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<title>Performance Tuning Macros</title>
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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.tuning"></a><a class="link" href="tuning.html" title="Performance Tuning Macros">Performance Tuning Macros</a>
</h2></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="math_toolkit.tuning.a_comparison_of_polynomial_evalu"></a><p class="title"><b>Table 15.3. 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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="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="pol_ref/internal_promotion.html" title="Internal Floating-point Promotion Policies">by
specifying a specific policy</a> when calling the special functions or distributions.
See also the <a class="link" href="pol_tutorial.html" title="Policy Tutorial">policy tutorial</a>.
</p>
<div class="table">
<a name="math_toolkit.tuning.performance_comparison_with_and_"></a><p class="title"><b>Table 15.4. 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="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="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="sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
and <a class="link" href="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="sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>
and <a class="link" href="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="sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
<a class="link" href="sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inva</a>,
<a class="link" href="sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_invb</a>
and <a class="link" href="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="sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a> and
<a class="link" href="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="sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>
and <a class="link" href="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="sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inva</a>
and <a class="link" href="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>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin 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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