376 lines
23 KiB
HTML
376 lines
23 KiB
HTML
<!DOCTYPE html>
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<html lang="zh" dir="ltr" class="client-nojs" xmlns="http://www.w3.org/1999/xhtml" xml:lang="zh">
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<head><meta http-equiv="x-ua-compatible" content="ie=edge">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
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<title>数学常数</title>
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<link rel="stylesheet" href="ext.css" />
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<link rel="stylesheet" href="site_modules.css" />
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</head>
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_constants skin-cppreference2 action-view cpp-navbar">
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<div id="cpp-content-base">
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<div id="content"><a id="top"></a>
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<h1 id="firstHeading" class="firstHeading">数学常数</h1>
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<div id="bodyContent">
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<div id="contentSub"><span class="subpages">< <a href="cpp.html">cpp</a>‎ | <a href="cpp-numeric.html">numeric</a></span></div>
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<div id="mw-content-text" lang="zh" dir="ltr" class="mw-content-ltr" xml:lang="zh">
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<h3><span class="mw-headline" id=".E5.B8.B8.E6.95.B0_.28C.2B.2B20_.E8.B5.B7.29">常数 <span class="t-mark-rev t-since-cxx20">(C++20 起)</span></span></h3>
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<table class="t-dsc-begin">
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<tr class="t-dsc-header">
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<td colspan="2">
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<div>定义于头文件 <code><a href="cpp-header-numbers.html"><numbers></a></code></div>
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</td>
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</tr>
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<tr class="t-dsc-header">
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<td colspan="2">
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<div>定义于命名空间 <code>std::numbers</code></div>
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</td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>e_v</span></span></div>
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</div>
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</td>
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<td><a rel="nofollow" class="external text" href="https://en.wikipedia.org/wiki/E_(mathematical_constant)">数学常数 <span class="mjax" style="display:none">\(\small e\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">e</span></a><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>log2e_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\log_{2}e\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">log<span class="t-su t-su-b"><br />
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2</span>e</span><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>log10e_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\log_{10}e\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">log<span class="t-su t-su-b"><br />
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10</span>e</span><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>pi_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\pi\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">π</span><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>inv_pi_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\frac1\pi\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>1</td>
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</tr>
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<tr>
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<td>π</td>
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</tr>
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</table>
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<br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>inv_sqrtpi_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\frac1{\sqrt\pi}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>1</td>
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</tr>
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<tr>
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<td><span class="t-mrad"><span>√</span><span>π</span></span></td>
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</tr>
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</table>
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<br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>ln2_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\ln{2}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">ln 2</span><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>ln10_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\ln{10}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;">ln 10</span><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>sqrt2_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\sqrt2\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mrad"><span>√</span><span>2</span></span></span><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>sqrt3_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\sqrt3\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mrad"><span>√</span><span>3</span></span></span><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>inv_sqrt3_v</span></span></div>
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</div>
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</td>
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<td><span class="mjax" style="display:none">\(\frac1{\sqrt3}\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>1</td>
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</tr>
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<tr>
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<td><span class="t-mrad"><span>√</span><span>3</span></span></td>
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</tr>
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</table>
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<br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>egamma_v</span></span></div>
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</div>
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</td>
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<td><a rel="nofollow" class="external text" href="https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant">欧拉-马歇罗尼常数</a><br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span>phi_v</span></span></div>
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</div>
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</td>
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<td><a rel="nofollow" class="external text" href="https://en.wikipedia.org/wiki/Golden_ratio">黄金比 Φ 常数</a> (<span class="mjax" style="display:none">\(\frac{1+\sqrt5}2\)</span><span class="mjax-fallback texhtml" style="white-space: nowrap;"><span class="t-mfrac"></span></span>
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<table>
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<tr>
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<td>1 + <span class="t-mrad"><span>√</span><span>5</span></span></td>
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</tr>
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<tr>
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<td>2</td>
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</tr>
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</table>
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)<br />
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<span class="t-mark">(变量模板)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> e</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">e_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> log2e</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">log2e_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> log10e</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">log10e_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> pi</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">pi_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> inv_pi</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">inv_pi_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> inv_sqrtpi</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">inv_sqrtpi_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> ln2</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">ln2_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> ln10</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">ln10_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> sqrt2</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">sqrt2_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> sqrt3</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">sqrt3_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> inv_sqrt3</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">inv_sqrt3_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> egamma</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">egamma_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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<tr class="t-dsc">
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<td>
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<div class="t-dsc-member-div">
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<div><span class="t-lines"><span><span class="t-dsc-small">inline constexpr double</span> phi</span></span></div>
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</div>
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</td>
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<td><span class="t-c"><span class="mw-geshi cpp source-cpp">phi_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span></span></span><br />
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<span class="t-mark">(常量)</span></td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E6.B3.A8.E8.A7.A3">注解</span></h3>
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<p>实例化数学常数变量模板的初等模板的程序为谬构。</p>
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<p>标准库对所有浮点类型(即 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">float</span></span></span> 、 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 与 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> )特化数学常数变量模板。</p>
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<p>程序可以部分或显式特化数学常数变量模板,只要该特化依赖程序定义的类型。</p>
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<h3><span class="mw-headline" id=".E7.A4.BA.E4.BE.8B">示例</span></h3>
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<div class="t-example">
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<div class="t-example-live-link"></div>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="cpp source-cpp">
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<pre class="de1"><span class="co2">#include <cmath></span>
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<span class="co2">#include <iomanip></span>
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<span class="co2">#include <iostream></span>
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<span class="co2">#include <limits></span>
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<span class="co2">#include <numbers></span>
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<span class="co2">#include <string_view></span>
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<span class="kw4">int</span> main<span class="br0">(</span><span class="br0">)</span>
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<span class="br0">{</span>
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<span class="kw1">using</span> <span class="kw1">namespace</span> std<span class="sy4">::</span><span class="me2">numbers</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a>
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<span class="sy1"><<</span> <a href="cpp-numeric-math-pow.html"><span class="kw1332">std::<span class="me2">pow</span></span></a><span class="br0">(</span>e, ln2<span class="br0">)</span> <span class="sy2">/</span> <span class="nu0">2</span> <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> <a href="cpp-numeric-math-pow.html"><span class="kw1332">std::<span class="me2">pow</span></span></a><span class="br0">(</span><a href="cpp-numeric-math-cosh.html"><span class="kw1341">std::<span class="me2">cosh</span></span></a><span class="br0">(</span>pi<span class="br0">)</span>, <span class="nu0">2</span><span class="br0">)</span> <span class="sy2">-</span> <a href="cpp-numeric-math-pow.html"><span class="kw1332">std::<span class="me2">pow</span></span></a><span class="br0">(</span><a href="cpp-numeric-math-sinh.html"><span class="kw1340">std::<span class="me2">sinh</span></span></a><span class="br0">(</span>pi<span class="br0">)</span>, <span class="nu0">2</span><span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> <a href="cpp-numeric-math-sqrt.html"><span class="kw1329">std::<span class="me2">sqrt</span></span></a><span class="br0">(</span>pi<span class="br0">)</span> <span class="sy2">*</span> inv_sqrtpi <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> <a href="cpp-numeric-math-pow.html"><span class="kw1332">std::<span class="me2">pow</span></span></a><span class="br0">(</span>sqrt2 <span class="sy2">*</span> sqrt3, <span class="nu0">2</span><span class="br0">)</span> <span class="sy2">/</span> <span class="nu0">6</span> <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> sqrt3 <span class="sy2">*</span> inv_sqrt3 <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> log2e <span class="sy2">*</span> ln2 <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> log10e <span class="sy2">*</span> ln10 <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> pi <span class="sy2">*</span> inv_pi <span class="sy1"><<</span> <span class="st0">' '</span>
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<span class="sy1"><<</span> phi <span class="sy2">*</span> phi <span class="sy2">-</span> phi <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="kw4">auto</span> egamma_aprox <span class="sy1">=</span> <span class="br0">[</span><span class="br0">]</span> <span class="br0">{</span>
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<span class="kw4">long</span> <span class="kw4">double</span> s <span class="sy1">=</span> <span class="nu0">0</span>, m <span class="sy1">=</span> <span class="nu16">2.0</span><span class="sy4">;</span>
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<span class="kw1">for</span> <span class="br0">(</span><span class="kw4">unsigned</span> c <span class="sy1">=</span> <span class="nu0">2</span><span class="sy4">;</span> c <span class="sy3">!</span><span class="sy1">=</span> <span class="nu0">1</span><span class="st0">'000'</span><span class="nu8">000</span><span class="sy4">;</span> <span class="sy2">++</span>c, <span class="sy2">++</span>m<span class="br0">)</span> <span class="br0">{</span>
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<span class="kw4">const</span> <span class="kw4">long</span> <span class="kw4">double</span> t <span class="sy1">=</span> <a href="cpp-numeric-special_math-riemann_zeta.html"><span class="kw1438">std::<span class="me2">riemann_zeta</span></span></a><span class="br0">(</span>m<span class="br0">)</span> <span class="sy2">/</span> m<span class="sy4">;</span>
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<span class="br0">(</span>c <span class="sy3">&</span> <span class="nu0">1</span><span class="br0">)</span> <span class="sy1">==</span> <span class="nu0">0</span> <span class="sy4">?</span> s <span class="sy2">+</span><span class="sy1">=</span> t <span class="sy4">:</span> s <span class="sy2">-</span><span class="sy1">=</span> t<span class="sy4">;</span>
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<span class="br0">}</span>
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<span class="kw1">return</span> s<span class="sy4">;</span>
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<span class="br0">}</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <a href="cpp-io-manip-fixed.html"><span class="kw1784">std::<span class="me2">fixed</span></span></a> <span class="sy1"><<</span> <span class="br0">(</span>egamma_v<span class="sy1"><</span><span class="kw4">long</span> <span class="kw4">double</span><span class="sy1">></span> <span class="sy2">-</span> egamma_aprox<span class="br0">(</span><span class="br0">)</span><span class="br0">)</span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="kw4">constexpr</span> <a href="cpp-string-basic_string_view.html"><span class="kw1248">std::<span class="me2">string_view</span></span></a> γ <span class="br0">{</span><span class="st0">"0.577215664901532860606512090082402"</span><span class="br0">}</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1758">std::<span class="me2">cout</span></span></a>
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<span class="sy1"><<</span> <span class="st0">"γ as egamma_v<float> = "</span>
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<span class="sy1"><<</span> <a href="cpp-io-manip-setprecision.html"><span class="kw1796">std::<span class="me2">setprecision</span></span></a><span class="br0">(</span><a href="cpp-types-numeric_limits.html"><span class="kw344">std::<span class="me2">numeric_limits</span></span></a><span class="sy1"><</span><span class="kw4">float</span><span class="sy1">></span><span class="sy4">::</span><span class="me2">digits10</span> <span class="sy2">+</span> <span class="nu0">1</span><span class="br0">)</span>
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<span class="sy1"><<</span> egamma_v<span class="sy1"><</span><span class="kw4">float</span><span class="sy1">></span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span>
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<span class="sy1"><<</span> <span class="st0">"γ as egamma_v<double> = "</span>
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<span class="sy1"><<</span> <a href="cpp-io-manip-setprecision.html"><span class="kw1796">std::<span class="me2">setprecision</span></span></a><span class="br0">(</span><a href="cpp-types-numeric_limits.html"><span class="kw344">std::<span class="me2">numeric_limits</span></span></a><span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span><span class="sy4">::</span><span class="me2">digits10</span> <span class="sy2">+</span> <span class="nu0">1</span><span class="br0">)</span>
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<span class="sy1"><<</span> egamma_v<span class="sy1"><</span><span class="kw4">double</span><span class="sy1">></span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span>
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<span class="sy1"><<</span> <span class="st0">"γ as egamma_v<long double> = "</span>
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<span class="sy1"><<</span> <a href="cpp-io-manip-setprecision.html"><span class="kw1796">std::<span class="me2">setprecision</span></span></a><span class="br0">(</span><a href="cpp-types-numeric_limits.html"><span class="kw344">std::<span class="me2">numeric_limits</span></span></a><span class="sy1"><</span><span class="kw4">long</span> <span class="kw4">double</span><span class="sy1">></span><span class="sy4">::</span><span class="me2">digits10</span> <span class="sy2">+</span> <span class="nu0">1</span><span class="br0">)</span>
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<span class="sy1"><<</span> egamma_v<span class="sy1"><</span><span class="kw4">long</span> <span class="kw4">double</span><span class="sy1">></span> <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span>
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<span class="sy1"><<</span> <span class="st0">"γ with "</span> <span class="sy1"><<</span> γ.<span class="me1">length</span><span class="br0">(</span><span class="br0">)</span> <span class="sy2">-</span> <span class="nu0">1</span> <span class="sy1"><<</span> <span class="st0">" digits precision = "</span> <span class="sy1"><<</span> γ <span class="sy1"><<</span> <span class="st0">'<span class="es1">\n</span>'</span><span class="sy4">;</span>
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<span class="br0">}</span></pre></div>
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</div>
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<p>可能的输出:</p>
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<div dir="ltr" class="mw-geshi" style="text-align: left;">
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<div class="text source-text">
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<pre class="de1">1 1 1 1 1 1 1 1 1
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0.000001
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γ as egamma_v<float> = 0.5772157
|
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γ as egamma_v<double> = 0.5772156649015329
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γ as egamma_v<long double> = 0.5772156649015328606
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γ with 34 digits precision = 0.577215664901532860606512090082402</pre></div>
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