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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.sf_erf.error_function"></a><a class="link" href="error_function.html" title="Error Functions"> Error |
| Functions</a> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.sf_erf.error_function.synopsis"></a><h5> |
| <a name="id1105415"></a> |
| <a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.synopsis">Synopsis</a> |
| </h5> |
| <p> |
| |
| </p> |
| <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> |
| </pre> |
| <p> |
| </p> |
| <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> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| 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. |
| </p> |
| <p> |
| </p> |
| <p> |
| The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument |
| is optional and can be used to control the behaviour of the function: |
| how it handles errors, what level of precision to use etc. Refer to the |
| <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. |
| </p> |
| <p> |
| </p> |
| <a name="math_toolkit.special.sf_erf.error_function.description"></a><h5> |
| <a name="id1105798"></a> |
| <a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.description">Description</a> |
| </h5> |
| <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> |
| <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the <a href="http://en.wikipedia.org/wiki/Error_function" target="_top">error |
| function</a> <a href="http://functions.wolfram.com/GammaBetaErf/Erf/" target="_top">erf</a> |
| of z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/erf1.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/erf.png" align="middle"></span> |
| </p> |
| <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> |
| <a class="link" href="../../main_overview/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">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="Policies">Policy</a><span class="special">></span> |
| <a class="link" href="../../main_overview/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="Policies">Policy</a><span class="special">&);</span> |
| </pre> |
| <p> |
| Returns the complement of the <a href="http://functions.wolfram.com/GammaBetaErf/Erfc/" target="_top">error |
| function</a> of z: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/erf2.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/erfc.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.special.sf_erf.error_function.accuracy"></a><h5> |
| <a name="id1106186"></a> |
| <a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows the peak errors (in units of epsilon) found on |
| various platforms with various floating point types, along with comparisons |
| 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 C Library</a> |
| and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. |
| Unless otherwise specified any floating point type that is narrower than |
| the one shown will have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.sf_erf.error_function.errors_in_the_function_erf_z_"></a><p class="title"><b>Table 27. Errors In the Function erf(z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function erf(z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| z < 0.5 |
| </p> |
| </th> |
| <th> |
| <p> |
| 0.5 < z < 8 |
| </p> |
| </th> |
| <th> |
| <p> |
| z > 8 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| GSL Peak=2.0 Mean=0.3 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=1.1 |
| Mean=0.7 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.9 Mean=0.09 |
| </p> |
| <p> |
| GSL Peak=2.3 Mean=0.3 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=1.3 |
| Mean=0.2 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| GSL Peak=0 Mean=0 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=0 |
| Mean=0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-3.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.7 Mean=0.07 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0.9 Mean=0.2 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.9 Mean=0.2 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0.9 Mean=0.07 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0 Mean=0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.7 Mean=0.07 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0 Mean=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.9 Mean=0.1 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0.5 Mean=0.03 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0 Mean=0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.8 Mean=0.1 |
| </p> |
| <p> |
| <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> Lib Peak=0.9 Mean=0.2 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.9 Mean=0.1 |
| </p> |
| <p> |
| <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> Lib Peak=0.5 Mean=0.02 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> Lib Peak=0 Mean=0 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.sf_erf.error_function.errors_in_the_function_erfc_z_"></a><p class="title"><b>Table 28. Errors In the Function erfc(z)</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In the Function erfc(z)"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| z < 0.5 |
| </p> |
| </th> |
| <th> |
| <p> |
| 0.5 < z < 8 |
| </p> |
| </th> |
| <th> |
| <p> |
| z > 8 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.7 Mean=0.06 |
| </p> |
| <p> |
| GSL Peak=1.0 Mean=0.4 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=0.7 |
| Mean=0.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0.99 Mean=0.3 |
| </p> |
| <p> |
| GSL Peak=2.6 Mean=0.6 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=3.6 |
| Mean=0.7 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.0 Mean=0.2 |
| </p> |
| <p> |
| GSL Peak=3.9 Mean=0.4 |
| </p> |
| <p> |
| <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2.7 |
| Mean=0.4 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-3.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0 Mean=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.4 Mean=0.3 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=1.3 Mean=0.3 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.6 Mean=0.4 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=1.3 Mean=0.4 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0 Mean=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.4 Mean=0.3 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0 Mean=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.5 Mean=0.4 |
| </p> |
| <p> |
| <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a> |
| Peak=0 Mean=0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=0 Mean=0 |
| </p> |
| <p> |
| <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> Peak=0 Mean=0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.5 Mean=0.3 |
| </p> |
| <p> |
| <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> Peak=0.9 Mean=0.08 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.6 Mean=0.4 |
| </p> |
| <p> |
| <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX |
| C Library</a> Peak=0.9 Mean=0.1 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><a name="math_toolkit.special.sf_erf.error_function.testing"></a><h5> |
| <a name="id1108310"></a> |
| <a class="link" href="error_function.html#math_toolkit.special.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> |
| <a name="math_toolkit.special.sf_erf.error_function.implementation"></a><h5> |
| <a name="id1108339"></a> |
| <a class="link" href="error_function.html#math_toolkit.special.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> |
| <pre class="programlisting"><span class="identifier">erf</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span> |
| |
| <span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">2</span> <span class="special">-</span> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span> <span class="comment">// preferred when -z < -0.5 |
| </span> |
| <span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">1</span> <span class="special">+</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span> <span class="comment">// preferred when -0.5 <= -z < 0 |
| </span></pre> |
| <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="../../backgrounders/implementation.html#math_toolkit.backgrounders.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> |
| <pre class="programlisting"><span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">z</span> <span class="special">*</span> <span class="special">(</span><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">z</span><span class="special">));</span> |
| </pre> |
| <p> |
| where the rational approximation R(z*z) 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 [a, b) then: |
| </p> |
| <pre class="programlisting"><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> <span class="special">~</span> <span class="identifier">c</span> |
| </pre> |
| <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 erfc using: |
| </p> |
| <pre class="programlisting"><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="special">(</span><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> <span class="special">/</span> <span class="identifier">z</span><span class="special">;</span> |
| </pre> |
| <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> |
| <pre class="programlisting"><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="special">(</span><span class="identifier">C</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="number">1</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">;</span> |
| </pre> |
| </div> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<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>) |
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