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<div class="section" lang="en">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.perf.tuning"></a><a class="link" href="tuning.html" title="Performance Tuning Macros"> Performance Tuning Macros</a>
</h3></div></div></div>
<p>
There are a small number of performance tuning options that are determined
by configuration macros. These should be set in boost/math/tools/user.hpp;
or else reported to the Boost-development mailing list so that the appropriate
option for a given compiler and OS platform can be set automatically in our
configuration setup.
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Macro
</p>
</th>
<th>
<p>
Meaning
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
BOOST_MATH_POLY_METHOD
</p>
</td>
<td>
<p>
Determines how polynomials and most rational functions are evaluated.
Define to one of the values 0, 1, 2 or 3: see below for the meaning
of these values.
</p>
</td>
</tr>
<tr>
<td>
<p>
BOOST_MATH_RATIONAL_METHOD
</p>
</td>
<td>
<p>
Determines how symmetrical rational functions are evaluated: mostly
this only effects how the Lanczos approximation is evaluated, and
how the <code class="computeroutput"><span class="identifier">evaluate_rational</span></code>
function behaves. Define to one of the values 0, 1, 2 or 3: see
below for the meaning of these values.
</p>
</td>
</tr>
<tr>
<td>
<p>
BOOST_MATH_MAX_POLY_ORDER
</p>
</td>
<td>
<p>
The maximum order of polynomial or rational function that will
be evaluated by a method other than 0 (a simple "for"
loop).
</p>
</td>
</tr>
<tr>
<td>
<p>
BOOST_MATH_INT_TABLE_TYPE(RT, IT)
</p>
</td>
<td>
<p>
Many of the coefficients to the polynomials and rational functions
used by this library are integers. Normally these are stored as
tables as integers, but if mixed integer / floating point arithmetic
is much slower than regular floating point arithmetic then they
can be stored as tables of floating point values instead. If mixed
arithmetic is slow then add:
</p>
<p>
#define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
</p>
<p>
to boost/math/tools/user.hpp, otherwise the default of:
</p>
<p>
#define BOOST_MATH_INT_TABLE_TYPE(RT, IT) IT
</p>
<p>
Set in boost/math/config.hpp is fine, and may well result in smaller
code.
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
The values to which <code class="computeroutput"><span class="identifier">BOOST_MATH_POLY_METHOD</span></code>
and <code class="computeroutput"><span class="identifier">BOOST_MATH_RATIONAL_METHOD</span></code>
may be set are as follows:
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Value
</p>
</th>
<th>
<p>
Effect
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
0
</p>
</td>
<td>
<p>
The polynomial or rational function is evaluated using Horner's
method, and a simple for-loop.
</p>
<p>
Note that if the order of the polynomial or rational function is
a runtime parameter, or the order is greater than the value of
<code class="computeroutput"><span class="identifier">BOOST_MATH_MAX_POLY_ORDER</span></code>,
then this method is always used, irrespective of the value of
<code class="computeroutput"><span class="identifier">BOOST_MATH_POLY_METHOD</span></code>
or <code class="computeroutput"><span class="identifier">BOOST_MATH_RATIONAL_METHOD</span></code>.
</p>
</td>
</tr>
<tr>
<td>
<p>
1
</p>
</td>
<td>
<p>
The polynomial or rational function is evaluated without the use
of a loop, and using Horner's method. This only occurs if the order
of the polynomial is known at compile time and is less than or
equal to <code class="computeroutput"><span class="identifier">BOOST_MATH_MAX_POLY_ORDER</span></code>.
</p>
</td>
</tr>
<tr>
<td>
<p>
2
</p>
</td>
<td>
<p>
The polynomial or rational function is evaluated without the use
of a loop, and using a second order Horner's method. In theory
this permits two operations to occur in parallel for polynomials,
and four in parallel for rational functions. This only occurs if
the order of the polynomial is known at compile time and is less
than or equal to <code class="computeroutput"><span class="identifier">BOOST_MATH_MAX_POLY_ORDER</span></code>.
</p>
</td>
</tr>
<tr>
<td>
<p>
3
</p>
</td>
<td>
<p>
The polynomial or rational function is evaluated without the use
of a loop, and using a second order Horner's method. In theory
this permits two operations to occur in parallel for polynomials,
and four in parallel for rational functions. This differs from
method "2" in that the code is carefully ordered to make
the parallelisation more obvious to the compiler: rather than relying
on the compiler's optimiser to spot the parallelisation opportunities.
This only occurs if the order of the polynomial is known at compile
time and is less than or equal to <code class="computeroutput"><span class="identifier">BOOST_MATH_MAX_POLY_ORDER</span></code>.
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
To determine which of these options is best for your particular compiler/platform
build the performance test application with your usual release settings,
and run the program with the --tune command line option.
</p>
<p>
In practice the difference between methods is rather small at present, as
the following table shows. However, parallelisation /vectorisation is likely
to become more important in the future: quite likely the methods currently
supported will need to be supplemented or replaced by ones more suited to
highly vectorisable processors in the future.
</p>
<div class="table">
<a name="id1278977"></a><p class="title"><b>Table 49. A Comparison of Polynomial Evaluation Methods</b></p>
<div class="table-contents"><table class="table" summary="A Comparison of Polynomial Evaluation Methods">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Compiler/platform
</p>
</th>
<th>
<p>
Method 0
</p>
</th>
<th>
<p>
Method 1
</p>
</th>
<th>
<p>
Method 2
</p>
</th>
<th>
<p>
Method 3
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Microsoft C++ 9.0, Polynomial evaluation
</p>
</td>
<td>
<p>
</p>
<p>1.26</p>
<p> </p>
<p>(7.421e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.22</p>
<p> </p>
<p>(7.226e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(5.901e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.04</p>
<p> </p>
<p>(6.115e-008s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
Microsoft C++ 9.0, Rational evaluation
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(1.008e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(1.008e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.43</p>
<p> </p>
<p>(1.445e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.40</p>
<p> </p>
<p>(1.409e-007s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
Intel C++ 11.1 (Windows), Polynomial evaluation
</p>
</td>
<td>
<p>
</p>
<p>1.18</p>
<p> </p>
<p>(6.517e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.18</p>
<p> </p>
<p>(6.505e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(5.516e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(5.516e-008s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
Intel C++ 11.1 (Windows), Rational evaluation
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(8.947e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.02</p>
<p> </p>
<p>(9.130e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.49</p>
<p> </p>
<p>(1.333e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.04</p>
<p> </p>
<p>(9.325e-008s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
GNU G++ 4.2 (Linux), Polynomial evaluation
</p>
</td>
<td>
<p>
</p>
<p>1.61</p>
<p> </p>
<p>(1.220e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.68</p>
<p> </p>
<p>(1.269e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.23</p>
<p> </p>
<p>(9.275e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(7.566e-008s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
GNU G++ 4.2 (Linux), Rational evaluation
</p>
</td>
<td>
<p>
</p>
<p>1.26</p>
<p> </p>
<p>(1.660e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.33</p>
<p> </p>
<p>(1.758e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(1.318e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.15</p>
<p> </p>
<p>(1.513e-007s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
Intel C++ 10.0 (Linux), Polynomial evaluation
</p>
</td>
<td>
<p>
</p>
<p>1.15</p>
<p> </p>
<p>(9.154e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.15</p>
<p> </p>
<p>(9.154e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(7.934e-008s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(7.934e-008s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
Intel C++ 10.0 (Linux), Rational evaluation
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(1.245e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(1.245e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.35</p>
<p> </p>
<p>(1.684e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p>1.04</p>
<p> </p>
<p>(1.294e-007s)</p>
<p>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
There is one final performance tuning option that is available as a compile
time <a class="link" href="../policy.html" title="Policies">policy</a>. Normally when evaluating
functions at <code class="computeroutput"><span class="keyword">double</span></code> precision,
these are actually evaluated at <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> precision internally: this
helps to ensure that as close to full <code class="computeroutput"><span class="keyword">double</span></code>
precision as possible is achieved, but may slow down execution in some environments.
The defaults for this policy can be changed by <a class="link" href="../policy/pol_ref/policy_defaults.html" title="Using macros to Change the Policy Defaults">defining
the macro <code class="computeroutput"><span class="identifier">BOOST_MATH_PROMOTE_DOUBLE_POLICY</span></code></a>
to <code class="computeroutput"><span class="keyword">false</span></code>, or <a class="link" href="../policy/pol_ref/internal_promotion.html" title="Internal Promotion Policies">by
specifying a specific policy</a> when calling the special functions or
distributions. See also the <a class="link" href="../policy/pol_tutorial.html" title="Policy Tutorial">policy
tutorial</a>.
</p>
<div class="table">
<a name="id1279676"></a><p class="title"><b>Table 50. Performance Comparison with and Without Internal Promotion to long
double</b></p>
<div class="table-contents"><table class="table" summary="Performance Comparison with and Without Internal Promotion to long
double">
<colgroup>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Function
</p>
</th>
<th>
<p>
GCC 4.2 , Linux
</p>
<p>
(with internal promotion of double to long double).
</p>
</th>
<th>
<p>
GCC 4.2, Linux
</p>
<p>
(without promotion of double).
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
<a class="link" href="../special/sf_erf/error_function.html" title="Error Functions">erf</a>
</p>
</td>
<td>
<p>
</p>
<p>1.48</p>
<p> </p>
<p>(1.387e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(9.377e-008s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../special/sf_erf/error_inv.html" title="Error Function Inverses">erf_inv</a>
</p>
</td>
<td>
<p>
</p>
<p>1.11</p>
<p> </p>
<p>(4.009e-007s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(3.598e-007s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
and <a class="link" href="../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>
</p>
</td>
<td>
<p>
</p>
<p>1.29</p>
<p> </p>
<p>(5.354e-006s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(4.137e-006s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>
and <a class="link" href="../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>
</p>
</td>
<td>
<p>
</p>
<p>1.44</p>
<p> </p>
<p>(2.220e-005s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(1.538e-005s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
<a class="link" href="../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inva</a>,
<a class="link" href="../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_invb</a>
and <a class="link" href="../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_invb</a>
</p>
</td>
<td>
<p>
</p>
<p>1.25</p>
<p> </p>
<p>(7.009e-005s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(5.607e-005s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>
and <a class="link" href="../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>
</p>
</td>
<td>
<p>
</p>
<p>1.26</p>
<p> </p>
<p>(3.116e-006s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(2.464e-006s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
and <a class="link" href="../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>
</p>
</td>
<td>
<p>
</p>
<p>1.27</p>
<p> </p>
<p>(1.178e-005s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(9.291e-006s)</p>
<p>
</p>
</td>
</tr>
<tr>
<td>
<p>
<a class="link" href="../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inva</a>
and <a class="link" href="../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inva</a>
</p>
</td>
<td>
<p>
</p>
<p>1.20</p>
<p> </p>
<p>(2.765e-005s)</p>
<p>
</p>
</td>
<td>
<p>
</p>
<p><span class="bold"><strong>1.00</strong></span></p>
<p> </p>
<p>(2.311e-005s)</p>
<p>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break">
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow,
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>)
</p>
</div></td>
</tr></table>
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