CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>bfloat16 floating-point format - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"56872998-b766-4706-ad9d-1c24ce111969","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Bfloat16_floating-point_format","wgTitle":"Bfloat16 floating-point format","wgCurRevisionId":1268317276,"wgRevisionId":1268317276,"wgArticleId":57499027,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 maint: multiple names: authors list","Articles with short description","Short description is different from Wikidata","Binary arithmetic","Floating point types"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Bfloat16_floating-point_format","wgRelevantArticleId":57499027,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q54083815","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGELevelingUpEnabledForUser":false}; RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.pygments":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.pygments.view","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.pygments%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.21"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="bfloat16 floating-point format - Wikipedia"> <meta property="og:type" content="website"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Bfloat16_floating-point_format"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Bfloat16_floating-point_format&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Bfloat16_floating-point_format"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Bfloat16_floating-point_format rootpage-Bfloat16_floating-point_format skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Bfloat16+floating-point+format" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Bfloat16+floating-point+format" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Bfloat16+floating-point+format" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Bfloat16+floating-point+format" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-bfloat16_floating-point_format" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#bfloat16_floating-point_format"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>bfloat16 floating-point format</span> </div> </a> <button aria-controls="toc-bfloat16_floating-point_format-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle bfloat16 floating-point format subsection</span> </button> <ul id="toc-bfloat16_floating-point_format-sublist" class="vector-toc-list"> <li id="toc-Exponent_encoding" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exponent_encoding"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Exponent encoding</span> </div> </a> <ul id="toc-Exponent_encoding-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rounding_and_conversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rounding_and_conversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Rounding and conversion</span> </div> </a> <ul id="toc-Rounding_and_conversion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Encoding_of_special_values" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Encoding_of_special_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Encoding of special values</span> </div> </a> <button aria-controls="toc-Encoding_of_special_values-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Encoding of special values subsection</span> </button> <ul id="toc-Encoding_of_special_values-sublist" class="vector-toc-list"> <li id="toc-Positive_and_negative_infinity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Positive_and_negative_infinity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Positive and negative infinity</span> </div> </a> <ul id="toc-Positive_and_negative_infinity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Not_a_Number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Not_a_Number"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Not a Number</span> </div> </a> <ul id="toc-Not_a_Number-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Range_and_precision" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Range_and_precision"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Range and precision</span> </div> </a> <ul id="toc-Range_and_precision-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Zeros_and_infinities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zeros_and_infinities"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Zeros and infinities</span> </div> </a> <ul id="toc-Zeros_and_infinities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Special values</span> </div> </a> <ul id="toc-Special_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-NaNs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#NaNs"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>NaNs</span> </div> </a> <ul id="toc-NaNs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">bfloat16 floating-point format</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 3 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-3" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">3 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bfloat16" title="Bfloat16 – Czech" lang="cs" hreflang="cs" data-title="Bfloat16" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bfloat16" title="Bfloat16 – German" lang="de" hreflang="de" data-title="Bfloat16" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/Bf16" title="Bf16 – Japanese" lang="ja" hreflang="ja" data-title="Bf16" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q54083815#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Bfloat16_floating-point_format" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Bfloat16_floating-point_format" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Bfloat16_floating-point_format"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bfloat16_floating-point_format&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Bfloat16_floating-point_format"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bfloat16_floating-point_format&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Bfloat16_floating-point_format" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Bfloat16_floating-point_format" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Bfloat16_floating-point_format&oldid=1268317276" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Bfloat16_floating-point_format&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Bfloat16_floating-point_format&id=1268317276&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FBfloat16_floating-point_format"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FBfloat16_floating-point_format"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Bfloat16_floating-point_format&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Bfloat16_floating-point_format&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q54083815" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">binary16</a>, a different 16-bit floating-point format.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Floating-point number format used in computer processors</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a> <a href="/wiki/Computer_number_format" title="Computer number format">formats</a></th></tr><tr><th class="sidebar-heading"> <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a></th></tr><tr><td class="sidebar-content"> <ul><li>16-bit: <a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half</a> (binary16)</li> <li>32-bit: <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single</a> (binary32), <a href="/wiki/Decimal32_floating-point_format" title="Decimal32 floating-point format">decimal32</a></li> <li>64-bit: <a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double</a> (binary64), <a href="/wiki/Decimal64_floating-point_format" title="Decimal64 floating-point format">decimal64</a></li> <li>128-bit: <a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple</a> (binary128), <a href="/wiki/Decimal128_floating-point_format" title="Decimal128 floating-point format">decimal128</a></li> <li>256-bit: <a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple</a> (binary256)</li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Other</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a></li> <li><a class="mw-selflink selflink">bfloat16</a></li> <li><a href="/wiki/TensorFloat-32" title="TensorFloat-32">TensorFloat-32</a></li> <li><a href="/wiki/Microsoft_Binary_Format" title="Microsoft Binary Format">Microsoft Binary Format</a></li> <li><a href="/wiki/IBM_hexadecimal_floating-point" title="IBM hexadecimal floating-point">IBM floating-point architecture</a></li> <li><a href="/wiki/Power_Management_Bus#Linear11_Floating-Point_Format" title="Power Management Bus">PMBus Linear-11</a></li> <li><a href="/wiki/G.711" title="G.711">G.711 8-bit floats</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Alternatives</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">Arbitrary precision</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Tapered_floating_point" title="Tapered floating point">Tapered floating point</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Unum_(number_format)" title="Unum (number format)">Posit</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Floating-point" title="Template:Floating-point"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Floating-point" title="Template talk:Floating-point"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Floating-point" title="Special:EditPage/Template:Floating-point"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>bfloat16</b> (<b>brain floating point</b>)<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point</a> format is a <a href="/wiki/Computer_number_format" title="Computer number format">computer number format</a> occupying <a href="/wiki/16-bit_computing" title="16-bit computing">16 bits</a> in <a href="/wiki/Computer_memory" title="Computer memory">computer memory</a>; it represents a wide <a href="/wiki/Dynamic_range" title="Dynamic range">dynamic range</a> of numeric values by using a floating radix point. This format is a shortened (16-bit) version of the 32-bit <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">IEEE 754 single-precision floating-point format</a> (binary32) with the intent of <a href="/wiki/Hardware_acceleration" title="Hardware acceleration">accelerating</a> <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a> and <a href="/wiki/Intelligent_sensor" title="Intelligent sensor">near-sensor computing</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It preserves the approximate dynamic range of 32-bit floating-point numbers by retaining 8 <a href="/wiki/Exponent_bias" title="Exponent bias">exponent bits</a>, but supports only an 8-bit precision rather than the 24-bit <a href="/wiki/Significand" title="Significand">significand</a> of the binary32 format. More so than single-precision 32-bit floating-point numbers, bfloat16 numbers are unsuitable for integer calculations, but this is not their intended use. Bfloat16 is used to reduce the storage requirements and increase the calculation speed of machine learning algorithms.<sup id="cite_ref-Why_4-0" class="reference"><a href="#cite_note-Why-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The bfloat16 format was developed by <a href="/wiki/Google_Brain" title="Google Brain">Google Brain</a>, an artificial intelligence research group at Google. It is utilized in many CPUs, GPUs, and <a href="/wiki/AI_accelerator" class="mw-redirect" title="AI accelerator">AI processors</a>, such as Intel <a href="/wiki/Xeon" title="Xeon">Xeon</a> processors (<a href="/wiki/AVX-512" title="AVX-512">AVX-512</a> BF16 extensions), Intel Data Center GPU, Intel <a href="/wiki/Nervana_Systems" title="Nervana Systems">Nervana</a> NNP-L1000, Intel <a href="/wiki/FPGA" class="mw-redirect" title="FPGA">FPGAs</a>,<sup id="cite_ref-vent_Inte_5-0" class="reference"><a href="#cite_note-vent_Inte-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-top5_Inte_6-0" class="reference"><a href="#cite_note-top5_Inte-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-toms_Inte_7-0" class="reference"><a href="#cite_note-toms_Inte-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Zen_(microarchitecture)" title="Zen (microarchitecture)">AMD Zen</a>, <a href="/wiki/AMD_Instinct" title="AMD Instinct">AMD Instinct</a>, NVIDIA GPUs, Google Cloud <a href="/wiki/Tensor_processing_unit" class="mw-redirect" title="Tensor processing unit">TPUs</a>,<sup id="cite_ref-clou_Avai_8-0" class="reference"><a href="#cite_note-clou_Avai-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-blog_Comp_9-0" class="reference"><a href="#cite_note-blog_Comp-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gith_tens_10-0" class="reference"><a href="#cite_note-gith_tens-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> AWS Inferentia, AWS Trainium, <a href="/wiki/ARM_architecture#ARMv8.6-A" class="mw-redirect" title="ARM architecture">ARMv8.6-A</a>,<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and Apple's <a href="/wiki/Apple_M2" title="Apple M2">M2</a><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and therefore <a href="/wiki/Apple_A15" title="Apple A15">A15</a> chips and later. Many libraries support bfloat16, such as <a href="/wiki/CUDA" title="CUDA">CUDA</a>,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Math_Kernel_Library" title="Math Kernel Library">Intel oneAPI Math Kernel Library</a>, <a href="/wiki/ROCm" title="ROCm">AMD ROCm</a>,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> AMD Optimizing CPU Libraries, <a href="/wiki/PyTorch" title="PyTorch">PyTorch</a>, and <a href="/wiki/TensorFlow" title="TensorFlow">TensorFlow</a>.<sup id="cite_ref-gith_tens_10-1" class="reference"><a href="#cite_note-gith_tens-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-arxiv_1711.10604_15-0" class="reference"><a href="#cite_note-arxiv_1711.10604-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> On these platforms, bfloat16 may also be used in <a href="/wiki/Mixed-precision_arithmetic" title="Mixed-precision arithmetic">mixed-precision arithmetic</a>, where bfloat16 numbers may be operated on and expanded to wider data types. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="bfloat16_floating-point_format">bfloat16 floating-point format</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=1" title="Edit section: bfloat16 floating-point format"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>bfloat16</b> has the following format: </p> <ul><li><a href="/wiki/Sign_bit" title="Sign bit">Sign bit</a>: 1 bit</li> <li><a href="/wiki/Exponent" class="mw-redirect" title="Exponent">Exponent</a> width: 8 bits</li> <li><a href="/wiki/Significand" title="Significand">Significand</a> <a href="/wiki/Precision_(arithmetic)" class="mw-redirect" title="Precision (arithmetic)">precision</a>: 8 bits (7 explicitly stored, with an <a href="/wiki/Implicit_leading_bit" class="mw-redirect" title="Implicit leading bit">implicit leading bit</a>), as opposed to 24 bits in a classical single-precision floating-point format</li></ul> <p>The bfloat16 format, being a shortened <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">IEEE 754 single-precision</a> 32-bit float, allows for fast <a href="/wiki/Type_conversion" title="Type conversion">conversion</a> to and from an IEEE 754 single-precision 32-bit float; in conversion to the bfloat16 format, the exponent bits are preserved while the significand field can be reduced by truncation (thus corresponding to <a href="/wiki/IEEE_754#Rounding_rules" title="IEEE 754">round toward 0</a>) or other <a href="#Rounding_and_conversion">rounding mechanisms</a>, ignoring the <a href="/wiki/NaN" title="NaN">NaN</a> special case. Preserving the exponent bits maintains the 32-bit float's range of ≈ 10<sup>−38</sup> to ≈ 3 × 10<sup>38</sup>.<sup id="cite_ref-googleio18-day1-time2575_16-0" class="reference"><a href="#cite_note-googleio18-day1-time2575-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The bits are laid out as follows: </p> <table class="wikitable" style="text-align:center; border-width:0; background:#FFFFFF;"> <tbody><tr> <td style="border-width:0; background:#FFFFFF; text-align:left" colspan="17"><b><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">IEEE half-precision</a> 16-bit float</b> </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">sign </td> <td style="border-width:0; background:#FFFFFF;" colspan="5">exponent (5 bit) </td> <td style="border-width:0; background:#FFFFFF;" colspan="10">fraction (10 bit) </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">  ┃ </td> <td style="border-width:0; background:#FFFFFF;" colspan="5"> <table class="wikitable" style="border-width:0; margin:0 auto; width:90%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td> <td style="border-width:0; background:#FFFFFF;" colspan="10"> <table class="wikitable" style="border-width:0; margin:0 auto; width:95%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;"></td> <td style="background:#C4FCFF;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 1 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0  </td></tr> <tr style="font-size:9pt;"> <td style="border-width:0; background:#FFFFFF;"></td> <td style="border-width:0; background:#FFFFFF;">15</td> <td style="border-width:0; background:#FFFFFF;">14</td> <td style="border-width:0; background:#FFFFFF;" colspan="3"></td> <td style="border-width:0; background:#FFFFFF;">10</td> <td style="border-width:0; background:#FFFFFF;">9</td> <td style="border-width:0; background:#FFFFFF;" colspan="8"></td> <td style="border-width:0; background:#FFFFFF;">0 </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;">  </td></tr> <tr> <td style="border-width:0; background:#FFFFFF; text-align:left" colspan="17"><b>bfloat16</b> </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">sign </td> <td style="border-width:0; background:#FFFFFF;" colspan="8">exponent (8 bit) </td> <td style="border-width:0; background:#FFFFFF;" colspan="7">fraction (7 bit) </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">  ┃ </td> <td style="border-width:0; background:#FFFFFF;" colspan="8"> <table class="wikitable" style="border-width:0; margin:0 auto; width:90%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td> <td style="border-width:0; background:#FFFFFF;" colspan="7"> <table class="wikitable" style="border-width:0; margin:0 auto; width:90%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;"></td> <td style="background:#C4FCFF;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 1 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0  </td></tr> <tr style="font-size:9pt;"> <td style="border-width:0; background:#FFFFFF;"></td> <td style="border-width:0; background:#FFFFFF;">15</td> <td style="border-width:0; background:#FFFFFF;">14</td> <td style="border-width:0; background:#FFFFFF;" colspan="6"></td> <td style="border-width:0; background:#FFFFFF;">7</td> <td style="border-width:0; background:#FFFFFF;">6</td> <td style="border-width:0; background:#FFFFFF;" colspan="5"></td> <td style="border-width:0; background:#FFFFFF;">0 </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;">  </td></tr> <tr> <td style="border-width:0; background:#FFFFFF; text-align:left" colspan="20"><b><a href="/wiki/TensorFloat-32" title="TensorFloat-32">Nvidia's TensorFloat-32</a> (19 bits)</b> </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">sign </td> <td style="border-width:0; background:#FFFFFF;" colspan="8">exponent (8 bit) </td> <td style="border-width:0; background:#FFFFFF;" colspan="10">fraction (10 bit) </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">  ┃ </td> <td style="border-width:0; background:#FFFFFF;" colspan="8"> <table class="wikitable" style="border-width:0; margin:0 auto; width:90%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td> <td style="border-width:0; background:#FFFFFF;" colspan="10"> <table class="wikitable" style="border-width:0; margin:0 auto; width:95%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;"></td> <td style="background:#C4FCFF;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 1 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0  </td></tr> <tr style="font-size:9pt;"> <td style="border-width:0; background:#FFFFFF;"></td> <td style="border-width:0; background:#FFFFFF;">18</td> <td style="border-width:0; background:#FFFFFF;">17</td> <td style="border-width:0; background:#FFFFFF;" colspan="6"></td> <td style="border-width:0; background:#FFFFFF;">10</td> <td style="border-width:0; background:#FFFFFF;">9</td> <td style="border-width:0; background:#FFFFFF;" colspan="8"></td> <td style="border-width:0; background:#FFFFFF;">0 </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;">  </td></tr> <tr> <td style="border-width:0; background:#FFFFFF; text-align:left" colspan="25"><b>AMD's fp24 format</b> </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">sign </td> <td style="border-width:0; background:#FFFFFF;" colspan="7">exponent (7 bit) </td> <td style="border-width:0; background:#FFFFFF;" colspan="16">fraction (16 bit) </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">  ┃ </td> <td style="border-width:0; background:#FFFFFF;" colspan="7"> <table class="wikitable" style="border-width:0; margin:0 auto; width:90%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td> <td style="border-width:0; background:#FFFFFF;" colspan="16"> <table class="wikitable" style="border-width:0; margin:0 auto; width:95%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;"></td> <td style="background:#C4FCFF;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 1 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0  </td></tr> <tr style="font-size:9pt;"> <td style="border-width:0; background:#FFFFFF;"></td> <td style="border-width:0; background:#FFFFFF;">23</td> <td style="border-width:0; background:#FFFFFF;">22</td> <td style="border-width:0; background:#FFFFFF;" colspan="5"></td> <td style="border-width:0; background:#FFFFFF;">16</td> <td style="border-width:0; background:#FFFFFF;">15</td> <td style="border-width:0; background:#FFFFFF;" colspan="14"></td> <td style="border-width:0; background:#FFFFFF;">0 </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;">  </td></tr> <tr> <td style="border-width:0; background:#FFFFFF; text-align:left" colspan="25"><b>Pixar's PXR24 format</b> </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">sign </td> <td style="border-width:0; background:#FFFFFF;" colspan="8">exponent (8 bit) </td> <td style="border-width:0; background:#FFFFFF;" colspan="15">fraction (15 bit) </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">  ┃ </td> <td style="border-width:0; background:#FFFFFF;" colspan="8"> <table class="wikitable" style="border-width:0; margin:0 auto; width:90%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td> <td style="border-width:0; background:#FFFFFF;" colspan="15"> <table class="wikitable" style="border-width:0; margin:0 auto; width:95%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;"></td> <td style="background:#C4FCFF;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#FFACAC;"> 1 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0  </td></tr> <tr style="font-size:9pt;"> <td style="border-width:0; background:#FFFFFF;"></td> <td style="border-width:0; background:#FFFFFF;">23</td> <td style="border-width:0; background:#FFFFFF;">22</td> <td style="border-width:0; background:#FFFFFF;" colspan="6"></td> <td style="border-width:0; background:#FFFFFF;">15</td> <td style="border-width:0; background:#FFFFFF;">14</td> <td style="border-width:0; background:#FFFFFF;" colspan="13"></td> <td style="border-width:0; background:#FFFFFF;">0 </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;">  </td></tr> <tr> <td style="border-width:0; background:#FFFFFF; text-align:left" colspan="33"><b><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">IEEE 754 single-precision</a> 32-bit float</b> </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">sign </td> <td style="border-width:0; background:#FFFFFF;" colspan="8">exponent (8 bit) </td> <td style="border-width:0; background:#FFFFFF;" colspan="23">fraction (23 bit) </td></tr> <tr style="line-height:70%;"> <td style="border-width:0; background:#FFFFFF;" colspan="2">  ┃ </td> <td style="border-width:0; background:#FFFFFF;" colspan="8"> <table class="wikitable" style="border-width:0; margin:0 auto; width:90%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td> <td style="border-width:0; background:#FFFFFF;" colspan="23"> <table class="wikitable" style="border-width:0; margin:0 auto; width:97%; display:table;"> <tbody><tr> <td style="border:2px solid black; border-bottom:0; background:#FFFFFF;"> </td></tr></tbody></table> </td></tr> <tr> <td style="border-width:0; background:#FFFFFF;"></td> <td style="background:#C4FCFF;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 1 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#9FFFAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 1 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0 </td> <td style="background:#FFACAC;"> 0  </td></tr> <tr style="font-size:9pt;"> <td style="border-width:0; background:#FFFFFF;"></td> <td style="border-width:0; background:#FFFFFF;">31</td> <td style="border-width:0; background:#FFFFFF;">30</td> <td style="border-width:0; background:#FFFFFF;" colspan="6"></td> <td style="border-width:0; background:#FFFFFF;">23</td> <td style="border-width:0; background:#FFFFFF;">22</td> <td style="border-width:0; background:#FFFFFF;" colspan="21"></td> <td style="border-width:0; background:#FFFFFF;">0 </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Exponent_encoding">Exponent encoding</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=2" title="Edit section: Exponent encoding"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bfloat16 binary floating-point exponent is encoded using an <a href="/wiki/Offset_binary" title="Offset binary">offset-binary</a> representation, with the zero offset being 127; also known as exponent bias in the IEEE 754 standard. </p> <ul><li>E<sub>min</sub> = 01<sub>H</sub>−7F<sub>H</sub> = −126</li> <li>E<sub>max</sub> = FE<sub>H</sub>−7F<sub>H</sub> = 127</li> <li><a href="/wiki/Exponent_bias" title="Exponent bias">Exponent bias</a> = 7F<sub>H</sub> = 127</li></ul> <p>Thus, in order to get the true exponent as defined by the offset-binary representation, the offset of 127 has to be subtracted from the value of the exponent field. </p><p>The minimum and maximum values of the exponent field (00<sub>H</sub> and FF<sub>H</sub>) are interpreted specially, like in the IEEE 754 standard formats. </p> <table class="wikitable" style="text-align:center"> <tbody><tr> <th>Exponent </th> <th>Significand zero </th> <th>Significand non-zero </th> <th>Equation </th></tr> <tr> <td>00<sub>H</sub> </td> <td><a href="/wiki/0_(number)" class="mw-redirect" title="0 (number)">zero</a>, <a href="/wiki/%E2%88%920" class="mw-redirect" title="−0">−0</a> </td> <td><a href="/wiki/Subnormal_number" title="Subnormal number">subnormal numbers</a> </td> <td>(−1)<sup>signbit</sup>×2<sup>−126</sup>× 0.significandbits </td></tr> <tr> <td>01<sub>H</sub>, ..., FE<sub>H</sub> </td> <td colspan="2">normalized value </td> <td>(−1)<sup>signbit</sup>×2<sup>exponentbits−127</sup>× 1.significandbits </td></tr> <tr> <td>FF<sub>H</sub> </td> <td>±<a href="/wiki/Infinity" title="Infinity">infinity</a> </td> <td><a href="/wiki/NaN" title="NaN">NaN</a> (quiet, signaling) </td> <td> </td></tr></tbody></table> <p>The minimum positive normal value is 2<sup>−126</sup> ≈ 1.18 × 10<sup>−38</sup> and the minimum positive (subnormal) value is 2<sup>−126−7</sup> = 2<sup>−133</sup> ≈ 9.2 × 10<sup>−41</sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Rounding_and_conversion">Rounding and conversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=3" title="Edit section: Rounding and conversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most common use case is the conversion between <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">IEEE 754 binary32</a> and bfloat16. The following section describes the conversion process and its rounding scheme in the conversion. Note that there are other possible scenarios of format conversions to or from bfloat16. For example, int16 and bfloat16. </p> <ul><li>From binary32 to bfloat16. When bfloat16 was first introduced as a storage format,<sup id="cite_ref-arxiv_1711.10604_15-1" class="reference"><a href="#cite_note-arxiv_1711.10604-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> the conversion from IEEE 754 binary32 (32-bit floating point) to bfloat16 is truncation (<a href="/wiki/IEEE_754#Rounding_rules" title="IEEE 754">round toward 0</a>). Later on, when it becomes the input of matrix multiplication units, the conversion can have various rounding mechanisms depending on the hardware platforms. For example, for Google TPU, the rounding scheme in the conversion is round-to-nearest-even;<sup id="cite_ref-google_TPU_17-0" class="reference"><a href="#cite_note-google_TPU-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> ARM uses the non-IEEE Round-to-Odd mode;<sup id="cite_ref-arm_product_18-0" class="reference"><a href="#cite_note-arm_product-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> for NVIDIA, it supports converting float number to bfloat16 precision in round-to-nearest-even mode.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li>From bfloat16 to binary32. Since binary32 can represent all exact values in bfloat16, the conversion simply pads 16 zeros in the significand bits.<sup id="cite_ref-google_TPU_17-1" class="reference"><a href="#cite_note-google_TPU-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Encoding_of_special_values">Encoding of special values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=4" title="Edit section: Encoding of special values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Positive_and_negative_infinity">Positive and negative infinity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=5" title="Edit section: Positive and negative infinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Just as in <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a>, positive and negative infinity are represented with their corresponding <a href="/wiki/Sign_bit" title="Sign bit">sign bits</a>, all 8 exponent bits set (FF<sub>hex</sub>) and all significand bits zero. Explicitly,</p><div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre><span></span>val s_exponent_signcnd +inf = 0_11111111_0000000 -inf = 1_11111111_0000000 </pre></div> <div class="mw-heading mw-heading3"><h3 id="Not_a_Number">Not a Number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=6" title="Edit section: Not a Number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Just as in <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a>, <a href="/wiki/NaN" title="NaN">NaN</a> values are represented with either sign bit, all 8 exponent bits set (FF<sub>hex</sub>) and not all significand bits zero. Explicitly,</p><div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre><span></span>val s_exponent_signcnd +NaN = 0_11111111_klmnopq -NaN = 1_11111111_klmnopq </pre></div><p>where at least one of <i>k, l, m, n, o, p,</i> or <i>q</i> is 1. As with IEEE 754, NaN values can be quiet or signaling, although there are no known uses of signaling bfloat16 NaNs as of September 2018. </p><div class="mw-heading mw-heading2"><h2 id="Range_and_precision">Range and precision</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=7" title="Edit section: Range and precision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bfloat16 is designed to maintain the number range from the 32-bit <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">IEEE 754 single-precision floating-point format</a> (binary32), while reducing the precision from 24 bits to 8 bits. This means that the precision is between two and three decimal digits, and bfloat16 can represent finite values up to about 3.4 × 10<sup>38</sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=8" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These examples are given in bit <i>representation</i>, in <a href="/wiki/Hexadecimal" title="Hexadecimal">hexadecimal</a> and <a href="/wiki/Binary_number" title="Binary number">binary</a>, of the floating-point value. This includes the sign, (biased) exponent, and significand. </p> <pre>3f80 = 0 01111111 0000000 = 1 c000 = 1 10000000 0000000 = −2 </pre> <pre>7f7f = 0 11111110 1111111 = (2<sup>8</sup> − 1) × 2<sup>−7</sup> × 2<sup>127</sup> ≈ 3.38953139 × 10<sup>38</sup> (max finite positive value in bfloat16 precision) 0080 = 0 00000001 0000000 = 2<sup>−126</sup> ≈ 1.175494351 × 10<sup>−38</sup> (min normalized positive value in bfloat16 precision and single-precision floating point) </pre> <p>The maximum positive finite value of a normal bfloat16 number is 3.38953139 × 10<sup>38</sup>, slightly below (2<sup>24</sup> − 1) × 2<sup>−23</sup> × 2<sup>127</sup> = 3.402823466 × 10<sup>38</sup>, the max finite positive value representable in single precision. </p> <div class="mw-heading mw-heading3"><h3 id="Zeros_and_infinities">Zeros and infinities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=9" title="Edit section: Zeros and infinities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>0000 = 0 00000000 0000000 = 0 8000 = 1 00000000 0000000 = −0 </pre> <pre>7f80 = 0 11111111 0000000 = infinity ff80 = 1 11111111 0000000 = −infinity </pre> <div class="mw-heading mw-heading3"><h3 id="Special_values">Special values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=10" title="Edit section: Special values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>4049 = 0 10000000 1001001 = 3.140625 ≈ π ( pi ) 3eab = 0 01111101 0101011 = 0.333984375 ≈ 1/3 </pre> <div class="mw-heading mw-heading3"><h3 id="NaNs">NaNs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=11" title="Edit section: NaNs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <pre>ffc1 = x 11111111 1000001 => <a href="/wiki/NaN#Quiet_NaN" title="NaN">qNaN</a> ff81 = x 11111111 0000001 => <a href="/wiki/NaN#Signaling_NaN" title="NaN">sNaN</a> </pre> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half-precision floating-point format</a>: 16-bit float w/ 1-bit sign, 5-bit exponent, and 11-bit significand, as defined by <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a></li> <li><a href="/wiki/ISO/IEC_10967" title="ISO/IEC 10967">ISO/IEC 10967</a>, Language Independent Arithmetic</li> <li><a href="/wiki/Primitive_data_type" title="Primitive data type">Primitive data type</a></li> <li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a></li> <li><a href="/wiki/Google_Brain" title="Google Brain">Google Brain</a></li> <li><a href="/wiki/Tensor_Processing_Unit#Lawsuit" title="Tensor Processing Unit">Lawsuit</a> against Google for its use of bfloat16 in TPU</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bfloat16_floating-point_format&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFTeich2018" class="citation web cs1">Teich, Paul (2018-05-10). <a rel="nofollow" class="external text" href="https://www.nextplatform.com/2018/05/10/tearing-apart-googles-tpu-3-0-ai-coprocessor/">"Tearing Apart Google's TPU 3.0 AI Coprocessor"</a>. <i>The Next Platform</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-11</span></span>. <q>Google invented its own internal floating point format called "bfloat" for "brain floating point" (after Google Brain).</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Next+Platform&rft.atitle=Tearing+Apart+Google%27s+TPU+3.0+AI+Coprocessor&rft.date=2018-05-10&rft.aulast=Teich&rft.aufirst=Paul&rft_id=https%3A%2F%2Fwww.nextplatform.com%2F2018%2F05%2F10%2Ftearing-apart-googles-tpu-3-0-ai-coprocessor%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWangKanwar2019" class="citation web cs1">Wang, Shibo; Kanwar, Pankaj (2019-08-23). <a rel="nofollow" class="external text" href="https://cloud.google.com/blog/products/ai-machine-learning/bfloat16-the-secret-to-high-performance-on-cloud-tpus">"BFloat16: The secret to high performance on Cloud TPUs"</a>. <i>Google Cloud</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-11</span></span>. <q>This custom floating point format is called "Brain Floating Point Format," or "bfloat16" for short. The name flows from "Google Brain", which is an artificial intelligence research group at Google where the idea for this format was conceived.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google+Cloud&rft.atitle=BFloat16%3A+The+secret+to+high+performance+on+Cloud+TPUs&rft.date=2019-08-23&rft.aulast=Wang&rft.aufirst=Shibo&rft.au=Kanwar%2C+Pankaj&rft_id=https%3A%2F%2Fcloud.google.com%2Fblog%2Fproducts%2Fai-machine-learning%2Fbfloat16-the-secret-to-high-performance-on-cloud-tpus&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTagliaviniMachRossiMarongiu2018" class="citation book cs1">Tagliavini, Giuseppe; Mach, Stefan; Rossi, Davide; Marongiu, Andrea; Benin, Luca (2018). "A transprecision floating-point platform for ultra-low power computing". <i>2018 Design, Automation & Test in Europe Conference & Exhibition (DATE)</i>. pp. <span class="nowrap">1051–</span>1056. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1711.10374">1711.10374</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.23919%2FDATE.2018.8342167">10.23919/DATE.2018.8342167</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-9819263-0-9" title="Special:BookSources/978-3-9819263-0-9"><bdi>978-3-9819263-0-9</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5067903">5067903</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+transprecision+floating-point+platform+for+ultra-low+power+computing&rft.btitle=2018+Design%2C+Automation+%26+Test+in+Europe+Conference+%26+Exhibition+%28DATE%29&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1051-%3C%2Fspan%3E1056&rft.date=2018&rft_id=info%3Aarxiv%2F1711.10374&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A5067903%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.23919%2FDATE.2018.8342167&rft.isbn=978-3-9819263-0-9&rft.aulast=Tagliavini&rft.aufirst=Giuseppe&rft.au=Mach%2C+Stefan&rft.au=Rossi%2C+Davide&rft.au=Marongiu%2C+Andrea&rft.au=Benin%2C+Luca&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-Why-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Why_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDr._Ian_Cutress2020" class="citation web cs1">Dr. Ian Cutress (2020-03-17). <a rel="nofollow" class="external text" href="https://www.anandtech.com/show/15631/intels-cooper-lake-plans-the-chip-that-wasnt-meant-to-exist-dies-for-you">"Intel': Cooper lake Plans: Why is BF16 Important?"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-05-12</span></span>. <q>The bfloat16 standard is a targeted way of representing numbers that give the range of a full 32-bit number, but in the data size of a 16-bit number, keeping the accuracy close to zero but being a bit more loose with the accuracy near the limits of the standard. The bfloat16 standard has a lot of uses inside machine learning algorithms, by offering better accuracy of values inside the algorithm while affording double the data in any given dataset (or doubling the speed in those calculation sections).</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Intel%27%3A+Cooper+lake+Plans%3A+Why+is+BF16+Important%3F&rft.date=2020-03-17&rft.au=Dr.+Ian+Cutress&rft_id=https%3A%2F%2Fwww.anandtech.com%2Fshow%2F15631%2Fintels-cooper-lake-plans-the-chip-that-wasnt-meant-to-exist-dies-for-you&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-vent_Inte-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-vent_Inte_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKhari_Johnson2018" class="citation web cs1">Khari Johnson (2018-05-23). <a rel="nofollow" class="external text" href="https://venturebeat.com/2018/05/23/intel-unveils-nervana-neural-net-l-1000-for-accelerated-ai-training/">"Intel unveils Nervana Neural Net L-1000 for accelerated AI training"</a>. <i>VentureBeat</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-23</span></span>. <q>...Intel will be extending bfloat16 support across our AI product lines, including Intel Xeon processors and Intel FPGAs.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=VentureBeat&rft.atitle=Intel+unveils+Nervana+Neural+Net+L-1000+for+accelerated+AI+training&rft.date=2018-05-23&rft.au=Khari+Johnson&rft_id=https%3A%2F%2Fventurebeat.com%2F2018%2F05%2F23%2Fintel-unveils-nervana-neural-net-l-1000-for-accelerated-ai-training%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-top5_Inte-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-top5_Inte_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMichael_Feldman2018" class="citation web cs1">Michael Feldman (2018-05-23). <a rel="nofollow" class="external text" href="https://www.top500.org/news/intel-lays-out-new-roadmap-for-ai-portfolio/">"Intel Lays Out New Roadmap for AI Portfolio"</a>. <i>TOP500 Supercomputer Sites</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-23</span></span>. <q>Intel plans to support this format across all their AI products, including the Xeon and FPGA lines</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=TOP500+Supercomputer+Sites&rft.atitle=Intel+Lays+Out+New+Roadmap+for+AI+Portfolio&rft.date=2018-05-23&rft.au=Michael+Feldman&rft_id=https%3A%2F%2Fwww.top500.org%2Fnews%2Fintel-lays-out-new-roadmap-for-ai-portfolio%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-toms_Inte-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-toms_Inte_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLucian_Armasu2018" class="citation web cs1">Lucian Armasu (2018-05-23). <a rel="nofollow" class="external text" href="https://www.tomshardware.com/news/intel-neural-network-processor-lake-crest,37105.html">"Intel To Launch Spring Crest, Its First Neural Network Processor, In 2019"</a>. <i>Tom's Hardware</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-23</span></span>. <q>Intel said that the NNP-L1000 would also support bfloat16, a numerical format that's being adopted by all the ML industry players for neural networks. The company will also support bfloat16 in its FPGAs, Xeons, and other ML products. The Nervana NNP-L1000 is scheduled for release in 2019.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Tom%27s+Hardware&rft.atitle=Intel+To+Launch+Spring+Crest%2C+Its+First+Neural+Network+Processor%2C+In+2019&rft.date=2018-05-23&rft.au=Lucian+Armasu&rft_id=https%3A%2F%2Fwww.tomshardware.com%2Fnews%2Fintel-neural-network-processor-lake-crest%2C37105.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-clou_Avai-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-clou_Avai_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://cloud.google.com/tpu/docs/tensorflow-ops">"Available TensorFlow Ops | Cloud TPU | Google Cloud"</a>. <i>Google Cloud</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-23</span></span>. <q>This page lists the TensorFlow Python APIs and graph operators available on Cloud TPU.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google+Cloud&rft.atitle=Available+TensorFlow+Ops+%7C+Cloud+TPU+%7C+Google+Cloud&rft_id=https%3A%2F%2Fcloud.google.com%2Ftpu%2Fdocs%2Ftensorflow-ops&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-blog_Comp-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-blog_Comp_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFElmar_Haußmann2018" class="citation web cs1">Elmar Haußmann (2018-04-26). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180426200043/https://blog.riseml.com/comparing-google-tpuv2-against-nvidia-v100-on-resnet-50-c2bbb6a51e5e">"Comparing Google's TPUv2 against Nvidia's V100 on ResNet-50"</a>. <i>RiseML Blog</i>. Archived from <a rel="nofollow" class="external text" href="https://blog.riseml.com/comparing-google-tpuv2-against-nvidia-v100-on-resnet-50-c2bbb6a51e5e">the original</a> on 2018-04-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-23</span></span>. <q>For the Cloud TPU, Google recommended we use the bfloat16 implementation from the official TPU repository with TensorFlow 1.7.0. Both the TPU and GPU implementations make use of mixed-precision computation on the respective architecture and store most tensors with half-precision.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=RiseML+Blog&rft.atitle=Comparing+Google%27s+TPUv2+against+Nvidia%27s+V100+on+ResNet-50&rft.date=2018-04-26&rft.au=Elmar+Hau%C3%9Fmann&rft_id=https%3A%2F%2Fblog.riseml.com%2Fcomparing-google-tpuv2-against-nvidia-v100-on-resnet-50-c2bbb6a51e5e&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-gith_tens-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-gith_tens_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-gith_tens_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTensorflow_Authors2018" class="citation web cs1">Tensorflow Authors (2018-07-23). <a rel="nofollow" class="external text" href="https://github.com/tensorflow/tpu/tree/0ece10f6f4e523eab79aba0247b513fe57d38ae6/models/experimental/resnet_bfloat16">"ResNet-50 using BFloat16 on TPU"</a>. <i>Google</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-11-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google&rft.atitle=ResNet-50+using+BFloat16+on+TPU&rft.date=2018-07-23&rft.au=Tensorflow+Authors&rft_id=https%3A%2F%2Fgithub.com%2Ftensorflow%2Ftpu%2Ftree%2F0ece10f6f4e523eab79aba0247b513fe57d38ae6%2Fmodels%2Fexperimental%2Fresnet_bfloat16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://community.arm.com/developer/ip-products/processors/b/ml-ip-blog/posts/bfloat16-processing-for-neural-networks-on-armv8_2d00_a">"BFloat16 extensions for Armv8-A"</a>. <i>community.arm.com</i>. 29 August 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-08-30</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=community.arm.com&rft.atitle=BFloat16+extensions+for+Armv8-A&rft.date=2019-08-29&rft_id=https%3A%2F%2Fcommunity.arm.com%2Fdeveloper%2Fip-products%2Fprocessors%2Fb%2Fml-ip-blog%2Fposts%2Fbfloat16-processing-for-neural-networks-on-armv8_2d00_a&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/llvm/llvm-project/commit/677da09d0259d7530d32e85cb561bee15f0066e2">"AArch64: add support for newer Apple CPUs · llvm/llvm-project@677da09"</a>. <i>GitHub</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-05-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=GitHub&rft.atitle=AArch64%3A+add+support+for+newer+Apple+CPUs+%C2%B7+llvm%2Fllvm-project%40677da09&rft_id=https%3A%2F%2Fgithub.com%2Fllvm%2Fllvm-project%2Fcommit%2F677da09d0259d7530d32e85cb561bee15f0066e2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.nvidia.com/cuda/cuda-math-api/group__CUDA__MATH__INTRINSIC__BFLOAT16.html#group__CUDA__MATH__INTRINSIC__BFLOAT16">"CUDA Library bloat16 Intrinsics"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=CUDA+Library+bloat16+Intrinsics&rft_id=https%3A%2F%2Fdocs.nvidia.com%2Fcuda%2Fcuda-math-api%2Fgroup__CUDA__MATH__INTRINSIC__BFLOAT16.html%23group__CUDA__MATH__INTRINSIC__BFLOAT16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/RadeonOpenCompute/ROCm/blob/8bd9a527405cb466d45b3b343a33434e79b1d387/version_history.md#miopen-20">"ROCm version history"</a>. <i>github.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-10-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=github.com&rft.atitle=ROCm+version+history&rft_id=https%3A%2F%2Fgithub.com%2FRadeonOpenCompute%2FROCm%2Fblob%2F8bd9a527405cb466d45b3b343a33434e79b1d387%2Fversion_history.md%23miopen-20&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-arxiv_1711.10604-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-arxiv_1711.10604_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-arxiv_1711.10604_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJoshua_V._Dillon,_Ian_Langmore,_Dustin_Tran,_Eugene_Brevdo,_Srinivas_Vasudevan,_Dave_Moore,_Brian_Patton,_Alex_Alemi,_Matt_Hoffman,_Rif_A._Saurous2017" class="citation report cs1">Joshua V. Dillon, Ian Langmore, Dustin Tran, Eugene Brevdo, Srinivas Vasudevan, Dave Moore, Brian Patton, Alex Alemi, Matt Hoffman, Rif A. Saurous (2017-11-28). TensorFlow Distributions (Report). <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1711.10604">1711.10604</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017arXiv171110604D">2017arXiv171110604D</a>. Accessed 2018-05-23. <q>All operations in TensorFlow Distributions are numerically stable across half, single, and double floating-point precisions (as TensorFlow dtypes: tf.bfloat16 (truncated floating point), tf.float16, tf.float32, tf.float64). Class constructors have a validate_args flag for numerical asserts</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=TensorFlow+Distributions&rft.date=2017-11-28&rft_id=info%3Aarxiv%2F1711.10604&rft_id=info%3Abibcode%2F2017arXiv171110604D&rft.au=Joshua+V.+Dillon%2C+Ian+Langmore%2C+Dustin+Tran%2C+Eugene+Brevdo%2C+Srinivas+Vasudevan%2C+Dave+Moore%2C+Brian+Patton%2C+Alex+Alemi%2C+Matt+Hoffman%2C+Rif+A.+Saurous&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_report" title="Template:Cite report">cite report</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-googleio18-day1-time2575-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-googleio18-day1-time2575_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=vm67WcLzfvc&t=2555">"Livestream Day 1: Stage 8 (Google I/O '18) - YouTube"</a>. <i>Google</i>. 2018-05-08<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-05-23</span></span>. <q>In many models this is a drop-in replacement for float-32</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google&rft.atitle=Livestream+Day+1%3A+Stage+8+%28Google+I%2FO+%2718%29+-+YouTube&rft.date=2018-05-08&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3Dvm67WcLzfvc%26t%3D2555&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-google_TPU-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-google_TPU_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-google_TPU_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://cloud.google.com/tpu/docs/bfloat16">"The bfloat16 numerical format"</a>. <i>Google Cloud</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-07-11</span></span>. <q>On TPU, the rounding scheme in the conversion is round to nearest even and overflow to inf.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Google+Cloud&rft.atitle=The+bfloat16+numerical+format&rft_id=https%3A%2F%2Fcloud.google.com%2Ftpu%2Fdocs%2Fbfloat16&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-arm_product-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-arm_product_18-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://developer.arm.com/documentation/ddi0596/2021-12/SVE-Instructions/BFDOT--vectors---BFloat16-floating-point-dot-product-">"Arm A64 Instruction Set Architecture"</a>. <i>developer.arm.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-07-26</span></span>. <q>Uses the non-IEEE Round-to-Odd rounding mode.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=developer.arm.com&rft.atitle=Arm+A64+Instruction+Set+Architecture&rft_id=https%3A%2F%2Fdeveloper.arm.com%2Fdocumentation%2Fddi0596%2F2021-12%2FSVE-Instructions%2FBFDOT--vectors---BFloat16-floating-point-dot-product-&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://docs.nvidia.com/cuda/pdf/CUDA_Math_API.pdf">"1.3.5. Bfloat16 Precision Conversion and Data Movement"</a> <span class="cs1-format">(PDF)</span>. <i>docs.nvidia.com</i>. p. 199<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-07-26</span></span>. <q>Converts float number to nv_bfloat16 precision in round-to-nearest-even mode and returns nv_bfloat16 with converted value.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.nvidia.com&rft.atitle=1.3.5.+Bfloat16+Precision+Conversion+and+Data+Movement&rft.pages=199&rft_id=https%3A%2F%2Fdocs.nvidia.com%2Fcuda%2Fpdf%2FCUDA_Math_API.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABfloat16+floating-point+format" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Data_types177" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Data_types" title="Template:Data types"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Data_types" title="Template talk:Data types"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Data_types" title="Special:EditPage/Template:Data types"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_types177" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_type" title="Data type">Data types</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Units_of_information" title="Units of information">Uninterpreted</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit" title="Bit">Bit</a></li> <li><a href="/wiki/Byte" title="Byte">Byte</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">Trit</a></li> <li><a href="/wiki/Ternary_numeral_system#Tryte" title="Ternary numeral system">Tryte</a></li> <li><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word</a></li> <li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">Arbitrary-precision or bignum</a></li> <li><a href="/wiki/Complex_data_type" title="Complex data type">Complex</a></li> <li><a href="/wiki/Decimal_data_type" title="Decimal data type">Decimal</a></li> <li><a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">Fixed point</a></li> <li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating point</a> <ul><li>Reduced precision <ul><li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a></li> <li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half precision</a></li> <li><a class="mw-selflink selflink">bfloat16</a></li></ul></li> <li><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single precision</a></li> <li><a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double precision</a></li> <li><a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple precision</a></li> <li><a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple precision</a></li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a> <ul><li><a href="/wiki/Long_double" title="Long double">Long double</a></li></ul></li></ul></li> <li><a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">Integer</a> <ul><li><a href="/wiki/Signedness" title="Signedness">signedness</a></li></ul></li> <li><a href="/wiki/Interval_arithmetic#Implementations" title="Interval arithmetic">Interval</a></li> <li><a href="/wiki/Rational_data_type" title="Rational data type">Rational</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pointer_(computer_programming)" title="Pointer (computer programming)">Pointer</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Memory_address" title="Memory address">Address</a> <ul><li><a href="/wiki/Physical_address" title="Physical address">physical</a></li> <li><a href="/wiki/Virtual_address_space" title="Virtual address space">virtual</a></li></ul></li> <li><a href="/wiki/Reference_(computer_science)" title="Reference (computer science)">Reference</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plain_text" title="Plain text">Text</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Character_(computing)" title="Character (computing)">Character</a></li> <li><a href="/wiki/String_(computer_science)" title="String (computer science)">String</a> <ul><li><a href="/wiki/Null-terminated_string" title="Null-terminated string">null-terminated</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_data_type" title="Composite data type">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_data_type" title="Algebraic data type">Algebraic data type</a> <ul><li><a href="/wiki/Generalized_algebraic_data_type" title="Generalized algebraic data type">generalized</a></li></ul></li> <li><a href="/wiki/Array_data_type" class="mw-redirect" title="Array data type">Array</a></li> <li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">Class</a></li> <li><a href="/wiki/Dependent_type" title="Dependent type">Dependent</a></li> <li><a href="/wiki/Intuitionistic_type_theory#Equality_type" title="Intuitionistic type theory">Equality</a></li> <li><a href="/wiki/Inductive_type" title="Inductive type">Inductive</a></li> <li><a href="/wiki/Intersection_type" title="Intersection type">Intersection</a></li> <li><a href="/wiki/List_(abstract_data_type)" title="List (abstract data type)">List</a></li> <li><a href="/wiki/Object_(computer_science)" title="Object (computer science)">Object</a> <ul><li><a href="/wiki/Metaobject" title="Metaobject">metaobject</a></li></ul></li> <li><a href="/wiki/Option_type" title="Option type">Option type</a></li> <li><a href="/wiki/Product_type" title="Product type">Product</a></li> <li><a href="/wiki/Record_(computer_science)" title="Record (computer science)">Record or Struct</a></li> <li><a href="/wiki/Refinement_type" title="Refinement type">Refinement</a></li> <li><a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">Set</a></li> <li><a href="/wiki/Union_type" title="Union type">Union</a> <ul><li><a href="/wiki/Tagged_union" title="Tagged union">tagged</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_data_type" title="Boolean data type">Boolean</a></li> <li><a href="/wiki/Bottom_type" title="Bottom type">Bottom type</a></li> <li><a href="/wiki/Container_(abstract_data_type)" title="Container (abstract data type)">Collection</a></li> <li><a href="/wiki/Enumerated_type" title="Enumerated type">Enumerated type</a></li> <li><a href="/wiki/Exception_handling" title="Exception handling">Exception</a></li> <li><a href="/wiki/Function_type" title="Function type">Function type</a></li> <li><a href="/wiki/Opaque_data_type" title="Opaque data type">Opaque data type</a></li> <li><a href="/wiki/Recursive_data_type" title="Recursive data type">Recursive data type</a></li> <li><a href="/wiki/Semaphore_(programming)" title="Semaphore (programming)">Semaphore</a></li> <li><a href="/wiki/Stream_(computing)" title="Stream (computing)">Stream</a></li> <li><a href="/wiki/Strongly_typed_identifier" title="Strongly typed identifier">Strongly typed identifier</a></li> <li><a href="/wiki/Top_type" title="Top type">Top type</a></li> <li><a href="/wiki/Type_class" title="Type class">Type class</a></li> <li><a href="/wiki/Empty_type" title="Empty type">Empty type</a></li> <li><a href="/wiki/Unit_type" title="Unit type">Unit type</a></li> <li><a href="/wiki/Void_type" title="Void type">Void</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_data_type" title="Abstract data type">Abstract data type</a></li> <li><a href="/wiki/Boxing_(computer_science)" class="mw-redirect" title="Boxing (computer science)">Boxing</a></li> <li><a href="/wiki/Data_structure" title="Data structure">Data structure</a></li> <li><a href="/wiki/Generic_programming" title="Generic programming">Generic</a></li> <li><a href="/wiki/Kind_(type_theory)" title="Kind (type theory)">Kind</a> <ul><li><a href="/wiki/Metaclass" title="Metaclass">metaclass</a></li></ul></li> <li><a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">Parametric polymorphism</a></li> <li><a href="/wiki/Primitive_data_type" title="Primitive data type">Primitive data type</a></li> <li><a href="/wiki/Interface_(object-oriented_programming)" title="Interface (object-oriented programming)">Interface</a></li> <li><a href="/wiki/Subtyping" title="Subtyping">Subtyping</a></li> <li><a href="/wiki/Type_constructor" title="Type constructor">Type constructor</a></li> <li><a href="/wiki/Type_conversion" title="Type conversion">Type conversion</a></li> <li><a href="/wiki/Type_system" title="Type system">Type system</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li> <li><a href="/wiki/Variable_(computer_science)" title="Variable (computer science)">Variable</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Bfloat16_floating-point_format&oldid=1268317276">https://en.wikipedia.org/w/index.php?title=Bfloat16_floating-point_format&oldid=1268317276</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Binary_arithmetic" title="Category:Binary arithmetic">Binary arithmetic</a></li><li><a href="/wiki/Category:Floating_point_types" title="Category:Floating point types">Floating point types</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 9 January 2025, at 04:59<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Bfloat16_floating-point_format&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://www.wikimedia.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" >bfloat16 floating-point format</div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>3 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.eqiad.main-dbf9b8f9d-4c4hs","wgBackendResponseTime":140,"wgPageParseReport":{"limitreport":{"cputime":"0.388","walltime":"0.579","ppvisitednodes":{"value":1302,"limit":1000000},"postexpandincludesize":{"value":61581,"limit":2097152},"templateargumentsize":{"value":704,"limit":2097152},"expansiondepth":{"value":8,"limit":100},"expensivefunctioncount":{"value":3,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":89177,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 502.726 1 -total"," 32.83% 165.054 1 Template:Reflist"," 24.98% 125.564 17 Template:Cite_web"," 19.62% 98.656 1 Template:Floating-point"," 18.35% 92.254 1 Template:Sidebar"," 14.18% 71.303 1 Template:Short_description"," 12.22% 61.412 1 Template:Confuse"," 7.02% 35.302 1 Template:Data_types"," 6.48% 32.555 2 Template:Pagetype"," 6.14% 30.888 1 Template:Navbox"]},"scribunto":{"limitreport-timeusage":{"value":"0.251","limit":"10.000"},"limitreport-memusage":{"value":5886462,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-8669bc5c8-d9gxz","timestamp":"20250318164147","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Bfloat16 floating-point format","url":"https:\/\/en.wikipedia.org\/wiki\/Bfloat16_floating-point_format","sameAs":"http:\/\/www.wikidata.org\/entity\/Q54083815","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q54083815","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2018-05-24T07:09:09Z","dateModified":"2025-01-09T04:59:57Z","headline":"floating-point number format optimized for machine learning"}</script> </body> </html>