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<div class="section" lang="en">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.sf_erf.error_function"></a><a class="link" href="error_function.html" title="Error Functions"> Error
Functions</a>
</h4></div></div></div>
<a name="math_toolkit.special.sf_erf.error_function.synopsis"></a><h5>
<a name="id1105300"></a>
<a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.synopsis">Synopsis</a>
</h5>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">erf</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<p>
</p>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
<span class="special">}}</span> <span class="comment">// namespaces
</span></pre>
<p>
The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
</p>
<p>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument
is optional and can be used to control the behaviour of the function:
how it handles errors, what level of precision to use etc. Refer to the
<a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>.
</p>
<p>
</p>
<a name="math_toolkit.special.sf_erf.error_function.description"></a><h5>
<a name="id1105683"></a>
<a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.description">Description</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the <a href="http://en.wikipedia.org/wiki/Error_function" target="_top">error
function</a> <a href="http://functions.wolfram.com/GammaBetaErf/Erf/" target="_top">erf</a>
of z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/erf1.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/erf.png" align="middle"></span>
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span>
<a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span>
</pre>
<p>
Returns the complement of the <a href="http://functions.wolfram.com/GammaBetaErf/Erfc/" target="_top">error
function</a> of z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/erf2.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../graphs/erfc.png" align="middle"></span>
</p>
<a name="math_toolkit.special.sf_erf.error_function.accuracy"></a><h5>
<a name="id1106070"></a>
<a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.accuracy">Accuracy</a>
</h5>
<p>
The following table shows the peak errors (in units of epsilon) found on
various platforms with various floating point types, along with comparisons
to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>,
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>, <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX C Library</a>
and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries.
Unless otherwise specified any floating point type that is narrower than
the one shown will have <a class="link" href="../../backgrounders/relative_error.html#zero_error">effectively zero error</a>.
</p>
<div class="table">
<a name="id1106108"></a><p class="title"><b>Table 27. Errors In the Function erf(z)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function erf(z)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
z < 0.5
</p>
</th>
<th>
<p>
0.5 < z < 8
</p>
</th>
<th>
<p>
z > 8
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
GSL Peak=2.0 Mean=0.3
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=1.1
Mean=0.7
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.09
</p>
<p>
GSL Peak=2.3 Mean=0.3
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=1.3
Mean=0.2
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
GSL Peak=0 Mean=0
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=0
Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=0.7 Mean=0.07
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.9 Mean=0.2
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.2
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.9 Mean=0.07
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=0.7 Mean=0.07
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.1
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.5 Mean=0.03
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=0.8 Mean=0.1
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Lib Peak=0.9 Mean=0.2
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.1
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Lib Peak=0.5 Mean=0.02
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Lib Peak=0 Mean=0
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="id1107811"></a><p class="title"><b>Table 28. Errors In the Function erfc(z)</b></p>
<div class="table-contents"><table class="table" summary="Errors In the Function erfc(z)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Significand Size
</p>
</th>
<th>
<p>
Platform and Compiler
</p>
</th>
<th>
<p>
z < 0.5
</p>
</th>
<th>
<p>
0.5 < z < 8
</p>
</th>
<th>
<p>
z > 8
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=0.7 Mean=0.06
</p>
<p>
GSL Peak=1.0 Mean=0.4
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=0.7
Mean=0.06
</p>
</td>
<td>
<p>
Peak=0.99 Mean=0.3
</p>
<p>
GSL Peak=2.6 Mean=0.6
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=3.6
Mean=0.7
</p>
</td>
<td>
<p>
Peak=1.0 Mean=0.2
</p>
<p>
GSL Peak=3.9 Mean=0.4
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2.7
Mean=0.4
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.4 Mean=0.3
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=1.3 Mean=0.3
</p>
</td>
<td>
<p>
Peak=1.6 Mean=0.4
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=1.3 Mean=0.4
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.4 Mean=0.3
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.5 Mean=0.4
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.5 Mean=0.3
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak=0.9 Mean=0.08
</p>
</td>
<td>
<p>
Peak=1.6 Mean=0.4
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak=0.9 Mean=0.1
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><a name="math_toolkit.special.sf_erf.error_function.testing"></a><h5>
<a name="id1108170"></a>
<a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.testing">Testing</a>
</h5>
<p>
The tests for these functions come in two parts: basic sanity checks use
spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf" target="_top">Mathworld's
online evaluator</a>, while accuracy checks use high-precision test
values calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
and this implementation. Note that the generic and type-specific versions
of these functions use differing implementations internally, so this gives
us reasonably independent test data. Using our test data to test other
"known good" implementations also provides an additional sanity
check.
</p>
<a name="math_toolkit.special.sf_erf.error_function.implementation"></a><h5>
<a name="id1108199"></a>
<a class="link" href="error_function.html#math_toolkit.special.sf_erf.error_function.implementation">Implementation</a>
</h5>
<p>
All versions of these functions first use the usual reflection formulas
to make their arguments positive:
</p>
<pre class="programlisting"><span class="identifier">erf</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
<span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">2</span> <span class="special">-</span> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span> <span class="comment">// preferred when -z < -0.5
</span>
<span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">1</span> <span class="special">+</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span> <span class="comment">// preferred when -0.5 <= -z < 0
</span></pre>
<p>
The generic versions of these functions are implemented in terms of the
incomplete gamma function.
</p>
<p>
When the significand (mantissa) size is recognised (currently for 53, 64
and 113-bit reals, plus single-precision 24-bit handled via promotion to
double) then a series of rational approximations <a class="link" href="../../backgrounders/implementation.html#math_toolkit.backgrounders.implementation.rational_approximations_used">devised
by JM</a> are used.
</p>
<p>
For <code class="computeroutput"><span class="identifier">z</span> <span class="special"><=</span>
<span class="number">0.5</span></code> then a rational approximation
to erf is used, based on the observation that erf is an odd function and
therefore erf is calculated using:
</p>
<pre class="programlisting"><span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">z</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">C</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">));</span>
</pre>
<p>
where the rational approximation R(z*z) is optimised for absolute error:
as long as its absolute error is small enough compared to the constant
C, then any round-off error incurred during the computation of R(z*z) will
effectively disappear from the result. As a result the error for erf and
erfc in this region is very low: the last bit is incorrect in only a very
small number of cases.
</p>
<p>
For <code class="computeroutput"><span class="identifier">z</span> <span class="special">></span>
<span class="number">0.5</span></code> we observe that over a small
interval [a, b) then:
</p>
<pre class="programlisting"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">z</span> <span class="special">~</span> <span class="identifier">c</span>
</pre>
<p>
for some constant c.
</p>
<p>
Therefore for <code class="computeroutput"><span class="identifier">z</span> <span class="special">></span>
<span class="number">0.5</span></code> we calculate erfc using:
</p>
<pre class="programlisting"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">C</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span> <span class="special">-</span> <span class="identifier">B</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">;</span>
</pre>
<p>
Again R(z - B) is optimised for absolute error, and the constant <code class="computeroutput"><span class="identifier">C</span></code> is the average of <code class="computeroutput"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span>
<span class="special">*</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span>
<span class="identifier">z</span></code> taken at the endpoints of the
range. Once again, as long as the absolute error in R(z - B) is small compared
to <code class="computeroutput"><span class="identifier">c</span></code> then <code class="computeroutput"><span class="identifier">c</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span>
<span class="special">-</span> <span class="identifier">B</span><span class="special">)</span></code> will be correctly rounded, and the error
in the result will depend only on the accuracy of the exp function. In
practice, in all but a very small number of cases, the error is confined
to the last bit of the result. The constant <code class="computeroutput"><span class="identifier">B</span></code>
is chosen so that the left hand end of the range of the rational approximation
is 0.
</p>
<p>
For large <code class="computeroutput"><span class="identifier">z</span></code> over a range
[a, +∞] the above approximation is modified to:
</p>
<pre class="programlisting"><span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">C</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="number">1</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">z</span><span class="special">;</span>
</pre>
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Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani
and Thijs van den Berg<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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