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- <title>Gamma</title>
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- </div>
- <div class="section">
- <div class="titlepage"><div><div><h3 class="title">
- <a name="math_toolkit.sf_gamma.tgamma"></a><a class="link" href="tgamma.html" title="Gamma">Gamma</a>
- </h3></div></div></div>
- <h5>
- <a name="math_toolkit.sf_gamma.tgamma.h0"></a>
- <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.synopsis"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.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">gamma</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">tgamma</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 20. 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">tgamma</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 20. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</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 20. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- <span class="special">}}</span> <span class="comment">// namespaces</span>
- </pre>
- <h5>
- <a name="math_toolkit.sf_gamma.tgamma.h1"></a>
- <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.description"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.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">tgamma</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 20. 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">tgamma</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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- </pre>
- <p>
- Returns the "true gamma" (hence name tgamma) of value z:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/gamm1.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/tgamma.svg" align="middle"></span>
- </p></blockquote></div>
- <p>
- The final <a class="link" href="../../policy.html" title="Chapter 20. 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 20. Policies: Controlling Precision, Error Handling etc">policy
- documentation for more details</a>.
- </p>
- <p>
- The return type of this function 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 result is <code class="computeroutput"><span class="keyword">double</span></code>
- when T is an integer type, and T otherwise.
- </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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</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 20. 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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- </pre>
- <p>
- Returns <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span></code>.
- Internally the implementation does not make use of the addition and subtraction
- implied by the definition, leading to accurate results even for very small
- <code class="computeroutput"><span class="identifier">dz</span></code>.
- </p>
- <p>
- The return type of this function 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 result is <code class="computeroutput"><span class="keyword">double</span></code>
- when T is an integer type, and T otherwise.
- </p>
- <p>
- The final <a class="link" href="../../policy.html" title="Chapter 20. 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 20. Policies: Controlling Precision, Error Handling etc">policy
- documentation for more details</a>.
- </p>
- <h5>
- <a name="math_toolkit.sf_gamma.tgamma.h2"></a>
- <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.accuracy"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.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 other common 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_gamma.tgamma.table_tgamma"></a><p class="title"><b>Table 8.1. Error rates for tgamma</b></p>
- <div class="table-contents"><table class="table" summary="Error rates for tgamma">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- factorials
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 3.95ε (Mean = 0.783ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 314ε (Mean = 93.4ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.67ε (Mean = 0.617ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 1.66ε (Mean = 0.584ε))<br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 1.66ε (Mean = 0.584ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 172ε (Mean = 41ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.85ε (Mean = 0.566ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 3.17ε (Mean = 0.928ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- near 0
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 4.51ε (Mean = 1.92ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 1ε (Mean = 0.335ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2ε (Mean = 0.608ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 1ε (Mean = 0.376ε))<br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.376ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2ε (Mean = 0.647ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0.5ε (Mean = 0.0791ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.5ε (Mean = 0.635ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.405ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- near 1
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 4.41ε (Mean = 1.81ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 1ε (Mean = 0.32ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.51ε (Mean = 1.02ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 0.918ε (Mean = 0.203ε))<br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0.918ε (Mean = 0.203ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.01ε (Mean = 1.06ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.175ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.1ε (Mean = 0.59ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 1ε (Mean = 0.4ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- near 2
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 7.95ε (Mean = 3.12ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 1ε (Mean = 0.191ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 4.1ε (Mean = 1.55ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 0.558ε (Mean = 0.298ε))<br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0.558ε (Mean = 0.298ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 5.01ε (Mean = 1.89ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2ε (Mean = 0.733ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- near -10
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 2.6ε (Mean = 1.05ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 34.9ε (Mean = 9.2ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.75ε (Mean = 0.895ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 2.26ε (Mean = 1.08ε))<br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 2.26ε (Mean = 1.08ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.75ε (Mean = 0.819ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.86ε (Mean = 0.881ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0.866ε (Mean = 0.445ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- near -55
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 1.8ε (Mean = 0.782ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 3.89e+04ε (Mean = 9.52e+03ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.69ε (Mean = 1.09ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 1.79ε (Mean = 0.75ε))<br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 1.79ε (Mean = 0.75ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 98.5ε (Mean = 53.4ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 0ε (Mean = 0ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.7ε (Mean = 1.35ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 3.87e+04ε (Mean = 6.71e+03ε))
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- </div>
- <br class="table-break"><div class="table">
- <a name="math_toolkit.sf_gamma.tgamma.table_tgamma1pm1"></a><p class="title"><b>Table 8.2. Error rates for tgamma1pm1</b></p>
- <div class="table-contents"><table class="table" summary="Error rates for tgamma1pm1">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody><tr>
- <td>
- <p>
- tgamma1pm1(dz)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.12ε (Mean = 0.49ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.61ε (Mean = 0.84ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.31ε (Mean = 0.517ε)</span>
- </p>
- </td>
- </tr></tbody>
- </table></div>
- </div>
- <br class="table-break"><p>
- The following error plot are based on an exhaustive search of the functions
- domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
- precision, and GCC-7.1/Ubuntu 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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/tgamma__double.svg" align="middle"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/tgamma__80_bit_long_double.svg" align="middle"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/tgamma____float128.svg" align="middle"></span>
- </p></blockquote></div>
- <h5>
- <a name="math_toolkit.sf_gamma.tgamma.h3"></a>
- <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.testing"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.testing">Testing</a>
- </h5>
- <p>
- The gamma is relatively easy to test: factorials and half-integer factorials
- can be calculated exactly by other means and compared with the gamma function.
- In addition, some accuracy tests in known tricky areas were computed at high
- precision using the generic version of this function.
- </p>
- <p>
- The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
- tested against values calculated very naively using the formula <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="number">1</span><span class="special">+</span><span class="identifier">dz</span><span class="special">)-</span><span class="number">1</span></code> with a
- lanczos approximation accurate to around 100 decimal digits.
- </p>
- <h5>
- <a name="math_toolkit.sf_gamma.tgamma.h4"></a>
- <span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.implementation"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.implementation">Implementation</a>
- </h5>
- <p>
- The generic version of the <code class="computeroutput"><span class="identifier">tgamma</span></code>
- function is implemented Sterling's approximation for <code class="computeroutput"><span class="identifier">lgamma</span></code>
- for large z:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
- </p></blockquote></div>
- <p>
- Following exponentiation, downward recursion is then used for small values
- of z.
- </p>
- <p>
- For types of known precision the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
- approximation</a> is used, a traits class <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lanczos</span><span class="special">::</span><span class="identifier">lanczos_traits</span></code>
- maps type T to an appropriate approximation.
- </p>
- <p>
- For z in the range -20 < z < 1 then recursion is used to shift to z
- > 1 via:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/gamm3.svg"></span>
- </p></blockquote></div>
- <p>
- For very small z, this helps to preserve the identity:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/gamm4.svg"></span>
- </p></blockquote></div>
- <p>
- For z < -20 the reflection formula:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/gamm5.svg"></span>
- </p></blockquote></div>
- <p>
- is used. Particular care has to be taken to evaluate the <code class="literal">z * sin(π *
- z)</code> part: a special routine is used to reduce z prior to multiplying
- by π to ensure that the result in is the range [0, π/2]. Without this an excessive
- amount of error occurs in this region (which is hard enough already, as the
- rate of change near a negative pole is <span class="emphasis"><em>exceptionally</em></span>
- high).
- </p>
- <p>
- Finally if the argument is a small integer then table lookup of the factorial
- is used.
- </p>
- <p>
- The function <code class="computeroutput"><span class="identifier">tgamma1pm1</span></code> is
- implemented using rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
- by JM</a> in the region <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special"><</span> <span class="identifier">dz</span>
- <span class="special"><</span> <span class="number">2</span></code>.
- These are the same approximations (and internal routines) that are used for
- <a class="link" href="lgamma.html" title="Log Gamma">lgamma</a>, and so aren't
- detailed further here. The result of the approximation is <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">))</span></code> which can
- fed into <a class="link" href="../powers/expm1.html" title="expm1">expm1</a> to give the
- desired result. Outside the range <code class="computeroutput"><span class="special">-</span><span class="number">0.5</span> <span class="special"><</span> <span class="identifier">dz</span>
- <span class="special"><</span> <span class="number">2</span></code>
- then the naive formula <code class="computeroutput"><span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">dz</span><span class="special">)</span>
- <span class="special">=</span> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span><span class="special">+</span><span class="number">1</span><span class="special">)-</span><span class="number">1</span></code>
- can be used directly.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, 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></td>
- </tr></table>
- <hr>
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