195 lines
21 KiB
HTML
195 lines
21 KiB
HTML
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<title>std::fma, std::fmaf, std::fmal</title>
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<body class="mediawiki ltr sitedir-ltr ns-0 ns-subject page-cpp_numeric_math_fma skin-cppreference2 action-view cpp-navbar">
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<h1 id="firstHeading" class="firstHeading"><span style="font-size:0.7em; line-height:130%">std::</span>fma, <span style="font-size:0.7em; line-height:130%">std::</span>fmaf, <span style="font-size:0.7em; line-height:130%">std::</span>fmal</h1>
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<div id="contentSub"><span class="subpages">< <a href="cpp.html">cpp</a>‎ | <a href="cpp-numeric.html">numeric</a>‎ | <a href="cpp-numeric-math.html">math</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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<div>定义于头文件 <code><a href="cpp-header-cmath.html"><cmath></a></code></div>
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<td></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">float</span> fma <span class="br0">(</span> <span class="kw4">float</span> x, <span class="kw4">float</span> y, <span class="kw4">float</span> z <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">float</span> fmaf<span class="br0">(</span> <span class="kw4">float</span> x, <span class="kw4">float</span> y, <span class="kw4">float</span> z <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(1)</td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span> fma <span class="br0">(</span> <span class="kw4">double</span> x, <span class="kw4">double</span> y, <span class="kw4">double</span> z <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(2)</td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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</tr>
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<tr class="t-dcl t-since-cxx11">
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<div><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span> fma <span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> x, <span class="kw4">long</span> <span class="kw4">double</span> y, <span class="kw4">long</span> <span class="kw4">double</span> z <span class="br0">)</span><span class="sy4">;</span><br />
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<span class="kw4">long</span> <span class="kw4">double</span> fmal<span class="br0">(</span> <span class="kw4">long</span> <span class="kw4">double</span> x, <span class="kw4">long</span> <span class="kw4">double</span> y, <span class="kw4">long</span> <span class="kw4">double</span> z <span class="br0">)</span><span class="sy4">;</span></span></div>
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</td>
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<td>(3)</td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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<tr class="t-dcl t-since-cxx11">
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<div><span class="mw-geshi cpp source-cpp">Promoted fma <span class="br0">(</span> Arithmetic1 x, Arithmetic2 y, Arithmetic3 z <span class="br0">)</span><span class="sy4">;</span></span></div>
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<td>(4)</td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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</tr>
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<tr class="t-dcl t-since-cxx11">
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<div><span class="mw-geshi cpp source-cpp"><span class="co2">#define FP_FAST_FMA /* implementation-defined */</span></span></div>
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<td>(5)</td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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<div><span class="mw-geshi cpp source-cpp"><span class="co2">#define FP_FAST_FMAF /* implementation-defined */</span></span></div>
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<td>(6)</td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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</tr>
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<tr class="t-dcl t-since-cxx11">
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<td>
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<div><span class="mw-geshi cpp source-cpp"><span class="co2">#define FP_FAST_FMAL /* implementation-defined */</span></span></div>
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</td>
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<td>(7)</td>
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<td><span class="t-mark-rev t-since-cxx11">(C++11 起)</span></td>
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</tr>
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<tr class="t-dcl-sep">
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<td></td>
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<td></td>
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<td></td>
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<div class="t-li1"><span class="t-li">1-3)</span> 计算 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="br0">(</span>x<span class="sy2">*</span>y<span class="br0">)</span> <span class="sy2">+</span> z</span></span> ，如同用无限精度，而仅舍入一次到结果类型。</div>
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<div class="t-li1"><span class="t-li">4)</span> 所有 1-3) 所不覆盖的<a href="cpp-types-is_arithmetic.html">算术类型</a>参数组合的重载集或函数模板。若任何参数拥有<a href="cpp-types-is_integral.html">整数类型</a>，则将它转型为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 。若任何其他参数为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> ，则返回类型为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> ，否则为 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">double</span></span></span> 。</div>
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<div class="t-li1"><span class="t-li">5-7)</span> 若定义宏常量 <code>FP_FAST_FMAF</code> 、 <code>FP_FAST_FMA</code> 或 <code>FP_FAST_FMAL</code> ，则函数 <code>std::fma</code> 分别求值快于（并且精度高于） <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> 和 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="kw4">long</span> <span class="kw4">double</span></span></span> 参数的表达式 <span class="t-c"><span class="mw-geshi cpp source-cpp">x<span class="sy2">*</span>y<span class="sy2">+</span>z</span></span> 。若定义，则这些宏求值为整数 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="nu0">1</span></span></span> 。</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E6.95.B0">参数</span></h3>
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<table class="t-par-begin">
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<tr class="t-par">
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<td>x, y, z</td>
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<td>-</td>
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<td>浮点或<a href="cpp-types-is_integral.html">整数类型</a>值</td>
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</tr>
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</table>
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<h3><span class="mw-headline" id=".E8.BF.94.E5.9B.9E.E5.80.BC">返回值</span></h3>
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<p>若成功，则返回 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="br0">(</span>x<span class="sy2">*</span>y<span class="br0">)</span> <span class="sy2">+</span> z</span></span> 的值，如同计算为无限精度再舍入一次以适合目标类型（或者说是作为单次三元浮点运算计算）。</p>
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<p>若出现上溢所致的值域错误，则返回 <code>±HUGE_VAL</code> 、 <code>±HUGE_VALF</code> 或 <code>±HUGE_VALL</code> 。</p>
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<p>若出现下溢所致的值域错误，则返回（舍入后的）正确结果。</p>
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<h3><span class="mw-headline" id=".E9.94.99.E8.AF.AF.E5.A4.84.E7.90.86">错误处理</span></h3>
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<p>报告 <span class="t-lc"><a href="cpp-numeric-math-math_errhandling.html">math_errhandling</a></span> 中指定的错误。</p>
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<p>若实现支持 IEEE 浮点算术（ IEC 60559 ），则</p>
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<ul>
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<li>若 x 为零而 y 为无穷大或 x 为无穷大而 y 为零，且 z 非 NaN ，则返回 NaN 并引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span></li>
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<li>若 x 为零而 y 为无穷大或 x 为无穷大而 y 为零，且 z 为 NaN ，则返回 NaN 并可能引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span></li>
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<li>若 x*y 为准确的无穷大且 z 为带相反符号的无穷大，则返回 NaN 并引发 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span></li>
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<li>若 x 或 y 为 NaN ，则返回 NaN</li>
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<li>若 z 为 NaN ，且 x*y 不是 0*Inf 或 Inf*0 ，则返回 NaN （而无 FE_INVALID ）</li>
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</ul>
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<h3><span class="mw-headline" id=".E6.B3.A8.E6.84.8F">注意</span></h3>
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<p>此运算经常在硬件中实现为<a href="https://en.wikipedia.org/wiki/Multiply%E2%80%93accumulate_operation" class="extiw" title="enwiki:Multiply–accumulate operation">融合乘加</a> CPU 指令。若硬件支持，则期待定义相应的 <code>FP_FAST_FMA*</code> 宏，但多数实现即使在不定义这些宏时也利用该 CPU 指令。</p>
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<p><a rel="nofollow" class="external text" href="http://pubs.opengroup.org/onlinepubs/9699919799/functions/fma.html">POSIX 另外指定</a>被指定为返回 <span class="t-lc"><a href="cpp-numeric-fenv-FE_exceptions.html">FE_INVALID</a></span> 的情形是定义域错误。</p>
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<p>由于其无限的中间精度， <code>fma</code> 是其他正确舍入数学运算，如 <span class="t-lc"><a href="cpp-numeric-math-sqrt.html">std::sqrt</a></span> 或甚至除法（在 CPU 不支持的平台上，例如 Itanium ）的常用构建块。</p>
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<p>同所有浮点表达式，表达式 <span class="t-c"><span class="mw-geshi cpp source-cpp"><span class="br0">(</span>x<span class="sy2">*</span>y<span class="br0">)</span> <span class="sy2">+</span> z</span></span> 可编译为融合乘加，除非 <a href="cpp-preprocessor-impl.html">#pragma</a> <span class="t-c"><span class="mw-geshi cpp source-cpp">STDC FP_CONTRACT</span></span> 为关闭。</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 <iostream></span>
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<span class="co2">#include <iomanip></span>
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<span class="co2">#include <cmath></span>
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<span class="co2">#include <cfenv></span>
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<span class="co2">#pragma STDC FENV_ACCESS ON</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="co1">// 演示 fma 与内建运算符间的差别</span>
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<span class="kw4">double</span> in <span class="sy1">=</span> <span class="nu16">0.1</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"0.1 double is "</span> <span class="sy1"><<</span> <a href="cpp-io-manip-setprecision.html"><span class="kw1795">std::<span class="me2">setprecision</span></span></a><span class="br0">(</span><span class="nu0">23</span><span class="br0">)</span> <span class="sy1"><<</span> in
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<span class="sy1"><<</span> <span class="st0">" ("</span> <span class="sy1"><<</span> <a href="cpp-io-manip-fixed.html"><span class="kw1785">std::<span class="me2">hexfloat</span></span></a> <span class="sy1"><<</span> in <span class="sy1"><<</span> <a href="cpp-io-manip-fixed.html"><span class="kw1786">std::<span class="me2">defaultfloat</span></span></a> <span class="sy1"><<</span> <span class="st0">")<span class="es1">\n</span>"</span>
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<span class="sy1"><<</span> <span class="st0">"0.1*10 is 1.0000000000000000555112 (0x8.0000000000002p-3), "</span>
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<span class="sy1"><<</span> <span class="st0">"or 1.0 if rounded to double<span class="es1">\n</span>"</span><span class="sy4">;</span>
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<span class="kw4">double</span> expr_result <span class="sy1">=</span> <span class="nu16">0.1</span> <span class="sy2">*</span> <span class="nu0">10</span> <span class="sy2">-</span> <span class="nu0">1</span><span class="sy4">;</span>
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<span class="kw4">double</span> fma_result <span class="sy1">=</span> fma<span class="br0">(</span><span class="nu16">0.1</span>, <span class="nu0">10</span>, <span class="sy2">-</span><span class="nu0">1</span><span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"0.1 * 10 - 1 = "</span> <span class="sy1"><<</span> expr_result
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<span class="sy1"><<</span> <span class="st0">" : 1 subtracted after intermediate rounding<span class="es1">\n</span>"</span>
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<span class="sy1"><<</span> <span class="st0">"fma(0.1, 10, -1) = "</span> <span class="sy1"><<</span> <a href="cpp-io-manip-setprecision.html"><span class="kw1795">std::<span class="me2">setprecision</span></span></a><span class="br0">(</span><span class="nu0">6</span><span class="br0">)</span> <span class="sy1"><<</span> fma_result <span class="sy1"><<</span> <span class="st0">" ("</span>
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<span class="sy1"><<</span> <a href="cpp-io-manip-fixed.html"><span class="kw1785">std::<span class="me2">hexfloat</span></span></a> <span class="sy1"><<</span> fma_result <span class="sy1"><<</span> <a href="cpp-io-manip-fixed.html"><span class="kw1786">std::<span class="me2">defaultfloat</span></span></a> <span class="sy1"><<</span> <span class="st0">")<span class="es1">\n</span><span class="es1">\n</span>"</span><span class="sy4">;</span>
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<span class="co1">// fma 用于 double-double 算术</span>
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<span class="kw4">double</span> high <span class="sy1">=</span> <span class="nu16">0.1</span> <span class="sy2">*</span> <span class="nu0">10</span><span class="sy4">;</span>
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<span class="kw4">double</span> low <span class="sy1">=</span> fma<span class="br0">(</span><span class="nu16">0.1</span>, <span class="nu0">10</span>, <span class="sy2">-</span>high<span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"in double-double arithmetic, 0.1 * 10 is representable as "</span>
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<span class="sy1"><<</span> high <span class="sy1"><<</span> <span class="st0">" + "</span> <span class="sy1"><<</span> low <span class="sy1"><<</span> <span class="st0">"<span class="es1">\n</span><span class="es1">\n</span>"</span><span class="sy4">;</span>
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<span class="co1">// 错误处理</span>
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<a href="cpp-numeric-fenv-feclearexcept.html"><span class="kw1449">std::<span class="me2">feclearexcept</span></span></a><span class="br0">(</span><a href="cpp-numeric-fenv-FE_exceptions.html"><span class="kw1460">FE_ALL_EXCEPT</span></a><span class="br0">)</span><span class="sy4">;</span>
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<a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">"fma(+Inf, 10, -Inf) = "</span> <span class="sy1"><<</span> std<span class="sy4">::</span><span class="me2">fma</span><span class="br0">(</span><a href="cpp-numeric-math-INFINITY.html"><span class="kw1376">INFINITY</span></a>, <span class="nu0">10</span>, <span class="sy2">-</span><a href="cpp-numeric-math-INFINITY.html"><span class="kw1376">INFINITY</span></a><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="kw1">if</span><span class="br0">(</span><a href="cpp-numeric-fenv-fetestexcept.html"><span class="kw1450">std::<span class="me2">fetestexcept</span></span></a><span class="br0">(</span><a href="cpp-numeric-fenv-FE_exceptions.html"><span class="kw1463">FE_INVALID</span></a><span class="br0">)</span><span class="br0">)</span>
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<a href="cpp-io-cout.html"><span class="kw1757">std::<span class="me2">cout</span></span></a> <span class="sy1"><<</span> <span class="st0">" FE_INVALID raised<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">0.1 double is 0.10000000000000000555112 (0x1.999999999999ap-4)
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0.1*10 is 1.0000000000000000555112 (0x8.0000000000002p-3), or 1.0 if rounded to double
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0.1 * 10 - 1 = 0 : 1 subtracted after intermediate rounding
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fma(0.1, 10, -1) = 5.55112e-17 (0x1p-54)
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in double-double arithmetic, 0.1 * 10 is representable as 1 + 5.55112e-17
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fma(+Inf, 10, -Inf) = -nan
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FE_INVALID raised</pre></div>
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</div>
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</div>
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<h3><span class="mw-headline" id=".E5.8F.82.E9.98.85">参阅</span></h3>
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<table class="t-dsc-begin">
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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><a href="cpp-numeric-math-remainder.html"><span class="t-lines"><span>remainder</span><span>remainderf</span><span>remainderl</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>除法运算的有符号余数<br />
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<span class="t-mark">(函数)</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><a href="cpp-numeric-math-remquo.html"><span class="t-lines"><span>remquo</span><span>remquof</span><span>remquol</span></span></a></div>
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<div><span class="t-lines"><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span><span><span class="t-mark-rev t-since-cxx11">(C++11)</span></span></span></div>
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</div>
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</td>
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<td>除法运算的有符号余数和最后三个二进制位<br />
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<span class="t-mark">(函数)</span></td>
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</tr>
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