CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>IEEE 754 - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-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":"a0f48544-eb26-4390-87ed-2dabd99ab011","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"IEEE_754","wgTitle":"IEEE 754","wgCurRevisionId":1254909127,"wgRevisionId":1254909127,"wgArticleId":405039,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","Use dmy dates from May 2019","Wikipedia articles needing clarification from July 2024","Articles containing potentially dated statements from 2019","All articles containing potentially dated statements","CS1 maint: numeric names: authors list","CS1 location test","Commons category link is on Wikidata","Computer arithmetic","IEEE standards","Floating point types","Binary arithmetic"],"wgPageViewLanguage":"en", "wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"IEEE_754","wgRelevantArticleId":405039,"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":60000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q828287", "wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":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.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.tablesorter.styles":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site", "mediawiki.page.ready","jquery.tablesorter","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","mmv.bootstrap","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.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</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.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cjquery.tablesorter.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.4"> <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="IEEE 754 - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/IEEE_754"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=IEEE_754&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/IEEE_754"> <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-IEEE_754 rootpage-IEEE_754 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" > <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> </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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_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=IEEE+754" 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=IEEE+754" 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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_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=IEEE+754" 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=IEEE+754" 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-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formats" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formats"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formats</span> </div> </a> <button aria-controls="toc-Formats-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 Formats subsection</span> </button> <ul id="toc-Formats-sublist" class="vector-toc-list"> <li id="toc-Representation_and_encoding_in_memory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representation_and_encoding_in_memory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Representation and encoding in memory</span> </div> </a> <ul id="toc-Representation_and_encoding_in_memory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_and_interchange_formats" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basic_and_interchange_formats"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Basic and interchange formats</span> </div> </a> <ul id="toc-Basic_and_interchange_formats-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extended_and_extendable_precision_formats" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extended_and_extendable_precision_formats"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Extended and extendable precision formats</span> </div> </a> <ul id="toc-Extended_and_extendable_precision_formats-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interchange_formats" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interchange_formats"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Interchange formats</span> </div> </a> <ul id="toc-Interchange_formats-sublist" class="vector-toc-list"> <li id="toc-Binary" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Binary"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.1</span> <span>Binary</span> </div> </a> <ul id="toc-Binary-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decimal" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Decimal"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4.2</span> <span>Decimal</span> </div> </a> <ul id="toc-Decimal-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Rounding_rules" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Rounding_rules"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rounding rules</span> </div> </a> <button aria-controls="toc-Rounding_rules-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 Rounding rules subsection</span> </button> <ul id="toc-Rounding_rules-sublist" class="vector-toc-list"> <li id="toc-Roundings_to_nearest" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Roundings_to_nearest"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Roundings to nearest</span> </div> </a> <ul id="toc-Roundings_to_nearest-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Directed_roundings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Directed_roundings"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Directed roundings</span> </div> </a> <ul id="toc-Directed_roundings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Required_operations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Required_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Required operations</span> </div> </a> <button aria-controls="toc-Required_operations-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 Required operations subsection</span> </button> <ul id="toc-Required_operations-sublist" class="vector-toc-list"> <li id="toc-Comparison_predicates" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparison_predicates"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Comparison predicates</span> </div> </a> <ul id="toc-Comparison_predicates-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total-ordering_predicate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total-ordering_predicate"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Total-ordering predicate</span> </div> </a> <ul id="toc-Total-ordering_predicate-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exception_handling" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Exception_handling"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Exception handling</span> </div> </a> <ul id="toc-Exception_handling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_values" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Special_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Special values</span> </div> </a> <button aria-controls="toc-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 Special values subsection</span> </button> <ul id="toc-Special_values-sublist" class="vector-toc-list"> <li id="toc-Signed_zero" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signed_zero"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Signed zero</span> </div> </a> <ul id="toc-Signed_zero-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subnormal_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subnormal_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Subnormal numbers</span> </div> </a> <ul id="toc-Subnormal_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinities"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Infinities</span> </div> </a> <ul id="toc-Infinities-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">6.4</span> <span>NaNs</span> </div> </a> <ul id="toc-NaNs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Design_rationale" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Design_rationale"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Design rationale</span> </div> </a> <ul id="toc-Design_rationale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Recommendations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Recommendations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Recommendations</span> </div> </a> <button aria-controls="toc-Recommendations-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 Recommendations subsection</span> </button> <ul id="toc-Recommendations-sublist" class="vector-toc-list"> <li id="toc-Alternate_exception_handling" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternate_exception_handling"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Alternate exception handling</span> </div> </a> <ul id="toc-Alternate_exception_handling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Recommended_operations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Recommended_operations"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Recommended operations</span> </div> </a> <ul id="toc-Recommended_operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Expression_evaluation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Expression_evaluation"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Expression evaluation</span> </div> </a> <ul id="toc-Expression_evaluation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reproducibility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reproducibility"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Reproducibility</span> </div> </a> <ul id="toc-Reproducibility-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Character_representation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Character_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Character representation</span> </div> </a> <button aria-controls="toc-Character_representation-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 Character representation subsection</span> </button> <ul id="toc-Character_representation-sublist" class="vector-toc-list"> <li id="toc-Hexadecimal_literals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hexadecimal_literals"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Hexadecimal literals</span> </div> </a> <ul id="toc-Hexadecimal_literals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <button aria-controls="toc-References-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 References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Standards" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Standards"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>Standards</span> </div> </a> <ul id="toc-Standards-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Secondary_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Secondary_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.2</span> <span>Secondary references</span> </div> </a> <ul id="toc-Secondary_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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" > <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"><span class="mw-page-title-main">IEEE 754</span></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 19 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-19" 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">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A3%D9%8A_%D8%A5%D9%8A_%D8%A5%D9%8A_%D8%A5%D9%8A_754" title="أي إي إي إي 754 – Arabic" lang="ar" hreflang="ar" data-title="أي إي إي إي 754" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Catalan" lang="ca" hreflang="ca" data-title="IEEE 754" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Czech" lang="cs" hreflang="cs" data-title="IEEE 754" 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/IEEE_754" title="IEEE 754 – German" lang="de" hreflang="de" data-title="IEEE 754" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Spanish" lang="es" hreflang="es" data-title="IEEE 754" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/IEEE_%DB%B7%DB%B5%DB%B4" title="IEEE ۷۵۴ – Persian" lang="fa" hreflang="fa" data-title="IEEE ۷۵۴" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – French" lang="fr" hreflang="fr" data-title="IEEE 754" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Korean" lang="ko" hreflang="ko" data-title="IEEE 754" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Italian" lang="it" hreflang="it" data-title="IEEE 754" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Lombard" lang="lmo" hreflang="lmo" data-title="IEEE 754" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/IEEE_lebeg%C5%91pontos_sz%C3%A1mform%C3%A1tum" title="IEEE lebegőpontos számformátum – Hungarian" lang="hu" hreflang="hu" data-title="IEEE lebegőpontos számformátum" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Dutch" lang="nl" hreflang="nl" data-title="IEEE 754" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Japanese" lang="ja" hreflang="ja" data-title="IEEE 754" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Polish" lang="pl" hreflang="pl" data-title="IEEE 754" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Russian" lang="ru" hreflang="ru" data-title="IEEE 754" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Serbian" lang="sr" hreflang="sr" data-title="IEEE 754" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Turkish" lang="tr" hreflang="tr" data-title="IEEE 754" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Ukrainian" lang="uk" hreflang="uk" data-title="IEEE 754" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/IEEE_754" title="IEEE 754 – Chinese" lang="zh" hreflang="zh" data-title="IEEE 754" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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/Q828287#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/IEEE_754" 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:IEEE_754" 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/IEEE_754"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=IEEE_754&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=IEEE_754&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/IEEE_754"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=IEEE_754&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=IEEE_754&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/IEEE_754" 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/IEEE_754" 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="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=IEEE_754&oldid=1254909127" 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=IEEE_754&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=IEEE_754&id=1254909127&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%2FIEEE_754"><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%2FIEEE_754"><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=IEEE_754&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=IEEE_754&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 class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:IEEE_754" hreflang="en"><span>Wikimedia Commons</span></a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Floating_Point" hreflang="en"><span>Wikibooks</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q828287" 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"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">IEEE standard for floating-point arithmetic</div> <p class="mw-empty-elt"> </p> <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 class="mw-selflink selflink">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 href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">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>IEEE Standard for Floating-Point Arithmetic</b> (<b>IEEE 754</b>) is a <a href="/wiki/Technical_standard" title="Technical standard">technical standard</a> for <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">floating-point arithmetic</a> originally established in 1985 by the <a href="/wiki/Institute_of_Electrical_and_Electronics_Engineers" title="Institute of Electrical and Electronics Engineers">Institute of Electrical and Electronics Engineers</a> (IEEE). The standard <a href="#Design_rationale">addressed many problems</a> found in the diverse floating-point implementations that made them difficult to use reliably and <a href="/wiki/Software_portability" title="Software portability">portably</a>. Many hardware <a href="/wiki/Floating-point_unit" title="Floating-point unit">floating-point units</a> use the IEEE 754 standard. </p><p>The standard defines: </p> <ul><li><i>arithmetic formats:</i> sets of <a href="/wiki/Binary_code" title="Binary code">binary</a> and <a href="/wiki/Decimal" title="Decimal">decimal</a> floating-point data, which consist of finite numbers (including <a href="/wiki/Signed_zero" title="Signed zero">signed zeros</a> and <a href="/wiki/Subnormal_number" title="Subnormal number">subnormal numbers</a>), <a href="/wiki/Infinity" title="Infinity">infinities</a>, and special "not a number" values (<a href="/wiki/NaN" title="NaN">NaNs</a>)</li> <li><i>interchange formats:</i> encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form</li> <li><i>rounding rules:</i> properties to be satisfied when rounding numbers during arithmetic and conversions</li> <li><i>operations:</i> arithmetic and other operations (such as <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a>) on arithmetic formats</li> <li><i>exception handling:</i> indications of exceptional conditions (such as <a href="/wiki/Division_by_zero" title="Division by zero">division by zero</a>, overflow, etc.)</li></ul> <p><a href="/wiki/IEEE_754-2008_revision" title="IEEE 754-2008 revision">IEEE 754-2008</a>, published in August 2008, includes nearly all of the original <a href="/wiki/IEEE_754-1985" title="IEEE 754-1985">IEEE 754-1985</a> standard, plus the <a href="/wiki/IEEE_854-1987" title="IEEE 854-1987">IEEE 854-1987 Standard for Radix-Independent Floating-Point Arithmetic</a>. <span class="anchor" id="2019"></span>The current version, IEEE 754-2019, was published in July 2019.<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> It is a minor revision of the previous version, incorporating mainly clarifications, defect fixes and new recommended operations. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The need for a floating-point standard arose from chaos in the business and scientific computing industry in the 1960s and 1970s. IBM used a <a href="/wiki/IBM_hexadecimal_floating-point" title="IBM hexadecimal floating-point">hexadecimal floating-point format</a> with <span class="cleanup-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">a longer significand and a shorter exponent</span><sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="Compared to which other hexadecimal format? (July 2024)">clarification needed</span></a></i>]</sup>. <a href="/wiki/Control_Data_Corporation" title="Control Data Corporation">CDC</a> and <a href="/wiki/Cray" title="Cray">Cray</a> computers used <a href="/wiki/Ones%27_complement" title="Ones' complement">ones' complement</a> representation, which admits a value of +0 and −0. CDC 60-bit computers did not have full 60-bit adders, so integer arithmetic was limited to 48 bits of precision from the floating-point unit. Exception processing from divide-by-zero was different on different computers. Moving data between systems and even repeating the same calculations on different systems was often difficult. </p><p>The first IEEE standard for floating-point arithmetic, <a href="/wiki/IEEE_754-1985" title="IEEE 754-1985">IEEE 754-1985</a>, was published in 1985. It covered only binary floating-point arithmetic. </p><p>A new version, <a href="/wiki/IEEE_754-2008_revision" title="IEEE 754-2008 revision">IEEE 754-2008</a>, was published in August 2008, following a seven-year revision process, chaired by Dan Zuras and edited by <a href="/wiki/Mike_Cowlishaw" title="Mike Cowlishaw">Mike Cowlishaw</a>. It replaced both IEEE 754-1985 (binary floating-point arithmetic) and <a href="/wiki/IEEE_854-1987" title="IEEE 854-1987">IEEE 854-1987 Standard for Radix-Independent Floating-Point Arithmetic</a>. The binary formats in the original standard are included in this new standard along with three new basic formats, one binary and two decimal. To conform to the current standard, an implementation must implement at least one of the basic formats as both an arithmetic format and an interchange format. </p><p>The international standard <b>ISO/IEC/IEEE 60559:2011</b> (with content identical to IEEE 754-2008) has been approved for adoption through <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">ISO</a>/<a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">IEC</a> <a href="/wiki/ISO/IEC_JTC_1" title="ISO/IEC JTC 1">JTC 1</a>/SC 25 under the ISO/IEEE PSDO Agreement<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><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> and published.<sup id="cite_ref-FOOTNOTEISO/IEC_JTC_1/SC_252011_4-0" class="reference"><a href="#cite_note-FOOTNOTEISO/IEC_JTC_1/SC_252011-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The current version, IEEE 754-2019 published in July 2019, is derived from and replaces IEEE 754-2008, following a revision process started in September 2015, chaired by David G. Hough and edited by Mike Cowlishaw. It incorporates mainly clarifications (e.g. <i>totalOrder</i>) and defect fixes (e.g. <i>minNum</i>), but also includes some new recommended operations (e.g. <i>augmentedAddition</i>).<sup id="cite_ref-IEEE754-errata_5-0" class="reference"><a href="#cite_note-IEEE754-errata-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The international standard <b>ISO/IEC 60559:2020</b> (with content identical to IEEE 754-2019) has been approved for adoption through ISO/IEC <a href="/wiki/ISO/IEC_JTC_1" title="ISO/IEC JTC 1">JTC 1</a>/SC 25 and published.<sup id="cite_ref-FOOTNOTEISO/IEC_JTC_1/SC_252020_7-0" class="reference"><a href="#cite_note-FOOTNOTEISO/IEC_JTC_1/SC_252020-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The next projected revision of the standard is in 2029.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formats">Formats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=2" title="Edit section: Formats"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An IEEE 754 <i>format</i> is a "set of representations of numerical values and symbols". A format may also include how the set is encoded.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>A floating-point format is specified by </p> <ul><li>a base (also called <i>radix</i>) <i>b</i>, which is either 2 (binary) or 10 (decimal) in IEEE 754;</li> <li>a precision <i>p</i>;</li> <li>an exponent range from <i>emin</i> to <i>emax</i>, with <i>emin</i> = 1 − <i>emax</i>, or equivalently <i>emin</i> = − (<i>emax</i> − 1), for all IEEE 754 formats.</li></ul> <p>A format comprises </p> <ul><li>Finite numbers, which can be described by three integers: <i>s</i> = a <i>sign</i> (zero or one), <i>c</i> = a <i><a href="/wiki/Significand" title="Significand">significand</a></i> (or <i>coefficient</i>) having no more than <i>p</i> digits when written in base <i>b</i> (i.e., an integer in the range through 0 to <i>b</i><sup><i>p</i></sup> − 1), and <i>q</i> = an <i>exponent</i> such that <i>emin</i> ≤ <i>q</i> + <i>p</i> − 1 ≤ <i>emax</i>. The numerical value of such a finite number is <span class="nowrap">(−1)<sup><i>s</i></sup> × <i>c</i> × <i>b</i><sup><i>q</i></sup></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> Moreover, there are two zero values, called <a href="/wiki/Signed_zero" title="Signed zero">signed zeros</a>: the sign bit specifies whether a zero is +0 (positive zero) or −0 (negative zero).</li> <li>Two infinities: +∞ and −∞.</li> <li>Two kinds of <a href="/wiki/NaN" title="NaN">NaN</a> (not-a-number): a quiet NaN (qNaN) and a signaling NaN (sNaN).</li></ul> <p>For example, if <i>b</i> = 10, <i>p</i> = 7, and <i>emax</i> = 96, then <i>emin</i> = −95, the significand satisfies 0 ≤ <i>c</i> ≤ <span class="nowrap"><span data-sort-value="7006999999900000000♠"></span>9<span style="margin-left:.25em;">999</span><span style="margin-left:.25em;">999</span></span>, and the exponent satisfies <span class="nowrap">−101 ≤ <i>q</i> ≤ 90</span>. Consequently, the smallest non-zero positive number that can be represented is 1×10<sup>−101</sup>, and the largest is 9999999×10<sup>90</sup> (9.999999×10<sup>96</sup>), so the full range of numbers is −9.999999×10<sup>96</sup> through 9.999999×10<sup>96</sup>. The numbers −<i>b</i><sup>1−<i>emax</i></sup> and <i>b</i><sup>1−<i>emax</i></sup> (here, −1×10<sup>−95</sup> and 1×10<sup>−95</sup>) are the smallest (in magnitude) <i>normal numbers</i>; non-zero numbers between these smallest numbers are called <a href="/wiki/Subnormal_number" title="Subnormal number">subnormal numbers</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Representation_and_encoding_in_memory">Representation and encoding in memory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=3" title="Edit section: Representation and encoding in memory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some numbers may have several possible floating-point representations. For instance, if <i>b</i> = 10, and <i>p</i> = 7, then −12.345 can be represented by −12345×10<sup>−3</sup>, −123450×10<sup>−4</sup>, and −1234500×10<sup>−5</sup>. However, for most operations, such as arithmetic operations, the result (value) does not depend on the representation of the inputs. </p><p>For the decimal formats, any representation is valid, and the set of these representations is called a <i>cohort</i>. When a result can have several representations, the standard specifies which member of the cohort is chosen. </p><p>For the binary formats, the representation is made unique by choosing the smallest representable exponent allowing the value to be represented exactly. Further, the exponent is not represented directly, but a <a href="/wiki/Exponent_bias" title="Exponent bias">bias</a> is added so that the smallest representable exponent is represented as 1, with 0 used for subnormal numbers. For numbers with an exponent in the normal range (the exponent field being neither all ones nor all zeros), the leading bit of the significand will always be 1. Consequently, a leading 1 can be implied rather than explicitly present in the memory encoding, and under the standard the explicitly represented part of the significand will lie between 0 and 1. This rule is called <i>leading bit convention</i>, <i>implicit bit convention</i>, or <i>hidden bit convention</i>. This rule allows the binary format to have an extra bit of precision. The leading bit convention cannot be used for the subnormal numbers as they have an exponent outside the normal exponent range and scale by the smallest represented exponent as used for the smallest normal numbers. </p><p>Due to the possibility of multiple encodings (at least in formats called <i>interchange formats</i>), a NaN may carry other information: a sign bit (which has no meaning, but may be used by some operations) and a <i>payload</i>, which is intended for diagnostic information indicating the source of the NaN (but the payload may have other uses, such as <i>NaN-boxing</i><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>). </p> <div class="mw-heading mw-heading3"><h3 id="Basic_and_interchange_formats">Basic and interchange formats</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=4" title="Edit section: Basic and interchange formats"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard defines five basic formats that are named for their numeric base and the number of bits used in their interchange encoding. There are three binary floating-point basic formats (encoded with 32, 64 or 128 bits) and two decimal floating-point basic formats (encoded with 64 or 128 bits). The <a href="/wiki/Binary32" class="mw-redirect" title="Binary32">binary32</a> and <a href="/wiki/Binary64" class="mw-redirect" title="Binary64">binary64</a> formats are the <i>single</i> and <i>double</i> formats of <a href="/wiki/IEEE_754-1985" title="IEEE 754-1985">IEEE 754-1985</a> respectively. A conforming implementation must fully implement at least one of the basic formats. </p><p>The standard also defines <i><a href="#Interchange_formats">interchange formats</a></i>, which generalize these basic formats.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> For the binary formats, the leading bit convention is required. The following table summarizes some of the possible interchange formats (including the basic formats). <style data-mw-deduplicate="TemplateStyles:r1226773818">.mw-parser-output .defaultleft{text-align:left}.mw-parser-output .defaultcenter{text-align:center}.mw-parser-output .defaultright{text-align:right}.mw-parser-output .col1left td:nth-child(1),.mw-parser-output .col2left td:nth-child(2),.mw-parser-output .col3left td:nth-child(3),.mw-parser-output .col4left td:nth-child(4),.mw-parser-output .col5left td:nth-child(5),.mw-parser-output .col6left td:nth-child(6),.mw-parser-output .col7left td:nth-child(7),.mw-parser-output .col8left td:nth-child(8),.mw-parser-output .col9left td:nth-child(9),.mw-parser-output .col10left td:nth-child(10),.mw-parser-output .col11left td:nth-child(11),.mw-parser-output .col12left td:nth-child(12),.mw-parser-output .col13left td:nth-child(13),.mw-parser-output .col14left td:nth-child(14),.mw-parser-output .col15left td:nth-child(15),.mw-parser-output .col16left td:nth-child(16),.mw-parser-output .col17left td:nth-child(17),.mw-parser-output .col18left td:nth-child(18),.mw-parser-output .col19left td:nth-child(19),.mw-parser-output .col20left td:nth-child(20),.mw-parser-output .col21left td:nth-child(21),.mw-parser-output .col22left td:nth-child(22),.mw-parser-output .col23left td:nth-child(23),.mw-parser-output .col24left td:nth-child(24),.mw-parser-output .col25left td:nth-child(25),.mw-parser-output .col26left td:nth-child(26),.mw-parser-output .col27left td:nth-child(27),.mw-parser-output .col28left td:nth-child(28),.mw-parser-output .col29left td:nth-child(29){text-align:left}.mw-parser-output .col1center td:nth-child(1),.mw-parser-output .col2center td:nth-child(2),.mw-parser-output .col3center td:nth-child(3),.mw-parser-output .col4center td:nth-child(4),.mw-parser-output .col5center td:nth-child(5),.mw-parser-output .col6center td:nth-child(6),.mw-parser-output .col7center td:nth-child(7),.mw-parser-output .col8center td:nth-child(8),.mw-parser-output .col9center td:nth-child(9),.mw-parser-output .col10center td:nth-child(10),.mw-parser-output .col11center td:nth-child(11),.mw-parser-output .col12center td:nth-child(12),.mw-parser-output .col13center td:nth-child(13),.mw-parser-output .col14center td:nth-child(14),.mw-parser-output .col15center td:nth-child(15),.mw-parser-output .col16center td:nth-child(16),.mw-parser-output .col17center td:nth-child(17),.mw-parser-output .col18center td:nth-child(18),.mw-parser-output .col19center td:nth-child(19),.mw-parser-output .col20center td:nth-child(20),.mw-parser-output .col21center td:nth-child(21),.mw-parser-output .col22center td:nth-child(22),.mw-parser-output .col23center td:nth-child(23),.mw-parser-output .col24center td:nth-child(24),.mw-parser-output .col25center td:nth-child(25),.mw-parser-output .col26center td:nth-child(26),.mw-parser-output .col27center td:nth-child(27),.mw-parser-output .col28center td:nth-child(28),.mw-parser-output .col29center td:nth-child(29){text-align:center}.mw-parser-output .col1right td:nth-child(1),.mw-parser-output .col2right td:nth-child(2),.mw-parser-output .col3right td:nth-child(3),.mw-parser-output .col4right td:nth-child(4),.mw-parser-output .col5right td:nth-child(5),.mw-parser-output .col6right td:nth-child(6),.mw-parser-output .col7right td:nth-child(7),.mw-parser-output .col8right td:nth-child(8),.mw-parser-output .col9right td:nth-child(9),.mw-parser-output .col10right td:nth-child(10),.mw-parser-output .col11right td:nth-child(11),.mw-parser-output .col12right td:nth-child(12),.mw-parser-output .col13right td:nth-child(13),.mw-parser-output .col14right td:nth-child(14),.mw-parser-output .col15right td:nth-child(15),.mw-parser-output .col16right td:nth-child(16),.mw-parser-output .col17right td:nth-child(17),.mw-parser-output .col18right td:nth-child(18),.mw-parser-output .col19right td:nth-child(19),.mw-parser-output .col20right td:nth-child(20),.mw-parser-output .col21right td:nth-child(21),.mw-parser-output .col22right td:nth-child(22),.mw-parser-output .col23right td:nth-child(23),.mw-parser-output .col24right td:nth-child(24),.mw-parser-output .col25right td:nth-child(25),.mw-parser-output .col26right td:nth-child(26),.mw-parser-output .col27right td:nth-child(27),.mw-parser-output .col28right td:nth-child(28),.mw-parser-output .col29right td:nth-child(29){text-align:right}</style> </p> <table class="wikitable defaultright col1left col2left col3center col12left"> <tbody><tr> <th scope="col" colspan="3"> </th> <th scope="col" colspan="2">Significand </th> <th scope="col" colspan="2">Exponent </th> <th scope="col" colspan="4">Properties<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> </th> <th scope="col" colspan="1"> </th></tr> <tr> <th scope="col">Name </th> <th scope="col">Common name </th> <th scope="col">Radix </th> <th scope="col">Digits<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> </th> <th scope="col" style="max-width:8em">Decimal digits<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </th> <th scope="col">Min </th> <th scope="col">Max </th> <th scope="col"><i>MAXVAL</i> </th> <th scope="col" style="min-width:8em">log<sub>10</sub> <i>MAXVAL</i> </th> <th scope="col" style="min-width:6em"><i>MINVAL</i>>0 (normal) </th> <th scope="col" style="min-width:6em"><i>MINVAL</i>>0 (subnorm) </th> <th scope="col">Notes </th></tr> <tr> <td><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">binary16</a> </td> <td>Half precision </td> <td>2 </td> <td>11 </td> <td>3.31 </td> <td>−14 </td> <td>15 </td> <td>65504 </td> <td>4.816 </td> <td>6.10·10<sup>−5</sup> </td> <td>5.96·10<sup>−8</sup> </td> <td>Interchange </td></tr> <tr> <td><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">binary32</a> </td> <td>Single precision </td> <td>2 </td> <td>24 </td> <td>7.22 </td> <td>−126 </td> <td>+127 </td> <td>3.40·10<sup>38</sup> </td> <td>38.532 </td> <td>1.18·10<sup>−38</sup> </td> <td>1.40·10<sup>−45</sup> </td> <td>Basic </td></tr> <tr> <td><a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">binary64</a> </td> <td>Double precision </td> <td>2 </td> <td>53 </td> <td>15.95 </td> <td>−1022 </td> <td>+1023 </td> <td>1.80·10<sup>308</sup> </td> <td>308.255 </td> <td>2.23·10<sup>−308</sup> </td> <td>4.94·10<sup>−324</sup> </td> <td>Basic </td></tr> <tr> <td><a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">binary128</a> </td> <td>Quadruple precision </td> <td>2 </td> <td>113 </td> <td>34.02 </td> <td>−16382 </td> <td>+16383 </td> <td>1.19·10<sup>4932</sup> </td> <td>4932.075 </td> <td>3.36·10<sup>−4932</sup> </td> <td>6.48·10<sup>−4966</sup> </td> <td>Basic </td></tr> <tr> <td><a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">binary256</a> </td> <td>Octuple precision </td> <td>2 </td> <td>237 </td> <td>71.34 </td> <td>−262142 </td> <td>+262143 </td> <td>1.61·10<sup>78913</sup> </td> <td>78913.207 </td> <td>2.48·10<sup>−78913</sup> </td> <td>2.25·10<sup>−78984</sup> </td> <td>Interchange </td></tr> <tr> <td><a href="/wiki/Decimal32_floating-point_format" title="Decimal32 floating-point format">decimal32</a> </td> <td> </td> <td>10 </td> <td>7 </td> <td>7 </td> <td>−95 </td> <td>+96 </td> <td>1.0·10<sup>97</sup> </td> <td>97 − 4.34·10<sup>−8</sup> </td> <td>1·10<sup>−95</sup> </td> <td>1·10<sup>−101</sup> </td> <td>Interchange </td></tr> <tr> <td><a href="/wiki/Decimal64_floating-point_format" title="Decimal64 floating-point format">decimal64</a> </td> <td> </td> <td>10 </td> <td>16 </td> <td>16 </td> <td>−383 </td> <td>+384 </td> <td>1.0·10<sup>385</sup> </td> <td>385 − 4.34·10<sup>−17</sup> </td> <td>1·10<sup>−383</sup> </td> <td>1·10<sup>−398</sup> </td> <td>Basic </td></tr> <tr> <td><a href="/wiki/Decimal128_floating-point_format" title="Decimal128 floating-point format">decimal128</a> </td> <td> </td> <td>10 </td> <td>34 </td> <td>34 </td> <td>−6143 </td> <td>+6144 </td> <td>1.0·10<sup>6145</sup> </td> <td>6145 − 4.34·10<sup>−35</sup> </td> <td>1·10<sup>−6143</sup> </td> <td>1·10<sup>−6176</sup> </td> <td>Basic </td></tr> </tbody></table> <p>In the table above, integer values are exact, whereas values in decimal notation (e.g. 1.0) are rounded values. The minimum exponents listed are for normal numbers; the special <a href="/wiki/Subnormal_number" title="Subnormal number">subnormal number</a> representation allows even smaller (in magnitude) numbers to be represented with some loss of precision. For example, the smallest positive number that can be represented in binary64 is 2<sup>−1074</sup>; contributions to the −1074 figure include the <i>emin</i> value −1022 and all but one of the 53 significand bits (2<sup>−1022 − (53 − 1)</sup> = 2<sup>−1074</sup>). </p><p>Decimal digits is the precision of the format expressed in terms of an equivalent number of decimal digits. It is computed as <i>digits</i> × log<sub>10</sub> <i>base</i>. E.g. binary128 has approximately the same precision as a 34 digit decimal number. </p><p>log<sub>10</sub> <i>MAXVAL</i> is a measure of the range of the encoding. Its integer part is the largest exponent shown on the output of a value in scientific notation with one leading digit in the significand before the decimal point (e.g. 1.698·10<sup>38</sup> is near the largest value in binary32, 9.999999·10<sup>96</sup> is the largest value in decimal32). </p><p>The binary32 (single) and binary64 (double) formats are two of the most common formats used today. The figure below shows the absolute precision for both formats over a range of values. This figure can be used to select an appropriate format given the expected value of a number and the required precision. </p> <figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:IEEE754.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/IEEE754.svg/550px-IEEE754.svg.png" decoding="async" width="550" height="411" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/IEEE754.svg/825px-IEEE754.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/IEEE754.svg/1100px-IEEE754.svg.png 2x" data-file-width="698" data-file-height="521" /></a><figcaption>Precision of binary32 and binary64 in the range 10<sup>−12</sup> to 10<sup>12</sup></figcaption></figure> <p>An example of a layout for <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">32-bit floating point</a> is </p> <figure class="mw-default-size mw-halign-none" typeof="mw:File"><a href="/wiki/File:Float_example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Float_example.svg/590px-Float_example.svg.png" decoding="async" width="590" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Float_example.svg/885px-Float_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Float_example.svg/1180px-Float_example.svg.png 2x" data-file-width="590" data-file-height="75" /></a><figcaption></figcaption></figure> <p>and the <a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">64 bit layout</a> is similar. </p> <div class="mw-heading mw-heading3"><h3 id="Extended_and_extendable_precision_formats">Extended and extendable precision formats</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=5" title="Edit section: Extended and extendable precision formats"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard specifies optional <a href="/wiki/Extended_precision" title="Extended precision">extended</a> and extendable precision formats, which provide greater precision than the basic formats.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> An extended precision format extends a basic format by using more precision and more exponent range. An extendable precision format allows the user to specify the precision and exponent range. An implementation may use whatever internal representation it chooses for such formats; all that needs to be defined are its parameters (<i>b</i>, <i>p</i>, and <i>emax</i>). These parameters uniquely describe the set of finite numbers (combinations of sign, significand, and exponent for the given radix) that it can represent. </p><p>The standard recommends that language standards provide a method of specifying <i>p</i> and <i>emax</i> for each supported base <i>b</i>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The standard recommends that language standards and implementations support an extended format which has a greater precision than the largest basic format supported for each radix <i>b</i>.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> For an extended format with a precision between two basic formats the exponent range must be as great as that of the next wider basic format. So for instance a 64-bit extended precision binary number must have an 'emax' of at least 16383. The <a href="/wiki/X87" title="X87">x87</a> <a href="/wiki/Extended_precision#x86_extended_precision_format" title="Extended precision">80-bit extended format</a> meets this requirement. </p><p>The original <a href="/wiki/IEEE_754-1985" title="IEEE 754-1985">IEEE 754-1985</a> standard also had the concept of <i>extended formats</i>, but without any mandatory relation between <i>emin</i> and <i>emax</i>. For example, the <a href="/wiki/Motorola_68881" title="Motorola 68881">Motorola 68881</a> 80-bit format,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> where <i>emin</i> = − <i>emax</i>, was a conforming extended format, but it became non-conforming in the 2008 revision. </p> <div class="mw-heading mw-heading3"><h3 id="Interchange_formats">Interchange formats</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=6" title="Edit section: Interchange formats"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Interchange formats are intended for the exchange of floating-point data using a bit string of fixed length for a given format. </p> <div class="mw-heading mw-heading4"><h4 id="Binary">Binary</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=7" title="Edit section: Binary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the exchange of binary floating-point numbers, interchange formats of length 16 bits, 32 bits, 64 bits, and any multiple of 32 bits ≥ 128<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>e<span class="cite-bracket">]</span></a></sup> are defined. The 16-bit format is intended for the exchange or storage of small numbers (e.g., for graphics). </p><p>The encoding scheme for these binary interchange formats is the same as that of IEEE 754-1985: a sign bit, followed by <i>w</i> exponent bits that describe the exponent offset by a <i><a href="/wiki/Exponent_bias" title="Exponent bias">bias</a></i>, and <i>p</i> − 1 bits that describe the significand. The width of the exponent field for a <i>k</i>-bit format is computed as <i>w</i> = round(4 log<sub>2</sub>(<i>k</i>)) − 13. The existing 64- and 128-bit formats follow this rule, but the 16- and 32-bit formats have more exponent bits (5 and 8 respectively) than this formula would provide (3 and 7 respectively). </p><p>As with IEEE 754-1985, the biased-exponent field is filled with all 1 bits to indicate either infinity (trailing significand field = 0) or a NaN (trailing significand field ≠ 0). For NaNs, quiet NaNs and signaling NaNs are distinguished by using the most significant bit of the trailing significand field exclusively,<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>f<span class="cite-bracket">]</span></a></sup> and the payload is carried in the remaining bits. </p> <div class="mw-heading mw-heading4"><h4 id="Decimal">Decimal</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=8" title="Edit section: Decimal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For the exchange of decimal floating-point numbers, interchange formats of any multiple of 32 bits are defined. As with binary interchange, the encoding scheme for the decimal interchange formats encodes the sign, exponent, and significand. Two different bit-level encodings are defined, and interchange is complicated by the fact that some external indicator of the encoding in use may be required. </p><p>The two options allow the significand to be encoded as a compressed sequence of decimal digits using <a href="/wiki/Densely_packed_decimal" title="Densely packed decimal">densely packed decimal</a> or, alternatively, as a <a href="/wiki/Binary_integer_decimal" title="Binary integer decimal">binary integer</a>. The former is more convenient for direct hardware implementation of the standard, while the latter is more suited to software emulation on a binary computer. In either case, the set of numbers (combinations of sign, significand, and exponent) that may be encoded is identical, and <a href="/wiki/Floating_point#Special_values" class="mw-redirect" title="Floating point">special values</a> (±zero with the minimum exponent, ±infinity, quiet NaNs, and signaling NaNs) have identical encodings. </p> <div class="mw-heading mw-heading2"><h2 id="Rounding_rules">Rounding rules</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=9" title="Edit section: Rounding rules"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard defines five rounding rules. The first two rules round to a nearest value; the others are called <i><a href="/wiki/Directed_rounding" class="mw-redirect" title="Directed rounding">directed roundings</a></i>: </p> <div class="mw-heading mw-heading3"><h3 id="Roundings_to_nearest">Roundings to nearest</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=10" title="Edit section: Roundings to nearest"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Rounding#Rounding_half_to_even" title="Rounding">Round to nearest, ties to even</a></b> – rounds to the nearest value; if the number falls midway, it is rounded to the nearest value with an even least significant digit.</li> <li><b><a href="/wiki/Rounding#Rounding_half_away_from_zero" title="Rounding">Round to nearest, ties away from zero</a></b> (or <b>ties to away</b>)  – rounds to the nearest value; if the number falls midway, it is rounded to the nearest value above (for positive numbers) or below (for negative numbers).</li></ul> <p>At the extremes, a value with a magnitude strictly less than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=b^{\text{emax}}\left(b-{\tfrac {1}{2}}b^{1-p}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>emax</mtext> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=b^{\text{emax}}\left(b-{\tfrac {1}{2}}b^{1-p}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a091178b3adde0b4a2d9c8f6be39fdee183764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.499ex; height:3.509ex;" alt="{\displaystyle k=b^{\text{emax}}\left(b-{\tfrac {1}{2}}b^{1-p}\right)}"></span> will be rounded to the minimum or maximum finite number (depending on the value's sign). Any numbers with exactly this magnitude are considered ties; this choice of tie may be conceptualized as the midpoint between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm b^{\text{emax}}(b-b^{1-p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>emax</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm b^{\text{emax}}(b-b^{1-p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/493ff088d26dd3759134d6d751ef39fcac11a1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.631ex; height:3.176ex;" alt="{\displaystyle \pm b^{\text{emax}}(b-b^{1-p})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm b^{{\text{emax}}+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>emax</mtext> </mrow> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm b^{{\text{emax}}+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cea6d7f20d377134d1261c5c9a37719896a9cd92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.927ex; height:2.676ex;" alt="{\displaystyle \pm b^{{\text{emax}}+1}}"></span>, which, were the exponent not limited, would be the next representable floating-point numbers larger in magnitude. Numbers with a magnitude strictly larger than <span class="texhtml mvar" style="font-style:italic;">k</span> are rounded to the corresponding infinity.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>"Round to nearest, ties to even" is the default for binary floating point and the recommended default for decimal. "Round to nearest, ties to away" is only required for decimal implementations.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Directed_roundings">Directed roundings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=11" title="Edit section: Directed roundings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Round toward 0</b> – directed rounding towards zero (also known as <i>truncation</i>).</li> <li><b>Round toward +∞</b> – directed rounding towards positive infinity (also known as <i>rounding up</i> or <i>ceiling</i>).</li> <li><b>Round toward −∞</b> – directed rounding towards negative infinity (also known as <i>rounding down</i> or <i>floor</i>).</li></ul> <table class="wikitable"> <caption>Example of rounding to integers using the IEEE 754 rules </caption> <tbody><tr> <th rowspan="2">Mode</th> <th colspan="4">Example value </th></tr> <tr> <th>+11.5 </th> <th>+12.5 </th> <th>−11.5 </th> <th>−12.5 </th></tr> <tr> <td>to nearest, ties to even </td> <td>+12.0 </td> <td>+12.0 </td> <td>−12.0 </td> <td>−12.0 </td></tr> <tr> <td>to nearest, ties away from zero </td> <td>+12.0 </td> <td>+13.0 </td> <td>−12.0 </td> <td>−13.0 </td></tr> <tr> <td>toward 0 </td> <td>+11.0 </td> <td>+12.0 </td> <td>−11.0 </td> <td>−12.0 </td></tr> <tr> <td>toward +∞ </td> <td>+12.0 </td> <td>+13.0 </td> <td>−11.0 </td> <td>−12.0 </td></tr> <tr> <td>toward −∞ </td> <td>+11.0 </td> <td>+12.0 </td> <td>−12.0 </td> <td>−13.0 </td></tr></tbody></table> <p>Unless specified otherwise, the floating-point result of an operation is determined by applying the rounding function on the infinitely precise (mathematical) result. Such an operation is said to be <i>correctly rounded</i>. This requirement is called <i>correct rounding</i>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Required_operations">Required operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=12" title="Edit section: Required operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Required operations for a supported arithmetic format (including the basic formats) include: </p> <ul><li>Conversions to and from integer<sup id="cite_ref-IEEE_754_2008_loc=§5.3.1_27-0" class="reference"><a href="#cite_note-IEEE_754_2008_loc=§5.3.1-27"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-IEEE_754_2008_loc=§5.4.1_28-0" class="reference"><a href="#cite_note-IEEE_754_2008_loc=§5.4.1-28"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>Previous and next consecutive values<sup id="cite_ref-IEEE_754_2008_loc=§5.3.1_27-1" class="reference"><a href="#cite_note-IEEE_754_2008_loc=§5.3.1-27"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li> <li>Arithmetic operations (add, subtract, multiply, divide, square root, <a href="/wiki/Multiply%E2%80%93accumulate_operation" title="Multiply–accumulate operation">fused multiply–add</a>, remainder, minimum, maximum)<sup id="cite_ref-IEEE_754_2008_loc=§5.3.1_27-2" class="reference"><a href="#cite_note-IEEE_754_2008_loc=§5.3.1-27"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-IEEE_754_2008_loc=§5.4.1_28-1" class="reference"><a href="#cite_note-IEEE_754_2008_loc=§5.4.1-28"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li> <li>Conversions (between formats, to and from strings, etc.)<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li> <li>Scaling and (for decimal) quantizing<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup></li> <li>Copying and manipulating the sign (abs, negate, etc.)<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li> <li>Comparisons and total ordering<sup id="cite_ref-total-ordering_34-0" class="reference"><a href="#cite_note-total-ordering-34"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li> <li>Classification of numbers (subnormal, finite, etc.) and testing for NaNs<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></li> <li>Testing and setting status flags<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Comparison_predicates">Comparison predicates</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=13" title="Edit section: Comparison predicates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard provides comparison predicates to compare one floating-point datum to another in the supported arithmetic format.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Any comparison with a NaN is treated as unordered. −0 and +0 compare as equal. </p> <div class="mw-heading mw-heading3"><h3 id="Total-ordering_predicate">Total-ordering predicate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=14" title="Edit section: Total-ordering predicate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard provides a predicate <i>totalOrder</i>, which defines a <a href="/wiki/Total_order" title="Total order">total ordering</a> on canonical members of the supported arithmetic format.<sup id="cite_ref-IEEE_754_2019_loc=§5.10_39-0" class="reference"><a href="#cite_note-IEEE_754_2019_loc=§5.10-39"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> The predicate agrees with the comparison predicates (see section <a href="#Comparison_predicates">§ Comparison predicates</a>) when one floating-point number is less than the other. The main differences are:<sup id="cite_ref-rust_total_cmp_40-0" class="reference"><a href="#cite_note-rust_total_cmp-40"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <ul><li>NaN is sortable. <ul><li>NaN is treated as if it had a larger absolute value than Infinity (or any other floating-point numbers). (−NaN < −Infinity; +Infinity < +NaN.)</li> <li>qNaN and sNaN are treated as if qNaN had a larger absolute value than sNaN. (−qNaN < −sNaN; +sNaN < +qNaN.)</li> <li>NaN is then sorted according to the payload. In IEEE 754-2008, a NaN with a lesser payload is treated as having a lesser absolute value. In IEEE 754-2019, any implementation-defined ordering is acceptable.</li></ul></li> <li>Negative zero is treated as smaller than positive zero.</li> <li>If both sides of the comparison refer to the same floating-point datum, the one with the lesser exponent is treated as having a lesser absolute value.<sup id="cite_ref-IEEE_754_2019_loc=§5.10_39-1" class="reference"><a href="#cite_note-IEEE_754_2019_loc=§5.10-39"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></li></ul> <p>The <i>totalOrder</i> predicate does not impose a total ordering on all encodings in a format. In particular, it does not distinguish among different encodings of the same floating-point representation, as when one or both encodings are non-canonical.<sup id="cite_ref-IEEE_754_2019_loc=§5.10_39-2" class="reference"><a href="#cite_note-IEEE_754_2019_loc=§5.10-39"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> IEEE 754-2019 incorporates clarifications of <i>totalOrder</i>. </p><p>For the binary interchange formats whose encoding follows the IEEE 754-2008 recommendation on <a href="/wiki/NaN#Encoding" title="NaN">placement of the NaN signaling bit</a>, the comparison is identical to one that <a href="/wiki/Type_punning" title="Type punning">type puns</a> the floating-point numbers to a sign–magnitude integer (assuming a payload ordering consistent with this comparison), an old trick for FP comparison without an FPU.<sup id="cite_ref-Herf_2001_41-0" class="reference"><a href="#cite_note-Herf_2001-41"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Exception_handling">Exception handling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=15" title="Edit section: Exception handling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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">See also: <a href="/wiki/Floating-point_arithmetic#Exception_handling" title="Floating-point arithmetic">Floating-point arithmetic § Exception handling</a></div> <p>The standard defines five exceptions, each of which returns a default value and has a corresponding status flag that is raised when the exception occurs.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>g<span class="cite-bracket">]</span></a></sup> No other exception handling is required, but additional non-default alternatives are recommended (see <a href="#Alternate_exception_handling">§ Alternate exception handling</a>). </p><p>The five possible exceptions are </p> <ul><li>Invalid operation: mathematically undefined, e.g., the square root of a negative number. By default, returns qNaN.</li> <li>Division by zero: an operation on finite operands gives an exact infinite result, e.g., 1/0 or log(0). By default, returns ±infinity.</li> <li>Overflow: a finite result is too large to be represented accurately (i.e., its exponent with an unbounded exponent range would be larger than <i>emax</i>). By default, returns ±infinity for the round-to-nearest modes (and follows the rounding rules for the directed rounding modes).</li> <li>Underflow: a result is very small (outside the normal range). By default, returns a number less than or equal to the minimum positive normal number in magnitude (following the rounding rules); a <a href="/wiki/Subnormal_number" title="Subnormal number">subnormal number</a> always implies an underflow exception, but by default, if it is exact, no flag is raised.</li> <li>Inexact: the exact (i.e., unrounded) result is not representable exactly. By default, returns the correctly rounded result.</li></ul> <p>These are the same five exceptions as were defined in IEEE 754-1985, but the <i>division by zero</i> exception has been extended to operations other than the division. </p><p>Some decimal floating-point implementations define additional exceptions,<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> which are not part of IEEE 754: </p> <ul><li>Clamped: a result's exponent is too large for the destination format. By default, trailing zeros will be added to the coefficient to reduce the exponent to the largest usable value. If this is not possible (because this would cause the number of digits needed to be more than the destination format) then an overflow exception occurs.</li> <li>Rounded: a result's coefficient requires more digits than the destination format provides. An inexact exception is signaled if any non-zero digits are discarded.</li></ul> <p>Additionally, operations like quantize when either operand is infinite, or when the result does not fit the destination format, will also signal invalid operation exception.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Special_values">Special values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=16" title="Edit section: Special values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Signed_zero">Signed zero</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=17" title="Edit section: Signed zero"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Signed_zero" title="Signed zero">Signed zero</a></div> <p>In the IEEE 754 standard, zero is signed, meaning that there exist both a "positive zero" (+0) and a "negative zero" (−0). In most <a href="/wiki/Run-time_environment" class="mw-redirect" title="Run-time environment">run-time environments</a>, positive zero is usually printed as "<code>0</code>" and the negative zero as "<code>-0</code>". The two values behave as equal in numerical comparisons, but some operations return different results for +0 and −0. For instance, 1/(−0) returns negative infinity, while 1/(+0) returns positive infinity (so that the identity <span class="texhtml">1/(1/±∞) = ±∞</span> is maintained). Other common <a href="/wiki/Discontinuous_function" class="mw-redirect" title="Discontinuous function">functions with a discontinuity</a> at <i>x</i>=0 which might treat +0 and −0 differently include <a href="/wiki/Gamma_function" title="Gamma function"><span class="texhtml">Γ</span></a>(<i>x</i>) and the <a href="/wiki/Square_root#Principal_square_root_of_a_complex_number" title="Square root">principal square root</a> of <span class="texhtml"><i>y</i> + <i>xi</i></span> for any negative number <i>y</i>. As with any approximation scheme, operations involving "negative zero" can occasionally cause confusion. For example, in IEEE 754, <span class="texhtml"><i>x</i> = <i>y</i></span> does not always imply <span class="texhtml">1/<i>x</i> = 1/<i>y</i></span>, as <span class="nowrap">0 = −0</span> but <span class="nowrap">1/0 ≠ 1/(−0)</span>.<sup id="cite_ref-FOOTNOTEGoldberg1991_46-0" class="reference"><a href="#cite_note-FOOTNOTEGoldberg1991-46"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Moreover, for the reciprocal square root, <span class="texhtml">rSqrt(±0)</span> is <span class="texhtml">±∞</span> while the mathematical function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1{\big /}{\sqrt {x{\vphantom {t}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mi>t</mi> </mphantom> </mpadded> </mrow> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1{\big /}{\sqrt {x{\vphantom {t}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c66dfb8a01ec56a4466acddaaec60a62381f1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.768ex; width:6.539ex; height:3.176ex;" alt="{\displaystyle 1{\big /}{\sqrt {x{\vphantom {t}}}}}"></span> over the real numbers does not have any negative value. </p> <div class="mw-heading mw-heading3"><h3 id="Subnormal_numbers">Subnormal numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=18" title="Edit section: Subnormal numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Subnormal_numbers" class="mw-redirect" title="Subnormal numbers">Subnormal numbers</a></div> <p>Subnormal values fill the <a href="/wiki/Arithmetic_underflow" title="Arithmetic underflow">underflow</a> gap with values where the absolute distance between them is the same as for adjacent values just outside the underflow gap. This is an improvement over the older practice to just have zero in the underflow gap, and where underflowing results were replaced by zero (flush to zero).<sup id="cite_ref-Muller_2010_47-0" class="reference"><a href="#cite_note-Muller_2010-47"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </p><p>Modern floating-point hardware usually handles subnormal values (as well as normal values), and does not require software emulation for subnormals. </p> <div class="mw-heading mw-heading3"><h3 id="Infinities">Infinities</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=19" title="Edit section: Infinities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information on the concept of infinite: <a href="/wiki/Infinity" title="Infinity">Infinity</a></div> <p>The infinities of the <a href="/wiki/Extended_real_number_line" title="Extended real number line">extended real number line</a> can be represented in IEEE floating-point datatypes, just like ordinary floating-point values like 1, 1.5, etc. They are not error values in any way, though they are often (depends on the rounding) used as replacement values when there is an overflow. Upon a divide-by-zero exception, a positive or negative infinity is returned as an exact result. An infinity can also be introduced as a numeral (like C's "INFINITY" macro, or "<span class="texhtml">∞</span>" if the programming language allows that syntax). </p><p>IEEE 754 requires infinities to be handled in a reasonable way, such as </p> <ul><li><span class="texhtml">(+∞) + (+7) = (+∞)</span></li> <li><span class="texhtml">(+∞) × (−2) = (−∞)</span></li> <li><span class="texhtml">(+∞) × 0 =</span> NaN – there is no meaningful thing to do</li></ul> <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=IEEE_754&action=edit&section=20" title="Edit section: NaNs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/NaN" title="NaN">NaN</a></div> <p>IEEE 754 specifies a special value called "Not a Number" (NaN) to be returned as the result of certain "invalid" operations, such as 0/0, <span class="texhtml">∞×0</span>, or sqrt(−1). In general, NaNs will be propagated, i.e. most operations involving a NaN will result in a NaN, although functions that would give some defined result for any given floating-point value will do so for NaNs as well, e.g. NaN ^ 0 = 1. There are two kinds of NaNs: the default <i>quiet</i> NaNs and, optionally, <i>signaling</i> NaNs. A signaling NaN in any arithmetic operation (including numerical comparisons) will cause an "invalid operation" <a href="#Exception_handling">exception</a> to be signaled. </p><p>The representation of NaNs specified by the standard has some unspecified bits that could be used to encode the type or source of error; but there is no standard for that encoding. In theory, signaling NaNs could be used by a <a href="/wiki/Runtime_system" title="Runtime system">runtime system</a> to flag uninitialized variables, or extend the floating-point numbers with other special values without slowing down the computations with ordinary values, although such extensions are not common. </p> <div class="mw-heading mw-heading2"><h2 id="Design_rationale">Design rationale</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=21" title="Edit section: Design rationale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:William_Kahan.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/William_Kahan.jpg/220px-William_Kahan.jpg" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/William_Kahan.jpg/330px-William_Kahan.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b1/William_Kahan.jpg 2x" data-file-width="367" data-file-height="279" /></a><figcaption><a href="/wiki/William_Kahan" title="William Kahan">William Kahan</a>. A primary architect of the Intel <a href="/wiki/80x87" class="mw-redirect" title="80x87">80x87</a> floating-point coprocessor and IEEE 754 floating-point standard.</figcaption></figure> <p>It is a common misconception that the more esoteric features of the IEEE 754 standard discussed here, such as extended formats, NaN, infinities, subnormals etc., are only of interest to <a href="/wiki/Numerical_analyst" class="mw-redirect" title="Numerical analyst">numerical analysts</a>, or for advanced numerical applications. In fact the opposite is true: these features are designed to give safe robust defaults for numerically unsophisticated programmers, in addition to supporting sophisticated numerical libraries by experts. The key designer of IEEE 754, <a href="/wiki/William_Kahan" title="William Kahan">William Kahan</a> notes that it is incorrect to "... [deem] features of IEEE Standard 754 for Binary Floating-Point Arithmetic that ...[are] not appreciated to be features usable by none but numerical experts. The facts are quite the opposite. In 1977 those features were designed into the Intel 8087 to serve the widest possible market... Error-analysis tells us how to design floating-point arithmetic, like IEEE Standard 754, moderately tolerant of well-meaning ignorance among programmers".<sup id="cite_ref-Kahan_2001_JavaHurt_48-0" class="reference"><a href="#cite_note-Kahan_2001_JavaHurt-48"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> </p> <ul><li>The special values such as infinity and NaN ensure that the floating-point arithmetic is algebraically complete: every floating-point operation produces a well-defined result and will not—by default—throw a machine interrupt or trap. Moreover, the choices of special values returned in exceptional cases were designed to give the correct answer in many cases. For instance, under IEEE 754 arithmetic, continued fractions such as R(z) := 7 − 3/[z − 2 − 1/(z − 7 + 10/[z − 2 − 2/(z − 3)])] will give the correct answer on all inputs, as the potential divide by zero, e.g. for <span class="nowrap">z = 3</span>, is correctly handled by giving +infinity, and so such exceptions can be safely ignored.<sup id="cite_ref-Kahan_1981_WhyIEEE_49-0" class="reference"><a href="#cite_note-Kahan_1981_WhyIEEE-49"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> As noted by Kahan, the unhandled trap consecutive to a floating-point to 16-bit integer conversion overflow that caused the <a href="/wiki/Ariane_flight_V88" title="Ariane flight V88">loss of an Ariane 5</a> rocket would not have happened under the default IEEE 754 floating-point policy.<sup id="cite_ref-Kahan_2001_JavaHurt_48-1" class="reference"><a href="#cite_note-Kahan_2001_JavaHurt-48"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup></li> <li>Subnormal numbers ensure that for <i>finite</i> floating-point numbers x and y, x − y = 0 if and only if x = y, as expected, but which did not hold under earlier floating-point representations.<sup id="cite_ref-Severance_1998_50-0" class="reference"><a href="#cite_note-Severance_1998-50"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup></li> <li>On the design rationale of the x87 <a href="/wiki/Extended_precision" title="Extended precision">80-bit format</a>, Kahan notes: "This Extended format is designed to be used, with negligible loss of speed, for all but the simplest arithmetic with float and double operands. For example, it should be used for scratch variables in loops that implement recurrences like polynomial evaluation, scalar products, partial and continued fractions. It often averts premature Over/Underflow or severe local cancellation that can spoil simple algorithms".<sup id="cite_ref-Kahan_1996_Baleful_51-0" class="reference"><a href="#cite_note-Kahan_1996_Baleful-51"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> Computing intermediate results in an extended format with high precision and extended exponent has precedents in the historical practice of scientific <a href="/wiki/Significant_figures#Arithmetic" title="Significant figures">calculation</a> and in the design of <a href="/wiki/Scientific_calculator" title="Scientific calculator">scientific calculators</a> e.g. <a href="/wiki/Hewlett-Packard" title="Hewlett-Packard">Hewlett-Packard</a>'s <a href="/wiki/Financial_calculator" title="Financial calculator">financial calculators</a> performed arithmetic and financial functions to three more significant decimals than they stored or displayed.<sup id="cite_ref-Kahan_1996_Baleful_51-1" class="reference"><a href="#cite_note-Kahan_1996_Baleful-51"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> The implementation of extended precision enabled standard elementary function libraries to be readily developed that normally gave double precision results within one <a href="/wiki/Unit_in_the_last_place" title="Unit in the last place">unit in the last place</a> (ULP) at high speed.</li> <li>Correct rounding of values to the nearest representable value avoids systematic biases in calculations and slows the growth of errors. Rounding ties to even removes the statistical bias that can occur in adding similar figures.</li> <li>Directed rounding was intended as an aid with checking error bounds, for instance in <a href="/wiki/Interval_arithmetic" title="Interval arithmetic">interval arithmetic</a>. It is also used in the implementation of some functions.</li> <li>The mathematical basis of the operations, in particular correct rounding, allows one to prove mathematical properties and design floating-point algorithms such as <a href="/wiki/2Sum" title="2Sum">2Sum, Fast2Sum</a> and <a href="/wiki/Kahan_summation_algorithm" title="Kahan summation algorithm">Kahan summation algorithm</a>, e.g. to improve accuracy or implement multiple-precision arithmetic subroutines relatively easily.</li></ul> <p>A property of the single- and double-precision formats is that their encoding allows one to easily sort them without using floating-point hardware, as if the bits represented <a href="/wiki/Sign-magnitude" class="mw-redirect" title="Sign-magnitude">sign-magnitude</a> integers, although it is unclear whether this was a design consideration (it seems noteworthy that the earlier <a href="/wiki/IBM_hexadecimal_floating-point" title="IBM hexadecimal floating-point">IBM hexadecimal floating-point</a> representation also had this property for normalized numbers). With the prevalent <a href="/wiki/Two%27s-complement" class="mw-redirect" title="Two's-complement">two's-complement</a> representation, <a href="/wiki/Type_punning" title="Type punning">interpreting</a> the bits as signed integers sorts the positives correctly, but with the negatives reversed; as one possible correction for that, with an <a href="/wiki/Exclusive_or" title="Exclusive or">xor</a> to flip the sign bit for positive values and all bits for negative values, all the values become sortable as unsigned integers (with <span class="nowrap">−0 < +0</span>).<sup id="cite_ref-Herf_2001_41-1" class="reference"><a href="#cite_note-Herf_2001-41"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Recommendations">Recommendations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=22" title="Edit section: Recommendations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Alternate_exception_handling">Alternate exception handling</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=23" title="Edit section: Alternate exception handling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard recommends optional exception handling in various forms, including presubstitution of user-defined default values, and traps (exceptions that change the flow of control in some way) and other exception handling models that interrupt the flow, such as try/catch. The traps and other exception mechanisms remain optional, as they were in IEEE 754-1985. </p> <div class="mw-heading mw-heading3"><h3 id="Recommended_operations">Recommended operations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=24" title="Edit section: Recommended operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Clause 9 in the standard recommends additional mathematical operations<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> that language standards should define.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> None are required in order to conform to the standard. </p><p>The following are recommended arithmetic operations, which must round correctly:<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> </p> <ul><li><a href="/wiki/Exp(x)" class="mw-redirect" title="Exp(x)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841c0d168e64191c45a45e54c7e447defd17ec6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.256ex; height:2.343ex;" alt="{\displaystyle e^{x}}"></span></a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5514b506cd3af97484bca3e166965ac4cef27a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.335ex; height:2.343ex;" alt="{\displaystyle 2^{x}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f9a6c2de625bd03ffd20e1fa89dad13da52eaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.497ex; height:2.343ex;" alt="{\displaystyle 10^{x}}"></span></li> <li><a href="/wiki/Exp(x)%E2%88%921" class="mw-redirect" title="Exp(x)−1"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9098adf157cfb3697609aaa10d9e3b4ec875d1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.259ex; height:2.509ex;" alt="{\displaystyle e^{x}-1}"></span></a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{x}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{x}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b48590c4c77174ac7363888b9c434668358813f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.338ex; height:2.509ex;" alt="{\displaystyle 2^{x}-1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{x}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{x}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/347b67eb9e0e08c17f6528c275da798b0c2bb7b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.5ex; height:2.509ex;" alt="{\displaystyle 10^{x}-1}"></span></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed172b0f5195382a3500c713941f945ad4db3898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.656ex; height:2.176ex;" alt="{\displaystyle \ln x}"></span></a>, <a href="/wiki/Binary_logarithm" title="Binary logarithm"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3cea09debff9219b7ac089e9b0ff0c4f411fa60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.743ex; height:2.676ex;" alt="{\displaystyle \log _{2}x}"></span></a>, <a href="/wiki/Common_logarithm" title="Common logarithm"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f174fa0b73f4f350aff3b5d44cb831dcabfcea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.565ex; height:2.676ex;" alt="{\displaystyle \log _{10}x}"></span></a></li> <li><a href="/wiki/Ln(1%2Bx)" class="mw-redirect" title="Ln(1+x)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(1+x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln(1+x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c16ccd97dc2eaef5ce6d8b7bc593a0e2dbd3dca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.081ex; height:2.843ex;" alt="{\displaystyle \ln(1+x)}"></span></a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{2}(1+x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{2}(1+x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb42831d75f7febea420c70a0a20591f71726265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.168ex; height:2.843ex;" alt="{\displaystyle \log _{2}(1+x)}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \log _{10}(1+x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \log _{10}(1+x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ff41b55edb9a580b11c5f559f37605edd29f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.99ex; height:2.843ex;" alt="{\displaystyle \log _{10}(1+x)}"></span></li> <li><a href="/wiki/Hypot" class="mw-redirect" title="Hypot"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {x^{2}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f4319b4270574ab98a1379e89a4848ac308417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.763ex; height:3.509ex;" alt="{\textstyle {\sqrt {x^{2}+y^{2}}}}"></span></a></li> <li><a href="/wiki/Reciprocal_square_root" class="mw-redirect" title="Reciprocal square root"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1{\big /}{\sqrt {x{\vphantom {t}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mpadded width="0"> <mphantom> <mi>t</mi> </mphantom> </mpadded> </mrow> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1{\big /}{\sqrt {x{\vphantom {t}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c66dfb8a01ec56a4466acddaaec60a62381f1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-right: -0.768ex; width:6.539ex; height:3.176ex;" alt="{\displaystyle 1{\big /}{\sqrt {x{\vphantom {t}}}}}"></span></a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+x)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+x)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04845cc4a998213c15d61b36512d27222fca3527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.36ex; height:2.843ex;" alt="{\displaystyle (1+x)^{n}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≥</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\geq -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c75aa35c23e4ad0bc0f0026153e47981b97e0776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x\geq -1}"></span> (named <i>compound</i> and used to compute an <a href="/wiki/Exponential_growth" title="Exponential growth">exponential growth</a>, whose rate cannot be less than −1)<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Nth_root" title="Nth root"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\frac {1}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\frac {1}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55c86c68cf18b5f2f38c440bee62562ff4f304f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.199ex; height:3.343ex;" alt="{\displaystyle x^{\frac {1}{n}}}"></span></a></li> <li><a href="/wiki/Exponentiation#Integer_exponents" title="Exponentiation"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/150d38e238991bc4d0689ffc9d2a852547d2658d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}"></span></a>, <a href="/wiki/Exponentiation" title="Exponentiation"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8561c712e86598255e8434a70affa18ffd7e0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.379ex; height:2.343ex;" alt="{\displaystyle x^{y}}"></span></a></li> <li><a href="/wiki/Sin_(trigonometry)" class="mw-redirect" title="Sin (trigonometry)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09b4b55580d6a821a07ad9fe35be88976917b10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.572ex; height:2.176ex;" alt="{\displaystyle \sin x}"></span></a>, <a href="/wiki/Cos_(trigonometry)" class="mw-redirect" title="Cos (trigonometry)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/184ba70c3a71df25a25c09f34cd7f8175a9b5280" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.828ex; height:1.676ex;" alt="{\displaystyle \cos x}"></span></a>, <a href="/wiki/Tan_(trigonometry)" class="mw-redirect" title="Tan (trigonometry)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tan x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3983a722002d77dd3d0babab871c50488aef9f4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.076ex; height:2.009ex;" alt="{\displaystyle \tan x}"></span></a></li> <li><a href="/wiki/Arcsin_(trigonometry)" class="mw-redirect" title="Arcsin (trigonometry)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arcsin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8941574c0f224adeedae0f72bd48098116cc2068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.679ex; height:2.176ex;" alt="{\displaystyle \arcsin x}"></span></a>, <a href="/wiki/Arccos_(trigonometry)" class="mw-redirect" title="Arccos (trigonometry)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arccos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arccos x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f5e84a5218f798ae5d01f85cec659fbb6e3df2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.934ex; height:1.676ex;" alt="{\displaystyle \arccos x}"></span></a>, <a href="/wiki/Arctan_(trigonometry)" class="mw-redirect" title="Arctan (trigonometry)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arctan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \arctan x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7315ab90c047f16ec0972c15e14988ed737c1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.183ex; height:2.009ex;" alt="{\displaystyle \arctan x}"></span></a>, <a href="/wiki/Atan2" title="Atan2"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2} (y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2} (y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf144c1f1dccdc6c0949ece514a77f3790530f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.013ex; height:2.843ex;" alt="{\displaystyle \operatorname {atan2} (y,x)}"></span></a></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinPi} x=\sin \pi x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sinPi</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sinPi} x=\sin \pi x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e52e7504e4f22e035f42626f484df8d5040e517b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.805ex; height:2.176ex;" alt="{\displaystyle \operatorname {sinPi} x=\sin \pi x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {cosPi} x=\cos \pi x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cosPi</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {cosPi} x=\cos \pi x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af83f85108f2e1546d5ea8c5e9caf756c8cbca1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.316ex; height:2.176ex;" alt="{\displaystyle \operatorname {cosPi} x=\cos \pi x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tanPi} x=\tan \pi x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanPi</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mi>tan</mi> <mo>⁡</mo> <mi>π</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tanPi} x=\tan \pi x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e736fa4f81b0ca7ecea4d4612804c8cfb9180d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.813ex; height:2.176ex;" alt="{\displaystyle \operatorname {tanPi} x=\tan \pi x}"></span> (see also: <a href="/wiki/Multiples_of_%CF%80" class="mw-redirect" title="Multiples of π">Multiples of π</a>)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {asinPi} x={\tfrac {1}{\pi }}\arcsin x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>asinPi</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mstyle> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {asinPi} x={\tfrac {1}{\pi }}\arcsin x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b55858d3faf8f28be967cacb9409a42f53f488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.907ex; height:3.343ex;" alt="{\displaystyle \operatorname {asinPi} x={\tfrac {1}{\pi }}\arcsin x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {acosPi} x={\tfrac {1}{\pi }}\arccos x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>acosPi</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mstyle> </mrow> <mi>arccos</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {acosPi} x={\tfrac {1}{\pi }}\arccos x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4c9eedd17787c5a8fcaf4ff5c5d6d5b6fc7b0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.418ex; height:3.343ex;" alt="{\displaystyle \operatorname {acosPi} x={\tfrac {1}{\pi }}\arccos x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atanPi} x={\tfrac {1}{\pi }}\arctan x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>atanPi</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mstyle> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {atanPi} x={\tfrac {1}{\pi }}\arctan x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb44a38d6d3ff23169c89338e6ade6e474eab5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.915ex; height:3.343ex;" alt="{\displaystyle \operatorname {atanPi} x={\tfrac {1}{\pi }}\arctan x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {atan2Pi} (y,x)={\tfrac {1}{\pi }}\operatorname {atan2} (y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>atan2Pi</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mstyle> </mrow> <mi>atan2</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {atan2Pi} (y,x)={\tfrac {1}{\pi }}\operatorname {atan2} (y,x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3603629d9d0849bb84e6db4b1707f26b1fb7dfe5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.519ex; height:3.343ex;" alt="{\displaystyle \operatorname {atan2Pi} (y,x)={\tfrac {1}{\pi }}\operatorname {atan2} (y,x)}"></span> (see also: <a href="/wiki/Multiples_of_%CF%80" class="mw-redirect" title="Multiples of π">Multiples of π</a>)</li> <li><a href="/wiki/Sinh_(mathematical_function)" class="mw-redirect" title="Sinh (mathematical function)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sinh x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sinh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sinh x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa855e59efeb97e8146142f9aa3414fca17f24a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.865ex; height:2.176ex;" alt="{\displaystyle \sinh x}"></span></a>, <a href="/wiki/Cosh_(mathematical_function)" class="mw-redirect" title="Cosh (mathematical function)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cosh x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cosh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cosh x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa23e796f0d4eae2439367502f387edfa6c4b559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.12ex; height:2.176ex;" alt="{\displaystyle \cosh x}"></span></a>, <a href="/wiki/Tanh_(mathematical_function)" class="mw-redirect" title="Tanh (mathematical function)"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tanh x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0be1dd47ba059a32307fa7dbcf98c0877931e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.369ex; height:2.176ex;" alt="{\displaystyle \tanh x}"></span></a></li> <li><a href="/wiki/Arsinh" class="mw-redirect" title="Arsinh"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arsinh} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arsinh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arsinh} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae4dd27fb2ea192a32d2dc919170dd6028efec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.939ex; height:2.176ex;" alt="{\displaystyle \operatorname {arsinh} x}"></span></a>, <a href="/wiki/Arcosh" class="mw-redirect" title="Arcosh"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {arcosh} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>arcosh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {arcosh} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f3d1be50bda0a00cb4354fce22fd95826dd8b73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.194ex; height:2.176ex;" alt="{\displaystyle \operatorname {arcosh} x}"></span></a>, <a href="/wiki/Artanh" class="mw-redirect" title="Artanh"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {artanh} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>artanh</mi> <mo>⁡</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {artanh} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e5332b77be1339db02748afcf70ce1814d5819" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.443ex; height:2.176ex;" alt="{\displaystyle \operatorname {artanh} x}"></span></a></li></ul> <p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {asinPi} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>asinPi</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {asinPi} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f15d434913cb9712434b45855f73f4d3103729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.248ex; height:2.176ex;" alt="{\displaystyle \operatorname {asinPi} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {acosPi} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>acosPi</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {acosPi} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3122a1aed0c91cb76e891a1ef44cb60423114a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.503ex; height:2.176ex;" alt="{\displaystyle \operatorname {acosPi} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tanPi} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tanPi</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tanPi} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faf274c5eef8b0a9d90b0f7458bbcea7b02c432e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.589ex; height:2.176ex;" alt="{\displaystyle \operatorname {tanPi} }"></span> functions were not part of the IEEE 754-2008 standard because they were deemed less necessary.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {asinPi} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>asinPi</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {asinPi} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f15d434913cb9712434b45855f73f4d3103729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.248ex; height:2.176ex;" alt="{\displaystyle \operatorname {asinPi} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {acosPi} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>acosPi</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {acosPi} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3122a1aed0c91cb76e891a1ef44cb60423114a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.503ex; height:2.176ex;" alt="{\displaystyle \operatorname {acosPi} }"></span> were mentioned, but this was regarded as an error.<sup id="cite_ref-IEEE754-errata_5-1" class="reference"><a href="#cite_note-IEEE754-errata-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> All three were added in the 2019 revision. </p><p>The recommended operations also include setting and accessing dynamic mode rounding direction,<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> and implementation-defined vector reduction operations such as sum, scaled product, and <a href="/wiki/Dot_product" title="Dot product">dot product</a>, whose accuracy is unspecified by the standard.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p><p><span class="anchor" id="Augmented_arithmetic_operation"></span> As of 2019<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=IEEE_754&action=edit">[update]</a></sup>, <i>augmented arithmetic operations</i><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> for the binary formats are also recommended. These operations, specified for addition, subtraction and multiplication, produce a pair of values consisting of a result correctly rounded to nearest in the format and the error term, which is representable exactly in the format. At the time of publication of the standard, no hardware implementations are known, but very similar operations were already implemented in software using well-known algorithms. The history and motivation for their standardization are explained in a background document.<sup id="cite_ref-Riedy_2018_60-0" class="reference"><a href="#cite_note-Riedy_2018-60"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Revision_2019_61-0" class="reference"><a href="#cite_note-Revision_2019-61"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> </p><p><span class="anchor" id="Minimum_and_maximum_operation"></span> As of 2019, the formerly required <i>minNum</i>, <i>maxNum</i>, <i>minNumMag</i>, and <i>maxNumMag</i> in IEEE 754-2008 are now <a href="/wiki/Deprecation" title="Deprecation">deprecated</a> due to their <a href="/wiki/Associative_property" title="Associative property">non-associativity</a>. Instead, two sets of new minimum and maximum operations are recommended.<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> The first set contains <i>minimum</i>, <i>minimumNumber</i>, <i>maximum</i> and <i>maximumNumber</i>. The second set contains <i>minimumMagnitude</i>, <i>minimumMagnitudeNumber</i>, <i>maximumMagnitude</i> and <i>maximumMagnitudeNumber</i>. The history and motivation for this change are explained in a background document.<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Expression_evaluation">Expression evaluation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=25" title="Edit section: Expression evaluation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard recommends how language standards should specify the semantics of sequences of operations, and points out the subtleties of literal meanings and optimizations that change the value of a result. By contrast, the previous <a href="/wiki/IEEE_754-1985" title="IEEE 754-1985">1985</a> version of the standard left aspects of the language interface unspecified, which led to inconsistent behavior between compilers, or different optimization levels in an <a href="/wiki/Optimizing_compiler" title="Optimizing compiler">optimizing compiler</a>. </p><p>Programming languages should allow a user to specify a minimum precision for intermediate calculations of expressions for each radix. This is referred to as <i>preferredWidth</i> in the standard, and it should be possible to set this on a per-block basis. Intermediate calculations within expressions should be calculated, and any temporaries saved, using the maximum of the width of the operands and the preferred width if set. Thus, for instance, a compiler targeting <a href="/wiki/X87" title="X87">x87</a> floating-point hardware should have a means of specifying that intermediate calculations must use the <a href="/wiki/Extended_precision#IEEE_754_extended_precision_formats" title="Extended precision">double-extended format</a>. The stored value of a variable must always be used when evaluating subsequent expressions, rather than any precursor from before rounding and assigning to the variable. </p> <div class="mw-heading mw-heading3"><h3 id="Reproducibility">Reproducibility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=26" title="Edit section: Reproducibility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The IEEE 754-1985 version of the standard allowed many variations in implementations (such as the encoding of some values and the detection of certain exceptions). IEEE 754-2008 has reduced these allowances, but a few variations still remain (especially for binary formats). The reproducibility clause recommends that language standards should provide a means to write reproducible programs (i.e., programs that will produce the same result in all implementations of a language) and describes what needs to be done to achieve reproducible results. </p> <div class="mw-heading mw-heading2"><h2 id="Character_representation">Character representation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=27" title="Edit section: Character representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Floating-point_arithmetic#Binary-to-decimal_conversion_with_minimal_number_of_digits" title="Floating-point arithmetic">Floating-point arithmetic § Binary-to-decimal conversion with minimal number of digits</a></div> <p>The standard requires operations to convert between basic formats and <i>external character sequence</i> formats.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> Conversions to and from a decimal character format are required for all formats. Conversion to an external character sequence must be such that conversion back using round to nearest, ties to even will recover the original number. There is no requirement to preserve the payload of a quiet NaN or signaling NaN, and conversion from the external character sequence may turn a signaling NaN into a quiet NaN. </p><p>The original binary value will be preserved by converting to decimal and back again using:<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> </p> <ul><li>5 decimal digits for binary16,</li> <li>9 decimal digits for binary32,</li> <li>17 decimal digits for binary64,</li> <li>36 decimal digits for binary128.</li></ul> <p>For other binary formats, the required number of decimal digits is<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>h<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+\lceil p\log _{10}(2)\rceil ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mo fence="false" stretchy="false">⌈</mo> <mi>p</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⌉</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+\lceil p\log _{10}(2)\rceil ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84d5104e56caed39e03d64e41bd6bae9a7909b14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.091ex; height:2.843ex;" alt="{\displaystyle 1+\lceil p\log _{10}(2)\rceil ,}"></span></dd></dl> <p>where <i>p</i> is the number of significant bits in the binary format, e.g. 237 bits for binary256. </p><p>When using a decimal floating-point format, the decimal representation will be preserved using: </p> <ul><li>7 decimal digits for decimal32,</li> <li>16 decimal digits for decimal64,</li> <li>34 decimal digits for decimal128.</li></ul> <p>Algorithms, with code, for correctly rounded conversion from binary to decimal and decimal to binary are discussed by Gay,<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> and for testing –  by Paxson and Kahan.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Hexadecimal_literals">Hexadecimal literals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=28" title="Edit section: Hexadecimal literals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The standard recommends providing conversions to and from <i>external hexadecimal-significand character sequences</i>, based on <a href="/wiki/C99" title="C99">C99</a>'s hexadecimal floating point literals. Such a literal consists of an optional sign (<code>+</code> or <code>-</code>), the indicator "0x", a hexadecimal number with or without a period, an exponent indicator "p", and a decimal exponent with optional sign. The syntax is not case-sensitive.<sup id="cite_ref-69" class="reference"><a href="#cite_note-69"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> The decimal exponent scales by powers of 2. For example, <code>0x0.1p0</code> is 1/16 and <code>0x0.1p-4</code> is 1/256.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> </p> <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=IEEE_754&action=edit&section=29" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">bfloat16 floating-point format</a></li> <li><a href="/wiki/Binade" title="Binade">Binade</a></li> <li><a href="/wiki/Coprocessor" title="Coprocessor">Coprocessor</a></li> <li><a href="/wiki/C99#IEEE_754_floating-point_support" title="C99">C99</a> for code examples demonstrating access and use of IEEE 754 features</li> <li><a href="/wiki/Floating-point_arithmetic#IEEE_754:_floating_point_in_modern_computers" title="Floating-point arithmetic">Floating-point arithmetic</a>, for history, design rationale and example usage of IEEE 754 features</li> <li><a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">Fixed-point arithmetic</a>, for an alternative approach at computation with rational numbers (especially beneficial when the exponent range is known, fixed, or bound at compile time)</li> <li><a href="/wiki/IBM_System_z9" title="IBM System z9">IBM System z9</a>, the first CPU to implement IEEE 754-2008 decimal arithmetic (using hardware microcode)</li> <li><a href="/wiki/IBM_z10" title="IBM z10">IBM z10</a>, <a href="/wiki/IBM_z196" title="IBM z196">IBM z196</a>, <a href="/wiki/IBM_zEC12_(microprocessor)" class="mw-redirect" title="IBM zEC12 (microprocessor)">IBM zEC12</a>, and <a href="/wiki/IBM_z13_(microprocessor)" class="mw-redirect" title="IBM z13 (microprocessor)">IBM z13</a>, CPUs that implement IEEE 754-2008 decimal arithmetic fully in hardware</li> <li><a href="/wiki/ISO/IEC_10967" title="ISO/IEC 10967">ISO/IEC 10967</a>, language-independent arithmetic (LIA)</li> <li><a href="/wiki/Minifloat" title="Minifloat">Minifloat</a>, low-precision binary floating-point formats following IEEE 754 principles</li> <li><a href="/wiki/POWER6" title="POWER6">POWER6</a>, <a href="/wiki/POWER7" title="POWER7">POWER7</a>, and <a href="/wiki/POWER8" title="POWER8">POWER8</a> CPUs that implement IEEE 754-2008 decimal arithmetic fully in hardware</li> <li><a href="/wiki/Strictfp" title="Strictfp">strictfp</a>, an obsolete keyword in the <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java programming language</a> that previously restricted arithmetic to IEEE 754 single and double precision to ensure reproducibility across common hardware platforms (as of Java 17, this behavior is required)</li> <li><a href="/wiki/Table-maker%27s_dilemma" class="mw-redirect" title="Table-maker's dilemma">Table-maker's dilemma</a> for more about the correct rounding of functions</li> <li><a href="/wiki/Standard_Apple_Numerics_Environment" title="Standard Apple Numerics Environment">Standard Apple Numerics Environment</a></li> <li><a href="/wiki/Tapered_floating_point" title="Tapered floating point">Tapered floating point</a></li> <li><a href="/wiki/Unum_(number_format)#Posit_(Type_III_Unum)" title="Unum (number format)">Posit</a>, an alternative number format</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=30" title="Edit section: Notes"><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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">For example, if the base is 10, the sign is 1 (indicating negative), the significand is 12345, and the exponent is −3, then the value of the number is <span class="nowrap">(−1)<sup>1</sup> × 12345 × 10<sup>−3</sup></span> = <span class="nowrap">−1 × 12345 × 0.001</span> = −12.345.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Approximative values. For exact values see each format's individual Wikipedia entry</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Number of digits in the radix used, including any implicit digit, but not counting the sign bit.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Corresponding number of decimal digits, see text for more details.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Contrary to decimal, there is no binary interchange format of 96-bit length. Such a format is still allowed as a non-interchange format, though.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">The standard recommends 0 for signaling NaNs, 1 for quiet NaNs, so that a signaling NaNs can be quieted by changing only this bit to 1, while the reverse could yield the encoding of an infinity.</span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text">No flag is raised in certain cases of underflow.</span> </li> <li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text">As an implementation limit, correct rounding is only guaranteed for the number of decimal digits required plus 3 for the largest supported binary format. For instance, if binary32 is the largest supported binary format, then a conversion from a decimal external sequence with 12 decimal digits is guaranteed to be correctly rounded when converted to binary32; but conversion of a sequence of 13 decimal digits is not; however, the standard recommends that implementations impose no such limit.</span> </li> </ol></div></div> <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=IEEE_754&action=edit&section=31" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><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"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a></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"><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="CITEREFHaasz" class="citation web cs1">Haasz, Jodi. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171027190846/http://grouper.ieee.org/groups/754/email/msg04167.html">"FW: ISO/IEC/IEEE 60559 (IEEE Std 754-2008)"</a>. <i><a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a></i>. Archived from <a rel="nofollow" class="external text" href="http://grouper.ieee.org/groups/754/email/msg04167.html">the original</a> on 2017-10-27<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-04-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IEEE&rft.atitle=FW%3A+ISO%2FIEC%2FIEEE+60559+%28IEEE+Std+754-2008%29&rft.aulast=Haasz&rft.aufirst=Jodi&rft_id=http%3A%2F%2Fgrouper.ieee.org%2Fgroups%2F754%2Femail%2Fmsg04167.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" 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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://grouper.ieee.org/groups/802/minutes/jul2008/opening_reports/psdo1.pdf">"ISO/IEEE Partner Standards Development Organization (PSDO) Cooperation Agreement"</a> <span class="cs1-format">(PDF)</span>. ISO. 2007-12-19<span class="reference-accessdate">. Retrieved <span class="nowrap">2021-12-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ISO%2FIEEE+Partner+Standards+Development+Organization+%28PSDO%29+Cooperation+Agreement&rft.pub=ISO&rft.date=2007-12-19&rft_id=https%3A%2F%2Fgrouper.ieee.org%2Fgroups%2F802%2Fminutes%2Fjul2008%2Fopening_reports%2Fpsdo1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEISO/IEC_JTC_1/SC_252011-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEISO/IEC_JTC_1/SC_252011_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFISO/IEC_JTC_1/SC_252011">ISO/IEC JTC 1/SC 25 2011</a>.</span> </li> <li id="cite_note-IEEE754-errata-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-IEEE754-errata_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IEEE754-errata_5-1"><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="CITEREFCowlishaw2013" class="citation web cs1">Cowlishaw, Mike (2013-11-13). <a rel="nofollow" class="external text" href="https://speleotrove.com/misc/IEEE754-errata.html">"IEEE 754-2008 errata"</a>. <i>speleotrove.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-01-24</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=speleotrove.com&rft.atitle=IEEE+754-2008+errata&rft.date=2013-11-13&rft.aulast=Cowlishaw&rft.aufirst=Mike&rft_id=https%3A%2F%2Fspeleotrove.com%2Fmisc%2FIEEE754-errata.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</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://754r.ucbtest.org/">"ANSI/IEEE Std 754-2019"</a>. <i>ucbtest.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ucbtest.org&rft.atitle=ANSI%2FIEEE+Std+754-2019&rft_id=https%3A%2F%2F754r.ucbtest.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEISO/IEC_JTC_1/SC_252020-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEISO/IEC_JTC_1/SC_252020_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFISO/IEC_JTC_1/SC_252020">ISO/IEC JTC 1/SC 25 2020</a>.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</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://grouper.ieee.org/groups/msc/ANSI_IEEE-Std-754-2019/background/future.txt">"Issues for the next revision of 754"</a>. <i><a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-08-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IEEE&rft.atitle=Issues+for+the+next+revision+of+754&rft_id=https%3A%2F%2Fgrouper.ieee.org%2Fgroups%2Fmsc%2FANSI_IEEE-Std-754-2019%2Fbackground%2Ffuture.txt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §2.1.27.</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://udn.realityripple.com/docs/Mozilla/Projects/SpiderMonkey/Internals">"SpiderMonkey Internals"</a>. <i>udn.realityripple.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=udn.realityripple.com&rft.atitle=SpiderMonkey+Internals&rft_id=https%3A%2F%2Fudn.realityripple.com%2Fdocs%2FMozilla%2FProjects%2FSpiderMonkey%2FInternals&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" 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 id="CITEREFKlemens2014" class="citation book cs1">Klemens, Ben (September 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ASuiBAAAQBAJ"><i>21st Century C: C Tips from the New School</i></a>. O'Reilly Media, Incorporated. p. 160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781491904442" title="Special:BookSources/9781491904442"><bdi>9781491904442</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-03-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=21st+Century+C%3A+C+Tips+from+the+New+School&rft.pages=160&rft.pub=O%27Reilly+Media%2C+Incorporated&rft.date=2014-09&rft.isbn=9781491904442&rft.aulast=Klemens&rft.aufirst=Ben&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DASuiBAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" 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://github.com/zuiderkwast/nanbox">"zuiderkwast/nanbox: NaN-boxing in C"</a>. <i><a href="/wiki/GitHub" title="GitHub">GitHub</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-03-11</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=zuiderkwast%2Fnanbox%3A+NaN-boxing+in+C&rft_id=https%3A%2F%2Fgithub.com%2Fzuiderkwast%2Fnanbox&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" 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"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §3.6.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §3.7.</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"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §3.7 states: "Language standards should define mechanisms supporting extendable precision for each supported radix."</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §3.7 states: "Language standards or implementations should support an extended precision format that extends the widest basic format that is supported in that radix."</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://www.nxp.com/docs/en/reference-manual/M68000PRM.pdf"><i>Motorola MC68000 Family</i></a> <span class="cs1-format">(PDF)</span>. Programmer's Reference Manual. NXP Semiconductors. 1992. pp. 1–16, 1–18, 1–23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Motorola+MC68000+Family&rft.series=Programmer%27s+Reference+Manual&rft.pages=1-16%2C+1-18%2C+1-23&rft.pub=NXP+Semiconductors&rft.date=1992&rft_id=https%3A%2F%2Fwww.nxp.com%2Fdocs%2Fen%2Freference-manual%2FM68000PRM.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §4.3.1. "In the following two rounding-direction attributes, an infinitely precise result with magnitude at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{\text{emax}}(b-{\tfrac {1}{2}}b^{1-p})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>emax</mtext> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{\text{emax}}(b-{\tfrac {1}{2}}b^{1-p})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9201d7d647ccfbc1d7f841a443ca1118156659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.481ex; height:3.509ex;" alt="{\displaystyle b^{\text{emax}}(b-{\tfrac {1}{2}}b^{1-p})}"></span> shall round to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span> with no change in sign."</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §4.3.3</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a>, §2.1</span> </li> <li id="cite_note-IEEE_754_2008_loc=§5.3.1-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-IEEE_754_2008_loc=§5.3.1_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IEEE_754_2008_loc=§5.3.1_27-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-IEEE_754_2008_loc=§5.3.1_27-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.3.1</span> </li> <li id="cite_note-IEEE_754_2008_loc=§5.4.1-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-IEEE_754_2008_loc=§5.4.1_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IEEE_754_2008_loc=§5.4.1_28-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.4.1</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.4.2</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.4.3</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.3.2</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.3.3</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.5.1</span> </li> <li id="cite_note-total-ordering-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-total-ordering_34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.10</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.11</span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.7.2</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.7.4</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a>, §5.11</span> </li> <li id="cite_note-IEEE_754_2019_loc=§5.10-39"><span class="mw-cite-backlink">^ <a href="#cite_ref-IEEE_754_2019_loc=§5.10_39-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IEEE_754_2019_loc=§5.10_39-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-IEEE_754_2019_loc=§5.10_39-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a>, §5.10</span> </li> <li id="cite_note-rust_total_cmp-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-rust_total_cmp_40-0">^</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://github.com/rust-lang/rust/pull/72568">"Implement total_cmp for f32, f64 by golddranks · Pull Request #72568 · rust-lang/rust"</a>. <i>GitHub</i>.</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=Implement+total_cmp+for+f32%2C+f64+by+golddranks+%C2%B7+Pull+Request+%2372568+%C2%B7+rust-lang%2Frust&rft_id=https%3A%2F%2Fgithub.com%2Frust-lang%2Frust%2Fpull%2F72568&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span> – contains relevant quotations from IEEE 754-2008 and -2019. Contains a type-pun implementation and explanation.</span> </li> <li id="cite_note-Herf_2001-41"><span class="mw-cite-backlink">^ <a href="#cite_ref-Herf_2001_41-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Herf_2001_41-1"><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="CITEREFHerf2001" class="citation web cs1">Herf, Michael (December 2001). <a rel="nofollow" class="external text" href="http://stereopsis.com/radix.html">"radix tricks"</a>. <i>stereopsis: graphics</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=stereopsis%3A+graphics&rft.atitle=radix+tricks&rft.date=2001-12&rft.aulast=Herf&rft.aufirst=Michael&rft_id=http%3A%2F%2Fstereopsis.com%2Fradix.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</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.python.org/library/decimal.html#signals">"9.4. decimal — Decimal fixed point and floating point arithmetic — Python 3.6.5 documentation"</a>. <i>docs.python.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-04-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=docs.python.org&rft.atitle=9.4.+decimal+%E2%80%94+Decimal+fixed+point+and+floating+point+arithmetic+%E2%80%94+Python+3.6.5+documentation&rft_id=https%3A%2F%2Fdocs.python.org%2Flibrary%2Fdecimal.html%23signals&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</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="http://speleotrove.com/decimal/daexcep.html">"Decimal Arithmetic - Exceptional conditions"</a>. <i>speleotrove.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2018-04-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=speleotrove.com&rft.atitle=Decimal+Arithmetic+-+Exceptional+conditions&rft_id=http%3A%2F%2Fspeleotrove.com%2Fdecimal%2Fdaexcep.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §7.2(h)</span> </li> <li id="cite_note-FOOTNOTEGoldberg1991-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGoldberg1991_46-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGoldberg1991">Goldberg 1991</a>.</span> </li> <li id="cite_note-Muller_2010-47"><span class="mw-cite-backlink"><b><a href="#cite_ref-Muller_2010_47-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="muller_et_al_pg_16" class="citation book cs1">Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; <a href="/wiki/Nathalie_Revol" title="Nathalie Revol">Revol, Nathalie</a>; Stehlé, Damien; Torres, Serge (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=baFvrIOPvncC&pg=PA16"><i>Handbook of Floating-Point Arithmetic</i></a> (1 ed.). <a href="/wiki/Birkh%C3%A4user" title="Birkhäuser">Birkhäuser</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-8176-4705-6">10.1007/978-0-8176-4705-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4704-9" title="Special:BookSources/978-0-8176-4704-9"><bdi>978-0-8176-4704-9</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2009939668">2009939668</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Floating-Point+Arithmetic&rft.edition=1&rft.pub=Birkh%C3%A4user&rft.date=2010&rft_id=info%3Alccn%2F2009939668&rft_id=info%3Adoi%2F10.1007%2F978-0-8176-4705-6&rft.isbn=978-0-8176-4704-9&rft.aulast=Muller&rft.aufirst=Jean-Michel&rft.au=Brisebarre%2C+Nicolas&rft.au=de+Dinechin%2C+Florent&rft.au=Jeannerod%2C+Claude-Pierre&rft.au=Lef%C3%A8vre%2C+Vincent&rft.au=Melquiond%2C+Guillaume&rft.au=Revol%2C+Nathalie&rft.au=Stehl%C3%A9%2C+Damien&rft.au=Torres%2C+Serge&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbaFvrIOPvncC%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-Kahan_2001_JavaHurt-48"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kahan_2001_JavaHurt_48-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kahan_2001_JavaHurt_48-1"><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="CITEREFKahanDarcy2001" class="citation web cs1"><a href="/wiki/William_Morton_Kahan" class="mw-redirect" title="William Morton Kahan">Kahan, William Morton</a>; Darcy, Joseph (2001) [1998-03-01]. <a rel="nofollow" class="external text" href="http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf">"How Java's floating-point hurts everyone everywhere"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20000816043653/http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2000-08-16<span class="reference-accessdate">. Retrieved <span class="nowrap">2003-09-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=How+Java%27s+floating-point+hurts+everyone+everywhere&rft.date=2001&rft.aulast=Kahan&rft.aufirst=William+Morton&rft.au=Darcy%2C+Joseph&rft_id=http%3A%2F%2Fwww.cs.berkeley.edu%2F~wkahan%2FJAVAhurt.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-Kahan_1981_WhyIEEE-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kahan_1981_WhyIEEE_49-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKahan1981" class="citation web cs1"><a href="/wiki/William_Morton_Kahan" class="mw-redirect" title="William Morton Kahan">Kahan, William Morton</a> (1981-02-12). <a rel="nofollow" class="external text" href="http://www.cs.berkeley.edu/~wkahan/ieee754status/why-ieee.pdf">"Why do we need a floating-point arithmetic standard?"</a> <span class="cs1-format">(PDF)</span>. p. 26. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041204070746/http://www.cs.berkeley.edu/~wkahan/ieee754status/why-ieee.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2004-12-04.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Why+do+we+need+a+floating-point+arithmetic+standard%3F&rft.pages=26&rft.date=1981-02-12&rft.aulast=Kahan&rft.aufirst=William+Morton&rft_id=http%3A%2F%2Fwww.cs.berkeley.edu%2F~wkahan%2Fieee754status%2Fwhy-ieee.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-Severance_1998-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-Severance_1998_50-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeverance1998" class="citation web cs1"><a href="/wiki/Charles_Severance_(computer_scientist)" title="Charles Severance (computer scientist)">Severance, Charles</a> (1998-02-20). <a rel="nofollow" class="external text" href="http://www.eecs.berkeley.edu/~wkahan/ieee754status/754story.html">"An Interview with the Old Man of Floating-Point"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=An+Interview+with+the+Old+Man+of+Floating-Point&rft.date=1998-02-20&rft.aulast=Severance&rft.aufirst=Charles&rft_id=http%3A%2F%2Fwww.eecs.berkeley.edu%2F~wkahan%2Fieee754status%2F754story.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-Kahan_1996_Baleful-51"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kahan_1996_Baleful_51-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kahan_1996_Baleful_51-1"><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="CITEREFKahan1996" class="citation web cs1"><a href="/wiki/William_Morton_Kahan" class="mw-redirect" title="William Morton Kahan">Kahan, William Morton</a> (1996-06-11). <a rel="nofollow" class="external text" href="http://www.cs.berkeley.edu/~wkahan/ieee754status/baleful.pdf">"The Baleful Effect of Computer Benchmarks upon Applied Mathematics, Physics and Chemistry"</a> <span class="cs1-format">(PDF)</span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131013011212/http://www.cs.berkeley.edu/~wkahan/ieee754status/baleful.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2013-10-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Baleful+Effect+of+Computer+Benchmarks+upon+Applied+Mathematics%2C+Physics+and+Chemistry&rft.date=1996-06-11&rft.aulast=Kahan&rft.aufirst=William+Morton&rft_id=http%3A%2F%2Fwww.cs.berkeley.edu%2F~wkahan%2Fieee754status%2Fbaleful.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a>, §9.2</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, Clause 9</span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a>, §9.2.</span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</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://grouper.ieee.org/groups/msc/ANSI_IEEE-Std-754-2019/background/power.txt">"Too much power - pow vs powr, powd, pown, rootn, compound"</a>. <i><a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-16</span></span>. <q>Since growth rates can't be less than -1, such rates signal invalid exceptions.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IEEE&rft.atitle=Too+much+power+-+pow+vs+powr%2C+powd%2C+pown%2C+rootn%2C+compound&rft_id=https%3A%2F%2Fgrouper.ieee.org%2Fgroups%2Fmsc%2FANSI_IEEE-Std-754-2019%2Fbackground%2Fpower.txt&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</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://web.archive.org/web/20170706053605/http://grouper.ieee.org/groups/754/email/msg03842.html">"Re: Missing functions tanPi, asinPi and acosPi"</a>. <i><a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a></i>. Archived from <a rel="nofollow" class="external text" href="http://grouper.ieee.org/groups/754/email/msg03842.html">the original</a> on 2017-07-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-04-04</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IEEE&rft.atitle=Re%3A+Missing+functions+tanPi%2C+asinPi+and+acosPi&rft_id=http%3A%2F%2Fgrouper.ieee.org%2Fgroups%2F754%2Femail%2Fmsg03842.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §9.3.</span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §9.4.</span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a>, §9.5</span> </li> <li id="cite_note-Riedy_2018-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-Riedy_2018_60-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiedyDemmel" class="citation web cs1">Riedy, Jason; Demmel, James. <a rel="nofollow" class="external text" href="http://www.ecs.umass.edu/arith-2018/pdf/arith25_34.pdf">"Augmented Arithmetic Operations Proposed for IEEE-754 2018"</a> <span class="cs1-format">(PDF)</span>. 25th IEEE Symbosium on Computer Arithmetic (ARITH 2018). pp. 49–56. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190723172615/http://www.ecs.umass.edu/arith-2018/pdf/arith25_34.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2019-07-23<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-07-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Augmented+Arithmetic+Operations+Proposed+for+IEEE-754+2018&rft.pages=49-56&rft.pub=25th+IEEE+Symbosium+on+Computer+Arithmetic+%28ARITH+2018%29&rft.aulast=Riedy&rft.aufirst=Jason&rft.au=Demmel%2C+James&rft_id=http%3A%2F%2Fwww.ecs.umass.edu%2Farith-2018%2Fpdf%2Farith25_34.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-Revision_2019-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-Revision_2019_61-0">^</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://grouper.ieee.org/groups/msc/ANSI_IEEE-Std-754-2019/background/">"ANSI/IEEE Std 754-2019 – Background Documents"</a>. <i><a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IEEE&rft.atitle=ANSI%2FIEEE+Std+754-2019+%E2%80%93+Background+Documents&rft_id=https%3A%2F%2Fgrouper.ieee.org%2Fgroups%2Fmsc%2FANSI_IEEE-Std-754-2019%2Fbackground%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542019">IEEE 754 2019</a>, §9.6.</span> </li> <li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen" class="citation web cs1">Chen, David. <a rel="nofollow" class="external text" href="https://grouper.ieee.org/groups/msc/ANSI_IEEE-Std-754-2019/background/minNum_maxNum_Removal_Demotion_v3.pdf">"The Removal/Demotion of MinNum and MaxNum Operations from IEEE 754-2018"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-01-16</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=IEEE&rft.atitle=The+Removal%2FDemotion+of+MinNum+and+MaxNum+Operations+from+IEEE+754-2018&rft.aulast=Chen&rft.aufirst=David&rft_id=https%3A%2F%2Fgrouper.ieee.org%2Fgroups%2Fmsc%2FANSI_IEEE-Std-754-2019%2Fbackground%2FminNum_maxNum_Removal_Demotion_v3.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.12.</span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.12.2.</span> </li> <li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGay1990" class="citation cs2">Gay, David M. (1990-11-30), <a rel="nofollow" class="external text" href="http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.4049"><i>Correctly rounded binary-decimal and decimal-binary conversions</i></a>, Numerical Analysis Manuscript, Murry Hill, NJ, US: AT&T Laboratories, 90-10</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Correctly+rounded+binary-decimal+and+decimal-binary+conversions&rft.place=Murry+Hill%2C+NJ%2C+US&rft.series=Numerical+Analysis+Manuscript&rft.pub=AT%26T+Laboratories&rft.date=1990-11-30&rft.aulast=Gay&rft.aufirst=David+M.&rft_id=http%3A%2F%2Fciteseer.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.31.4049&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaxsonKahan1991" class="citation cs2">Paxson, Vern; <a href="/wiki/William_Kahan" title="William Kahan">Kahan, William</a> (1991-05-22), <i>A Program for Testing IEEE Decimal–Binary Conversion</i>, Manuscript, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.144.5889">10.1.1.144.5889</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Program+for+Testing+IEEE+Decimal%E2%80%93Binary+Conversion&rft.series=Manuscript&rft.date=1991-05-22&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.144.5889%23id-name%3DCiteSeerX&rft.aulast=Paxson&rft.aufirst=Vern&rft.au=Kahan%2C+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> <li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text"><a href="#CITEREFIEEE_7542008">IEEE 754 2008</a>, §5.12.3</span> </li> <li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</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://ghc.gitlab.haskell.org/ghc/doc/users_guide/exts/hex_float_literals.html">"6.9.3. Hexadecimal floating point literals — Glasgow Haskell Compiler 9.3.20220129 User's Guide"</a>. <i>ghc.gitlab.haskell.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-01-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ghc.gitlab.haskell.org&rft.atitle=6.9.3.+Hexadecimal+floating+point+literals+%E2%80%94+Glasgow+Haskell+Compiler+9.3.20220129+User%27s+Guide&rft_id=https%3A%2F%2Fghc.gitlab.haskell.org%2Fghc%2Fdoc%2Fusers_guide%2Fexts%2Fhex_float_literals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Standards">Standards</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=32" title="Edit section: Standards"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>IEEE Standard for Binary Floating-Point Arithmetic</i>. ANSI/IEEE STD 754-1985. IEEE. 1985-10-12. pp. 1–20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIEEESTD.1985.82928">10.1109/IEEESTD.1985.82928</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7381-1165-1" title="Special:BookSources/0-7381-1165-1"><bdi>0-7381-1165-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IEEE+Standard+for+Binary+Floating-Point+Arithmetic&rft.series=ANSI%2FIEEE+STD+754-1985&rft.pages=1-20&rft.pub=IEEE&rft.date=1985-10-12&rft_id=info%3Adoi%2F10.1109%2FIEEESTD.1985.82928&rft.isbn=0-7381-1165-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIEEE_7542008" class="citation book cs1">IEEE Computer Society (2008-08-29). <i>IEEE Standard for Floating-Point Arithmetic</i>. IEEE STD 754-2008. IEEE. pp. 1–70. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIEEESTD.2008.4610935">10.1109/IEEESTD.2008.4610935</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7381-5753-5" title="Special:BookSources/978-0-7381-5753-5"><bdi>978-0-7381-5753-5</bdi></a>. IEEE Std 754-2008.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IEEE+Standard+for+Floating-Point+Arithmetic&rft.series=IEEE+STD+754-2008&rft.pages=1-70&rft.pub=IEEE&rft.date=2008-08-29&rft_id=info%3Adoi%2F10.1109%2FIEEESTD.2008.4610935&rft.isbn=978-0-7381-5753-5&rft.au=IEEE+Computer+Society&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIEEE_7542019" class="citation book cs1">IEEE Computer Society (2019-07-22). <i>IEEE Standard for Floating-Point Arithmetic</i>. IEEE STD 754-2019. IEEE. pp. 1–84. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIEEESTD.2019.8766229">10.1109/IEEESTD.2019.8766229</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-5044-5924-2" title="Special:BookSources/978-1-5044-5924-2"><bdi>978-1-5044-5924-2</bdi></a>. IEEE Std 754-2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IEEE+Standard+for+Floating-Point+Arithmetic&rft.series=IEEE+STD+754-2019&rft.pages=1-84&rft.pub=IEEE&rft.date=2019-07-22&rft_id=info%3Adoi%2F10.1109%2FIEEESTD.2019.8766229&rft.isbn=978-1-5044-5924-2&rft.au=IEEE+Computer+Society&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFISO/IEC_JTC_1/SC_252011" class="citation book cs1">ISO/IEC JTC 1/SC 25 (June 2011). <a rel="nofollow" class="external text" href="https://www.iso.org/standard/57469.html"><i>ISO/IEC/IEEE 60559:2011 — Information technology — Microprocessor Systems — Floating-Point arithmetic</i></a>. ISO. pp. 1–58.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=ISO%2FIEC%2FIEEE+60559%3A2011+%E2%80%94+Information+technology+%E2%80%94+Microprocessor+Systems+%E2%80%94+Floating-Point+arithmetic&rft.pages=1-58&rft.pub=ISO&rft.date=2011-06&rft.au=ISO%2FIEC+JTC+1%2FSC+25&rft_id=https%3A%2F%2Fwww.iso.org%2Fstandard%2F57469.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFISO/IEC_JTC_1/SC_252020" class="citation book cs1">ISO/IEC JTC 1/SC 25 (May 2020). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.iso.org/standard/80985.html"><i>ISO/IEC 60559:2020 — Information technology — Microprocessor Systems — Floating-Point arithmetic</i></a></span>. ISO. pp. 1–74.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=ISO%2FIEC+60559%3A2020+%E2%80%94+Information+technology+%E2%80%94+Microprocessor+Systems+%E2%80%94+Floating-Point+arithmetic&rft.pages=1-74&rft.pub=ISO&rft.date=2020-05&rft.au=ISO%2FIEC+JTC+1%2FSC+25&rft_id=https%3A%2F%2Fwww.iso.org%2Fstandard%2F80985.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: numeric names: authors list (<a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">link</a>)</span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Secondary_references">Secondary references</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=33" title="Edit section: Secondary references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://speleotrove.com/decimal">Decimal floating-point</a> arithmetic, FAQs, bibliography, and links</li> <li><a rel="nofollow" class="external text" href="http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm">Comparing binary floats</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20111201211023/http://babbage.cs.qc.cuny.edu/IEEE-754.old/References.xhtml">IEEE 754 Reference Material</a></li> <li><a rel="nofollow" class="external text" href="http://speleotrove.com/decimal/854mins.html">IEEE 854-1987</a> – History and minutes</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20171230124220/http://grouper.ieee.org/groups/754/reading.html">Supplementary readings for IEEE 754</a>. Includes historical perspectives.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=34" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldberg1991" class="citation journal cs1">Goldberg, David (March 1991). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F103162.103163">"What Every Computer Scientist Should Know About Floating-Point Arithmetic"</a>. <i><a href="/wiki/ACM_Computing_Surveys" title="ACM Computing Surveys">ACM Computing Surveys</a></i>. <b>23</b> (1): 5–48. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F103162.103163">10.1145/103162.103163</a></span>. <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:222008826">222008826</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Computing+Surveys&rft.atitle=What+Every+Computer+Scientist+Should+Know+About+Floating-Point+Arithmetic&rft.volume=23&rft.issue=1&rft.pages=5-48&rft.date=1991-03&rft_id=info%3Adoi%2F10.1145%2F103162.103163&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A222008826%23id-name%3DS2CID&rft.aulast=Goldberg&rft.aufirst=David&rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F103162.103163&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span> (With the addendum "Differences Among IEEE 754 Implementations": <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20171011072644/http://www.cse.msu.edu/~cse320/Documents/FloatingPoint.pdf">[1]</a>, <a rel="nofollow" class="external autonumber" href="https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html">[2]</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a href="/wiki/Chris_Hecker" title="Chris Hecker">Hecker, Chris</a> (February 1996). <a rel="nofollow" class="external text" href="http://chrishecker.com/images/f/fb/Gdmfp.pdf">"Let's Get To The (Floating) Point"</a> <span class="cs1-format">(PDF)</span>. <i>Game Developer Magazine</i>: 19–24. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1073-922X">1073-922X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Game+Developer+Magazine&rft.atitle=Let%27s+Get+To+The+%28Floating%29+Point&rft.pages=19-24&rft.date=1996-02&rft.issn=1073-922X&rft.aulast=Hecker&rft.aufirst=Chris&rft_id=http%3A%2F%2Fchrishecker.com%2Fimages%2Ff%2Ffb%2FGdmfp.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1">Severance, Charles (March 1998). <a rel="nofollow" class="external text" href="http://www.dr-chuck.com/dr-chuck/papers/columns/r3114.pdf">"IEEE 754: An Interview with William Kahan"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/IEEE_Computer" class="mw-redirect" title="IEEE Computer">IEEE Computer</a></i>. <b>31</b> (3): 114–115. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMC.1998.660194">10.1109/MC.1998.660194</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:33291145">33291145</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-03-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Computer&rft.atitle=IEEE+754%3A+An+Interview+with+William+Kahan&rft.volume=31&rft.issue=3&rft.pages=114-115&rft.date=1998-03&rft_id=info%3Adoi%2F10.1109%2FMC.1998.660194&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A33291145%23id-name%3DS2CID&rft.aulast=Severance&rft.aufirst=Charles&rft_id=http%3A%2F%2Fwww.dr-chuck.com%2Fdr-chuck%2Fpapers%2Fcolumns%2Fr3114.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a href="/wiki/Mike_Cowlishaw" title="Mike Cowlishaw">Cowlishaw, Mike</a> (June 2003). "Decimal floating-point: Algorism for computers". <a rel="nofollow" class="external text" href="http://speleotrove.com/decimal/IEEE-cowlishaw-arith16.pdf"><i>16th IEEE Symposium on Computer Arithmetic, 2003. Proceedings</i></a> <span class="cs1-format">(PDF)</span>. Los Alamitos, Calif.: IEEE Computer Society. pp. 104–111. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FARITH.2003.1207666">10.1109/ARITH.2003.1207666</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7695-1894-7" title="Special:BookSources/978-0-7695-1894-7"><bdi>978-0-7695-1894-7</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:18713046">18713046</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2014-11-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Decimal+floating-point%3A+Algorism+for+computers&rft.btitle=16th+IEEE+Symposium+on+Computer+Arithmetic%2C+2003.+Proceedings.&rft.place=Los+Alamitos%2C+Calif.&rft.pages=104-111&rft.pub=IEEE+Computer+Society&rft.date=2003-06&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18713046%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FARITH.2003.1207666&rft.isbn=978-0-7695-1894-7&rft.aulast=Cowlishaw&rft.aufirst=Mike&rft_id=http%3A%2F%2Fspeleotrove.com%2Fdecimal%2FIEEE-cowlishaw-arith16.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span>. (Note: <i>Algorism</i> is not a misspelling of the title; see also <a href="/wiki/Algorism" title="Algorism">algorism</a>.)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1">Monniaux, David (May 2008). <a rel="nofollow" class="external text" href="https://hal.science/hal-00128124/en/">"The pitfalls of verifying floating-point computations"</a>. <i><a href="/wiki/ACM_Transactions_on_Programming_Languages_and_Systems" title="ACM Transactions on Programming Languages and Systems">ACM Transactions on Programming Languages and Systems</a></i>. <b>30</b> (3): 1–41. <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/cs/0701192">cs/0701192</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/10.1145%2F1353445.1353446">10.1145/1353445.1353446</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0164-0925">0164-0925</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:218578808">218578808</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Transactions+on+Programming+Languages+and+Systems&rft.atitle=The+pitfalls+of+verifying+floating-point+computations&rft.volume=30&rft.issue=3&rft.pages=1-41&rft.date=2008-05&rft_id=info%3Aarxiv%2Fcs%2F0701192&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A218578808%23id-name%3DS2CID&rft.issn=0164-0925&rft_id=info%3Adoi%2F10.1145%2F1353445.1353446&rft.aulast=Monniaux&rft.aufirst=David&rft_id=https%3A%2F%2Fhal.science%2Fhal-00128124%2Fen%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span>: A compendium of non-intuitive behaviours of floating-point on popular architectures, with implications for program verification and testing.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">Muller, Jean-Michel; Brunie, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume; <a href="/wiki/Nathalie_Revol" title="Nathalie Revol">Revol, Nathalie</a>; Torres, Serge (2018) [2010]. <a rel="nofollow" class="external text" href="https://cds.cern.ch/record/1315760"><i>Handbook of Floating-Point Arithmetic</i></a> (2 ed.). <a href="/wiki/Birkh%C3%A4user" title="Birkhäuser">Birkhäuser</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-76526-6">10.1007/978-3-319-76526-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-76525-9" title="Special:BookSources/978-3-319-76525-9"><bdi>978-3-319-76525-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Floating-Point+Arithmetic&rft.edition=2&rft.pub=Birkh%C3%A4user&rft.date=2018&rft_id=info%3Adoi%2F10.1007%2F978-3-319-76526-6&rft.isbn=978-3-319-76525-9&rft.aulast=Muller&rft.aufirst=Jean-Michel&rft.au=Brunie%2C+Nicolas&rft.au=de+Dinechin%2C+Florent&rft.au=Jeannerod%2C+Claude-Pierre&rft.au=Joldes%2C+Mioara&rft.au=Lef%C3%A8vre%2C+Vincent&rft.au=Melquiond%2C+Guillaume&rft.au=Revol%2C+Nathalie&rft.au=Torres%2C+Serge&rft_id=https%3A%2F%2Fcds.cern.ch%2Frecord%2F1315760&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1 cs1-prop-location-test">Overton, Michael L. (2001). Written at <a href="/wiki/Courant_Institute_of_Mathematical_Sciences" title="Courant Institute of Mathematical Sciences">Courant Institute of Mathematical Sciences</a>, <a href="/wiki/New_York_University" title="New York University">New York University</a>, New York, US. <i>Numerical Computing with IEEE Floating Point Arithmetic</i> (1 ed.). Philadelphia, US: <a href="/wiki/Society_for_Industrial_and_Applied_Mathematics" title="Society for Industrial and Applied Mathematics">SIAM</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9780898718072">10.1137/1.9780898718072</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-482-1" title="Special:BookSources/978-0-89871-482-1"><bdi>978-0-89871-482-1</bdi></a>. 978-0-89871-571-2, 0-89871-571-7.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Computing+with+IEEE+Floating+Point+Arithmetic&rft.place=Philadelphia%2C+US&rft.edition=1&rft.pub=SIAM&rft.date=2001&rft_id=info%3Adoi%2F10.1137%2F1.9780898718072&rft.isbn=978-0-89871-482-1&rft.aulast=Overton&rft.aufirst=Michael+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://blogs.mathworks.com/cleve/2014/07/07/floating-point-numbers/">Cleve Moler on Floating Point numbers</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">Beebe, Nelson H. F. (2017-08-22). <i>The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library</i> (1 ed.). Salt Lake City, UT, US: <a href="/wiki/Springer_International_Publishing_AG" class="mw-redirect" title="Springer International Publishing AG">Springer International Publishing AG</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-64110-2">10.1007/978-3-319-64110-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-64109-6" title="Special:BookSources/978-3-319-64109-6"><bdi>978-3-319-64109-6</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2017947446">2017947446</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:30244721">30244721</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical-Function+Computation+Handbook+-+Programming+Using+the+MathCW+Portable+Software+Library&rft.place=Salt+Lake+City%2C+UT%2C+US&rft.edition=1&rft.pub=Springer+International+Publishing+AG&rft.date=2017-08-22&rft_id=info%3Adoi%2F10.1007%2F978-3-319-64110-2&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A30244721%23id-name%3DS2CID&rft_id=info%3Alccn%2F2017947446&rft.isbn=978-3-319-64109-6&rft.aulast=Beebe&rft.aufirst=Nelson+H.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1">Hough, David G. (December 2019). <a rel="nofollow" class="external text" href="https://www.computer.org/csdl/magazine/co/2019/12/08909942/1f8KFWxbTCU">"The IEEE Standard 754: One for the History Books"</a>. <i>Computer</i>. <b>52</b> (12). <a href="/wiki/IEEE" class="mw-redirect" title="IEEE">IEEE</a>: 109–112. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMC.2019.2926614">10.1109/MC.2019.2926614</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:208281213">208281213</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer&rft.atitle=The+IEEE+Standard+754%3A+One+for+the+History+Books&rft.volume=52&rft.issue=12&rft.pages=109-112&rft.date=2019-12&rft_id=info%3Adoi%2F10.1109%2FMC.2019.2926614&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A208281213%23id-name%3DS2CID&rft.aulast=Hough&rft.aufirst=David+G.&rft_id=https%3A%2F%2Fwww.computer.org%2Fcsdl%2Fmagazine%2Fco%2F2019%2F12%2F08909942%2F1f8KFWxbTCU&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=IEEE_754&action=edit&section=35" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></span></span></div> <div class="side-box-text plainlist">The Wikibook <i><a href="https://en.wikibooks.org/wiki/Floating_Point" class="extiw" title="wikibooks:Floating Point">Floating Point</a></i> has a page on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Floating_Point/Special_Numbers" class="extiw" title="wikibooks:Floating Point/Special Numbers">special numbers specified in the IEEE 754 standard</a></b></i></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:IEEE_754" class="extiw" title="commons:Category:IEEE 754">IEEE 754</a></span>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1"><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ATCpecsyPE8"><i>Kahan on creating IEEE Standard Floating Point</i></a>. <i>Turing Awardee Clips</i>. 2020-11-16. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211108/ATCpecsyPE8">Archived</a> from the original on 2021-11-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Kahan+on+creating+IEEE+Standard+Floating+Point&rft.date=2020-11-16&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DATCpecsyPE8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIEEE+754" class="Z3988"></span></li></ul> <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="IEEE_standards" 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:IEEE_standards" title="Template:IEEE standards"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:IEEE_standards" title="Template talk:IEEE standards"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:IEEE_standards" title="Special:EditPage/Template:IEEE standards"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="IEEE_standards" style="font-size:114%;margin:0 4em"><a href="/wiki/IEEE_Standards_Association" title="IEEE Standards Association">IEEE standards</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Current</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/IEEE-488" class="mw-redirect" title="IEEE-488">488</a></li> <li><a href="/wiki/IEEE_693" title="IEEE 693">693</a></li> <li><a href="/wiki/Software_quality_assurance" title="Software quality assurance">730</a></li> <li><a class="mw-selflink selflink">754</a> <ul><li><a href="/wiki/IEEE_754-2008_revision" title="IEEE 754-2008 revision">Revision</a></li></ul></li> <li><a href="/wiki/IEEE_854-1987" title="IEEE 854-1987">854</a></li> <li><a href="/wiki/Software_configuration_management" title="Software configuration management">828</a></li> <li><a href="/wiki/Software_test_documentation" title="Software test documentation">829</a></li> <li><a href="/wiki/Futurebus" title="Futurebus">896</a></li> <li><a href="/wiki/Single_UNIX_Specification" title="Single UNIX Specification">1003</a></li> <li><a href="/wiki/VMEbus" title="VMEbus">1014</a></li> <li><a href="/wiki/Software_design_description" title="Software design description">1016</a></li> <li><a href="/wiki/VHDL" title="VHDL">1076</a></li> <li><a href="/wiki/JTAG" title="JTAG">1149.1</a></li> <li><a href="/wiki/PILOT" title="PILOT">1154</a></li> <li><a href="/wiki/IEEE_1164" title="IEEE 1164">1164</a></li> <li><a href="/wiki/Open_Firmware" title="Open Firmware">1275</a></li> <li><a href="/wiki/Distributed_Interactive_Simulation" title="Distributed Interactive Simulation">1278</a></li> <li><a href="/wiki/IEEE_1284" title="IEEE 1284">1284</a></li> <li><a href="/wiki/IEEE_1355" title="IEEE 1355">1355</a></li> <li><a href="/wiki/IEEE_1394" title="IEEE 1394">1394</a></li> <li><a href="/wiki/IEEE_1451" title="IEEE 1451">1451</a></li> <li><a href="/wiki/Standard_Delay_Format" title="Standard Delay Format">1497</a></li> <li><a href="/wiki/High-level_architecture" class="mw-redirect" title="High-level architecture">1516</a></li> <li><a href="/wiki/IEEE_1541-2002" class="mw-redirect" title="IEEE 1541-2002">1541</a></li> <li><a href="/wiki/IEEE_1547" title="IEEE 1547">1547</a></li> <li><a href="/wiki/IEEE_1584" title="IEEE 1584">1584</a></li> <li><a href="/wiki/Precision_Time_Protocol" title="Precision Time Protocol">1588</a></li> <li><a href="/wiki/Scalable_Coherent_Interface" title="Scalable Coherent Interface">1596</a></li> <li><a href="/wiki/Advanced_Library_Format" title="Advanced Library Format">1603</a></li> <li><a href="/wiki/IEEE_1613" title="IEEE 1613">1613</a></li> <li><a href="/wiki/IEEE_1619" class="mw-redirect" title="IEEE 1619">1619</a></li> <li><a href="/wiki/SystemC" title="SystemC">1666</a></li> <li><a href="/wiki/IEEE_1667" title="IEEE 1667">1667</a></li> <li><a href="/wiki/IEEE_1675-2008" title="IEEE 1675-2008">1675</a></li> <li><a href="/wiki/IP-XACT" title="IP-XACT">1685</a></li> <li><a href="/wiki/IEEE_1722" class="mw-redirect" title="IEEE 1722">1722</a></li> <li><a href="/wiki/IEEE_1733" class="mw-redirect" title="IEEE 1733">1733</a></li> <li><a href="/wiki/SystemVerilog" title="SystemVerilog">1800</a></li> <li><a href="/wiki/Unified_Power_Format" title="Unified Power Format">1801</a></li> <li><a href="/wiki/DNP3" title="DNP3">1815</a></li> <li><a href="/wiki/IEEE_1849" title="IEEE 1849">1849</a></li> <li><a href="/wiki/Property_Specification_Language" title="Property Specification Language">1850</a></li> <li><a href="/wiki/IEEE_1855" title="IEEE 1855">1855</a></li> <li><a href="/wiki/DySPAN" title="DySPAN">1900</a></li> <li><a href="/wiki/IEEE_1901" title="IEEE 1901">1901</a></li> <li><a href="/wiki/RuBee" title="RuBee">1902</a></li> <li><a href="/wiki/Service_Interoperability_in_Ethernet_Passive_Optical_Networks" title="Service Interoperability in Ethernet Passive Optical Networks">1904</a></li> <li><a href="/wiki/IEEE_1905" title="IEEE 1905">1905</a></li> <li><a href="/wiki/IEEE_2030" title="IEEE 2030">2030</a></li> <li><a href="/wiki/Micro_T-Kernel" title="Micro T-Kernel">2050</a></li> <li><a href="/wiki/ISO/IEEE_11073" title="ISO/IEEE 11073">11073</a></li> <li><a href="/wiki/ISO/IEC_12207" title="ISO/IEC 12207">12207</a></li> <li><a href="/wiki/Software_maintenance" title="Software maintenance">14764</a></li> <li><a href="/wiki/Risk_management" title="Risk management">16085</a></li> <li><a href="/wiki/Project_management" title="Project management">16326</a></li> <li><a href="/wiki/Requirements_engineering" title="Requirements engineering">29148</a></li> <li><a href="/wiki/ISO/IEC_42010" title="ISO/IEC 42010">42010</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/IEEE_802" title="IEEE 802">802 series</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/IEEE_802" title="IEEE 802">802</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/IEEE_802.2" title="IEEE 802.2">.2</a></li> <li><a href="/wiki/Token_bus_network" title="Token bus network">.4</a></li> <li><a href="/wiki/Token_Ring" title="Token Ring">.5</a></li> <li><a href="/wiki/IEEE_802.6" title="IEEE 802.6">.6</a></li> <li><a href="/wiki/IEEE_802.7" title="IEEE 802.7">.7</a></li> <li><a href="/wiki/IEEE_802.8" title="IEEE 802.8">.8</a></li> <li><a href="/wiki/IEEE_802.9" title="IEEE 802.9">.9</a></li> <li><a href="/wiki/IEEE_802.10" title="IEEE 802.10">.10</a></li> <li><a href="/wiki/100BaseVG" title="100BaseVG">.12</a></li> <li><a href="/wiki/Cable_modem#IEEE_802.14" title="Cable modem">.14</a></li> <li><a href="/wiki/IEEE_802.16" title="IEEE 802.16">.16</a> <ul><li><a href="/wiki/WiMAX" title="WiMAX">WiMAX · d · e</a></li></ul></li> <li><a href="/wiki/Resilient_Packet_Ring" title="Resilient Packet Ring">.17</a></li> <li><a href="/wiki/IEEE_802.18" title="IEEE 802.18">.18</a></li> <li><a href="/wiki/IEEE_802.20" title="IEEE 802.20">.20</a></li> <li><a href="/wiki/IEEE_802.21" title="IEEE 802.21">.21</a></li> <li><a href="/wiki/IEEE_802.22" title="IEEE 802.22">.22</a></li> <li><a href="/w/index.php?title=IEEE_802.24&action=edit&redlink=1" class="new" title="IEEE 802.24 (page does not exist)">.24</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/IEEE_802.1" title="IEEE 802.1">802.1</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/IEEE_802.1D" title="IEEE 802.1D">D</a></li> <li><a href="/wiki/IEEE_P802.1p" title="IEEE P802.1p">p</a></li> <li><a href="/wiki/IEEE_802.1Q" title="IEEE 802.1Q">Q</a></li> <li><a href="/wiki/IEEE_802.1Qav" class="mw-redirect" title="IEEE 802.1Qav">Qav</a></li> <li><a href="/wiki/Stream_Reservation_Protocol" title="Stream Reservation Protocol">Qat</a></li> <li><a href="/wiki/Provider_Backbone_Bridge_Traffic_Engineering" title="Provider Backbone Bridge Traffic Engineering">Qay</a></li> <li><a href="/wiki/Data_center_bridging#IEEE_Task_Group" title="Data center bridging">Qaz</a></li> <li><a href="/wiki/IEEE_802.1Qbb" class="mw-redirect" title="IEEE 802.1Qbb">Qbb</a></li> <li><a href="/wiki/Spanning_Tree_Protocol" title="Spanning Tree Protocol">w</a></li> <li><a href="/wiki/IEEE_802.1X" title="IEEE 802.1X">X</a></li> <li><a href="/wiki/Link_Layer_Discovery_Protocol" title="Link Layer Discovery Protocol">AB</a></li> <li><a href="/wiki/IEEE_802.1ad" title="IEEE 802.1ad">ad</a></li> <li><a href="/wiki/IEEE_802.1AE" title="IEEE 802.1AE">AE</a></li> <li><a href="/wiki/IEEE_802.1ag" title="IEEE 802.1ag">ag</a></li> <li><a href="/wiki/IEEE_802.1ah-2008" class="mw-redirect" title="IEEE 802.1ah-2008">ah</a></li> <li><a href="/wiki/Multiple_Registration_Protocol" title="Multiple Registration Protocol">ak</a></li> <li><a href="/wiki/IEEE_802.1aq" title="IEEE 802.1aq">aq</a></li> <li><a href="/wiki/IEEE_802.1AS" class="mw-redirect" title="IEEE 802.1AS">AS</a></li> <li><a href="/wiki/Link_aggregation" title="Link aggregation">AX</a> (<a href="/wiki/Link_Aggregation_Control_Protocol" class="mw-redirect" title="Link Aggregation Control Protocol">LACP</a>)</li> <li><a href="/wiki/Audio_Video_Bridging" title="Audio Video Bridging">BA</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/IEEE_802.3" title="IEEE 802.3">802.3</a> <br />(<a href="/wiki/Ethernet" title="Ethernet">Ethernet</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/10BASE5" title="10BASE5">-1983</a></li> <li><a href="/wiki/802.3a" class="mw-redirect" title="802.3a">a</a></li> <li><a href="/wiki/802.3b" class="mw-redirect" title="802.3b">b</a></li> <li><a href="/wiki/802.3d" class="mw-redirect" title="802.3d">d</a></li> <li><a href="/wiki/802.3e" class="mw-redirect" title="802.3e">e</a></li> <li><a href="/wiki/802.3i" class="mw-redirect" title="802.3i">i</a></li> <li><a href="/wiki/802.3j" class="mw-redirect" title="802.3j">j</a></li> <li><a href="/wiki/802.3u" class="mw-redirect" title="802.3u">u</a></li> <li><a href="/wiki/IEEE_802.3x" class="mw-redirect" title="IEEE 802.3x">x</a></li> <li><a href="/wiki/802.3y" class="mw-redirect" title="802.3y">y</a></li> <li><a href="/wiki/802.3z" class="mw-redirect" title="802.3z">z</a></li> <li><a href="/wiki/802.3ab" class="mw-redirect" title="802.3ab">ab</a></li> <li><a href="/wiki/802.3ac" class="mw-redirect" title="802.3ac">ac</a></li> <li><a href="/wiki/802.3ad" class="mw-redirect" title="802.3ad">ad</a></li> <li><a href="/wiki/802.3ae" class="mw-redirect" title="802.3ae">ae</a></li> <li><b><a href="/wiki/802.3af" class="mw-redirect" title="802.3af">af</a></b></li> <li><a href="/wiki/802.3ah" class="mw-redirect" title="802.3ah">ah</a></li> <li><a href="/wiki/802.3ak" class="mw-redirect" title="802.3ak">ak</a></li> <li><a href="/wiki/802.3an" class="mw-redirect" title="802.3an">an</a></li> <li><a href="/wiki/802.3aq" class="mw-redirect" title="802.3aq">aq</a></li> <li><b><a href="/wiki/802.3at" class="mw-redirect" title="802.3at">at</a></b></li> <li><a href="/wiki/802.3au" class="mw-redirect" title="802.3au">au</a></li> <li><a href="/wiki/802.3av" class="mw-redirect" title="802.3av">av</a></li> <li><a href="/wiki/802.3az" class="mw-redirect" title="802.3az">az</a></li> <li><a href="/wiki/802.3ba" class="mw-redirect" title="802.3ba">ba</a></li> <li><b><a href="/wiki/802.3bt" class="mw-redirect" title="802.3bt">bt</a></b></li> <li><a href="/wiki/802.3bu" class="mw-redirect" title="802.3bu">bu</a></li> <li><a href="/wiki/802.3by" class="mw-redirect" title="802.3by">by</a></li> <li><a href="/wiki/802.3bz" class="mw-redirect" title="802.3bz">bz</a></li> <li><a href="/w/index.php?title=802.3ca&action=edit&redlink=1" class="new" title="802.3ca (page does not exist)">ca</a></li> <li><a href="/w/index.php?title=802.3cb&action=edit&redlink=1" class="new" title="802.3cb (page does not exist)">cb</a></li> <li><a href="/w/index.php?title=802.3cc&action=edit&redlink=1" class="new" title="802.3cc (page does not exist)">cc</a></li> <li><a href="/w/index.php?title=802.3cd&action=edit&redlink=1" class="new" title="802.3cd (page does not exist)">cd</a></li> <li><a href="/w/index.php?title=802.3ce&action=edit&redlink=1" class="new" title="802.3ce (page does not exist)">ce</a></li> <li><a href="/wiki/802.3cg" class="mw-redirect" title="802.3cg">cg</a></li> <li><a href="/wiki/802.3ch" class="mw-redirect" title="802.3ch">ch</a></li> <li><a href="/w/index.php?title=802.3ck&action=edit&redlink=1" class="new" title="802.3ck (page does not exist)">ck</a></li> <li><a href="/w/index.php?title=802.3cm&action=edit&redlink=1" class="new" title="802.3cm (page does not exist)">cm</a></li> <li><a href="/w/index.php?title=802.3cn&action=edit&redlink=1" class="new" title="802.3cn (page does not exist)">cn</a></li> <li><a href="/w/index.php?title=802.3cp&action=edit&redlink=1" class="new" title="802.3cp (page does not exist)">cp</a></li> <li><a href="/wiki/802.3cq" class="mw-redirect" title="802.3cq">cq</a></li> <li><a href="/w/index.php?title=802.3cr&action=edit&redlink=1" class="new" title="802.3cr (page does not exist)">cr</a></li> <li><a href="/w/index.php?title=802.3cs&action=edit&redlink=1" class="new" title="802.3cs (page does not exist)">cs</a></li> <li><a href="/w/index.php?title=802.3ct&action=edit&redlink=1" class="new" title="802.3ct (page does not exist)">ct</a></li> <li><a href="/w/index.php?title=802.3cu&action=edit&redlink=1" class="new" title="802.3cu (page does not exist)">cu</a></li> <li><a href="/wiki/802.3cv" class="mw-redirect" title="802.3cv">cv</a></li> <li><a href="/w/index.php?title=802.3cw&action=edit&redlink=1" class="new" title="802.3cw (page does not exist)">cw</a></li> <li><a href="/w/index.php?title=802.3cx&action=edit&redlink=1" class="new" title="802.3cx (page does not exist)">cx</a></li> <li><a href="/w/index.php?title=802.3cy&action=edit&redlink=1" class="new" title="802.3cy (page does not exist)">cy</a></li> <li><a href="/w/index.php?title=802.3cz&action=edit&redlink=1" class="new" title="802.3cz (page does not exist)">cz</a></li> <li><a href="/w/index.php?title=802.3da&action=edit&redlink=1" class="new" title="802.3da (page does not exist)">da</a></li> <li><a href="/w/index.php?title=802.3db&action=edit&redlink=1" class="new" title="802.3db (page does not exist)">db</a></li> <li><a href="/wiki/802.3dd" class="mw-redirect" title="802.3dd">dd</a></li> <li><a href="/w/index.php?title=802.3de&action=edit&redlink=1" class="new" title="802.3de (page does not exist)">de</a></li> <li><a href="/w/index.php?title=802.3df&action=edit&redlink=1" class="new" title="802.3df (page does not exist)">df</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/IEEE_802.11" title="IEEE 802.11">802.11</a> <br />(<a href="/wiki/Wi-Fi" title="Wi-Fi">Wi-Fi</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/IEEE_802.11-1997" class="mw-redirect" title="IEEE 802.11-1997">-1997</a></li> <li><a href="/wiki/IEEE_802.11_(legacy_mode)" title="IEEE 802.11 (legacy mode)">legacy mode</a></li> <li><a href="/wiki/IEEE_802.11a-1999" title="IEEE 802.11a-1999">a</a></li> <li><a href="/wiki/IEEE_802.11b-1999" title="IEEE 802.11b-1999">b</a></li> <li><a href="/wiki/IEEE_802.11c" title="IEEE 802.11c">c</a></li> <li><a href="/wiki/IEEE_802.11d-2001" title="IEEE 802.11d-2001">d</a></li> <li><a href="/wiki/IEEE_802.11e-2005" title="IEEE 802.11e-2005">e</a></li> <li><a href="/wiki/Inter-Access_Point_Protocol" title="Inter-Access Point Protocol">f</a></li> <li><a href="/wiki/IEEE_802.11g-2003" title="IEEE 802.11g-2003">g</a></li> <li><a href="/wiki/IEEE_802.11h-2003" title="IEEE 802.11h-2003">h</a></li> <li><a href="/wiki/IEEE_802.11i-2004" title="IEEE 802.11i-2004">i</a></li> <li><a href="/wiki/IEEE_802.11j-2004" title="IEEE 802.11j-2004">j</a></li> <li><a href="/wiki/IEEE_802.11k-2008" title="IEEE 802.11k-2008">k</a></li> <li><b><a href="/wiki/IEEE_802.11n-2009" title="IEEE 802.11n-2009">n</a></b> (<a href="/wiki/Wi-Fi_4" class="mw-redirect" title="Wi-Fi 4">Wi-Fi 4</a>)</li> <li><a href="/wiki/IEEE_802.11p" title="IEEE 802.11p">p</a></li> <li><a href="/wiki/IEEE_802.11r-2008" title="IEEE 802.11r-2008">r</a></li> <li><a href="/wiki/IEEE_802.11s" title="IEEE 802.11s">s</a></li> <li><a href="/wiki/IEEE_802.11u" title="IEEE 802.11u">u</a></li> <li><a href="/wiki/IEEE_802.11v" class="mw-redirect" title="IEEE 802.11v">v</a></li> <li><a href="/wiki/IEEE_802.11w-2009" title="IEEE 802.11w-2009">w</a></li> <li><a href="/wiki/IEEE_802.11y-2008" title="IEEE 802.11y-2008">y</a></li> <li><a href="/wiki/IEEE_802.11z" class="mw-redirect" title="IEEE 802.11z">z</a></li> <li>aa</li> <li><b><a href="/wiki/IEEE_802.11ac" class="mw-redirect" title="IEEE 802.11ac">ac</a></b> (<a href="/wiki/Wi-Fi_5" class="mw-redirect" title="Wi-Fi 5">Wi-Fi 5</a>)</li> <li><a href="/wiki/IEEE_802.11ad" title="IEEE 802.11ad">ad</a> (<a href="/wiki/WiGig" title="WiGig">WiGig</a>)</li> <li>ae</li> <li><a href="/wiki/IEEE_802.11af" title="IEEE 802.11af">af</a></li> <li><a href="/wiki/IEEE_802.11ah" title="IEEE 802.11ah">ah</a></li> <li><a href="/wiki/IEEE_802.11ai" title="IEEE 802.11ai">ai</a></li> <li><a href="/wiki/IEEE_802.11aj" class="mw-redirect" title="IEEE 802.11aj">aj</a></li> <li>ak</li> <li>aq</li> <li><b><a href="/wiki/IEEE_802.11ax" class="mw-redirect" title="IEEE 802.11ax">ax</a></b> (<a href="/wiki/Wi-Fi_6" title="Wi-Fi 6">Wi-Fi 6</a>)</li> <li><a href="/wiki/IEEE_802.11ay" title="IEEE 802.11ay">ay</a></li> <li>az</li> <li>ba</li> <li><a href="/wiki/IEEE_802.11bb" title="IEEE 802.11bb">bb</a></li> <li>bc</li> <li>bd</li> <li><b><a href="/wiki/IEEE_802.11be" class="mw-redirect" title="IEEE 802.11be">be</a></b> (<a href="/wiki/Wi-Fi_7" title="Wi-Fi 7">Wi-Fi 7</a>)</li> <li>bf</li> <li>bh</li> <li>bi</li> <li>bk</li> <li><b><a href="/wiki/IEEE_802.11bn" title="IEEE 802.11bn">bn</a></b> (<a href="/w/index.php?title=Wi-Fi_8&action=edit&redlink=1" class="new" title="Wi-Fi 8 (page does not exist)">Wi-Fi 8</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/IEEE_802.15" title="IEEE 802.15">802.15</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/IEEE_802.15.1" class="mw-redirect" title="IEEE 802.15.1">.1</a> (<a href="/wiki/Bluetooth" title="Bluetooth">Bluetooth</a>)</li> <li><a href="/wiki/IEEE_802.15.2" class="mw-redirect" title="IEEE 802.15.2">.2</a></li> <li><a href="/wiki/IEEE_802.15.3" class="mw-redirect" title="IEEE 802.15.3">.3</a></li> <li><a href="/wiki/IEEE_802.15.4" title="IEEE 802.15.4">.4</a> (<a href="/wiki/Zigbee" title="Zigbee">Zigbee</a>)</li> <li><a href="/wiki/IEEE_802.15.4a" title="IEEE 802.15.4a">.4a</a></li> <li><a href="/w/index.php?title=IEEE_802.15.4b&action=edit&redlink=1" class="new" title="IEEE 802.15.4b (page does not exist)">.4b</a></li> <li><a href="/w/index.php?title=IEEE_802.15.4c&action=edit&redlink=1" class="new" title="IEEE 802.15.4c (page does not exist)">.4c</a></li> <li><a href="/w/index.php?title=IEEE_802.15.4d&action=edit&redlink=1" class="new" title="IEEE 802.15.4d (page does not exist)">.4d</a></li> <li><a href="/w/index.php?title=IEEE_802.15.4e&action=edit&redlink=1" class="new" title="IEEE 802.15.4e (page does not exist)">.4e</a></li> <li><a href="/w/index.php?title=IEEE_802.15.4f&action=edit&redlink=1" class="new" title="IEEE 802.15.4f (page does not exist)">.4f</a></li> <li><a href="/w/index.php?title=IEEE_802.15.4g&action=edit&redlink=1" class="new" title="IEEE 802.15.4g (page does not exist)">.4g</a></li> <li><a href="/w/index.php?title=IEEE_802.15.4z&action=edit&redlink=1" class="new" title="IEEE 802.15.4z (page does not exist)">.4z</a></li> <li><a href="/wiki/IEEE_802.15.5" class="mw-redirect" title="IEEE 802.15.5">.5</a></li> <li><a href="/wiki/IEEE_802.15.6" title="IEEE 802.15.6">.6</a></li> <li><a href="/wiki/IEEE_802.15.7" class="mw-redirect" title="IEEE 802.15.7">.7</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Proposed</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/IEEE_P1363" title="IEEE P1363">P1363</a></li> <li><a href="/wiki/IEEE_P1619" title="IEEE P1619">P1619</a></li> <li><a href="/wiki/Rosetta-lang" title="Rosetta-lang">P1699</a></li> <li><a href="/wiki/Universal_Power_Adapter_for_Mobile_Devices" title="Universal Power Adapter for Mobile Devices">P1823</a></li> <li><a href="/wiki/IEEE_P1906.1" title="IEEE P1906.1">P1906.1</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Superseded</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/IEEE_754-1985" title="IEEE 754-1985">754-1985</a></li> <li><a href="/wiki/Software_requirements_specification" title="Software requirements specification">830</a></li> <li><a href="/wiki/IEEE_1219" class="mw-redirect" title="IEEE 1219">1219</a></li> <li><a href="/wiki/Software_requirements_specification" title="Software requirements specification">1233</a></li> <li><a href="/wiki/Concept_of_operations" title="Concept of operations">1362</a></li> <li><a href="/wiki/Verilog" title="Verilog">1364</a></li> <li><a href="/wiki/IEEE_1471" title="IEEE 1471">1471</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <dl><dt><i>See also</i></dt> <dd><a href="/wiki/IEEE_Standards_Association" title="IEEE Standards Association">IEEE Standards Association</a></dd> <dd><a href="/wiki/Category:IEEE_standards" title="Category:IEEE standards">Category:IEEE standards</a></dd></dl> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="IEC_standards" 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:List_of_IEC_standards" title="Template:List of IEC standards"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:List_of_IEC_standards" title="Template talk:List of IEC standards"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:List_of_IEC_standards" title="Special:EditPage/Template:List of IEC standards"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="IEC_standards" style="font-size:114%;margin:0 4em"><a href="/wiki/List_of_IEC_standards" title="List of IEC standards">IEC standards</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">IEC</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/IEC_60027" title="IEC 60027">60027</a></li> <li><a href="/wiki/IEC_60034" title="IEC 60034">60034</a></li> <li><a href="/wiki/IEC_60038" title="IEC 60038">60038</a></li> <li><a href="/wiki/RKM_code" title="RKM code">60062</a></li> <li><a href="/wiki/E_series_of_preferred_numbers" title="E series of preferred numbers">60063</a></li> <li><a href="/wiki/IEC_60068" title="IEC 60068">60068</a></li> <li><a href="/wiki/Comparative_Tracking_Index" title="Comparative Tracking Index">60112</a></li> <li><a href="/wiki/IEC_60228" title="IEC 60228">60228</a></li> <li><a href="/wiki/IEC_60269" title="IEC 60269">60269</a></li> <li><a href="/wiki/19-inch_rack" title="19-inch rack">60297</a></li> <li><a href="/wiki/IEC_60309" title="IEC 60309">60309</a></li> <li><a href="/wiki/IEC_60320" title="IEC 60320">60320</a></li> <li><a href="/wiki/IEC_60364" title="IEC 60364">60364</a></li> <li><a href="/wiki/IEC_60446" title="IEC 60446">60446</a></li> <li><a class="mw-selflink selflink">60559</a></li> <li><a href="/wiki/IEC_60601" title="IEC 60601">60601</a></li> <li><a href="/wiki/IEC_60870" title="IEC 60870">60870</a> <ul><li><a href="/wiki/IEC_60870-5" title="IEC 60870-5">60870-5</a></li> <li><a href="/wiki/IEC_60870-6" title="IEC 60870-6">60870-6</a></li></ul></li> <li><a href="/wiki/IEC_60906-1" title="IEC 60906-1">60906-1</a></li> <li><a href="/wiki/Compact_Disc_Digital_Audio" title="Compact Disc Digital Audio">60908</a></li> <li><a href="/wiki/IEC_60929" title="IEC 60929">60929</a></li> <li><a href="/wiki/AES3" title="AES3">60958</a></li> <li><a href="/w/index.php?title=IEC_60980-344&action=edit&redlink=1" class="new" title="IEC 60980-344 (page does not exist)">60980-344</a></li> <li><a href="/wiki/IEC_61030" title="IEC 61030">61030</a></li> <li><a href="/wiki/IEC_61131" title="IEC 61131">61131</a> <ul><li><a href="/wiki/IEC_61131-3" title="IEC 61131-3">61131-3</a></li> <li><a href="/wiki/IO-Link" title="IO-Link">61131-9</a></li></ul></li> <li><a href="/wiki/Fieldbus" title="Fieldbus">61158</a></li> <li><a href="/wiki/IEC_61162" title="IEC 61162">61162</a></li> <li><a href="/wiki/IEC_61334" title="IEC 61334">61334</a></li> <li><a href="/wiki/IEC_61355" title="IEC 61355">61355</a></li> <li><a href="/wiki/IEC_61360" title="IEC 61360">61360</a></li> <li><a href="/wiki/IEC_61400" title="IEC 61400">61400</a></li> <li><a href="/wiki/IEC_61499" title="IEC 61499">61499</a></li> <li><a href="/wiki/IEC_61508" title="IEC 61508">61508</a></li> <li><a href="/wiki/IEC_61511" title="IEC 61511">61511</a></li> <li><a href="/wiki/Fieldbus" title="Fieldbus">61784</a></li> <li><a href="/wiki/IEC_61850" title="IEC 61850">61850</a></li> <li><a href="/wiki/IEC_61851" title="IEC 61851">61851</a></li> <li><a href="/wiki/IEC_61883" title="IEC 61883">61883</a></li> <li><a href="/wiki/Battery_nomenclature" title="Battery nomenclature">61960</a></li> <li><a href="/wiki/IEC_61968" title="IEC 61968">61968</a></li> <li><a href="/wiki/IEC_61970" title="IEC 61970">61970</a></li> <li><a href="/wiki/IP-XACT" title="IP-XACT">62014-4</a></li> <li><a href="/wiki/Fieldbus" title="Fieldbus">62026</a></li> <li><a href="/wiki/IEC_62056" title="IEC 62056">62056</a></li> <li><a href="/wiki/IEC_62061" title="IEC 62061">62061</a></li> <li><a href="/wiki/IEC_62196" title="IEC 62196">62196</a></li> <li><a href="/wiki/EN_62262" title="EN 62262">62262</a></li> <li><a href="/wiki/IEC_62264" title="IEC 62264">62264</a></li> <li><a href="/wiki/IEC_62304" title="IEC 62304">62304</a></li> <li><a href="/wiki/IEC_62325" title="IEC 62325">62325</a></li> <li><a href="/wiki/IEC_62351" title="IEC 62351">62351</a></li> <li><a href="/wiki/AES47" title="AES47">62365</a></li> <li><a href="/wiki/IEC_62366" title="IEC 62366">62366</a></li> <li><a href="/wiki/IEC_62379" title="IEC 62379">62379</a></li> <li><a href="/wiki/Digital_Addressable_Lighting_Interface" title="Digital Addressable Lighting Interface">62386</a></li> <li><a href="/wiki/IEC_62455" title="IEC 62455">62455</a></li> <li><a href="/wiki/USB" title="USB">62680</a></li> <li><a href="/wiki/IEC_62682" title="IEC 62682">62682</a></li> <li><a href="/wiki/IEC_62700" title="IEC 62700">62700</a></li> <li><a href="/wiki/IEC_63110" title="IEC 63110">63110</a></li> <li><a href="/wiki/IEC_63119" title="IEC 63119">63119</a></li> <li><a href="/wiki/IEC_63382" title="IEC 63382">63382</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">ISO/IEC</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/ISO/IEC_646" title="ISO/IEC 646">646</a></li> <li><a href="/wiki/COBOL" title="COBOL">1989</a></li> <li><a href="/wiki/ISO/IEC_2022" title="ISO/IEC 2022">2022</a></li> <li><a href="/wiki/ISO/IEC_4909" title="ISO/IEC 4909">4909</a></li> <li><a href="/wiki/ISO/IEC_5218" title="ISO/IEC 5218">5218</a></li> <li><a href="/wiki/ANSI_escape_code" title="ANSI escape code">6429</a></li> <li><a href="/wiki/ISO/IEC_6523" title="ISO/IEC 6523">6523</a></li> <li><a href="/wiki/ISO/IEC_7810" title="ISO/IEC 7810">7810</a></li> <li><a href="/wiki/ISO/IEC_7811" title="ISO/IEC 7811">7811</a></li> <li><a href="/wiki/ISO/IEC_7812" title="ISO/IEC 7812">7812</a></li> <li><a href="/wiki/ISO/IEC_7813" title="ISO/IEC 7813">7813</a></li> <li><a href="/wiki/ISO/IEC_7816" title="ISO/IEC 7816">7816</a></li> <li><a href="/wiki/Graphical_Kernel_System" title="Graphical Kernel System">7942</a></li> <li><a href="/wiki/Open_Document_Architecture" title="Open Document Architecture">8613</a></li> <li><a href="/wiki/Computer_Graphics_Metafile" title="Computer Graphics Metafile">8632</a></li> <li><a href="/wiki/ISO/IEC_8652" title="ISO/IEC 8652">8652</a></li> <li><a href="/wiki/ISO/IEC_8859" title="ISO/IEC 8859">8859</a></li> <li><a href="/wiki/ISO/IEC_9126" title="ISO/IEC 9126">9126</a></li> <li><a href="/wiki/File_Allocation_Table" title="File Allocation Table">9293</a></li> <li><a href="/wiki/CHILL" title="CHILL">9496</a></li> <li><a href="/wiki/ISO/IEC_9529" title="ISO/IEC 9529">9529</a></li> <li><a href="/wiki/PHIGS" title="PHIGS">9592</a></li> <li><a href="/wiki/PHIGS" title="PHIGS">9593</a></li> <li><a href="/wiki/ANSI_C" title="ANSI C">9899</a></li> <li><a href="/wiki/POSIX" title="POSIX">9945</a></li> <li><a href="/wiki/ISO/IEC_9995" title="ISO/IEC 9995">9995</a></li> <li><a href="/wiki/ISO/IEC_10021" title="ISO/IEC 10021">10021</a></li> <li><a href="/wiki/ISO/IEC_10116" title="ISO/IEC 10116">10116</a></li> <li><a href="/wiki/Guidelines_for_the_Definition_of_Managed_Objects" title="Guidelines for the Definition of Managed Objects">10165</a></li> <li><a href="/wiki/Document_Style_Semantics_and_Specification_Language" title="Document Style Semantics and Specification Language">10179</a></li> <li><a href="/wiki/Full_BASIC" title="Full BASIC">10279</a></li> <li><a href="/wiki/Universal_Coded_Character_Set" title="Universal Coded Character Set">10646</a></li> <li><a href="/wiki/ISO/IEC_10967" title="ISO/IEC 10967">10967</a></li> <li><a href="/wiki/MPEG-1" title="MPEG-1">11172</a></li> <li><a href="/wiki/ISO/IEC_11179" title="ISO/IEC 11179">11179</a></li> <li><a href="/wiki/ISO/IEC_11404" title="ISO/IEC 11404">11404</a></li> <li><a href="/wiki/JBIG" title="JBIG">11544</a></li> <li><a href="/wiki/ISO/IEC_11801" title="ISO/IEC 11801">11801</a></li> <li><a href="/wiki/ISO/IEC_12207" title="ISO/IEC 12207">12207</a></li> <li><a href="/wiki/Topic_map" title="Topic map">13250</a></li> <li><a href="/wiki/Universal_Disk_Format" title="Universal Disk Format">13346</a></li> <li><a href="/wiki/MHEG-5" title="MHEG-5">13522-5</a></li> <li><a href="/wiki/Z_notation" title="Z notation">13568</a></li> <li><a href="/wiki/ISLISP" title="ISLISP">13816</a></li> <li><a href="/wiki/MPEG-2" title="MPEG-2">13818</a></li> <li><a href="/wiki/ISO/IEC_14443" title="ISO/IEC 14443">14443</a></li> <li><a href="/wiki/MPEG-4" title="MPEG-4">14496</a></li> <li><a href="/wiki/ISO/IEC_14651" title="ISO/IEC 14651">14651</a></li> <li><a href="/wiki/C%2B%2B" title="C++">14882</a></li> <li><a href="/wiki/ISO/IEC_15288" title="ISO/IEC 15288">15288</a></li> <li><a href="/wiki/Ada_Semantic_Interface_Specification" title="Ada Semantic Interface Specification">15291</a></li> <li><a href="/wiki/Common_Criteria" title="Common Criteria">15408</a></li> <li><a href="/wiki/JPEG_2000" title="JPEG 2000">15444</a></li> <li><a href="/wiki/HTML" title="HTML">15445</a></li> <li><a href="/wiki/ISO/IEC_15504" title="ISO/IEC 15504">15504</a></li> <li><a href="/wiki/International_Standard_Identifier_for_Libraries_and_Related_Organizations" title="International Standard Identifier for Libraries and Related Organizations">15511</a></li> <li><a href="/wiki/ISO/IEC_15693" title="ISO/IEC 15693">15693</a></li> <li><a href="/wiki/ISO/IEC_15897" title="ISO/IEC 15897">15897</a></li> <li><a href="/wiki/MPEG-7" title="MPEG-7">15938</a></li> <li><a href="/wiki/ECMAScript" title="ECMAScript">16262</a></li> <li><a href="/wiki/Mixed_raster_content" title="Mixed raster content">16485</a></li> <li><a href="/wiki/ISO/IEC_17024" title="ISO/IEC 17024">17024</a></li> <li><a href="/wiki/ISO/IEC_17025" title="ISO/IEC 17025">17025</a></li> <li><a href="/wiki/QR_code" title="QR code">18004</a></li> <li><a href="/wiki/ISO/IEC_18014" title="ISO/IEC 18014">18014</a></li> <li><a href="/wiki/JPEG_XL" title="JPEG XL">18181</a></li> <li><a href="/wiki/ISO/IEC_19752" title="ISO/IEC 19752">19752</a></li> <li><a href="/wiki/RELAX_NG" title="RELAX NG">19757</a></li> <li><a href="/wiki/ISO/IEC_19770" title="ISO/IEC 19770">19770</a></li> <li><a href="/wiki/ISO/IEC_19788" title="ISO/IEC 19788">19788</a></li> <li><a href="/wiki/ISO/IEC_20000" title="ISO/IEC 20000">20000</a></li> <li><a href="/wiki/Open_Data_Protocol" title="Open Data Protocol">20802</a></li> <li><a href="/wiki/MPEG-21" title="MPEG-21">21000</a></li> <li><a href="/wiki/ISO/IEC_21827" title="ISO/IEC 21827">21827</a></li> <li><a href="/wiki/ECMAScript" title="ECMAScript">22275</a></li> <li><a href="/wiki/ECMAScript_for_XML" title="ECMAScript for XML">22537</a></li> <li><a href="/wiki/MPEG-A" title="MPEG-A">23000</a></li> <li><a href="/wiki/MPEG-D" title="MPEG-D">23003</a></li> <li><a href="/wiki/MPEG-H" title="MPEG-H">23008</a></li> <li><a href="/wiki/C_Sharp_(programming_language)" title="C Sharp (programming language)">23270</a></li> <li><a href="/wiki/Linux_Standard_Base" title="Linux Standard Base">23360</a></li> <li><a href="/wiki/Common_Logic" title="Common Logic">24707</a></li> <li><a href="/wiki/ISO/IEC_24727" title="ISO/IEC 24727">24727</a></li> <li><a href="/wiki/ISO/IEC_24744" title="ISO/IEC 24744">24744</a></li> <li><a href="/wiki/Universal_Remote_Console" title="Universal Remote Console">24752</a></li> <li><a href="/wiki/OpenDocument" title="OpenDocument">26300</a></li> <li><a href="/wiki/ISO/IEC_27000" title="ISO/IEC 27000">27000</a></li> <li><a href="/wiki/ISO/IEC_27000-series" class="mw-redirect" title="ISO/IEC 27000-series">27000-series</a></li> <li><a href="/wiki/ISO/IEC_27002" title="ISO/IEC 27002">27002</a></li> <li><a href="/wiki/ISO/IEC_27040" title="ISO/IEC 27040">27040</a></li> <li>29110</li> <li><a href="/wiki/ISO/IEC_29119" title="ISO/IEC 29119">29119</a></li> <li><a href="/wiki/ISO/IEC_33001" title="ISO/IEC 33001">33001</a></li> <li><a href="/wiki/ISO/IEC_38500" title="ISO/IEC 38500">38500</a></li> <li><a href="/wiki/Graph_Query_Language" title="Graph Query Language">39075</a></li> <li><a href="/wiki/ISO/IEC_42010" title="ISO/IEC 42010">42010</a></li> <li><a href="/wiki/ISO/IEC_80000" title="ISO/IEC 80000">80000</a></li> <li><a href="/wiki/IEC_81346" title="IEC 81346">81346</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/International_Electrotechnical_Commission" title="International Electrotechnical Commission">International Electrotechnical Commission</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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=IEEE_754&oldid=1254909127">https://en.wikipedia.org/w/index.php?title=IEEE_754&oldid=1254909127</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:Computer_arithmetic" title="Category:Computer arithmetic">Computer arithmetic</a></li><li><a href="/wiki/Category:IEEE_standards" title="Category:IEEE standards">IEEE standards</a></li><li><a href="/wiki/Category:Floating_point_types" title="Category:Floating point types">Floating point types</a></li><li><a href="/wiki/Category:Binary_arithmetic" title="Category:Binary arithmetic">Binary arithmetic</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:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_May_2019" title="Category:Use dmy dates from May 2019">Use dmy dates from May 2019</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_July_2024" title="Category:Wikipedia articles needing clarification from July 2024">Wikipedia articles needing clarification from July 2024</a></li><li><a href="/wiki/Category:Articles_containing_potentially_dated_statements_from_2019" title="Category:Articles containing potentially dated statements from 2019">Articles containing potentially dated statements from 2019</a></li><li><a href="/wiki/Category:All_articles_containing_potentially_dated_statements" title="Category:All articles containing potentially dated statements">All articles containing potentially dated statements</a></li><li><a href="/wiki/Category:CS1_maint:_numeric_names:_authors_list" title="Category:CS1 maint: numeric names: authors list">CS1 maint: numeric names: authors list</a></li><li><a href="/wiki/Category:CS1_location_test" title="Category:CS1 location test">CS1 location test</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on 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 2 November 2024, at 07:56<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=IEEE_754&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://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></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"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-6b7f745dd4-zzkzm","wgBackendResponseTime":144,"wgPageParseReport":{"limitreport":{"cputime":"1.093","walltime":"1.523","ppvisitednodes":{"value":5411,"limit":1000000},"postexpandincludesize":{"value":152668,"limit":2097152},"templateargumentsize":{"value":5395,"limit":2097152},"expansiondepth":{"value":15,"limit":100},"expensivefunctioncount":{"value":9,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":181877,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 1057.015 1 -total"," 26.84% 283.737 2 Template:Reflist"," 15.96% 168.713 21 Template:Cite_web"," 11.09% 117.258 1 Template:Floating-point"," 10.78% 113.985 1 Template:Sidebar"," 10.04% 106.085 1 Template:Short_description"," 6.83% 72.172 12 Template:Cite_book"," 6.68% 70.556 2 Template:Pagetype"," 5.65% 59.771 3 Template:Sfn"," 5.05% 53.409 2 Template:Sister_project"]},"scribunto":{"limitreport-timeusage":{"value":"0.571","limit":"10.000"},"limitreport-memusage":{"value":11134009,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"2019\"] = 1,\n [\"Augmented_arithmetic_operation\"] = 1,\n [\"CITEREFChen\"] = 1,\n [\"CITEREFCowlishaw2013\"] = 1,\n [\"CITEREFGay1990\"] = 1,\n [\"CITEREFGoldberg1991\"] = 1,\n [\"CITEREFHaasz\"] = 1,\n [\"CITEREFHerf2001\"] = 1,\n [\"CITEREFIEEE_7542008\"] = 1,\n [\"CITEREFIEEE_7542019\"] = 1,\n [\"CITEREFISO/IEC_JTC_1/SC_252011\"] = 1,\n [\"CITEREFISO/IEC_JTC_1/SC_252020\"] = 1,\n [\"CITEREFKahan1981\"] = 1,\n [\"CITEREFKahan1996\"] = 1,\n [\"CITEREFKahanDarcy2001\"] = 1,\n [\"CITEREFKlemens2014\"] = 1,\n [\"CITEREFPaxsonKahan1991\"] = 1,\n [\"CITEREFRiedyDemmel\"] = 1,\n [\"CITEREFSeverance1998\"] = 1,\n [\"Minimum_and_maximum_operation\"] = 1,\n [\"muller_et_al_pg_16\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 2,\n [\"Anchor\"] = 3,\n [\"As of\"] = 1,\n [\"Cbignore\"] = 1,\n [\"Citation\"] = 2,\n [\"Cite book\"] = 12,\n [\"Cite journal\"] = 5,\n [\"Cite video\"] = 1,\n [\"Cite web\"] = 21,\n [\"Clarify\"] = 1,\n [\"Commons category\"] = 1,\n [\"Efn\"] = 8,\n [\"Floating-point\"] = 1,\n [\"Further\"] = 1,\n [\"Harvnb\"] = 33,\n [\"IEEE standards\"] = 1,\n [\"List of IEC standards\"] = 1,\n [\"Main\"] = 3,\n [\"Math\"] = 12,\n [\"Mvar\"] = 1,\n [\"Nobr\"] = 2,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 4,\n [\"Nowrap begin\"] = 2,\n [\"Nowrap end\"] = 2,\n [\"Reflist\"] = 1,\n [\"Section link\"] = 1,\n [\"See also\"] = 2,\n [\"Sfn\"] = 3,\n [\"Short description\"] = 1,\n [\"Slink\"] = 1,\n [\"Snd\"] = 1,\n [\"Table alignment\"] = 1,\n [\"Use dmy dates\"] = 1,\n [\"Val\"] = 1,\n [\"Wikibooks\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-856t2","timestamp":"20241125133342","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"IEEE 754","url":"https:\/\/en.wikipedia.org\/wiki\/IEEE_754","sameAs":"http:\/\/www.wikidata.org\/entity\/Q828287","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q828287","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":"2003-12-17T01:35:20Z","dateModified":"2024-11-02T07:56:20Z","headline":"IEEE Standard for Floating-Point Arithmetic"}</script> </body> </html>