CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Minifloat - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"02aca7e1-373f-4e7e-853a-3e0cd6180039","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Minifloat","wgTitle":"Minifloat","wgCurRevisionId":1273110295,"wgRevisionId":1273110295,"wgArticleId":5334646,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["All articles with bare URLs for citations","Articles with bare URLs for citations from August 2024","Articles with PDF format bare URLs for citations","CS1 maint: bot: original URL status unknown","Articles with short description","Short description is different from Wikidata","Use dmy dates from January 2025","All articles with unsourced statements","Articles with unsourced statements from May 2023","Floating point types","Computer arithmetic"],"wgPageViewLanguage":"en", "wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Minifloat","wgRelevantArticleId":5334646,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q449270","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","mediawiki.page.gallery.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js", "ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","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"];</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%7Cmediawiki.page.gallery.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.15"> <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="Minifloat - 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/Minifloat"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Minifloat&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/Minifloat"> <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-Minifloat rootpage-Minifloat skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Minifloat" 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=Minifloat" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Minifloat" 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=Minifloat" 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-Notation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Notation</span> </div> </a> <ul id="toc-Notation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_8-bit_float_(1.4.3)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example_8-bit_float_(1.4.3)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Example 8-bit float (1.4.3)</span> </div> </a> <button aria-controls="toc-Example_8-bit_float_(1.4.3)-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 Example 8-bit float (1.4.3) subsection</span> </button> <ul id="toc-Example_8-bit_float_(1.4.3)-sublist" class="vector-toc-list"> <li id="toc-Representation_of_zero" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representation_of_zero"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Representation of zero</span> </div> </a> <ul id="toc-Representation_of_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">2.2</span> <span>Subnormal numbers</span> </div> </a> <ul id="toc-Subnormal_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normalized_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normalized_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Normalized numbers</span> </div> </a> <ul id="toc-Normalized_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Infinity</span> </div> </a> <ul id="toc-Infinity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Not_a_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Not_a_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Not a number</span> </div> </a> <ul id="toc-Not_a_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Table_of_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Table_of_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Table of values</span> </div> </a> <ul id="toc-Table_of_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Alternative_bias_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_bias_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Alternative bias values</span> </div> </a> <ul id="toc-Alternative_bias_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Different_bit_allocations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Different_bit_allocations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Different bit allocations</span> </div> </a> <ul id="toc-Different_bit_allocations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Arithmetic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Arithmetic</span> </div> </a> <button aria-controls="toc-Arithmetic-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 Arithmetic subsection</span> </button> <ul id="toc-Arithmetic-sublist" class="vector-toc-list"> <li id="toc-Addition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Addition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Addition</span> </div> </a> <ul id="toc-Addition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subtraction,_multiplication_and_division" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subtraction,_multiplication_and_division"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Subtraction, multiplication and division</span> </div> </a> <ul id="toc-Subtraction,_multiplication_and_division-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_sizes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_sizes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other sizes</span> </div> </a> <button aria-controls="toc-Other_sizes-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 Other sizes subsection</span> </button> <ul id="toc-Other_sizes-sublist" class="vector-toc-list"> <li id="toc-4_bits_and_fewer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#4_bits_and_fewer"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>4 bits and fewer</span> </div> </a> <ul id="toc-4_bits_and_fewer-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-In_embedded_devices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_embedded_devices"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In embedded devices</span> </div> </a> <ul id="toc-In_embedded_devices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Minifloat</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 3 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-3" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">3 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Minifloat" title="Minifloat – German" lang="de" hreflang="de" data-title="Minifloat" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%9F%E3%83%8B%E3%83%95%E3%83%AD%E3%83%BC%E3%83%88" title="ミニフロート – Japanese" lang="ja" hreflang="ja" data-title="ミニフロート" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%BF%B7%E4%BD%A0%E6%B5%AE%E9%BB%9E%E6%95%B8" title="迷你浮點數 – Chinese" lang="zh" hreflang="zh" data-title="迷你浮點數" 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/Q449270#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/Minifloat" 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:Minifloat" 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/Minifloat"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Minifloat&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=Minifloat&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/Minifloat"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Minifloat&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=Minifloat&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/Minifloat" 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/Minifloat" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-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=Minifloat&oldid=1273110295" 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=Minifloat&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=Minifloat&id=1273110295&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%2FMinifloat"><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%2FMinifloat"><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=Minifloat&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=Minifloat&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:Minifloats" hreflang="en"><span>Wikimedia Commons</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/Q449270" 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">Floating-point values coded as few bits</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a> <a href="/wiki/Computer_number_format" title="Computer number format">formats</a></th></tr><tr><th class="sidebar-heading"> <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a></th></tr><tr><td class="sidebar-content"> <ul><li>16-bit: <a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half</a> (binary16)</li> <li>32-bit: <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single</a> (binary32), <a href="/wiki/Decimal32_floating-point_format" title="Decimal32 floating-point format">decimal32</a></li> <li>64-bit: <a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double</a> (binary64), <a href="/wiki/Decimal64_floating-point_format" title="Decimal64 floating-point format">decimal64</a></li> <li>128-bit: <a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple</a> (binary128), <a href="/wiki/Decimal128_floating-point_format" title="Decimal128 floating-point format">decimal128</a></li> <li>256-bit: <a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple</a> (binary256)</li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Other</th></tr><tr><td class="sidebar-content"> <ul><li><a class="mw-selflink selflink">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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size: 120%;">Computer architecture bit widths</th></tr><tr><th class="sidebar-heading" style="background-color: #ccccff;"> <a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Bit</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/1-bit_computing" title="1-bit computing">1</a></li><li><a href="/wiki/4-bit_computing" title="4-bit computing">4</a></li><li><a href="/wiki/8-bit_computing" title="8-bit computing">8</a></li><li><a href="/wiki/12-bit_computing" title="12-bit computing">12</a></li><li><a href="/wiki/16-bit_computing" title="16-bit computing">16</a></li><li><a href="/wiki/18-bit_computing" title="18-bit computing">18</a></li><li><a href="/wiki/24-bit_computing" title="24-bit computing">24</a></li><li><a href="/wiki/26-bit_computing" title="26-bit computing">26</a></li><li><a href="/wiki/UNIVAC_490" title="UNIVAC 490">30</a></li><li><a href="/wiki/31-bit_computing" title="31-bit computing">31</a></li><li><a href="/wiki/32-bit_computing" title="32-bit computing">32</a></li><li><a href="/wiki/36-bit_computing" title="36-bit computing">36</a></li><li><a href="/wiki/45-bit_computing" title="45-bit computing">45</a></li><li><a href="/wiki/48-bit_computing" title="48-bit computing">48</a></li><li><a href="/wiki/60-bit_computing" title="60-bit computing">60</a></li><li><a href="/wiki/64-bit_computing" title="64-bit computing">64</a></li><li><a href="/wiki/128-bit_computing" title="128-bit computing">128</a></li><li><a href="/wiki/256-bit_computing" title="256-bit computing">256</a></li><li><a href="/wiki/512-bit_computing" title="512-bit computing">512</a></li><li><a href="/wiki/Bit_slicing" title="Bit slicing">bit slicing</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background-color: #ccccff;"> <a href="/wiki/Application_software" title="Application software">Application</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/8-bit_computing" title="8-bit computing">8</a></li><li><a href="/wiki/16-bit_computing#16-bit_application" title="16-bit computing">16</a></li><li><a href="/wiki/32-bit_computing#32-bit_application" title="32-bit computing">32</a></li><li><a href="/wiki/64-bit_computing#64-bit_applications" title="64-bit computing">64</a></li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background-color: #ccccff;"> <a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Binary floating-point</a> <a href="/wiki/Precision_(computer_science)" title="Precision (computer science)">precision</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">16</a> (×½)</li><li><a class="mw-selflink selflink">24</a></li><li><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">32</a> (×1)</li><li><a href="/wiki/Extended_precision" title="Extended precision">40</a></li><li><a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">64</a> (×2)</li><li><a href="/wiki/Extended_precision" title="Extended precision">80</a></li><li><a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">128</a> (×4)</li><li><a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">256</a> (×8)</li></ul></div></td> </tr><tr><th class="sidebar-heading" style="background-color: #ccccff;"> <a href="/wiki/Decimal_floating_point" title="Decimal floating point">Decimal floating-point</a> <a href="/wiki/Precision_(computer_science)" title="Precision (computer science)">precision</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Decimal32_floating-point_format" title="Decimal32 floating-point format">32</a></li><li><a href="/wiki/Decimal64_floating-point_format" title="Decimal64 floating-point format">64</a></li><li><a href="/wiki/Decimal128_floating-point_format" title="Decimal128 floating-point format">128</a></li></ul></div></td> </tr><tr><td class="sidebar-navbar"><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:Computer_architecture_bit_widths" title="Template:Computer architecture bit widths"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Computer_architecture_bit_widths" title="Template talk:Computer architecture bit widths"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Computer_architecture_bit_widths" title="Special:EditPage/Template:Computer architecture bit widths"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p> In <a href="/wiki/Computing" title="Computing">computing</a>, <b>minifloats</b> are <a href="/wiki/Floating-point" class="mw-redirect" title="Floating-point">floating-point</a> values represented with very few <a href="/wiki/Bit" title="Bit">bits</a>. This reduced precision makes them ill-suited for general-purpose numerical calculations, but they are useful for special purposes such as: </p> <ul><li>Computer graphics, where iterations are small and precision has aesthetic effects.<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></li> <li><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a>, which can be relatively insensitive to numeric precision. <a href="/wiki/Bfloat16" class="mw-redirect" title="Bfloat16">bfloat16</a> and fp8 are common formats.<sup id="cite_ref-fp8-ml_2-0" class="reference"><a href="#cite_note-fp8-ml-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <p>Additionally, they are frequently encountered as a pedagogical tool in computer-science courses to demonstrate the properties and structures of <a href="/wiki/Floating-point_number" class="mw-redirect" title="Floating-point number">floating-point</a> arithmetic and <a href="/wiki/IEEE_floating-point_standard" class="mw-redirect" title="IEEE floating-point standard">IEEE 754</a> numbers. </p><p>Minifloats with 16 <a href="/wiki/Bit" title="Bit">bits</a> are <a href="/wiki/Half_precision" class="mw-redirect" title="Half precision">half-precision</a> numbers (opposed to <a href="/wiki/Single_precision" class="mw-redirect" title="Single precision">single</a> and <a href="/wiki/Double_precision" class="mw-redirect" title="Double precision">double precision</a>). There are also minifloats with 8 bits or even fewer.<sup id="cite_ref-fp8-ml_2-1" class="reference"><a href="#cite_note-fp8-ml-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Minifloats can be designed following the principles of the <a href="/wiki/IEEE_floating-point_standard" class="mw-redirect" title="IEEE floating-point standard">IEEE 754</a> standard. In this case they must obey the (not explicitly written) rules for the frontier between <a href="/wiki/Subnormal_number" title="Subnormal number">subnormal and normal numbers</a> and must have special patterns for <a href="/wiki/Infinity" title="Infinity">infinity</a> and <a href="/wiki/NaN" title="NaN">NaN</a>. Normalized numbers are stored with a <a href="/wiki/Exponent_bias" title="Exponent bias">biased exponent</a>. The new revision of the standard, <a href="/wiki/IEEE_754-2008" class="mw-redirect" title="IEEE 754-2008">IEEE 754-2008</a>, has <a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">16-bit binary minifloats</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=1" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A minifloat is usually described using a tuple of four numbers, (<i>S</i>, <i>E</i>, <i>M</i>, <i>B</i>): </p> <ul><li><i>S</i> is the length of the sign field. It is usually either 0 or 1.</li> <li><i>E</i> is the length of the exponent field.</li> <li><i>M</i> is the length of the mantissa (significand) field.</li> <li><i>B</i> is the <a href="/wiki/Exponent_bias" title="Exponent bias">exponent bias</a>.</li></ul> <p>A minifloat format denoted by (<i>S</i>, <i>E</i>, <i>M</i>, <i>B</i>) is, therefore, <span class="texhtml"><i>S</i> + <i>E</i> + <i>M</i></span> bits long. The (<i>S</i>, <i>E</i>, <i>M</i>, <i>B</i>) notation can be converted to a <a href="/wiki/Floating-point_arithmetic#Range_of_floating-point_numbers" title="Floating-point arithmetic">(<i>B</i>, <i>P</i>, <i>L</i>, <i>U</i>)</a> format as <span class="texhtml">(2, <i>M</i> + 1, <i>B</i> + 1, 2<sup><i>S</i></sup> − <i>B</i>)</span> (with IEEE use of exponents). </p> <div class="mw-heading mw-heading2"><h2 id="Example_8-bit_float_(1.4.3)"><span id="Example_8-bit_float_.281.4.3.29"></span>Example 8-bit float (1.4.3)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=2" title="Edit section: Example 8-bit float (1.4.3)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="text-align: center"> <caption>Layout of an example 8-bit minifloat (1.4.3) </caption> <tbody><tr> <th width="12.5%">sign </th> <th width="50%" colspan="4">exponent </th> <th width="37.5%" colspan="3">significand </th></tr> <tr> <td style="background: #d5ffff">0 </td> <td style="background: #a4ffb4">0 </td> <td style="background: #a4ffb4">0 </td> <td style="background: #a4ffb4">0 </td> <td style="background: #a4ffb4">0 </td> <td style="background: #ffb2b4">0 </td> <td style="background: #ffb2b4">0 </td> <td style="background: #ffb2b4">0 </td></tr></tbody></table> <p>A minifloat in 1 byte (8 bit) with 1 sign bit, 4 exponent bits and 3 significand bits (in short, a 1.4.3 minifloat) is demonstrated here. The <a href="/wiki/Exponent_bias" title="Exponent bias">exponent bias</a> is defined as 7 to center the values around 1 to match other IEEE 754 floats<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><sup id="cite_ref-Bryant_2010_4-0" class="reference"><a href="#cite_note-Bryant_2010-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> so (for most values) the actual multiplier for exponent <span class="texhtml mvar" style="font-style:italic;">x</span> is <span class="texhtml">2<sup><i>x</i>−7</sup></span>. All IEEE 754 principles should be valid.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Numbers in a different base are marked as ...<sub>base</sub>, for example, 101<sub>2</sub> = 5. The bit patterns have spaces to visualize their parts. </p> <div class="mw-heading mw-heading3"><h3 id="Representation_of_zero">Representation of zero</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=3" title="Edit section: Representation of zero"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zero is represented as zero exponent with a zero mantissa. The zero exponent means zero is a subnormal number with a leading "0." prefix, and with the zero mantissa all bits after the decimal point are zero, meaning this value is interpreted as <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 0.000_{2}\times 2^{-6}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0.000</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.000_{2}\times 2^{-6}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a1723d8f46705f8795bd018ff6da7333bd217b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.947ex; height:3.009ex;" alt="{\displaystyle 0.000_{2}\times 2^{-6}=0}"></span>. Floating point numbers use a <a href="/wiki/Signed_zero" title="Signed zero">signed zero</a>, so <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 -0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6d0e9fffdcc10786302e1fb8e4ea3d031d1598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -0}"></span> is also available and is equal to positive <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 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>. </p> <pre>0 0000 000 = 0 1 0000 000 = −0 </pre> <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=Minifloat&action=edit&section=4" title="Edit section: Subnormal numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The significand is extended with "0." and the exponent value is treated as 1 higher like the least normalized number: </p> <pre>0 0000 001 = 0.001<sub>2</sub> × 2<sup>1 - 7</sup> = 0.125 × 2<sup>−6</sup> = 0.001953125 (least subnormal number) ... 0 0000 111 = 0.111<sub>2</sub> × 2<sup>1 - 7</sup> = 0.875 × 2<sup>−6</sup> = 0.013671875 (greatest subnormal number) </pre> <div class="mw-heading mw-heading3"><h3 id="Normalized_numbers">Normalized numbers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=5" title="Edit section: Normalized numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The significand is extended with "1.": </p> <pre>0 0001 000 = 1.000<sub>2</sub> × 2<sup>1 - 7</sup> = 1 × 2<sup>−6</sup> = 0.015625 (least normalized number) 0 0001 001 = 1.001<sub>2</sub> × 2<sup>1 - 7</sup> = 1.125 × 2<sup>−6</sup> = 0.017578125 ... 0 0111 000 = 1.000<sub>2</sub> × 2<sup>7 - 7</sup> = 1 × 2<sup>0</sup> = 1 0 0111 001 = 1.001<sub>2</sub> × 2<sup>7 - 7</sup> = 1.125 × 2<sup>0</sup> = 1.125 (least value above 1) ... 0 1110 000 = 1.000<sub>2</sub> × 2<sup>14 - 7</sup> = 1.000 × 2<sup>7</sup> = 128 0 1110 001 = 1.001<sub>2</sub> × 2<sup>14 - 7</sup> = 1.125 × 2<sup>7</sup> = 144 ... 0 1110 110 = 1.110<sub>2</sub> × 2<sup>14 - 7</sup> = 1.750 × 2<sup>7</sup> = 224 0 1110 111 = 1.111<sub>2</sub> × 2<sup>14 - 7</sup> = 1.875 × 2<sup>7</sup> = 240 (greatest normalized number) </pre> <div class="mw-heading mw-heading3"><h3 id="Infinity">Infinity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=6" title="Edit section: Infinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Infinity values have the highest exponent, with the mantissa set to zero. The sign bit can be either positive or negative. </p> <pre>0 1111 000 = +infinity 1 1111 000 = −infinity </pre> <div class="mw-heading mw-heading3"><h3 id="Not_a_number">Not a number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=7" title="Edit section: Not a number"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>NaN values have the highest exponent, with a non-zero value for the mantissa. A float with 1-bit sign and 3-bit mantissa has <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\times (2^{3}-1)=14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times (2^{3}-1)=14}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7c86951a42ca1a251fc3617a05d17c715af2e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.455ex; height:3.176ex;" alt="{\displaystyle 2\times (2^{3}-1)=14}"></span> NaN values. </p> <pre><i>s</i> 1111 <i>mmm</i> = NaN (if <i>mmm</i> ≠ 000) </pre> <div class="mw-heading mw-heading3"><h3 id="Table_of_values">Table of values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=8" title="Edit section: Table of values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is a chart of all possible values for this example 8-bit float. </p> <table class="wikitable" style="text-align:right; font-size:small"> <tbody><tr> <th> </th> <th>… 000</th> <th>… 001</th> <th>… 010</th> <th>… 011</th> <th>… 100</th> <th>… 101</th> <th>… 110</th> <th>… 111 </th></tr> <tr> <th>0 0000 … </th> <td>0</td> <td>0.001953125</td> <td>0.00390625</td> <td>0.005859375</td> <td>0.0078125</td> <td>0.009765625</td> <td>0.01171875</td> <td>0.013671875 </td></tr> <tr> <th>0 0001 … </th> <td>0.015625</td> <td>0.017578125</td> <td>0.01953125</td> <td>0.021484375</td> <td>0.0234375</td> <td>0.025390625</td> <td>0.02734375</td> <td>0.029296875 </td></tr> <tr> <th>0 0010 … </th> <td>0.03125</td> <td>0.03515625</td> <td>0.0390625</td> <td>0.04296875</td> <td>0.046875</td> <td>0.05078125</td> <td>0.0546875</td> <td>0.05859375 </td></tr> <tr> <th>0 0011 … </th> <td>0.0625</td> <td>0.0703125</td> <td>0.078125</td> <td>0.0859375</td> <td>0.09375</td> <td>0.1015625</td> <td>0.109375</td> <td>0.1171875 </td></tr> <tr> <th>0 0100 … </th> <td>0.125</td> <td>0.140625</td> <td>0.15625</td> <td>0.171875</td> <td>0.1875</td> <td>0.203125</td> <td>0.21875</td> <td>0.234375 </td></tr> <tr> <th>0 0101 … </th> <td>0.25</td> <td>0.28125</td> <td>0.3125</td> <td>0.34375</td> <td>0.375</td> <td>0.40625</td> <td>0.4375</td> <td>0.46875 </td></tr> <tr> <th>0 0110 … </th> <td>0.5</td> <td>0.5625</td> <td>0.625</td> <td>0.6875</td> <td>0.75</td> <td>0.8125</td> <td>0.875</td> <td>0.9375 </td></tr> <tr> <th>0 0111 … </th> <td>1</td> <td>1.125</td> <td>1.25</td> <td>1.375</td> <td>1.5</td> <td>1.625</td> <td>1.75</td> <td>1.875 </td></tr> <tr> <th>0 1000 … </th> <td>2</td> <td>2.25</td> <td>2.5</td> <td>2.75</td> <td>3</td> <td>3.25</td> <td>3.5</td> <td>3.75 </td></tr> <tr> <th>0 1001 … </th> <td>4</td> <td>4.5</td> <td>5</td> <td>5.5</td> <td>6</td> <td>6.5</td> <td>7</td> <td>7.5 </td></tr> <tr> <th>0 1010 … </th> <td>8</td> <td>9</td> <td>10</td> <td>11</td> <td>12</td> <td>13</td> <td>14</td> <td>15 </td></tr> <tr> <th>0 1011 … </th> <td>16</td> <td>18</td> <td>20</td> <td>22</td> <td>24</td> <td>26</td> <td>28</td> <td>30 </td></tr> <tr> <th>0 1100 … </th> <td>32</td> <td>36</td> <td>40</td> <td>44</td> <td>48</td> <td>52</td> <td>56</td> <td>60 </td></tr> <tr> <th>0 1101 … </th> <td>64</td> <td>72</td> <td>80</td> <td>88</td> <td>96</td> <td>104</td> <td>112</td> <td>120 </td></tr> <tr> <th>0 1110 … </th> <td>128</td> <td>144</td> <td>160</td> <td>176</td> <td>192</td> <td>208</td> <td>224</td> <td>240 </td></tr> <tr> <th>0 1111 … </th> <td>Inf</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN </td></tr> <tr> <th>1 0000 … </th> <td>−0</td> <td>−0.001953125</td> <td>−0.00390625</td> <td>−0.005859375</td> <td>−0.0078125</td> <td>−0.009765625</td> <td>−0.01171875</td> <td>−0.013671875 </td></tr> <tr> <th>1 0001 … </th> <td>−0.015625</td> <td>−0.017578125</td> <td>−0.01953125</td> <td>−0.021484375</td> <td>−0.0234375</td> <td>−0.025390625</td> <td>−0.02734375</td> <td>−0.029296875 </td></tr> <tr> <th>1 0010 … </th> <td>−0.03125</td> <td>−0.03515625</td> <td>−0.0390625</td> <td>−0.04296875</td> <td>−0.046875</td> <td>−0.05078125</td> <td>−0.0546875</td> <td>−0.05859375 </td></tr> <tr> <th>1 0011 … </th> <td>−0.0625</td> <td>−0.0703125</td> <td>−0.078125</td> <td>−0.0859375</td> <td>−0.09375</td> <td>−0.1015625</td> <td>−0.109375</td> <td>−0.1171875 </td></tr> <tr> <th>1 0100 … </th> <td>−0.125</td> <td>−0.140625</td> <td>−0.15625</td> <td>−0.171875</td> <td>−0.1875</td> <td>−0.203125</td> <td>−0.21875</td> <td>−0.234375 </td></tr> <tr> <th>1 0101 … </th> <td>−0.25</td> <td>−0.28125</td> <td>−0.3125</td> <td>−0.34375</td> <td>−0.375</td> <td>−0.40625</td> <td>−0.4375</td> <td>−0.46875 </td></tr> <tr> <th>1 0110 … </th> <td>−0.5</td> <td>−0.5625</td> <td>−0.625</td> <td>−0.6875</td> <td>−0.75</td> <td>−0.8125</td> <td>−0.875</td> <td>−0.9375 </td></tr> <tr> <th>1 0111 … </th> <td>−1</td> <td>−1.125</td> <td>−1.25</td> <td>−1.375</td> <td>−1.5</td> <td>−1.625</td> <td>−1.75</td> <td>−1.875 </td></tr> <tr> <th>1 1000 … </th> <td>−2</td> <td>−2.25</td> <td>−2.5</td> <td>−2.75</td> <td>−3</td> <td>−3.25</td> <td>−3.5</td> <td>−3.75 </td></tr> <tr> <th>1 1001 … </th> <td>−4</td> <td>−4.5</td> <td>−5</td> <td>−5.5</td> <td>−6</td> <td>−6.5</td> <td>−7</td> <td>−7.5 </td></tr> <tr> <th>1 1010 … </th> <td>−8</td> <td>−9</td> <td>−10</td> <td>−11</td> <td>−12</td> <td>−13</td> <td>−14</td> <td>−15 </td></tr> <tr> <th>1 1011 … </th> <td>−16</td> <td>−18</td> <td>−20</td> <td>−22</td> <td>−24</td> <td>−26</td> <td>−28</td> <td>−30 </td></tr> <tr> <th>1 1100 … </th> <td>−32</td> <td>−36</td> <td>−40</td> <td>−44</td> <td>−48</td> <td>−52</td> <td>−56</td> <td>−60 </td></tr> <tr> <th>1 1101 … </th> <td>−64</td> <td>−72</td> <td>−80</td> <td>−88</td> <td>−96</td> <td>−104</td> <td>−112</td> <td>−120 </td></tr> <tr> <th>1 1110 … </th> <td>−128</td> <td>−144</td> <td>−160</td> <td>−176</td> <td>−192</td> <td>−208</td> <td>−224</td> <td>−240 </td></tr> <tr> <th>1 1111 … </th> <td>−Inf</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN </td></tr></tbody></table> <p>There are only 242 different non-NaN values (if +0 and −0 are regarded as different), because 14 of the bit patterns represent NaNs. </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_bias_values">Alternative bias values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=9" title="Edit section: Alternative bias values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At these small sizes other bias values may be interesting, for instance a bias of -2 will make the numbers 0-16 have the same bit representation as the integers 0-16, with the loss that no non-integer values can be represented. </p> <pre>0 0000 000 = 0.000<sub>2</sub> × 2<sup>1 - (-2)</sup> = 0.0 × 2<sup>3</sup> = 0 (subnormal number) 0 0000 001 = 0.001<sub>2</sub> × 2<sup>1 - (-2)</sup> = 0.125 × 2<sup>3</sup> = 1 (subnormal number) 0 0000 111 = 0.111<sub>2</sub> × 2<sup>1 - (-2)</sup> = 0.875 × 2<sup>3</sup> = 7 (subnormal number) 0 0001 000 = 1.000<sub>2</sub> × 2<sup>1 - (-2)</sup> = 1.000 × 2<sup>3</sup> = 8 (normalized number) 0 0001 111 = 1.111<sub>2</sub> × 2<sup>1 - (-2)</sup> = 1.875 × 2<sup>3</sup> = 15 (normalized number) 0 0010 000 = 1.000<sub>2</sub> × 2<sup>2 - (-2)</sup> = 1.000 × 2<sup>4</sup> = 16 (normalized number) </pre> <div class="mw-heading mw-heading3"><h3 id="Different_bit_allocations">Different bit allocations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=10" title="Edit section: Different bit allocations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above describes an example 8-bit float with 1 sign bit, 4 exponent bits, and 3 significand bits, which is a nice balance. However, any bit allocation is possible. A format could choose to give more of the bits to the exponent if they need more dynamic range with less precision, or give more of the bits to the significand if they need more precision with less dynamic range. At the extreme, it is possible to allocate all bits to the exponent, or all but one of the bits to the significand, leaving the exponent with only one bit. The exponent must be given at least one bit, or else it no longer makes sense as a float, it just becomes a <a href="/wiki/Signed_number_representations" title="Signed number representations">signed number</a>. </p><p>Here is a chart of all possible values for a different 8-bit float with 1 sign bit, 3 exponent bits and 4 significand bits. Having 1 more significand bit than exponent bits ensures that the precision remains at least 0.5 throughout the entire range.<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> <table class="wikitable" style="text-align:right; font-size:67%"> <tbody><tr> <th> </th> <th>… 0000</th> <th>… 0001</th> <th>… 0010</th> <th>… 0011</th> <th>… 0100</th> <th>… 0101</th> <th>… 0110</th> <th>… 0111</th> <th>… 1000</th> <th>… 1001</th> <th>… 1010</th> <th>… 1011</th> <th>… 1100</th> <th>… 1101</th> <th>… 1110</th> <th>… 1111 </th></tr> <tr> <th>0 000 … </th> <td>0</td> <td>0.015625</td> <td>0.03125</td> <td>0.046875</td> <td>0.0625</td> <td>0.078125</td> <td>0.09375</td> <td>0.109375</td> <td>0.125</td> <td>0.140625</td> <td>0.15625</td> <td>0.171875</td> <td>0.1875</td> <td>0.203125</td> <td>0.21875</td> <td>0.234375 </td></tr> <tr> <th>0 001 … </th> <td>0.25</td> <td>0.265625</td> <td>0.28125</td> <td>0.296875</td> <td>0.3125</td> <td>0.328125</td> <td>0.34375</td> <td>0.359375</td> <td>0.375</td> <td>0.390625</td> <td>0.40625</td> <td>0.421875</td> <td>0.4375</td> <td>0.453125</td> <td>0.46875</td> <td>0.484375 </td></tr> <tr> <th>0 010 … </th> <td>0.5</td> <td>0.53125</td> <td>0.5625</td> <td>0.59375</td> <td>0.625</td> <td>0.65625</td> <td>0.6875</td> <td>0.71875</td> <td>0.75</td> <td>0.78125</td> <td>0.8125</td> <td>0.84375</td> <td>0.875</td> <td>0.90625</td> <td>0.9375</td> <td>0.96875 </td></tr> <tr> <th>0 011 … </th> <td>1</td> <td>1.0625</td> <td>1.125</td> <td>1.1875</td> <td>1.25</td> <td>1.3125</td> <td>1.375</td> <td>1.4375</td> <td>1.5</td> <td>1.5625</td> <td>1.625</td> <td>1.6875</td> <td>1.75</td> <td>1.8125</td> <td>1.875</td> <td>1.9375 </td></tr> <tr> <th>0 100 … </th> <td>2</td> <td>2.125</td> <td>2.25</td> <td>2.375</td> <td>2.5</td> <td>2.625</td> <td>2.75</td> <td>2.875</td> <td>3</td> <td>3.125</td> <td>3.25</td> <td>3.375</td> <td>3.5</td> <td>3.625</td> <td>3.75</td> <td>3.875 </td></tr> <tr> <th>0 101 … </th> <td>4</td> <td>4.25</td> <td>4.5</td> <td>4.75</td> <td>5</td> <td>5.25</td> <td>5.5</td> <td>5.75</td> <td>6</td> <td>6.25</td> <td>6.5</td> <td>6.75</td> <td>7</td> <td>7.25</td> <td>7.5</td> <td>7.75 </td></tr> <tr> <th>0 110 … </th> <td>8</td> <td>8.5</td> <td>9</td> <td>9.5</td> <td>10</td> <td>10.5</td> <td>11</td> <td>11.5</td> <td>12</td> <td>12.5</td> <td>13</td> <td>13.5</td> <td>14</td> <td>14.5</td> <td>15</td> <td>15.5 </td></tr> <tr> <th>0 111 … </th> <td>Inf</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN </td></tr> <tr> <th>1 000 … </th> <td>−0</td> <td>−0.015625</td> <td>−0.03125</td> <td>−0.046875</td> <td>−0.0625</td> <td>−0.078125</td> <td>−0.09375</td> <td>−0.109375</td> <td>−0.125</td> <td>−0.140625</td> <td>−0.15625</td> <td>−0.171875</td> <td>−0.1875</td> <td>−0.203125</td> <td>−0.21875</td> <td>−0.234375 </td></tr> <tr> <th>1 001 … </th> <td>−0.25</td> <td>−0.265625</td> <td>−0.28125</td> <td>−0.296875</td> <td>−0.3125</td> <td>−0.328125</td> <td>−0.34375</td> <td>−0.359375</td> <td>−0.375</td> <td>−0.390625</td> <td>−0.40625</td> <td>−0.421875</td> <td>−0.4375</td> <td>−0.453125</td> <td>−0.46875</td> <td>−0.484375 </td></tr> <tr> <th>1 010 … </th> <td>−0.5</td> <td>−0.53125</td> <td>−0.5625</td> <td>−0.59375</td> <td>−0.625</td> <td>−0.65625</td> <td>−0.6875</td> <td>−0.71875</td> <td>−0.75</td> <td>−0.78125</td> <td>−0.8125</td> <td>−0.84375</td> <td>−0.875</td> <td>−0.90625</td> <td>−0.9375</td> <td>−0.96875 </td></tr> <tr> <th>1 011 … </th> <td>−1</td> <td>−1.0625</td> <td>−1.125</td> <td>−1.1875</td> <td>−1.25</td> <td>−1.3125</td> <td>−1.375</td> <td>−1.4375</td> <td>−1.5</td> <td>−1.5625</td> <td>−1.625</td> <td>−1.6875</td> <td>−1.75</td> <td>−1.8125</td> <td>−1.875</td> <td>−1.9375 </td></tr> <tr> <th>1 100 … </th> <td>−2</td> <td>−2.125</td> <td>−2.25</td> <td>−2.375</td> <td>−2.5</td> <td>−2.625</td> <td>−2.75</td> <td>−2.875</td> <td>−3</td> <td>−3.125</td> <td>−3.25</td> <td>−3.375</td> <td>−3.5</td> <td>−3.625</td> <td>−3.75</td> <td>−3.875 </td></tr> <tr> <th>1 101 … </th> <td>−4</td> <td>−4.25</td> <td>−4.5</td> <td>−4.75</td> <td>−5</td> <td>−5.25</td> <td>−5.5</td> <td>−5.75</td> <td>−6</td> <td>−6.25</td> <td>−6.5</td> <td>−6.75</td> <td>−7</td> <td>−7.25</td> <td>−7.5</td> <td>−7.75 </td></tr> <tr> <th>1 110 … </th> <td>−8</td> <td>−8.5</td> <td>−9</td> <td>−9.5</td> <td>−10</td> <td>−10.5</td> <td>−11</td> <td>−11.5</td> <td>−12</td> <td>−12.5</td> <td>−13</td> <td>−13.5</td> <td>−14</td> <td>−14.5</td> <td>−15</td> <td>−15.5 </td></tr> <tr> <th>1 111 … </th> <td>−Inf</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN</td> <td>NaN </td></tr></tbody></table> <p>Tables like the above can be generated for any combination of SEMB (sign, exponent, mantissa/significand, and bias) values using a script <a rel="nofollow" class="external text" href="https://gist.github.com/aaronfranke/0d1217e521c4ec784d39e92b5f039115">in Python</a> or <a rel="nofollow" class="external text" href="https://gist.github.com/aaronfranke/8b2c39bfbe48068c574e4daf2710c875">in GDScript</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Arithmetic">Arithmetic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=11" title="Edit section: Arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Addition">Addition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=12" title="Edit section: Addition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:MinifloatAddition_1_3_2_3_72.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/MinifloatAddition_1_3_2_3_72.png/220px-MinifloatAddition_1_3_2_3_72.png" decoding="async" width="220" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/MinifloatAddition_1_3_2_3_72.png/330px-MinifloatAddition_1_3_2_3_72.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/MinifloatAddition_1_3_2_3_72.png/440px-MinifloatAddition_1_3_2_3_72.png 2x" data-file-width="538" data-file-height="521" /></a><figcaption>Addition of (1.3.2.3)-minifloats</figcaption></figure> <p>The graphic demonstrates the addition of even smaller (1.3.2.3)-minifloats with 6 bits. This floating-point system follows the rules of IEEE 754 exactly. NaN as operand produces always NaN results. Inf − Inf and (−Inf) + Inf results in NaN too (green area). Inf can be augmented and decremented by finite values without change. Sums with finite operands can give an infinite result (i.e. 14.0 + 3.0 = +Inf as a result is the cyan area, −Inf is the magenta area). The range of the finite operands is filled with the curves <i>x</i> + <i>y</i> = <i>c</i>, where <i>c</i> is always one of the representable float values (blue and red for positive and negative results respectively). </p> <div class="mw-heading mw-heading3"><h3 id="Subtraction,_multiplication_and_division"><span id="Subtraction.2C_multiplication_and_division"></span>Subtraction, multiplication and division</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=13" title="Edit section: Subtraction, multiplication and division"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The other arithmetic operations can be illustrated similarly: </p> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:MinifloatSubtraction_1_3_2_3_72.png" class="mw-file-description" title="Subtraction"><img alt="Subtraction" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/MinifloatSubtraction_1_3_2_3_72.png/120px-MinifloatSubtraction_1_3_2_3_72.png" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/MinifloatSubtraction_1_3_2_3_72.png/180px-MinifloatSubtraction_1_3_2_3_72.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/MinifloatSubtraction_1_3_2_3_72.png/240px-MinifloatSubtraction_1_3_2_3_72.png 2x" data-file-width="538" data-file-height="521" /></a></span></div> <div class="gallerytext">Subtraction</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:MinifloatMultiplication_1_3_2_3_72.png" class="mw-file-description" title="Multiplication"><img alt="Multiplication" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/MinifloatMultiplication_1_3_2_3_72.png/120px-MinifloatMultiplication_1_3_2_3_72.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/MinifloatMultiplication_1_3_2_3_72.png/180px-MinifloatMultiplication_1_3_2_3_72.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/MinifloatMultiplication_1_3_2_3_72.png/240px-MinifloatMultiplication_1_3_2_3_72.png 2x" data-file-width="538" data-file-height="523" /></a></span></div> <div class="gallerytext">Multiplication</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:MinifloatDivision_1_3_2_3_72.png" class="mw-file-description" title="Division"><img alt="Division" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/MinifloatDivision_1_3_2_3_72.png/120px-MinifloatDivision_1_3_2_3_72.png" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/MinifloatDivision_1_3_2_3_72.png/180px-MinifloatDivision_1_3_2_3_72.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/49/MinifloatDivision_1_3_2_3_72.png/240px-MinifloatDivision_1_3_2_3_72.png 2x" data-file-width="538" data-file-height="521" /></a></span></div> <div class="gallerytext">Division</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Other_sizes">Other sizes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=14" title="Edit section: Other sizes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Radeon_R300" class="mw-redirect" title="Radeon R300">Radeon R300</a> and <a href="/wiki/Radeon_R420" class="mw-redirect" title="Radeon R420">R420</a> GPUs used an "fp24" floating-point format with 7 bits of exponent and 16 bits (+1 implicit) of mantissa.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> "Full Precision" in Direct3D 9.0 is a proprietary 24-bit floating-point format. Microsoft's D3D9 (Shader Model 2.0) graphics <a href="/wiki/API" title="API">API</a> initially supported both FP24 (as in ATI's R300 chip) and FP32 (as in Nvidia's NV30 chip) as "Full Precision", as well as FP16 as "Partial Precision" for vertex and pixel shader calculations performed by the graphics hardware. </p><p>Khronos defines 10-bit and 11-bit float formats for use with Vulkan. Both formats have no sign bit and a 5-bit exponent. The 10-bit format has a 5-bit mantissa, and the 11-bit format has a 6-bit mantissa.<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><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>IEEE SA Working Group P3109 is currently working on a standard for 8-bit minifloats optimized for machine learning. The current draft defines not one format, but a family of 7 different formats, named "binary8pP", where "P" is a number from 1 to 7. These floats are designed to be compact and efficient, but do not follow the same semantics as other IEEE floats, and are missing features such as negative zero and multiple NaN values. Infinity is defined as both the exponent and significand having all ones, instead of other IEEE floats where the exponent is all ones and the significand is all zeroes.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="4_bits_and_fewer">4 bits and fewer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=15" title="Edit section: 4 bits and fewer"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The smallest possible float size that follows all IEEE principles, including normalized numbers, subnormal numbers, signed zero, signed infinity, and multiple NaN values, is a 4-bit float with 1-bit sign, 2-bit exponent, and 1-bit mantissa.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In the table below, the columns have different values for the sign and mantissa bits, and the rows are different values for the exponent bits. </p> <table class="wikitable" style="text-align:right; font-size:small"> <tbody><tr> <th> </th> <th>0 … 0</th> <th>0 … 1</th> <th>1 … 0</th> <th>1 … 1 </th></tr> <tr> <th>… 00 … </th> <td>0</td> <td>0.5</td> <td>−0</td> <td>−0.5 </td></tr> <tr> <th>… 01 … </th> <td>1</td> <td>1.5</td> <td>−1</td> <td>−1.5 </td></tr> <tr> <th>… 10 … </th> <td>2</td> <td>3</td> <td>−2</td> <td>−3 </td></tr> <tr> <th>… 11 … </th> <td>Inf</td> <td>NaN</td> <td>−Inf</td> <td>NaN </td></tr></tbody></table> <p>If normalized numbers are not required, the size can be reduced to 3-bit by reducing the exponent down to 1. </p> <table class="wikitable" style="text-align:right; font-size:small"> <tbody><tr> <th> </th> <th>0 … 0</th> <th>0 … 1</th> <th>1 … 0</th> <th>1 … 1 </th></tr> <tr> <th>… 0 … </th> <td>0</td> <td>1</td> <td>−0</td> <td>−1 </td></tr> <tr> <th>… 1 … </th> <td>Inf</td> <td>NaN</td> <td>−Inf</td> <td>NaN </td></tr></tbody></table> <p>In situations where the sign bit can be excluded, each of the above examples can be reduced by 1 bit further, keeping only the left half of the above tables. A 2-bit float with 1-bit exponent and 1-bit mantissa would only have 0, 1, Inf, NaN values. </p><p>If the mantissa is allowed to be 0-bit, a 1-bit float format would have a 1-bit exponent, and the only two values would be 0 and Inf. The exponent must be at least 1 bit or else it no longer makes sense as a float (it would just be a <a href="/wiki/Signed_number_representations" title="Signed number representations">signed number</a>). </p><p>4-bit floating point numbers — without the four special IEEE values — have found use in accelerating <a href="/wiki/Large_language_model" title="Large language model">large language models</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_embedded_devices">In embedded devices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=16" title="Edit section: In embedded devices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Minifloats are also commonly used in embedded devices,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2023)">citation needed</span></a></i>]</sup> especially on <a href="/wiki/Microcontrollers" class="mw-redirect" title="Microcontrollers">microcontrollers</a> where floating-point will need to be emulated in software. To speed up the computation, the mantissa typically occupies exactly half of the bits, so the register boundary automatically addresses the parts without shifting. </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=Minifloat&action=edit&section=17" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">Fixed-point arithmetic</a></li> <li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half-precision floating-point format</a></li> <li><a href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">bfloat16 floating-point format</a></li> <li><a href="/wiki/G.711#A-Law" title="G.711">G.711 A-Law</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minifloat&action=edit&section=18" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMocerinoCalimera2021" class="citation journal cs1">Mocerino, Luca; Calimera, Andrea (24 November 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fapp112311164">"AxP: A HW-SW Co-Design Pipeline for Energy-Efficient Approximated ConvNets via Associative Matching"</a>. <i>Applied Sciences</i>. <b>11</b> (23): 11164. <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/10.3390%2Fapp112311164">10.3390/app112311164</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Sciences&rft.atitle=AxP%3A+A+HW-SW+Co-Design+Pipeline+for+Energy-Efficient+Approximated+ConvNets+via+Associative+Matching&rft.volume=11&rft.issue=23&rft.pages=11164&rft.date=2021-11-24&rft_id=info%3Adoi%2F10.3390%2Fapp112311164&rft.aulast=Mocerino&rft.aufirst=Luca&rft.au=Calimera%2C+Andrea&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fapp112311164&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span></span> </li> <li id="cite_note-fp8-ml-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-fp8-ml_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-fp8-ml_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external free" href="https://developer.nvidia.com/blog/nvidia-arm-and-intel-publish-fp8-specification-for-standardization-as-an-interchange-format-for-ai/">https://developer.nvidia.com/blog/nvidia-arm-and-intel-publish-fp8-specification-for-standardization-as-an-interchange-format-for-ai/</a> (joint announcement by Intel, NVIDIA, Arm); <a rel="nofollow" class="external free" href="https://arxiv.org/abs/2209.05433">https://arxiv.org/abs/2209.05433</a> (preprint paper jointly written by researchers from aforementioned 3 companies)</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"><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">IEEE half-precision</a> has 5 exponent bits with bias 15 (<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^{5-1}-1=15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{5-1}-1=15}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c9843e78dc1ccd6bb4871dff646861dddd1707" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.743ex; height:2.843ex;" alt="{\displaystyle 2^{5-1}-1=15}"></span>), <a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">IEEE single-precision</a> has 8 exponent bits with bias 127 (<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^{8-1}-1=127}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>127</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{8-1}-1=127}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4dba2fd2f8c7d7cae2cbe7c3a608dde232bad78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.906ex; height:2.843ex;" alt="{\displaystyle 2^{8-1}-1=127}"></span>), <a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">IEEE double-precision</a> has 11 exponent bits with bias 1023 (<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^{11-1}-1=1023}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1023</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{11-1}-1=1023}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8541d81e4043ebbbe0cfd8dac1a5ccf86a6e90a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\displaystyle 2^{11-1}-1=1023}"></span>), and <a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">IEEE quadruple-precision</a> has 15 exponent bits with bias 16383 (<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^{15-1}-1=16383}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>16383</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{15-1}-1=16383}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc604a73a7f1d0c65f2851acc162a488f249197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.053ex; height:2.843ex;" alt="{\displaystyle 2^{15-1}-1=16383}"></span>). See the <a href="/wiki/Exponent_bias" title="Exponent bias">Exponent bias</a> article for more detail.</span> </li> <li id="cite_note-Bryant_2010-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bryant_2010_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'HallaronBryant2010" class="citation book cs1">O'Hallaron, David R.; <a href="/wiki/Randal_E._Bryant" class="mw-redirect" title="Randal E. Bryant">Bryant, Randal E.</a> (2010). <i>Computer systems: a programmer's perspective</i> (2 ed.). Boston, Massachusetts, USA: <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-610804-7" title="Special:BookSources/978-0-13-610804-7"><bdi>978-0-13-610804-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+systems%3A+a+programmer%27s+perspective&rft.place=Boston%2C+Massachusetts%2C+USA&rft.edition=2&rft.pub=Prentice+Hall&rft.date=2010&rft.isbn=978-0-13-610804-7&rft.aulast=O%27Hallaron&rft.aufirst=David+R.&rft.au=Bryant%2C+Randal+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurch" class="citation web cs1">Burch, Carl. <a rel="nofollow" class="external text" href="http://www.cburch.com/books/float/">"Floating-point representation"</a>. Hendrix College<span class="reference-accessdate">. Retrieved <span class="nowrap">29 August</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Floating-point+representation&rft.pub=Hendrix+College&rft.aulast=Burch&rft.aufirst=Carl&rft_id=http%3A%2F%2Fwww.cburch.com%2Fbooks%2Ffloat%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" 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"><a rel="nofollow" class="external free" href="https://people.cs.umass.edu/~verts/cmpsci145/8-Bit_Floating_Point.pdf">https://people.cs.umass.edu/~verts/cmpsci145/8-Bit_Floating_Point.pdf</a> <sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Bare_URLs" title="Wikipedia:Bare URLs"><span title="A full citation of this PDF document is required to prevent link rot. (August 2024)">bare URL PDF</span></a></i>]</sup></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuck2005" class="citation cs2">Buck, Ian (13 March 2005), <a rel="nofollow" class="external text" href="http://developer.nvidia.com/gpugems/GPUGems2/gpugems2_chapter32.html">"Chapter 32. Taking the Plunge into GPU Computing"</a>, in Pharr, Matt (ed.), <i>GPU Gems</i>, Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-321-33559-7" title="Special:BookSources/0-321-33559-7"><bdi>0-321-33559-7</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">5 April</span> 2018</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+32.+Taking+the+Plunge+into+GPU+Computing&rft.btitle=GPU+Gems&rft.pub=Addison-Wesley&rft.date=2005-03-13&rft.isbn=0-321-33559-7&rft.aulast=Buck&rft.aufirst=Ian&rft_id=http%3A%2F%2Fdeveloper.nvidia.com%2Fgpugems%2FGPUGems2%2Fgpugems2_chapter32.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span>.</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 id="CITEREFGarrard" class="citation web cs1">Garrard, Andrew. <a rel="nofollow" class="external text" href="https://www.khronos.org/registry/DataFormat/specs/1.2/dataformat.1.2.html#10bitfp">"10.3. Unsigned 10-bit floating-point numbers"</a>. <i>Khronos Data Format Specification v1.2 rev 1</i>. Khronos Group<span class="reference-accessdate">. Retrieved <span class="nowrap">10 August</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Khronos+Data+Format+Specification+v1.2+rev+1&rft.atitle=10.3.+Unsigned+10-bit+floating-point+numbers&rft.aulast=Garrard&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fwww.khronos.org%2Fregistry%2FDataFormat%2Fspecs%2F1.2%2Fdataformat.1.2.html%2310bitfp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarrard" class="citation web cs1">Garrard, Andrew. <a rel="nofollow" class="external text" href="https://www.khronos.org/registry/DataFormat/specs/1.2/dataformat.1.2.html#11bitfp">"10.2. Unsigned 11-bit floating-point numbers"</a>. <i>Khronos Data Format Specification v1.2 rev 1</i>. Khronos Group<span class="reference-accessdate">. Retrieved <span class="nowrap">10 August</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Khronos+Data+Format+Specification+v1.2+rev+1&rft.atitle=10.2.+Unsigned+11-bit+floating-point+numbers&rft.aulast=Garrard&rft.aufirst=Andrew&rft_id=https%3A%2F%2Fwww.khronos.org%2Fregistry%2FDataFormat%2Fspecs%2F1.2%2Fdataformat.1.2.html%2311bitfp&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</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/20240507011540/https://github.com/P3109/Public/blob/main/Shared%20Reports/P3109%20WG%20Interim%20Report.pdf">"IEEE Working Group P3109 Interim Report on 8-bit Binary Floating-point Formats"</a> <span class="cs1-format">(PDF)</span>. <i>GitHub</i>. IEEE Working Group P3109. Archived from the original on 7 May 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">7 May</span> 2024</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=IEEE+Working+Group+P3109+Interim+Report+on+8-bit+Binary+Floating-point+Formats&rft_id=https%3A%2F%2Fgithub.com%2FP3109%2FPublic%2Fblob%2Fmain%2FShared%2520Reports%2FP3109%2520WG%2520Interim%2520Report.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: bot: original URL status unknown (<a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">link</a>)</span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShaneyfelt" class="citation web cs1">Shaneyfelt, Dr. Ted. <a rel="nofollow" class="external text" href="https://www2.hawaii.edu/~tvs/gizmo.html">"Dr. Shaneyfelt's Floating Point Construction Gizmo"</a>. Dr. Ted Shaneyfelt<span class="reference-accessdate">. Retrieved <span class="nowrap">29 August</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Dr.+Shaneyfelt%27s+Floating+Point+Construction+Gizmo&rft.pub=Dr.+Ted+Shaneyfelt&rft.aulast=Shaneyfelt&rft.aufirst=Dr.+Ted&rft_id=https%3A%2F%2Fwww2.hawaii.edu%2F~tvs%2Fgizmo.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.intel.com/content/www/us/en/developer/articles/technical/accelerating-language-model-inference-on-your-pc.html">"Accelerate LLM Inference on Your Local PC"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Accelerate+LLM+Inference+on+Your+Local+PC&rft_id=https%3A%2F%2Fwww.intel.com%2Fcontent%2Fwww%2Fus%2Fen%2Fdeveloper%2Farticles%2Ftechnical%2Faccelerating-language-model-inference-on-your-pc.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span></span> </li> </ol></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunafo2016" class="citation web cs1">Munafo, Robert (15 May 2016). <a rel="nofollow" class="external text" href="http://www.mrob.com/pub/math/floatformats.html">"Survey of Floating-Point Formats"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">8 August</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Survey+of+Floating-Point+Formats&rft.date=2016-05-15&rft.aulast=Munafo&rft.aufirst=Robert&rft_id=http%3A%2F%2Fwww.mrob.com%2Fpub%2Fmath%2Ffloatformats.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinifloat" class="Z3988"></span></li></ul> </div> <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=Minifloat&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150702114550/http://oss.sgi.com/projects/ogl-sample/registry/ARB/half_float_pixel.txt">OpenGL half float pixel</a></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="Data_types177" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Data_types" title="Template:Data types"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Data_types" title="Template talk:Data types"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Data_types" title="Special:EditPage/Template:Data types"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_types177" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_type" title="Data type">Data types</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Units_of_information" title="Units of information">Uninterpreted</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit" title="Bit">Bit</a></li> <li><a href="/wiki/Byte" title="Byte">Byte</a></li> <li><a href="/wiki/Ternary_numeral_system" title="Ternary numeral system">Trit</a></li> <li><a href="/wiki/Ternary_numeral_system#Tryte" title="Ternary numeral system">Tryte</a></li> <li><a href="/wiki/Word_(computer_architecture)" title="Word (computer architecture)">Word</a></li> <li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arbitrary-precision_arithmetic" title="Arbitrary-precision arithmetic">Arbitrary-precision or bignum</a></li> <li><a href="/wiki/Complex_data_type" title="Complex data type">Complex</a></li> <li><a href="/wiki/Decimal_data_type" title="Decimal data type">Decimal</a></li> <li><a href="/wiki/Fixed-point_arithmetic" title="Fixed-point arithmetic">Fixed point</a></li> <li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating point</a> <ul><li>Reduced precision <ul><li><a class="mw-selflink selflink">Minifloat</a></li> <li><a href="/wiki/Half-precision_floating-point_format" title="Half-precision floating-point format">Half precision</a></li> <li><a href="/wiki/Bfloat16_floating-point_format" title="Bfloat16 floating-point format">bfloat16</a></li></ul></li> <li><a href="/wiki/Single-precision_floating-point_format" title="Single-precision floating-point format">Single precision</a></li> <li><a href="/wiki/Double-precision_floating-point_format" title="Double-precision floating-point format">Double precision</a></li> <li><a href="/wiki/Quadruple-precision_floating-point_format" title="Quadruple-precision floating-point format">Quadruple precision</a></li> <li><a href="/wiki/Octuple-precision_floating-point_format" title="Octuple-precision floating-point format">Octuple precision</a></li> <li><a href="/wiki/Extended_precision" title="Extended precision">Extended precision</a> <ul><li><a href="/wiki/Long_double" title="Long double">Long double</a></li></ul></li></ul></li> <li><a href="/wiki/Integer_(computer_science)" title="Integer (computer science)">Integer</a> <ul><li><a href="/wiki/Signedness" title="Signedness">signedness</a></li></ul></li> <li><a href="/wiki/Interval_arithmetic#Implementations" title="Interval arithmetic">Interval</a></li> <li><a href="/wiki/Rational_data_type" title="Rational data type">Rational</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Pointer_(computer_programming)" title="Pointer (computer programming)">Pointer</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Memory_address" title="Memory address">Address</a> <ul><li><a href="/wiki/Physical_address" title="Physical address">physical</a></li> <li><a href="/wiki/Virtual_address_space" title="Virtual address space">virtual</a></li></ul></li> <li><a href="/wiki/Reference_(computer_science)" title="Reference (computer science)">Reference</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plain_text" title="Plain text">Text</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Character_(computing)" title="Character (computing)">Character</a></li> <li><a href="/wiki/String_(computer_science)" title="String (computer science)">String</a> <ul><li><a href="/wiki/Null-terminated_string" title="Null-terminated string">null-terminated</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composite_data_type" title="Composite data type">Composite</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_data_type" title="Algebraic data type">Algebraic data type</a> <ul><li><a href="/wiki/Generalized_algebraic_data_type" title="Generalized algebraic data type">generalized</a></li></ul></li> <li><a href="/wiki/Array_data_type" class="mw-redirect" title="Array data type">Array</a></li> <li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Class_(computer_programming)" title="Class (computer programming)">Class</a></li> <li><a href="/wiki/Dependent_type" title="Dependent type">Dependent</a></li> <li><a href="/wiki/Intuitionistic_type_theory#Equality_type" title="Intuitionistic type theory">Equality</a></li> <li><a href="/wiki/Inductive_type" title="Inductive type">Inductive</a></li> <li><a href="/wiki/Intersection_type" title="Intersection type">Intersection</a></li> <li><a href="/wiki/List_(abstract_data_type)" title="List (abstract data type)">List</a></li> <li><a href="/wiki/Object_(computer_science)" title="Object (computer science)">Object</a> <ul><li><a href="/wiki/Metaobject" title="Metaobject">metaobject</a></li></ul></li> <li><a href="/wiki/Option_type" title="Option type">Option type</a></li> <li><a href="/wiki/Product_type" title="Product type">Product</a></li> <li><a href="/wiki/Record_(computer_science)" title="Record (computer science)">Record or Struct</a></li> <li><a href="/wiki/Refinement_type" title="Refinement type">Refinement</a></li> <li><a href="/wiki/Set_(abstract_data_type)" title="Set (abstract data type)">Set</a></li> <li><a href="/wiki/Union_type" title="Union type">Union</a> <ul><li><a href="/wiki/Tagged_union" title="Tagged union">tagged</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean_data_type" title="Boolean data type">Boolean</a></li> <li><a href="/wiki/Bottom_type" title="Bottom type">Bottom type</a></li> <li><a href="/wiki/Container_(abstract_data_type)" title="Container (abstract data type)">Collection</a></li> <li><a href="/wiki/Enumerated_type" title="Enumerated type">Enumerated type</a></li> <li><a href="/wiki/Exception_handling" title="Exception handling">Exception</a></li> <li><a href="/wiki/Function_type" title="Function type">Function type</a></li> <li><a href="/wiki/Opaque_data_type" title="Opaque data type">Opaque data type</a></li> <li><a href="/wiki/Recursive_data_type" title="Recursive data type">Recursive data type</a></li> <li><a href="/wiki/Semaphore_(programming)" title="Semaphore (programming)">Semaphore</a></li> <li><a href="/wiki/Stream_(computing)" title="Stream (computing)">Stream</a></li> <li><a href="/wiki/Strongly_typed_identifier" title="Strongly typed identifier">Strongly typed identifier</a></li> <li><a href="/wiki/Top_type" title="Top type">Top type</a></li> <li><a href="/wiki/Type_class" title="Type class">Type class</a></li> <li><a href="/wiki/Empty_type" title="Empty type">Empty type</a></li> <li><a href="/wiki/Unit_type" title="Unit type">Unit type</a></li> <li><a href="/wiki/Void_type" title="Void type">Void</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />topics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_data_type" title="Abstract data type">Abstract data type</a></li> <li><a href="/wiki/Boxing_(computer_science)" class="mw-redirect" title="Boxing (computer science)">Boxing</a></li> <li><a href="/wiki/Data_structure" title="Data structure">Data structure</a></li> <li><a href="/wiki/Generic_programming" title="Generic programming">Generic</a></li> <li><a href="/wiki/Kind_(type_theory)" title="Kind (type theory)">Kind</a> <ul><li><a href="/wiki/Metaclass" title="Metaclass">metaclass</a></li></ul></li> <li><a href="/wiki/Parametric_polymorphism" title="Parametric polymorphism">Parametric polymorphism</a></li> <li><a href="/wiki/Primitive_data_type" title="Primitive data type">Primitive data type</a></li> <li><a href="/wiki/Interface_(object-oriented_programming)" title="Interface (object-oriented programming)">Interface</a></li> <li><a href="/wiki/Subtyping" title="Subtyping">Subtyping</a></li> <li><a href="/wiki/Type_constructor" title="Type constructor">Type constructor</a></li> <li><a href="/wiki/Type_conversion" title="Type conversion">Type conversion</a></li> <li><a href="/wiki/Type_system" title="Type system">Type system</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li> <li><a href="/wiki/Variable_(computer_science)" title="Variable (computer science)">Variable</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Minifloat&oldid=1273110295">https://en.wikipedia.org/w/index.php?title=Minifloat&oldid=1273110295</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:Floating_point_types" title="Category:Floating point types">Floating point types</a></li><li><a href="/wiki/Category:Computer_arithmetic" title="Category:Computer arithmetic">Computer 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:All_articles_with_bare_URLs_for_citations" title="Category:All articles with bare URLs for citations">All articles with bare URLs for citations</a></li><li><a href="/wiki/Category:Articles_with_bare_URLs_for_citations_from_August_2024" title="Category:Articles with bare URLs for citations from August 2024">Articles with bare URLs for citations from August 2024</a></li><li><a href="/wiki/Category:Articles_with_PDF_format_bare_URLs_for_citations" title="Category:Articles with PDF format bare URLs for citations">Articles with PDF format bare URLs for citations</a></li><li><a href="/wiki/Category:CS1_maint:_bot:_original_URL_status_unknown" title="Category:CS1 maint: bot: original URL status unknown">CS1 maint: bot: original URL status unknown</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_January_2025" title="Category:Use dmy dates from January 2025">Use dmy dates from January 2025</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2023" title="Category:Articles with unsourced statements from May 2023">Articles with unsourced statements from May 2023</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 31 January 2025, at 19:41<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=Minifloat&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" lang="en" 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-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Minifloat</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>3 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-68cdd48dd4-lrk5z","wgBackendResponseTime":132,"wgPageParseReport":{"limitreport":{"cputime":"0.543","walltime":"0.721","ppvisitednodes":{"value":1645,"limit":1000000},"postexpandincludesize":{"value":58007,"limit":2097152},"templateargumentsize":{"value":2197,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":76410,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 566.293 1 -total"," 20.27% 114.790 2 Template:Sidebar"," 17.97% 101.775 1 Template:Floating-point"," 14.74% 83.470 1 Template:Short_description"," 14.54% 82.316 1 Template:Cite_journal"," 10.12% 57.315 1 Template:Data_types"," 9.35% 52.949 1 Template:Navbox"," 8.98% 50.877 2 Template:Pagetype"," 8.48% 48.016 7 Template:Cite_web"," 8.42% 47.669 2 Template:Fix"]},"scribunto":{"limitreport-timeusage":{"value":"0.329","limit":"10.000"},"limitreport-memusage":{"value":6024058,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-7878cd4448-rwsxb","timestamp":"20250211200703","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Minifloat","url":"https:\/\/en.wikipedia.org\/wiki\/Minifloat","sameAs":"http:\/\/www.wikidata.org\/entity\/Q449270","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q449270","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":"2006-05-29T10:59:54Z","dateModified":"2025-01-31T19:41:47Z","headline":"floating-point values represented with very few bits"}</script> </body> </html>