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<div class="section math_toolkit_sf_erf_error_function">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.sf_erf.error_function"></a><a class="link" href="error_function.html" title="Error Functions">Error Functions</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.sf_erf.error_function.h0"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.synopsis"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.synopsis">Synopsis</a>
</h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&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;13.&#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;13.&#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">&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;13.&#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;13.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;13.&#160;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;13.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<h5>
<a name="math_toolkit.sf_erf.error_function.h1"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.description"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.description">Description</a>
</h5>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&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;13.&#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;13.&#160;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.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">&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;13.&#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;13.&#160;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.png"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/erfc.png" align="middle"></span>
</p>
<h5>
<a name="math_toolkit.sf_erf.error_function.h2"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.accuracy"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.accuracy">Accuracy</a>
</h5>
<p>
The following table shows the peak errors (in units of epsilon) found on
various platforms with various floating point types, along with comparisons
to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>, <a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>, <a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
libraries. Unless otherwise specified any floating point type that is narrower
than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
zero error</a>.
</p>
<div class="table">
<a name="math_toolkit.sf_erf.error_function.errors_in_the_function_erf_z"></a><p class="title"><b>Table&#160;3.12.&#160;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 &lt; 0.5
</p>
</th>
<th>
<p>
0.5 &lt; z &lt; 8
</p>
</th>
<th>
<p>
z &gt; 8
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
GSL Peak=2.0 Mean=0.3
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=1.1
Mean=0.7
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.09
</p>
<p>
GSL Peak=2.3 Mean=0.3
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=1.3
Mean=0.2
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
GSL Peak=0 Mean=0
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=0
Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=0.7 Mean=0.07
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.9 Mean=0.2
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.2
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.9 Mean=0.07
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=0.7 Mean=0.07
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.1
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0.5 Mean=0.03
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=0.8 Mean=0.1
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Lib Peak=0.9 Mean=0.2
</p>
</td>
<td>
<p>
Peak=0.9 Mean=0.1
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Lib Peak=0.5 Mean=0.02
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Lib Peak=0 Mean=0
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_erf.error_function.errors_in_the_function_erfc_z"></a><p class="title"><b>Table&#160;3.13.&#160;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 &lt; 0.5
</p>
</th>
<th>
<p>
0.5 &lt; z &lt; 8
</p>
</th>
<th>
<p>
z &gt; 8
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
53
</p>
</td>
<td>
<p>
Win32, Visual C++ 8
</p>
</td>
<td>
<p>
Peak=0.7 Mean=0.06
</p>
<p>
GSL Peak=1.0 Mean=0.4
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=0.7
Mean=0.06
</p>
</td>
<td>
<p>
Peak=0.99 Mean=0.3
</p>
<p>
GSL Peak=2.6 Mean=0.6
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=3.6
Mean=0.7
</p>
</td>
<td>
<p>
Peak=1.0 Mean=0.2
</p>
<p>
GSL Peak=3.9 Mean=0.4
</p>
<p>
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> Peak=2.7
Mean=0.4
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
RedHat Linux IA32, gcc-3.3
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.4 Mean=0.3
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=1.3 Mean=0.3
</p>
</td>
<td>
<p>
Peak=1.6 Mean=0.4
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=1.3 Mean=0.4
</p>
</td>
</tr>
<tr>
<td>
<p>
64
</p>
</td>
<td>
<p>
Redhat Linux IA64, gcc-3.4.4
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.4 Mean=0.3
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.5 Mean=0.4
</p>
<p>
<a href="http://www.gnu.org/software/libc/" target="_top">GNU C Lib</a>
Peak=0 Mean=0
</p>
</td>
</tr>
<tr>
<td>
<p>
113
</p>
</td>
<td>
<p>
HPUX IA64, aCC A.06.06
</p>
</td>
<td>
<p>
Peak=0 Mean=0
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak=0 Mean=0
</p>
</td>
<td>
<p>
Peak=1.5 Mean=0.3
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak=0.9 Mean=0.08
</p>
</td>
<td>
<p>
Peak=1.6 Mean=0.4
</p>
<p>
<a href="http://docs.hp.com/en/B9106-90010/index.html" target="_top">HP-UX
C Library</a> Peak=0.9 Mean=0.1
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<a name="math_toolkit.sf_erf.error_function.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.testing"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.testing">Testing</a>
</h5>
<p>
The tests for these functions come in two parts: basic sanity checks use
spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf" target="_top">Mathworld's
online evaluator</a>, while accuracy checks use high-precision test values
calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
and this implementation. Note that the generic and type-specific versions
of these functions use differing implementations internally, so this gives
us reasonably independent test data. Using our test data to test other "known
good" implementations also provides an additional sanity check.
</p>
<h5>
<a name="math_toolkit.sf_erf.error_function.h4"></a>
<span class="phrase"><a name="math_toolkit.sf_erf.error_function.implementation"></a></span><a class="link" href="error_function.html#math_toolkit.sf_erf.error_function.implementation">Implementation</a>
</h5>
<p>
All versions of these functions first use the usual reflection formulas to
make their arguments positive:
</p>
<pre class="programlisting"><span class="identifier">erf</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span>
<span class="identifier">erfc</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">=</span> <span class="number">2</span> <span class="special">-</span> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">z</span><span class="special">);</span> <span class="comment">// preferred when -z &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>
<p>
The generic versions of these functions are implemented in terms of the incomplete
gamma function.
</p>
<p>
When the significand (mantissa) size is recognised (currently for 53, 64
and 113-bit reals, plus single-precision 24-bit handled via promotion to
double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
by JM</a> are used.
</p>
<p>
For <code class="computeroutput"><span class="identifier">z</span> <span class="special">&lt;=</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">&gt;</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">&gt;</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, +&#8734;] 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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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012, 2013 Paul A. Bristow, Christopher Kormanyos,
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Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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