[DEV] add v1.76.0

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2021-10-05 21:37:46 +02:00
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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Error Functions</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Error Function erf and complement erfc</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_erf.html" title="Error Functions">
<link rel="prev" href="../sf_erf.html" title="Error Functions">
<link rel="next" href="error_inv.html" title="Error Function Inverses">
@@ -24,7 +24,8 @@
</div>
<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>
<a name="math_toolkit.sf_erf.error_function"></a><a class="link" href="error_function.html" title="Error Function erf and complement erfc">Error Function erf
and complement erfc</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.sf_erf.error_function.h0"></a>
@@ -37,14 +38,14 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -53,9 +54,9 @@
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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<h5>
@@ -65,51 +66,55 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/erf1.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/erf.svg" align="middle"></span>
</p></blockquote></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/erf2.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/erfc.svg" align="middle"></span>
</p></blockquote></div>
<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
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
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.table_erf"></a><p class="title"><b>Table&#160;6.28.&#160;Error rates for erf</b></p>
<a name="math_toolkit.sf_erf.error_function.table_erf"></a><p class="title"><b>Table 8.28. Error rates for erf</b></p>
<div class="table-contents"><table class="table" summary="Error rates for erf">
<colgroup>
<col>
@@ -123,22 +128,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -151,28 +156,27 @@
</td>
<td>
<p>
<span class="blue">Max = 0.996&#949; (Mean = 0.182&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.57&#949; (Mean = 0.317&#949;))
<span class="blue">Max = 0.925ε (Mean = 0.193ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.944ε (Mean = 0.191ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.944ε (Mean = 0.191ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.925&#949; (Mean = 0.193&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.944&#949; (Mean = 0.191&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.944&#949; (Mean = 0.191&#949;))
<span class="blue">Max = 0.841ε (Mean = 0.0687ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.06ε (Mean = 0.319ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.841&#949; (Mean = 0.0687&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.06&#949; (Mean = 0.319&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 1.13&#949; (Mean = 0.442&#949;))
<span class="blue">Max = 0.925ε (Mean = 0.193ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.944ε (Mean = 0.194ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.925&#949; (Mean = 0.193&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.944&#949; (Mean = 0.194&#949;))
<span class="blue">Max = 0.996ε (Mean = 0.182ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.57ε (Mean = 0.317ε))
</p>
</td>
</tr>
@@ -184,28 +188,27 @@
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.169&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.19&#949; (Mean = 0.244&#949;))
<span class="blue">Max = 1.5ε (Mean = 0.193ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.921ε (Mean = 0.0723ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.921ε (Mean = 0.0723ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.5&#949; (Mean = 0.193&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.921&#949; (Mean = 0.0723&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.921&#949; (Mean = 0.0723&#949;))
<span class="blue">Max = 1ε (Mean = 0.119ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.31ε (Mean = 0.368ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.119&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.31&#949; (Mean = 0.368&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 1.34&#949; (Mean = 0.279&#949;))
<span class="blue">Max = 1.5ε (Mean = 0.197ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.921ε (Mean = 0.071ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.5&#949; (Mean = 0.202&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.921&#949; (Mean = 0.071&#949;))
<span class="blue">Max = 1ε (Mean = 0.171ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.19ε (Mean = 0.244ε))
</p>
</td>
</tr>
@@ -217,28 +220,27 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span>
Max = 0ε (Mean = 0ε))<br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))<br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 0&#949; (Mean = 0&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0ε (Mean = 0ε))
</p>
</td>
</tr>
@@ -246,7 +248,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_erf.error_function.table_erfc"></a><p class="title"><b>Table&#160;6.29.&#160;Error rates for erfc</b></p>
<a name="math_toolkit.sf_erf.error_function.table_erfc"></a><p class="title"><b>Table 8.29. Error rates for erfc</b></p>
<div class="table-contents"><table class="table" summary="Error rates for erfc">
<colgroup>
<col>
@@ -260,22 +262,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -288,28 +290,27 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span>
Max = 0ε (Mean = 0ε))<br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))<br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 0.658ε (Mean = 0.0537ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.01ε (Mean = 0.485ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.658&#949; (Mean = 0.0537&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.01&#949; (Mean = 0.485&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 0.786&#949; (Mean = 0.0642&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span>
Max = 0ε (Mean = 0ε))
</p>
</td>
</tr>
@@ -321,28 +322,27 @@
</td>
<td>
<p>
<span class="blue">Max = 1.65&#949; (Mean = 0.373&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.36&#949; (Mean = 0.539&#949;))
<span class="blue">Max = 1.76ε (Mean = 0.365ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.3 (Mean = 0.307ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.35ε (Mean = 0.307ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.76&#949; (Mean = 0.365&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.35&#949; (Mean = 0.307&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.35&#949; (Mean = 0.307&#949;))
<span class="blue">Max = 0.983ε (Mean = 0.213ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.64ε (Mean = 0.662ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.983&#949; (Mean = 0.213&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.64&#949; (Mean = 0.662&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 3.59&#949; (Mean = 0.779&#949;))
<span class="blue">Max = 1.76ε (Mean = 0.38ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.81ε (Mean = 0.739ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.76&#949; (Mean = 0.383&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.81&#949; (Mean = 0.739&#949;))
<span class="blue">Max = 1.65ε (Mean = 0.373ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.36ε (Mean = 0.539ε))
</p>
</td>
</tr>
@@ -354,35 +354,72 @@
</td>
<td>
<p>
<span class="blue">Max = 1.14&#949; (Mean = 0.248&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.84&#949; (Mean = 0.331&#949;))
<span class="blue">Max = 1.57ε (Mean = 0.542ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.26ε (Mean = 0.441ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.26ε (Mean = 0.441ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.57&#949; (Mean = 0.542&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.26&#949; (Mean = 0.441&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.26&#949; (Mean = 0.441&#949;))
<span class="blue">Max = 0.868ε (Mean = 0.147ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 3.9ε (Mean = 0.472ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.868&#949; (Mean = 0.147&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 3.9&#949; (Mean = 0.472&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 2.74&#949; (Mean = 0.413&#949;))
<span class="blue">Max = 1.57ε (Mean = 0.564ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 4.91ε (Mean = 1.54ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.57&#949; (Mean = 0.564&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 4.91&#949; (Mean = 1.54&#949;))
<span class="blue">Max = 1.14ε (Mean = 0.248ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.84ε (Mean = 0.331ε))
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<br class="table-break"><p>
The following error plots are based on an exhaustive search of the functions
domain, MSVC-16.7.1 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-10/Mingw64 for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_double.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_long_double.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_float128.png"></span>
</p>
<p>
In the erfc case, error rates are almost entirely the error in calculating
<code class="computeroutput"><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">)</span></code>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_double.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_long_double.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_float128.png"></span>
</p>
<p>
Multiprecision error rates are similar, albeit with a much larger error in
calculating the exponent term for erfc:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erf_errors_cpp_bin_float_50.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../reporting/accuracy/erfc_errors_cpp_bin_float_50.png"></span>
</p>
<h5>
<a name="math_toolkit.sf_erf.error_function.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.testing"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.testing">Testing</a>
</h5>
@@ -404,12 +441,18 @@
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 &lt; -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 &lt;= -z &lt; 0</span>
</pre>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>erf(-z) = 1 - erf(z);</em></span>
</span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>erfc(-z) = 2 - erfc(z);</em></span>
// preferred when -z &lt; -0.5</span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>erfc(-z) = 1 + erf(z);</em></span>
// preferred when -0.5 &lt;= -z &lt; 0</span>
</p></blockquote></div>
<p>
The generic versions of these functions are implemented in terms of the incomplete
gamma function.
@@ -426,32 +469,37 @@
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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>erf(z) = z * (C + R(z*z));</em></span></span>
</p></blockquote></div>
<p>
where the rational approximation 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.
where the rational approximation <span class="emphasis"><em>R(z*z)</em></span> is optimised
for absolute error: as long as its absolute error is small enough compared
to the constant C, then any round-off error incurred during the computation
of R(z*z) will effectively disappear from the result. As a result the error
for erf and erfc in this region is very low: the last bit is incorrect in
only a very small number of cases.
</p>
<p>
For <code class="computeroutput"><span class="identifier">z</span> <span class="special">&gt;</span>
<span class="number">0.5</span></code> we observe that over a small interval
[a, b) then:
[<span class="emphasis"><em>a, b)</em></span> 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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>erfc(z) * exp(z*z) * z ~ c</em></span></span>
</p></blockquote></div>
<p>
for some constant c.
</p>
<p>
Therefore for <code class="computeroutput"><span class="identifier">z</span> <span class="special">&gt;</span>
<span class="number">0.5</span></code> we calculate erfc using:
<span class="number">0.5</span></code> we calculate <code class="computeroutput"><span class="identifier">erfc</span></code>
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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>erfc(z) = exp(-z*z) * (C + R(z -
B)) / z;</em></span></span>
</p></blockquote></div>
<p>
Again R(z - B) is optimised for absolute error, and the constant <code class="computeroutput"><span class="identifier">C</span></code> is the average of <code class="computeroutput"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span>
<span class="special">*</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span>
@@ -468,18 +516,20 @@
</p>
<p>
For large <code class="computeroutput"><span class="identifier">z</span></code> over a range
[a, +&#8734;] the above approximation is modified to:
[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 class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>erfc(z) = exp(-z*z) * (C + R(1 /
z)) / z;</em></span></span>
</p></blockquote></div>
</div>
<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 &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>

View File

@@ -1,12 +1,12 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Error Function Inverses</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_erf.html" title="Error Functions">
<link rel="prev" href="error_function.html" title="Error Functions">
<link rel="prev" href="error_function.html" title="Error Function erf and complement erfc">
<link rel="next" href="../sf_poly.html" title="Polynomials">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
@@ -37,14 +37,14 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -53,9 +53,9 @@
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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. 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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<h5>
@@ -65,33 +65,37 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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_inv</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">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the <a href="http://functions.wolfram.com/GammaBetaErf/InverseErf/" target="_top">inverse
error function</a> of z, that is a value x such that:
</p>
<pre class="programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/erf_inv.svg" align="middle"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>p = erf(x);</em></span></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/erf_inv.svg" align="middle"></span>
</p></blockquote></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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_inv</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">&lt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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_inv</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the inverse of the complement of the error function of z, that is
a value x such that:
</p>
<pre class="programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/erfc_inv.svg" align="middle"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>p = erfc(x);</em></span></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/erfc_inv.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.sf_erf.error_inv.h2"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_inv.accuracy"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.accuracy">Accuracy</a>
@@ -99,11 +103,11 @@
<p>
For types up to and including 80-bit long doubles the approximations used
are accurate to less than ~ 2 epsilon. For higher precision types these functions
have the same accuracy as the <a class="link" href="error_function.html" title="Error Functions">forward
have the same accuracy as the <a class="link" href="error_function.html" title="Error Function erf and complement erfc">forward
error functions</a>.
</p>
<div class="table">
<a name="math_toolkit.sf_erf.error_inv.table_erf_inv"></a><p class="title"><b>Table&#160;6.30.&#160;Error rates for erf_inv</b></p>
<a name="math_toolkit.sf_erf.error_inv.table_erf_inv"></a><p class="title"><b>Table 8.30. Error rates for erf_inv</b></p>
<div class="table-contents"><table class="table" summary="Error rates for erf_inv">
<colgroup>
<col>
@@ -117,22 +121,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -144,29 +148,29 @@
</td>
<td>
<p>
<span class="blue">Max = 1.09&#949; (Mean = 0.502&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 0.996ε (Mean = 0.389ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.996&#949; (Mean = 0.389&#949;)</span>
<span class="blue">Max = 1.08ε (Mean = 0.395ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.996&#949; (Mean = 0.385&#949;)</span>
<span class="blue">Max = 1.09ε (Mean = 0.502ε)</span>
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_erf.error_inv.table_erfc_inv"></a><p class="title"><b>Table&#160;6.31.&#160;Error rates for erfc_inv</b></p>
<a name="math_toolkit.sf_erf.error_inv.table_erfc_inv"></a><p class="title"><b>Table 8.31. Error rates for erfc_inv</b></p>
<div class="table-contents"><table class="table" summary="Error rates for erfc_inv">
<colgroup>
<col>
@@ -180,22 +184,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -208,22 +212,22 @@
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.491&#949;)</span>
<span class="blue">Max = (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 0.996ε (Mean = 0.397ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.996&#949; (Mean = 0.397&#949;)</span>
<span class="blue">Max = 1.08ε (Mean = 0.403ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.996&#949; (Mean = 0.397&#949;)</span>
<span class="blue">Max = 1ε (Mean = 0.491ε)</span>
</p>
</td>
</tr>
@@ -235,23 +239,41 @@
</td>
<td>
</td>
<td>
</td>
<td>
<p>
<span class="blue">Max = 1.62&#949; (Mean = 0.383&#949;)</span>
<span class="blue">Max = 1.62ε (Mean = 0.383ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.62&#949; (Mean = 0.385&#949;)</span>
<span class="blue">Max = 1.62ε (Mean = 0.38)</span>
</p>
</td>
<td>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<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/erfc__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/erfc__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/erfc____float128.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.sf_erf.error_inv.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_inv.testing"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.testing">Testing</a>
</h5>
@@ -299,8 +321,9 @@
For <span class="emphasis"><em>p &lt; 0.5</em></span> the inverse erf function is reasonably
smooth and the approximation:
</p>
<pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="identifier">p</span><span class="special">(</span><span class="identifier">p</span> <span class="special">+</span> <span class="number">10</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">p</span><span class="special">))</span>
</pre>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>x = p(p + 10)(Y + R(p))</em></span></span>
</p></blockquote></div>
<p>
Gives a good result for a constant Y, and R(p) optimised for low absolute
error compared to |Y|.
@@ -309,18 +332,21 @@
For q &lt; 0.5 things get trickier, over the interval <span class="emphasis"><em>0.5 &gt;
q &gt; 0.25</em></span> the following approximation works well:
</p>
<pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(-</span><span class="number">2l</span><span class="identifier">og</span><span class="special">(</span><span class="identifier">q</span><span class="special">))</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">q</span><span class="special">))</span>
</pre>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>x = sqrt(-2log(q)) / (Y + R(q))</em></span></span>
</p></blockquote></div>
<p>
While for q &lt; 0.25, let
</p>
<pre class="programlisting"><span class="identifier">z</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(-</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">q</span><span class="special">))</span>
</pre>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>z = sqrt(-log(q))</em></span></span>
</p></blockquote></div>
<p>
Then the result is given by:
</p>
<pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="identifier">z</span><span class="special">(</span><span class="identifier">Y</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>
</pre>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic"><span class="emphasis"><em>x = z(Y + R(z - B))</em></span></span>
</p></blockquote></div>
<p>
As before Y is a constant and the rational function R is optimised for low
absolute error compared to |Y|. B is also a constant: it is the smallest
@@ -334,11 +360,11 @@
</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 &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
</p>