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423 lines
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<title>Ratios of Gamma Functions</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.sf_gamma.gamma_ratios"></a><a class="link" href="gamma_ratios.html" title="Ratios of Gamma Functions">Ratios of Gamma Functions</a>
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</h3></div></div></div>
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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">gamma</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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>
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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">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</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">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</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>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.sf_gamma.gamma_ratios.h0"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.description"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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>
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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">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</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>
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</pre>
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<p>
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Returns the ratio of gamma functions:
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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/gamma_ratio0.svg"></span>
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</p></blockquote></div>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. 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 20. 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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<p>
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Internally this just calls <code class="computeroutput"><span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
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<span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span></code>.
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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>
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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">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</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>
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</pre>
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<p>
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Returns the ratio of gamma functions:
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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/gamma_ratio1.svg"></span>
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</p></blockquote></div>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 20. 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 20. 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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<p>
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Note that the result is calculated accurately even when <span class="emphasis"><em>delta</em></span>
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is small compared to <span class="emphasis"><em>a</em></span>: indeed even if <span class="emphasis"><em>a+delta
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~ a</em></span>. The function is typically used when <span class="emphasis"><em>a</em></span>
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is large and <span class="emphasis"><em>delta</em></span> is very small.
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</p>
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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> when T1 and T2 are different types,
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otherwise the result type is simple T1.
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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="../../../graphs/tgamma_delta_ratio.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_gamma.gamma_ratios.h1"></a>
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<span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.accuracy"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.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. Unless otherwise specified
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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>.
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</p>
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<div class="table">
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<a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_delta_ratio"></a><p class="title"><b>Table 8.7. Error rates for tgamma_delta_ratio</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for tgamma_delta_ratio">
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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> 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> 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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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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tgamma + small delta ratios
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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>
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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 = 5.83ε (Mean = 1.3ε)</span>
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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 = 15.4ε (Mean = 2.09ε)</span>
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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 = 7.56ε (Mean = 1.31ε)</span>
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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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tgamma + small delta ratios (negative delta)
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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>
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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 = 7.94ε (Mean = 1.4ε)</span>
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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 = 18.3ε (Mean = 2.03ε)</span>
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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 = 7.43ε (Mean = 1.42ε)</span>
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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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tgamma + small integer ratios
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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>
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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 = 1.96ε (Mean = 0.677ε)</span>
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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 = 1.96ε (Mean = 0.677ε)</span>
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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 = 2.74ε (Mean = 0.736ε)</span>
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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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tgamma + small integer ratios (negative delta)
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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>
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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 = 1.62ε (Mean = 0.451ε)</span>
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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 = 1.62ε (Mean = 0.451ε)</span>
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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 = 2.15ε (Mean = 0.685ε)</span>
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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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integer tgamma ratios
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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>
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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 = 0.997ε (Mean = 0.4ε)</span>
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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 = 0.997ε (Mean = 0.4ε)</span>
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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 = 0.968ε (Mean = 0.386ε)</span>
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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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integer tgamma ratios (negative delta)
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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>
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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 = 0.853ε (Mean = 0.176ε)</span>
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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 = 0.853ε (Mean = 0.176ε)</span>
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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 = 0.974ε (Mean = 0.175ε)</span>
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</p>
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</td>
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</tr>
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</tbody>
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</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_gamma.gamma_ratios.table_tgamma_ratio"></a><p class="title"><b>Table 8.8. Error rates for tgamma_ratio</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for tgamma_ratio">
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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> 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> long double
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</p>
|
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</th>
|
|
<th>
|
|
<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>
|
|
<th>
|
|
<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><tr>
|
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<td>
|
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<p>
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tgamma ratios
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</p>
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</td>
|
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<td>
|
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<p>
|
|
<span class="blue">Max = 0.694ε (Mean = 0.0347ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.99ε (Mean = 1.15ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 174ε (Mean = 61.2ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.28ε (Mean = 1.12ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr></tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><h5>
|
|
<a name="math_toolkit.sf_gamma.gamma_ratios.h2"></a>
|
|
<span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.testing"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.testing">Testing</a>
|
|
</h5>
|
|
<p>
|
|
Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL
|
|
RR class</a> set at 1000-bit precision: about 300 decimal digits) and
|
|
a deliberately naive calculation of Γ(x)/Γ(y).
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.sf_gamma.gamma_ratios.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.implementation"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.implementation">Implementation</a>
|
|
</h5>
|
|
<p>
|
|
The implementation of these functions is very similar to that of <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>,
|
|
and is based on combining similar power terms to improve accuracy and avoid
|
|
spurious overflow/underflow.
|
|
</p>
|
|
<p>
|
|
In addition there are optimisations for the situation where <span class="emphasis"><em>delta</em></span>
|
|
is a small integer: in which case this function is basically the reciprocal
|
|
of a rising factorial, or where both arguments are smallish integers: in
|
|
which case table lookup of factorials can be used to calculate the ratio.
|
|
</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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