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<a accesskey="p" href="../sf_erf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_erf.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="error_inv.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_erf.error_function"></a><a class="link" href="error_function.html" title="Error Function erf and complement erfc">Error Function erf
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and complement erfc</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.sf_erf.error_function.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_erf.error_function.synopsis"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">erf</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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||
|
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
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type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<h5>
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<a name="math_toolkit.sf_erf.error_function.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_erf.error_function.description"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.description">Description</a>
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</h5>
|
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns the <a href="http://en.wikipedia.org/wiki/Error_function" target="_top">error
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function</a> <a href="http://functions.wolfram.com/GammaBetaErf/Erf/" target="_top">erf</a>
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of z:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/erf1.svg"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/erf.svg" align="middle"></span>
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</p></blockquote></div>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<p>
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Returns the complement of the <a href="http://functions.wolfram.com/GammaBetaErf/Erfc/" target="_top">error
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function</a> of z:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/erf2.svg"></span>
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</p></blockquote></div>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../graphs/erfc.svg" align="middle"></span>
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</p></blockquote></div>
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<h5>
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<a name="math_toolkit.sf_erf.error_function.h2"></a>
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<span class="phrase"><a name="math_toolkit.sf_erf.error_function.accuracy"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.accuracy">Accuracy</a>
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</h5>
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<p>
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The following table shows the peak errors (in units of epsilon) found on
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various platforms with various floating-point types, along with comparisons
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to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>, <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>, <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
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C Library</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
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libraries. Unless otherwise specified any floating-point type that is narrower
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than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
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zero error</a>.
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</p>
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<div class="table">
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<a name="math_toolkit.sf_erf.error_function.table_erf"></a><p class="title"><b>Table 8.28. Error rates for erf</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for erf">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 7.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
|
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</p>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 14.1<br> Win32<br> double
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Erf Function: Small Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.925ε (Mean = 0.193ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 0.944ε (Mean = 0.191ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.944ε (Mean = 0.191ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.841ε (Mean = 0.0687ε)</span><br>
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.06ε (Mean = 0.319ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.925ε (Mean = 0.193ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.944ε (Mean = 0.194ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.996ε (Mean = 0.182ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.57ε (Mean = 0.317ε))
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Erf Function: Medium Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.5ε (Mean = 0.193ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 0.921ε (Mean = 0.0723ε))<br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.921ε (Mean = 0.0723ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1ε (Mean = 0.119ε)</span><br> <br>
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(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.31ε (Mean = 0.368ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1.5ε (Mean = 0.197ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 0.921ε (Mean = 0.071ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 1ε (Mean = 0.171ε)</span><br> <br>
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.19ε (Mean = 0.244ε))
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Erf Function: Large Values
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><cmath>:</em></span>
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Max = 0ε (Mean = 0ε))<br> (<span class="emphasis"><em><math.h>:</em></span>
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Max = 0ε (Mean = 0ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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2.1:</em></span> Max = 0ε (Mean = 0ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span>
|
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Max = 0ε (Mean = 0ε))
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</p>
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</td>
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<td>
|
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<p>
|
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span>
|
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Max = 0ε (Mean = 0ε))
|
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</p>
|
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</td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
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</div>
|
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<br class="table-break"><div class="table">
|
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<a name="math_toolkit.sf_erf.error_function.table_erfc"></a><p class="title"><b>Table 8.29. Error rates for erfc</b></p>
|
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<div class="table-contents"><table class="table" summary="Error rates for erfc">
|
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<colgroup>
|
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<col>
|
||
<col>
|
||
<col>
|
||
<col>
|
||
<col>
|
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</colgroup>
|
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<thead><tr>
|
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<th>
|
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</th>
|
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<th>
|
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<p>
|
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GNU C++ version 7.1.0<br> linux<br> long double
|
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</p>
|
||
</th>
|
||
<th>
|
||
<p>
|
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GNU C++ version 7.1.0<br> linux<br> double
|
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</p>
|
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</th>
|
||
<th>
|
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<p>
|
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Sun compiler version 0x5150<br> Sun Solaris<br> long double
|
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</p>
|
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</th>
|
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<th>
|
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<p>
|
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Microsoft Visual C++ version 14.1<br> Win32<br> double
|
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</p>
|
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</th>
|
||
</tr></thead>
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
Erf Function: Small Values
|
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</p>
|
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</td>
|
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<td>
|
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<p>
|
||
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><cmath>:</em></span>
|
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Max = 0ε (Mean = 0ε))<br> (<span class="emphasis"><em><math.h>:</em></span>
|
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Max = 0ε (Mean = 0ε))
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</p>
|
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</td>
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<td>
|
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<p>
|
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<span class="blue">Max = 0.658ε (Mean = 0.0537ε)</span><br>
|
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<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.01ε (Mean = 0.485ε))
|
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</p>
|
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</td>
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<td>
|
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<p>
|
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span>
|
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Max = 0ε (Mean = 0ε))
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em><math.h>:</em></span>
|
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Max = 0ε (Mean = 0ε))
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Erf Function: Medium Values
|
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</p>
|
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</td>
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<td>
|
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<p>
|
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<span class="blue">Max = 1.76ε (Mean = 0.365ε)</span><br> <br>
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(<span class="emphasis"><em><cmath>:</em></span> Max = 1.35ε (Mean = 0.307ε))<br>
|
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(<span class="emphasis"><em><math.h>:</em></span> Max = 1.35ε (Mean = 0.307ε))
|
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</p>
|
||
</td>
|
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<td>
|
||
<p>
|
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<span class="blue">Max = 0.983ε (Mean = 0.213ε)</span><br> <br>
|
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(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.64ε (Mean = 0.662ε))
|
||
</p>
|
||
</td>
|
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<td>
|
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<p>
|
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<span class="blue">Max = 1.76ε (Mean = 0.38ε)</span><br> <br>
|
||
(<span class="emphasis"><em><math.h>:</em></span> Max = 2.81ε (Mean = 0.739ε))
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 1.65ε (Mean = 0.373ε)</span><br> <br>
|
||
(<span class="emphasis"><em><math.h>:</em></span> Max = 2.36ε (Mean = 0.539ε))
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
Erf Function: Large Values
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 1.57ε (Mean = 0.542ε)</span><br> <br>
|
||
(<span class="emphasis"><em><cmath>:</em></span> Max = 1.26ε (Mean = 0.441ε))<br>
|
||
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.26ε (Mean = 0.441ε))
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 0.868ε (Mean = 0.147ε)</span><br> <br>
|
||
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.9ε (Mean = 0.472ε))
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 1.57ε (Mean = 0.564ε)</span><br> <br>
|
||
(<span class="emphasis"><em><math.h>:</em></span> Max = 4.91ε (Mean = 1.54ε))
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<span class="blue">Max = 1.14ε (Mean = 0.248ε)</span><br> <br>
|
||
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.84ε (Mean = 0.331ε))
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
</div>
|
||
<br class="table-break"><p>
|
||
The following error plots are based on an exhaustive search of the functions
|
||
domain, MSVC-16.7.1 at <code class="computeroutput"><span class="keyword">double</span></code>
|
||
precision, and GCC-10/Mingw64 for <code class="computeroutput"><span class="keyword">long</span>
|
||
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_double.png"></span>
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_long_double.png"></span>
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_float128.png"></span>
|
||
</p>
|
||
<p>
|
||
In the erfc case, error rates are almost entirely the error in calculating
|
||
<code class="computeroutput"><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">)</span></code>:
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_double.png"></span>
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_long_double.png"></span>
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_float128.png"></span>
|
||
</p>
|
||
<p>
|
||
Multiprecision error rates are similar, albeit with a much larger error in
|
||
calculating the exponent term for erfc:
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_cpp_bin_float_50.png"></span>
|
||
</p>
|
||
<p>
|
||
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_cpp_bin_float_50.png"></span>
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.sf_erf.error_function.h3"></a>
|
||
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.testing"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.testing">Testing</a>
|
||
</h5>
|
||
<p>
|
||
The tests for these functions come in two parts: basic sanity checks use
|
||
spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf" target="_top">Mathworld's
|
||
online evaluator</a>, while accuracy checks use high-precision test values
|
||
calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
|
||
and this implementation. Note that the generic and type-specific versions
|
||
of these functions use differing implementations internally, so this gives
|
||
us reasonably independent test data. Using our test data to test other "known
|
||
good" implementations also provides an additional sanity check.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.sf_erf.error_function.h4"></a>
|
||
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.implementation"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.implementation">Implementation</a>
|
||
</h5>
|
||
<p>
|
||
All versions of these functions first use the usual reflection formulas to
|
||
make their arguments positive:
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>erf(-z) = 1 - erf(z);</em></span>
|
||
</span>
|
||
</p></blockquote></div>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>erfc(-z) = 2 - erfc(z);</em></span>
|
||
// preferred when -z < -0.5</span>
|
||
</p></blockquote></div>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>erfc(-z) = 1 + erf(z);</em></span>
|
||
// preferred when -0.5 <= -z < 0</span>
|
||
</p></blockquote></div>
|
||
<p>
|
||
The generic versions of these functions are implemented in terms of the incomplete
|
||
gamma function.
|
||
</p>
|
||
<p>
|
||
When the significand (mantissa) size is recognised (currently for 53, 64
|
||
and 113-bit reals, plus single-precision 24-bit handled via promotion to
|
||
double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
|
||
by JM</a> are used.
|
||
</p>
|
||
<p>
|
||
For <code class="computeroutput"><span class="identifier">z</span> <span class="special"><=</span>
|
||
<span class="number">0.5</span></code> then a rational approximation to
|
||
erf is used, based on the observation that erf is an odd function and therefore
|
||
erf is calculated using:
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>erf(z) = z * (C + R(z*z));</em></span></span>
|
||
</p></blockquote></div>
|
||
<p>
|
||
where the rational approximation <span class="emphasis"><em>R(z*z)</em></span> is optimised
|
||
for absolute error: as long as its absolute error is small enough compared
|
||
to the constant C, then any round-off error incurred during the computation
|
||
of R(z*z) will effectively disappear from the result. As a result the error
|
||
for erf and erfc in this region is very low: the last bit is incorrect in
|
||
only a very small number of cases.
|
||
</p>
|
||
<p>
|
||
For <code class="computeroutput"><span class="identifier">z</span> <span class="special">></span>
|
||
<span class="number">0.5</span></code> we observe that over a small interval
|
||
[<span class="emphasis"><em>a, b)</em></span> then:
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>erfc(z) * exp(z*z) * z ~ c</em></span></span>
|
||
</p></blockquote></div>
|
||
<p>
|
||
for some constant c.
|
||
</p>
|
||
<p>
|
||
Therefore for <code class="computeroutput"><span class="identifier">z</span> <span class="special">></span>
|
||
<span class="number">0.5</span></code> we calculate <code class="computeroutput"><span class="identifier">erfc</span></code>
|
||
using:
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>erfc(z) = exp(-z*z) * (C + R(z -
|
||
B)) / z;</em></span></span>
|
||
</p></blockquote></div>
|
||
<p>
|
||
Again R(z - B) is optimised for absolute error, and the constant <code class="computeroutput"><span class="identifier">C</span></code> is the average of <code class="computeroutput"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span>
|
||
<span class="special">*</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span>
|
||
<span class="identifier">z</span></code> taken at the endpoints of the
|
||
range. Once again, as long as the absolute error in R(z - B) is small compared
|
||
to <code class="computeroutput"><span class="identifier">c</span></code> then <code class="computeroutput"><span class="identifier">c</span>
|
||
<span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span> <span class="special">-</span>
|
||
<span class="identifier">B</span><span class="special">)</span></code>
|
||
will be correctly rounded, and the error in the result will depend only on
|
||
the accuracy of the exp function. In practice, in all but a very small number
|
||
of cases, the error is confined to the last bit of the result. The constant
|
||
<code class="computeroutput"><span class="identifier">B</span></code> is chosen so that the left
|
||
hand end of the range of the rational approximation is 0.
|
||
</p>
|
||
<p>
|
||
For large <code class="computeroutput"><span class="identifier">z</span></code> over a range
|
||
[a, +∞] the above approximation is modified to:
|
||
</p>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="serif_italic"><span class="emphasis"><em>erfc(z) = exp(-z*z) * (C + R(1 /
|
||
z)) / z;</em></span></span>
|
||
</p></blockquote></div>
|
||
</div>
|
||
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
|
||
Walker and Xiaogang Zhang<p>
|
||
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
||
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
|
||
</p>
|
||
</div>
|
||
<hr>
|
||
<div class="spirit-nav">
|
||
<a accesskey="p" href="../sf_erf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_erf.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="error_inv.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
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|
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|
||
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|