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| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.sf_erf.error_function"></a><a class="link" href="error_function.html" title="Error Functions">Error Functions</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_function.h0"></a> |
| <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> |
| </h5> |
| <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> |
| <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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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 14. Policies: Controlling Precision, Error Handling etc">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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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 14. Policies: Controlling Precision, Error Handling etc">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="../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> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_function.h1"></a> |
| <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> |
| </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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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 14. Policies: Controlling Precision, Error Handling etc">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.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/erf.svg" 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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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 14. Policies: Controlling Precision, Error Handling etc">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.svg"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../graphs/erfc.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_function.h2"></a> |
| <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> |
| </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="../relative_error.html#math_toolkit.relative_error.zero_error">effectively |
| zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.sf_erf.error_function.errors_in_the_function_erf_z"></a><p class="title"><b>Table 6.12. 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.sf_erf.error_function.errors_in_the_function_erfc_z"></a><p class="title"><b>Table 6.13. 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"><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> |
| <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="../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> |
| <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 © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| 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>) |
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