

.outdated-warning {
	position: fixed;
	bottom: 50%;
	left: 0;
	right: 0;
	margin: 0 auto;
	width: 50%;
	background: maroon;
	background: var(--outdated-bg);
	color: white;
	color: var(--outdated-text);
	border-radius: 1em;
	box-shadow: 0 0 1em red;
	box-shadow: 0 0 1em var(--outdated-shadow);
	padding: 2em;
	text-align: center;
	z-index: 2;
}

.outdated-warning a {
	color: currentcolor;
	background: transparent;
}

.edited-rec-warning {
	background: darkorange;
	background: var(--editedrec-bg);
	box-shadow: 0 0 1em;
}

.outdated-warning button {
	color: var(--outdated-text);
	border-radius: 1em;
	box-shadow: 0 0 1em red;
	box-shadow: 0 0 1em var(--outdated-shadow);
	padding: 2em;
	text-align: center;
	z-index: 2;
}

.outdated-warning a {
	color: currentcolor;
	background: transparent;
}

.edited-rec-warning {
	background: darkorange;
	background: var(--editedrec-bg);
	box-shadow: 0 0 1em;
}

.outdated-warning button {
	position: absolute;
	top: 0;
	right:0;
	margin: 0;
	border: 0;
	padding: 0.25em 0.5em;
	background: transparent;
	color: white;
	color: var(--outdated-text);
	font:1em sans-serif;
	text-align:center;
}

.outdated-warning span {
	display: block;
}

.outdated-collapsed {
	bottom: 0;
	border-radius: 0;
	width: 100%;
	padding: 0;
}

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

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

		.outdated-warning {
			position: absolute;
			border-style: solid;
			border-color: red;
		}

		.outdated-warning input {
			display: none;
		}
	}
	@page {
		margin: 1.5cm 1.1cm;
	}



/******************************************************************************/
/*                             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 item positioning:
		  "content column" is 50ems wide at max; less on smaller screens.
		  Extra space (after ToC + content) is empty on the right.

		  1. When item < content column, centers item in column.
		  2. When content < item < available, left-aligns.
		  3. When item > 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>
    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 3383f409d, updated Fri Apr 28 15:49:22 2023 -0700" name="generator">
  <link href="http://wg21.link/P2880R0" rel="canonical">
  <link href="https://isocpp.org/favicon.ico" rel="icon">
<style>/* style-autolinks */
.css.css, .property.property, .descriptor.descriptor {
    color: var(--a-normal-text);
    font-size: inherit;
    font-family: inherit;
}
.css::before, .property::before, .descriptor::before {
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  <div class="head">
   <p data-fill-with="logo"></p>
   <h1 class="p-name no-ref" id="title">P2880R0<br>Algorithm-like vs std::simd based RNG API</h1>
   <h2 class="no-num no-toc no-ref heading settled" id="profile-and-date"><span class="content">Published Proposal, <time class="dt-updated" datetime="2023-05-18">2023-05-18</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt>This version:
     <dd><a class="u-url" href="http://wg21.link/P2880R0">http://wg21.link/P2880R0</a>
     <dt class="editor">Authors:
     <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:<burylov@gmail.com>">Ilya Burylov</a>
     <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:<ruslan.arutyunyan@intel.com>">Ruslan Arutyunyan</a> (<span class="p-org org">Intel</span>)
     <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:<af.nikolaev@gmail.com>">Andrey Nikolaev</a>
     <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:<alina.elizarova@intel.com>">Alina Elizarova</a> (<span class="p-org org">Intel</span>)
     <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:<pavel.dyakov@intel.com>">Pavel Dyakov</a> (<span class="p-org org">Intel</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>The paper compares the approaches of high-level (algorithm-like) vector API for random

          number generation (RNG) vs low-level (<code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> based) one.
          The main goal of this paper is to show the need for co-existance of both
          API levels, support the <a data-link-type="biblio" href="#biblio-p1068r7" title="Vector API for random number generation">[P1068R7]</a> paper and pave the way for <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> in RNG.</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="#intro"><span class="secno">1</span> <span class="content">Introduction</span></a>
    <li>
     <a href="#example"><span class="secno">2</span> <span class="content">Example</span></a>
     <ol class="toc">
      <li><a href="#high_level_api"><span class="secno">2.1</span> <span class="content">High-level API with <span title="Vector API for random number generation">[P1068R7]</span></span></a>
      <li>
       <a href="#simd_level_api"><span class="secno">2.2</span> <span class="content">Low-level API with <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code></span></a>
       <ol class="toc">
        <li><a href="#engine_template_parameter"><span class="secno">2.2.1</span> <span class="content">Engine template parameter</span></a>
        <li><a href="#template_parameter_rebind"><span class="secno">2.2.2</span> <span class="content">Engine template parameter + rebind constructor</span></a>
        <li><a href="#algorithm_function_template"><span class="secno">2.2.3</span> <span class="content">Algorithm-like function template parameter</span></a>
       </ol>
     </ol>
    <li><a href="#high_level_api_on_simd_based"><span class="secno">3</span> <span class="content">SIMD-level API vs High-level API</span></a>
    <li><a href="#summary"><span class="secno">4</span> <span class="content">Summary</span></a>
    <li>
     <a href="#references"><span class="secno"></span> <span class="content">References</span></a>
     <ol class="toc">
      <li><a href="#informative"><span class="secno"></span> <span class="content">Informative References</span></a>
     </ol>
   </ol>
  </nav>
  <main>
   <h2 class="heading settled" data-level="1" id="intro"><span class="secno">1. </span><span class="content">Introduction</span><a class="self-link" href="#intro"></a></h2>
    There is a <a data-link-type="biblio" href="#biblio-p1068r7" title="Vector API for random number generation">[P1068R7]</a> paper on the flight, which discusses a higher level API to enable
(RNG) in batches. The main purpose of the new API is to enable vectorization under the hood
in the implementation. Given some of us live in anticipation of final <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> <a data-link-type="biblio" href="#biblio-p1928r3" title="Merge data-parallel types from the Parallelism TS 2">[P1928R3]</a> standardization, a natural questions arise: 
   <ul>
    <li data-md>
     <p>Do we need a dedicated high-level API for RNG vectorization?</p>
    <li data-md>
     <p>Can <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code>-based low-level API with some generic mechanism for making it high level be sufficient?</p>
   </ul>
   <h2 class="heading settled" data-level="2" id="example"><span class="secno">2. </span><span class="content">Example</span><a class="self-link" href="#example"></a></h2>
   <p>When does RNG performance matter? An algorithm should consume many random numbers.
Additional computation on top of generation should be relatively lightweight.
The "European options pricing" benchmark can show the purpose:</p>
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>mt19937</c-> <c- nf>engine</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-> <c- nf>distribution</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->

<c- b>double</c-> <c- n>v0</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- n>v1</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->
<c- k>for</c-> <c- p>(</c-><c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>p</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c-> <c- n>p</c-> <c- o>&lt;</c-> <c- n>npath</c-><c- p>;</c-> <c- o>++</c-><c- n>p</c-><c- p>)</c-> <c- p>{</c-> <c- c1>//e.g., npath=1,000,000</c->
    <c- b>double</c-> <c- n>rand</c-> <c- o>=</c-> <c- n>distribution</c-><c- p>(</c-><c- n>engine</c-><c- p>);</c->
    <c- b>double</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
    <c- n>v0</c-> <c- o>+=</c-> <c- n>res</c-><c- p>;</c->
    <c- n>v1</c-> <c- o>+=</c-> <c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>;</c->
<c- p>}</c->
<c- n>result</c->     <c- o>=</c-> <c- n>v0</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
<c- n>confidence</c-> <c- o>=</c-> <c- n>v1</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
</pre>
   <p>One should have millions of iterations of the loop to get an accurate
estimation of the price: many random numbers will be consumed with a handful of operations on top.</p>
   <p>The loop will not be auto-vectorized by compilers because of RNG - lack of performance out of the box.
Production code is tending to be more complicated: there can be more than one random number
per computational block with potentially different distributions, random number paths may need to be
correlated, etc. but the "European option pricing" benchmark is good enough to show different approaches.</p>
   <h3 class="heading settled" data-level="2.1" id="high_level_api"><span class="secno">2.1. </span><span class="content">High-level API with <a data-link-type="biblio" href="#biblio-p1068r7" title="Vector API for random number generation">[P1068R7]</a></span><a class="self-link" href="#high_level_api"></a></h3>
   <p>Let’s consider the straightforward example</p>
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>mt19937</c-> <c- nf>engine</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-> <c- nf>distribution</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>array</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- n>npath</c-><c- o>></c-> <c- n>rand</c-><c- p>;</c-> <c- c1>// npath=1,000,000 -> sizeof(rand)=8 MB</c->

<c- n>std</c-><c- o>::</c-><c- n>ranges</c-><c- o>::</c-><c- n>generate_random</c-><c- p>(</c-><c- n>rand</c-><c- p>,</c-> <c- n>engine</c-><c- p>,</c-> <c- n>distribution</c-><c- p>);</c->

<c- b>double</c-> <c- n>v0</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- n>v1</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->

<c- k>for</c-><c- p>(</c-><c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>p</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c-> <c- n>p</c-> <c- o>&lt;</c-> <c- n>npath</c-><c- p>;</c-> <c- o>++</c-><c- n>p</c-><c- p>)</c-> <c- p>{</c->
    <c- b>double</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand</c-><c- p>[</c-><c- n>p</c-><c- p>]</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
    <c- n>v0</c-> <c- o>+=</c-> <c- n>res</c-><c- p>;</c->
    <c- n>v1</c-> <c- o>+=</c-> <c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>;</c->
<c- p>}</c->

<c- n>result</c->     <c- o>=</c-> <c- n>v0</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
<c- n>confidence</c-> <c- o>=</c-> <c- n>v1</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
</pre>
   <p>We got vectorization both in <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>ranges</c-><c- o>::</c-><c- n>generate_random</c-></code> and in the loop.
But we are forced to allocate a temporary storage for the random numbers. Given the size of <code class="highlight"><c- n>npath</c-></code>,
we run into L1 or L2 cache misses with sub-optimal performance at the end.</p>
   <p>Let us apply bufferization:</p>
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>mt19937</c-> <c- nf>engine</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-> <c- nf>distribution</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>array</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- n>nbuffer</c-><c- o>></c-> <c- n>rand</c-><c- p>;</c-> <c- c1>// e.g., nbuffer=128</c->

<c- b>double</c-> <c- n>v0</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- n>v1</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->

<c- k>for</c-><c- p>(</c-><c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>p</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c-> <c- n>p</c-> <c- o>&lt;</c-> <c- n>npath</c-><c- p>;</c-> <c- n>p</c-> <c- o>+=</c-> <c- n>nbuffer</c-><c- p>)</c-> <c- p>{</c->
    <c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>local_size</c-> <c- o>=</c-> <c- p>(</c-><c- n>p</c-> <c- o>+</c-> <c- n>nbuffer</c-> <c- o>&lt;=</c-> <c- n>npath</c-><c- p>)</c-> <c- o>?</c-> <c- n>nbuffer</c-> <c- o>:</c-> <c- p>(</c-><c- n>npath</c-> <c- o>-</c-> <c- n>p</c-><c- p>);</c-> <c- c1>// dealing with tail</c->

    <c- n>std</c-><c- o>::</c-><c- n>ranges</c-><c- o>::</c-><c- n>generate_random</c-><c- p>(</c-><c- n>std</c-><c- o>::</c-><c- n>span</c-><c- p>(</c-><c- n>rand</c-><c- p>.</c-><c- n>begin</c-><c- p>(),</c-> <c- n>local_size</c-><c- p>),</c-> <c- n>engine</c-><c- p>,</c-> <c- n>distribution</c-><c- p>);</c->

    <c- k>for</c-><c- p>(</c-><c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>b</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c-> <c- n>b</c-> <c- o>&lt;</c-> <c- n>local_size</c-><c- p>;</c-> <c- o>++</c-><c- n>b</c-><c- p>)</c-> <c- p>{</c->
        <c- b>double</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand</c-><c- p>[</c-><c- n>p</c-><c- p>]</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
        <c- n>v0</c-> <c- o>+=</c-> <c- n>res</c-><c- p>;</c->
        <c- n>v1</c-> <c- o>+=</c-> <c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>;</c->
    <c- p>}</c->
<c- p>}</c->

<c- n>result</c->     <c- o>=</c-> <c- n>v0</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
<c- n>confidence</c-> <c- o>=</c-> <c- n>v1</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
</pre>
   <p>The code is more complex, but is likely faster, and allows one to tune its performance by changing the <code class="highlight"><c- n>nbuffer</c-></code> parameter.</p>
   <h3 class="heading settled" data-level="2.2" id="simd_level_api"><span class="secno">2.2. </span><span class="content">Low-level API with <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code></span><a class="self-link" href="#simd_level_api"></a></h3>
   <p>Let’s investigate the opportunities for low-level SIMD based API in the same benchmark.
The API for SIMD-level RNG is not defined yet, but there are several possible approaches
with their own pros and cons, which we may consider. The key distinguishing property in
these approaches is the level at which the API gets the information about <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> type
being requested by the user. An engine, pseudo-random number generator that produces integer
sequence with uniform distribution, holds the state for the RNG generation including extra
intermediate structures for efficient calculation. With the knowledge about SIMD size, it may
store internal structures in the most efficient format and this format does differ in practice
for different SIMD sizes.</p>
   <p>In the ideal world, distribution objects should be almost stateless (except for the parameters
of the distribution), but in practice they are not. For example, Box-Muller method for
Gaussian random number generation generates two variates using two independent samples
from the uniform distribution. Thus, in the scalar use one already generated value can be
stored in the distribution object to be returned on the next call.</p>
   <p>This results in an amusing corner-case:</p>
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>mt19937</c-> <c- nf>E1</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>mt19937</c-> <c- nf>E2</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-> <c- nf>D</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c-> <c- c1>// Box-Muller</c->
<c- k>for</c-><c- p>(;;)</c-> <c- p>{</c->
    <c- b>double</c-> <c- n>res1</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E1</c-><c- p>);</c->
    <c- b>double</c-> <c- n>res2</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E2</c-><c- p>);</c->
<c- p>}</c->
</pre>
   <p>One may naively expect that <code class="highlight"><c- n>res1</c-> <c- o>==</c-> <c- n>res2</c-></code>, because the values are expected to be consumed
from 2 engines in the same starting state. But in fact all numbers will be generated using <code class="highlight"><c- n>E1</c-></code>, while the <code class="highlight"><c- n>E2</c-></code> engine will never be used in the most existing C++ standard library
implementations. The current standard wording specifies the output of a distribution object
when used with "successive invocations with the same [engine] objects", thus the result above
is unspecified and observed behavior is valid.</p>
   <h4 class="heading settled" data-level="2.2.1" id="engine_template_parameter"><span class="secno">2.2.1. </span><span class="content">Engine template parameter</span><a class="self-link" href="#engine_template_parameter"></a></h4>
   <p>In our first example, both engine and distribution will be constructed with the knowledge
about the <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> type, which will be requested from them.</p>
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>mt19937</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- b>uint_fast32_t</c-><c- p>,</c-> <c- mi>16</c-><c- o>>></c-> <c- n>E</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- mi>16</c-><c- o>>></c-> <c- n>D</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->
<c- k>auto</c-> <c- n>rand</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E</c-><c- p>);</c->
</pre>
   <p><code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> type in the engine and <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> type in the distribution are different.
The number of values produced by the engine per value required by distribution is currently
unspecified in the standard (even in scalar case), but in most existing implementations 2 values
of <code class="highlight"><c- n>std</c-><c- o>::</c-><c- b>uint_fast32_t</c-></code> will be consumed from the engine per double return result of the <code class="highlight"><c- n>normal_distribution</c-></code>.</p>
   <p>Implementation wide it may (or may not) be more optimal to define it as:</p>
<pre class="language-cpp highlight"><c- c1>// 32 SIMD size passed to engine</c->
<c- n>std</c-><c- o>::</c-><c- n>mt19937</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- b>uint_fast32_t</c-><c- p>,</c-> <c- mi>32</c-><c- o>>></c-> <c- n>E</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- c1>// 16 SIMD size passed to distribution</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- mi>16</c-><c- o>>></c-> <c- n>D</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->
<c- k>auto</c-> <c- n>rand</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E</c-><c- p>);</c->
</pre>
   <p>Going towards more sophisticated distributions acceptance-rejection type of methods are becoming
more common, which consume a varying number of engine values per resulting value.</p>
   <p>Thus, there is no universal good answer for the proper <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> size of the engine output,
which would be the best fit for the given distribution <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> output.</p>
   <p>If engine is capable of generating values only in the packs of fixed size (defined by the type
of the simd parameter) and distribution consumed this pack only partially, then the question
about the use of the remaining numbers is raised:</p>
   <ol>
    <li data-md>
     <p>Remaining part can be stored in the distribution object in either raw engine values form
or in the form of already produced, but not returned to the application values.
That brings two more oddities:</p>
     <ol>
      <li data-md>
       <p>Distribution is unaware of the pack size returned from the engine at construction point -
  it gets the engine type in <code class="highlight"><c- k>operator</c-><c- p>()</c-></code>. The size of potential tail is unknown thus
  static allocation of the storage for it is problematic.</p>
      <li data-md>
       <p>It extends the corner-case example above into a more complicated form, where it is unclear
  what was actually the source of the randomness - current function argument or something
  from the past.</p>
     </ol>
    <li data-md>
     <p>Remaining part can be discarded. While not entirely thrifty, it might not be that bad choice.</p>
     <ol>
      <li data-md>
       <p class="note" role="note"><span class="marker">Note:</span> There is no solid reason for distribution to know about <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> type at construction
       point in this case, except for providing corresponding member function in the API</p>
     </ol>
   </ol>
   <p>Having all those implementation implications in mind, let us look at our example:</p>
<pre class="language-cpp highlight"><c- k>constexpr</c-> <c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>size</c-> <c- o>=</c-> <c- mi>16</c-><c- p>;</c->
<c- n>std</c-><c- o>::</c-><c- n>mt19937</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- b>uint_fast32_t</c-><c- p>,</c-> <c- n>size</c-><c- o>>></c-> <c- n>E</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- n>size</c-><c- o>>></c-> <c- n>D</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->

<c- b>double</c-> <c- n>v0</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- n>v1</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->
<c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>p</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->

<c- k>for</c-><c- p>(;</c-> <c- n>p</c-> <c- o>+</c-> <c- n>size</c-> <c- o>&lt;=</c-> <c- n>npath</c-><c- p>;</c-> <c- n>p</c-> <c- o>+=</c-> <c- n>size</c-><c- p>)</c-> <c- p>{</c->
    <c- k>auto</c-> <c- n>rand</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E</c-><c- p>);</c-> <c- c1>// std::fixed_size_simd&lt;double, size></c->
    <c- k>auto</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
    <c- n>v0</c-> <c- o>+=</c-> <c- n>std</c-><c- o>::</c-><c- n>reduce</c-><c- p>(</c-><c- n>res</c-><c- p>);</c->
    <c- n>v1</c-> <c- o>+=</c-> <c- n>std</c-><c- o>::</c-><c- n>reduce</c-><c- p>(</c-><c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>);</c->
<c- p>}</c->

<c- c1>// npath is runtime parameter - it may not be divisible by 16</c->
<c- k>if</c-> <c- p>(</c-><c- n>p</c-> <c- o>!=</c-> <c- n>npath</c-><c- p>)</c-> <c- p>{</c->
    <c- k>auto</c-> <c- n>rand_tail</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E</c-><c- p>);</c->
    <c- k>auto</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand_tail</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
    <c- k>for</c-><c- p>(</c-><c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>i</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c-> <c- n>p</c-> <c- o>+</c-> <c- n>i</c-> <c- o>&lt;</c-> <c- n>npath</c-><c- p>;</c-> <c- o>++</c-><c- n>i</c-><c- p>)</c-> <c- p>{</c->
        <c- n>v0</c-> <c- o>+=</c-> <c- n>res</c-><c- p>[</c-><c- n>i</c-><c- p>];</c->
        <c- n>v1</c-> <c- o>+=</c-> <c- n>res</c-><c- p>[</c-><c- n>i</c-><c- p>]</c-> <c- o>*</c-> <c- n>res</c-><c- p>[</c-><c- n>i</c-><c- p>];</c->
    <c- p>}</c->
    <c- c1>// discarding unused part of res</c->
<c- p>}</c->
<c- n>result</c->     <c- o>=</c-> <c- n>v0</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
<c- n>confidence</c-> <c- o>=</c-> <c- n>v1</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
</pre>
   <p>If <code class="highlight"><c- n>npath</c-></code> is not a multiple of <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> size, then this example generates more random numbers
than actually needed in <code class="highlight"><c- n>rand_tail</c-></code> and throws away unused part. That makes the example not an
absolute match to our initial example, but it is as close as we can get.</p>
   <p>There is also an open question what <code class="highlight"><c- n>E</c-><c- p>.</c-><c- n>discard</c-><c- p>(</c-><c- mi>1</c-><c- p>)</c-></code> means in this case - whether it discards a <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> pack of 16 values or only one element.</p>
   <p class="note" role="note"><span class="marker">Note:</span> We do not want to combine the logic of the tail with the main loop in order to make our hot path
      as optimized as possible.</p>
   <h4 class="heading settled" data-level="2.2.2" id="template_parameter_rebind"><span class="secno">2.2.2. </span><span class="content">Engine template parameter + rebind constructor</span><a class="self-link" href="#template_parameter_rebind"></a></h4>
   <p>One may overcome the main limitation of the approach by allowing rebind construction of the
engine with the different <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> widths and scalar type.</p>
<pre class="language-cpp highlight"><c- k>constexpr</c-> <c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>size</c-> <c- o>=</c-> <c- mi>16</c-><c- p>;</c->
<c- n>std</c-><c- o>::</c-><c- n>mt19937</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- b>uint_fast32_t</c-><c- p>,</c-> <c- n>size</c-><c- o>>></c-> <c- n>E</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- n>size</c-><c- o>>></c-> <c- n>D</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->

<c- b>double</c-> <c- n>v0</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- n>v1</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->
<c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>p</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->

<c- k>for</c-> <c- p>(;</c-> <c- n>p</c-> <c- o>+</c-> <c- n>size</c-> <c- o>&lt;=</c-> <c- n>npath</c-><c- p>;</c-> <c- n>p</c-> <c- o>+=</c-> <c- n>size</c-><c- p>)</c-> <c- p>{</c->
    <c- k>auto</c-> <c- n>rand</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E</c-><c- p>);</c->
    <c- k>auto</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-><c- o>*</c-><c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-><c- o>-</c-><c- n>X</c-><c- p>);</c->
    <c- n>v0</c-> <c- o>+=</c-> <c- n>std</c-><c- o>::</c-><c- n>reduce</c-><c- p>(</c-><c- n>res</c-><c- p>);</c->
    <c- n>v1</c-> <c- o>+=</c-> <c- n>std</c-><c- o>::</c-><c- n>reduce</c-><c- p>(</c-><c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>);</c->
<c- p>}</c->

<c- k>if</c-> <c- p>(</c-><c- n>p</c-> <c- o>!=</c-> <c- n>npath</c-><c- p>)</c-> <c- p>{</c->
    <c- n>std</c-><c- o>::</c-><c- n>mt19937</c-> <c- nf>E_tail</c-><c- p>(</c-><c- n>E</c-><c- p>);</c-> <c- c1>// rebinding to scalar type</c->
    <c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-> <c- nf>D_tail</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c-> <c- c1>// getting scalar distribution</c->
    <c- k>for</c-><c- p>(;</c-> <c- n>p</c-> <c- o>&lt;</c-> <c- n>npath</c-><c- p>;</c-> <c- o>++</c-><c- n>p</c-><c- p>)</c-> <c- p>{</c->
        <c- k>auto</c-> <c- n>rand_tail</c-> <c- o>=</c-> <c- n>D_tail</c-><c- p>(</c-><c- n>E_tail</c-><c- p>);</c->
        <c- k>auto</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand_tail</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
        <c- n>v0</c-> <c- o>+=</c-> <c- n>res</c-><c- p>;</c->
        <c- n>v1</c-> <c- o>+=</c-> <c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>;</c->
    <c- p>}</c->
    <c- n>E</c-> <c- o>=</c-> <c- n>E_tail</c-><c- p>;</c-> <c- c1>// rebinding back to keep original engine in the expected state</c->
<c- p>}</c->

<c- n>result</c->     <c- o>=</c-> <c- n>v0</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
<c- n>confidence</c-> <c- o>=</c-> <c- n>v1</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
</pre>
   <p>While user-level flexibility is obtained, a set of redundant copy operations were introduced
into the code, which will bring visible overheads especially in case of engines with a larger state.</p>
   <h4 class="heading settled" data-level="2.2.3" id="algorithm_function_template"><span class="secno">2.2.3. </span><span class="content">Algorithm-like function template parameter</span><a class="self-link" href="#algorithm_function_template"></a></h4>
   <p>An alternative solution would be to shift the knowledge of the <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> size from
engine/distribution construction point to the point of usage. This makes the engine unaware
of its main usage mode and thus internal layout would be chosen by implementation to some
balanced form, which allows both packed and scalar generation.</p>
<pre class="language-cpp highlight"><c- n>std</c-><c- o>::</c-><c- n>mt19937</c->             <c- nf>E</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-> <c- nf>D</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->
<c- k>auto</c-> <c- n>rand</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>generate_random_simd</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- mi>16</c-><c- o>>></c-><c- p>(</c-><c- n>E</c-><c- p>,</c-> <c- n>D</c-><c- p>);</c->
</pre>
   <p>While it limits bare metal optimizations of the engine layout, it provides a significant freedom
of implementation to the standard library - the way of consuming base random numbers out of an
engine may vary from platform to platform. It also brings in more flexibility on the usage side.</p>
<pre class="language-cpp highlight"><c- k>constexpr</c-> <c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>size</c-> <c- o>=</c-> <c- mi>16</c-><c- p>;</c->
<c- n>std</c-><c- o>::</c-><c- n>mt19937</c-> <c- nf>E</c-><c- p>(</c-><c- mi>777</c-><c- p>);</c->
<c- n>std</c-><c- o>::</c-><c- n>normal_distribution</c-> <c- nf>D</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- mf>1.</c-><c- p>);</c->

<c- b>double</c-> <c- n>v0</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- n>v1</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->
<c- n>std</c-><c- o>::</c-><c- b>size_t</c-> <c- n>p</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->

<c- k>for</c-> <c- p>(;</c-> <c- n>p</c-><c- o>+</c-><c- n>size</c-> <c- o>&lt;=</c-> <c- n>npath</c-><c- p>;</c-> <c- n>p</c-> <c- o>+=</c-> <c- n>size</c-><c- p>)</c-> <c- p>{</c->
    <c- k>auto</c-> <c- n>rand</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>get_random_simd</c-><c- o>&lt;</c-><c- n>std</c-><c- o>::</c-><c- n>fixed_size_simd</c-><c- o>&lt;</c-><c- b>double</c-><c- p>,</c-> <c- n>size</c-><c- o>>></c-><c- p>(</c-><c- n>E</c-><c- p>,</c-> <c- n>D</c-><c- p>);</c->
    <c- k>auto</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
    <c- n>v0</c-> <c- o>+=</c-> <c- n>std</c-><c- o>::</c-><c- n>reduce</c-><c- p>(</c-><c- n>res</c-><c- p>);</c->
    <c- n>v1</c-> <c- o>+=</c-> <c- n>std</c-><c- o>::</c-><c- n>reduce</c-><c- p>(</c-><c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>);</c->
<c- p>}</c->

<c- c1>// scalar tail matches original example implementation</c->
<c- k>for</c-> <c- p>(;</c-> <c- n>p</c-> <c- o>&lt;</c-> <c- n>npath</c-><c- p>;</c-> <c- o>++</c-><c- n>p</c-><c- p>)</c-> <c- p>{</c->
    <c- k>auto</c-> <c- n>rand_tail</c-> <c- o>=</c-> <c- n>D</c-><c- p>(</c-><c- n>E</c-><c- p>);</c->
    <c- k>auto</c-> <c- n>res</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>max</c-><c- p>(</c-><c- mf>0.</c-><c- p>,</c-> <c- n>S</c-> <c- o>*</c-> <c- n>exp</c-><c- p>(</c-><c- n>sqrt</c-><c- p>(</c-><c- n>T</c-><c- p>)</c-> <c- o>*</c-> <c- n>vol</c-> <c- o>*</c-> <c- n>rand_tail</c-> <c- o>+</c-> <c- n>T</c-> <c- o>*</c-> <c- n>mu</c-><c- p>)</c-> <c- o>-</c-> <c- n>X</c-><c- p>);</c->
    <c- n>v0</c-> <c- o>+=</c-> <c- n>res</c-><c- p>;</c->
    <c- n>v1</c-> <c- o>+=</c-> <c- n>res</c-> <c- o>*</c-> <c- n>res</c-><c- p>;</c->
<c- p>}</c->

<c- n>result</c->     <c- o>=</c-> <c- n>v0</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
<c- n>confidence</c-> <c- o>=</c-> <c- n>v1</c-> <c- o>/</c-> <c- n>npath</c-><c- p>;</c->
</pre>
   <p>This example matches the initial example exactly and the user is not forced to copy engine
state to deal with tails. Moreover, having some belief in performance outcome, one may modify
example to optimize the tail computation more, by generating a sequence of decreasing <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> sizes of 8, 4, 2 and 1.</p>
   <h2 class="heading settled" data-level="3" id="high_level_api_on_simd_based"><span class="secno">3. </span><span class="content">SIMD-level API vs High-level API</span><a class="self-link" href="#high_level_api_on_simd_based"></a></h2>
   <p>There is no single answer on whether High-level or SIMD-level API should be used to optimize
a given scalar code with random number generation.</p>
   <p>High-level API requires additional buffers and may need additional blocking to achieve good performance.</p>
   <p>Low-level API may provide top performance but at the same time, SIMD-level API brings in
a number of implications on the user code - more efforts for dealing with tails, more code restructuring
to make all surrounding code <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> fashioned.</p>
   <p>At the same time any discussed Low-level API can be used under the hood of higher level API.</p>
   <ol>
    <li data-md>
     <p>Engine template parameter approach will require more API tuning under the hood
 of High-level API. Some kind of rebind operation will be needed to transform incoming legacy
 engine into another one with <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code> template parameter.</p>
    <li data-md>
     <p>Algorithm-like approach can be seamlessly used without additional modifications.</p>
   </ol>
   <p>With that we believe that both API levels make sense for the standard library.</p>
   <h2 class="heading settled" data-level="4" id="summary"><span class="secno">4. </span><span class="content">Summary</span><a class="self-link" href="#summary"></a></h2>
   <p>With the analysis of RNG API for support of vectorization, we conclude that:</p>
   <ol>
    <li data-md>
     <p>High level API aims at the support of the majority of the RNG based applications created by a
regular C++ developer. With opportunity to enable HW vectorization capability features by vendors
under the hood of the API implementation it would further enhance performance of C++ RNG based applications.
The blocking techniques for the further improvement of the performance are also possible upon
the same API. The implementation of the API might rely on SIMD capability.</p>
    <li data-md>
     <p>Low-level API aims at the advanced developers who need to code closer to HW still staying
in C++ convention. Though, the definition of SIMD based RNG API should avoid making the developers
understand the computational SIMD related specifics of RNG based on engine and distribution concepts.
The key to that might be a single point of SIMD configuration in the API, for example,
a SIMD-parameterized generation function. How this approach correlates with the SIMD use in other
domains, for example, mathematical functions, - open question.</p>
   </ol>
   <p>Based on those observations having both high- and low-level vectorized API for RNG in the C++ standard
library looks reasonable.</p>
  </main>
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  <h2 class="no-num no-ref heading settled" id="references"><span class="content">References</span><a class="self-link" href="#references"></a></h2>
  <h3 class="no-num no-ref heading settled" id="informative"><span class="content">Informative References</span><a class="self-link" href="#informative"></a></h3>
  <dl>
   <dt id="biblio-p1068r7">[P1068R7]
   <dd>Ilya Burylov; et al. <a href="http://wg21.link/P1068R7"><cite>Vector API for random number generation</cite></a>. URL: <a href="http://wg21.link/P1068R7">http://wg21.link/P1068R7</a>
   <dt id="biblio-p1928r3">[P1928R3]
   <dd>Matthias Kretz. <a href="https://wg21.link/p1928r3"><cite>Merge data-parallel types from the Parallelism TS 2</cite></a>. 3 February 2023. URL: <a href="https://wg21.link/p1928r3">https://wg21.link/p1928r3</a>
  </dl>