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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Appropriate Uses · DoubleFloats.jl</title><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">DoubleFloats.jl</a></span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Overview</a></li><li><a class="tocitem" href="../construction/">Construction</a></li><li><a class="tocitem" href="../characteristics/">Characteristics</a></li><li><a class="tocitem" href="../stringshowparse/">Show, String, Parse</a></li><li><a class="tocitem" href="../capabilities/">Capabilities</a></li><li><a class="tocitem" href="../special/">Special Functions</a></li><li><a class="tocitem" href="../linearalgebra/">Linear Algebra</a></li><li><a class="tocitem" href="../random/">Random Numbers</a></li><li class="is-active"><a class="tocitem" href>Appropriate Uses</a><ul class="internal"><li><a class="tocitem" href="#Double64"><span>Double64</span></a></li></ul></li><li><a class="tocitem" href="../references/">References</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Appropriate Uses</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Appropriate Uses</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaMath/DoubleFloats.jl/blob/main/docs/src/appropriate.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Types"><a class="docs-heading-anchor" href="#The-Types">The Types</a><a id="The-Types-1"></a><a class="docs-heading-anchor-permalink" href="#The-Types" title="Permalink"></a></h1><h2 id="Double64"><a class="docs-heading-anchor" href="#Double64">Double64</a><a id="Double64-1"></a><a class="docs-heading-anchor-permalink" href="#Double64" title="Permalink"></a></h2><p><code>Double64</code> is the accuracy stalwart.  Very good values are likely to result.</p><p>When used with reasonably sized values, the computations should limit the accrual of relative error to 10⋅𝘂², where 𝘂 is the relative rounding unit, the unit<em>in</em>the<em>last</em>place of the significand, often <code>eps(x)/2</code>. It is possible to accrue relative error steadily; so some experimentation has guided algorithmic selection. At worst, a sequence of 100<em>000 arithmetic and elementary operations might admit a relative error of 100</em>000 * 10⋅𝘂². The worst is unlikely.</p><p>One right way to use this type is</p><ol><li>map your input from Float64s to Double64s</li><li>compute with Double64s</li><li>map your resultant values from Double64s to Float64s</li></ol><p>The values obtained with cascaded arithimetic and composed elementary functions are reliable and their utility is desireable.</p><h3 id="what-it-is"><a class="docs-heading-anchor" href="#what-it-is">what it is</a><a id="what-it-is-1"></a><a class="docs-heading-anchor-permalink" href="#what-it-is" title="Permalink"></a></h3><p>What is that? 𝘂 is the last bit of the significand as a quantity, so a result that has a relative error of 1<em>000</em>000⋅𝘂 is as a report that the final <code>ceil(Int, log2(1_000_000))</code> bits of the result&#39;s significand are to be treated as inexactness rather than quantification. That means, for a Float64 value (with a 53-bit significand) 53-20 bits remain reliable, while ~38% of the precision has become unavailable to applications involve other&#39;s health, wealth, and well-being that are shepherded by responsible persons.</p><h3 id="how-it-is-used"><a class="docs-heading-anchor" href="#how-it-is-used">how it is used</a><a id="how-it-is-used-1"></a><a class="docs-heading-anchor-permalink" href="#how-it-is-used" title="Permalink"></a></h3><p>Were one working with Float32s (a 24-bit significand), the entire result would have become unreliable. With <code>Double</code>, the relative error accompanying any basic arithmetic operation is 10⋅𝘂² (10×𝘂^𝟐). It is reasonable to see this squaring as pulling in almost second significand&#39;s bits.  Our possibly accrued relative error does eat into the number of reliable bits in this second order significance. Nonetheless, if your use requires less than one million successive arithmetic operations, the result obtained is reliable as a Float64 or as a Float32.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../random/">« Random Numbers</a><a class="docs-footer-nextpage" href="../references/">References »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 5 February 2024 17:17">Monday 5 February 2024</span>. Using Julia version 1.6.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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		<!DOCTYPE html>
		<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Appropriate Uses · DoubleFloats.jl</title><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">DoubleFloats.jl</a></span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Overview</a></li><li><a class="tocitem" href="../construction/">Construction</a></li><li><a class="tocitem" href="../characteristics/">Characteristics</a></li><li><a class="tocitem" href="../stringshowparse/">Show, String, Parse</a></li><li><a class="tocitem" href="../capabilities/">Capabilities</a></li><li><a class="tocitem" href="../special/">Special Functions</a></li><li><a class="tocitem" href="../linearalgebra/">Linear Algebra</a></li><li><a class="tocitem" href="../random/">Random Numbers</a></li><li class="is-active"><a class="tocitem" href>Appropriate Uses</a><ul class="internal"><li><a class="tocitem" href="#Double64"><span>Double64</span></a></li></ul></li><li><a class="tocitem" href="../references/">References</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Appropriate Uses</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Appropriate Uses</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/JuliaMath/DoubleFloats.jl/blob/main/docs/src/appropriate.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="The-Types"><a class="docs-heading-anchor" href="#The-Types">The Types</a><a id="The-Types-1"></a><a class="docs-heading-anchor-permalink" href="#The-Types" title="Permalink"></a></h1><h2 id="Double64"><a class="docs-heading-anchor" href="#Double64">Double64</a><a id="Double64-1"></a><a class="docs-heading-anchor-permalink" href="#Double64" title="Permalink"></a></h2><p><code>Double64</code> is the accuracy stalwart. Very good values are likely to result.</p><p>When used with reasonably sized values, the computations should limit the accrual of relative error to 10⋅𝘂², where 𝘂 is the relative rounding unit, the unit<em>in</em>the<em>last</em>place of the significand, often <code>eps(x)/2</code>. It is possible to accrue relative error steadily; so some experimentation has guided algorithmic selection. At worst, a sequence of 100<em>000 arithmetic and elementary operations might admit a relative error of 100</em>000 * 10⋅𝘂². The worst is unlikely.</p><p>One right way to use this type is</p><ol><li>map your input from Float64s to Double64s</li><li>compute with Double64s</li><li>map your resultant values from Double64s to Float64s</li></ol><p>The values obtained with cascaded arithimetic and composed elementary functions are reliable and their utility is desireable.</p><h3 id="what-it-is"><a class="docs-heading-anchor" href="#what-it-is">what it is</a><a id="what-it-is-1"></a><a class="docs-heading-anchor-permalink" href="#what-it-is" title="Permalink"></a></h3><p>What is that? 𝘂 is the last bit of the significand as a quantity, so a result that has a relative error of 1<em>000</em>000⋅𝘂 is as a report that the final <code>ceil(Int, log2(1_000_000))</code> bits of the result's significand are to be treated as inexactness rather than quantification. That means, for a Float64 value (with a 53-bit significand) 53-20 bits remain reliable, while ~38% of the precision has become unavailable to applications involve other's health, wealth, and well-being that are shepherded by responsible persons.</p><h3 id="how-it-is-used"><a class="docs-heading-anchor" href="#how-it-is-used">how it is used</a><a id="how-it-is-used-1"></a><a class="docs-heading-anchor-permalink" href="#how-it-is-used" title="Permalink"></a></h3><p>Were one working with Float32s (a 24-bit significand), the entire result would have become unreliable. With <code>Double</code>, the relative error accompanying any basic arithmetic operation is 10⋅𝘂² (10×𝘂^𝟐). It is reasonable to see this squaring as pulling in almost second significand's bits. Our possibly accrued relative error does eat into the number of reliable bits in this second order significance. Nonetheless, if your use requires less than one million successive arithmetic operations, the result obtained is reliable as a Float64 or as a Float32.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../random/">« Random Numbers</a><a class="docs-footer-nextpage" href="../references/">References »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Monday 5 February 2024 17:17">Monday 5 February 2024</span>. Using Julia version 1.6.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>