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<html>
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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>
The performance test suite generates a report for your particular compiler
showing which method is likely to work best, the following tables show the
results for MSVC-14.0 and GCC-5.1.0 (Linux). There's not much to choose between
the various methods, but generally loop-unrolled methods perform better. Interestingly,
ordering the code to try and "second guess" possible optimizations
seems not to be such a good idea (method 3 below).
</p>
<div class="table">
<a name="math_toolkit.tuning.table_Polynomial_Method_Comparison_with_Microsoft_Visual_C_version_14_0_on_Windows_x64"></a><p class="title"><b>Table&#160;16.3.&#160;Polynomial Method Comparison with Microsoft Visual C++ version 14.0
on Windows x64</b></p>
<div class="table-contents"><table class="table" summary="Polynomial Method Comparison with Microsoft Visual C++ version 14.0
on Windows x64">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Function
</p>
</th>
<th>
<p>
Method 0<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 0<br> (Integer Coefficients)
</p>
</th>
<th>
<p>
Method 1<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 1<br> (Integer Coefficients)
</p>
</th>
<th>
<p>
Method 2<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 2<br> (Integer Coefficients)
</p>
</th>
<th>
<p>
Method 3<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 3<br> (Integer Coefficients)
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Order 2
</p>
</td>
<td>
<p>
<span class="grey">-</span>
</p>
</td>
<td>
<p>
<span class="grey">-</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (9ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (9ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (9ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (9ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (9ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (9ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 3
</p>
</td>
<td>
<p>
<span class="red">2.08<br> (25ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.75<br> (33ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (13ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (13ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (13ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (13ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (13ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (12ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 4
</p>
</td>
<td>
<p>
<span class="red">2.06<br> (35ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.71<br> (46ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.06<br> (18ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (17ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.06<br> (18ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.06<br> (18ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (17ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (17ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 5
</p>
</td>
<td>
<p>
<span class="blue">1.32<br> (29ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">2.00<br> (44ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (22ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (22ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (23ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (23ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (23ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (23ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 6
</p>
</td>
<td>
<p>
<span class="blue">1.38<br> (36ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.04<br> (53ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (28ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (26ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (28ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (28ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.35<br> (35ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.38<br> (36ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 7
</p>
</td>
<td>
<p>
<span class="blue">1.43<br> (43ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.13<br> (64ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.03<br> (31ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (30ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.10<br> (33ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.03<br> (31ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.10<br> (33ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.13<br> (34ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 8
</p>
</td>
<td>
<p>
<span class="blue">1.65<br> (61ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.22<br> (82ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (37ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (40ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.14<br> (42ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (39ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (40ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.11<br> (41ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 9
</p>
</td>
<td>
<p>
<span class="blue">1.39<br> (57ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.05<br> (84ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.17<br> (48ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.17<br> (48ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (41ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (43ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.15<br> (47ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.12<br> (46ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 10
</p>
</td>
<td>
<p>
<span class="blue">1.37<br> (63ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.20<br> (101ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (56ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.24<br> (57ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (46ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (46ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.17<br> (54ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.17<br> (54ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 11
</p>
</td>
<td>
<p>
<span class="blue">1.59<br> (78ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.24<br> (110ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.37<br> (67ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.29<br> (63ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (60ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (49ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (60ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (60ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 12
</p>
</td>
<td>
<p>
<span class="blue">1.46<br> (83ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.16<br> (123ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.28<br> (73ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.26<br> (72ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (58ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (57ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.07<br> (61ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (60ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 13
</p>
</td>
<td>
<p>
<span class="blue">1.61<br> (90ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.55<br> (143ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.32<br> (74ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.39<br> (78ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.04<br> (58ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (56ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.11<br> (62ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.07<br> (60ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 14
</p>
</td>
<td>
<p>
<span class="blue">1.61<br> (106ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.23<br> (147ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (96ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (96ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (67ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (67ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (66ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.09<br> (72ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 15
</p>
</td>
<td>
<p>
<span class="blue">1.49<br> (119ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.10<br> (168ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.35<br> (108ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.35<br> (108ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (80ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (80ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (80ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (82ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 16
</p>
</td>
<td>
<p>
<span class="blue">1.54<br> (129ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.99<br> (167ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.49<br> (125ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (122ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.07<br> (90ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (84ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (91ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (86ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 17
</p>
</td>
<td>
<p>
<span class="blue">1.51<br> (133ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.02<br> (178ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.57<br> (138ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.50<br> (132ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (90ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (88ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.07<br> (94ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.06<br> (93ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 18
</p>
</td>
<td>
<p>
<span class="blue">1.53<br> (148ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.16<br> (210ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.49<br> (145ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.57<br> (152ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.11<br> (108ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.09<br> (106ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (97ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (105ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 19
</p>
</td>
<td>
<p>
<span class="blue">1.90<br> (194ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.27<br> (232ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.62<br> (165ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.62<br> (165ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (110ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (102ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.17<br> (119ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.19<br> (121ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 20
</p>
</td>
<td>
<p>
<span class="blue">1.65<br> (206ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.08<br> (260ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (181ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.44<br> (180ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (125ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (125ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.01<br> (126ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.03<br> (129ns)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.tuning.table_Rational_Method_Comparison_with_Microsoft_Visual_C_version_14_0_on_Windows_x64"></a><p class="title"><b>Table&#160;16.4.&#160;Rational Method Comparison with Microsoft Visual C++ version 14.0 on
Windows x64</b></p>
<div class="table-contents"><table class="table" summary="Rational Method Comparison with Microsoft Visual C++ version 14.0 on
Windows x64">
<colgroup>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Function
</p>
</th>
<th>
<p>
Method 0<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 0<br> (Integer Coefficients)
</p>
</th>
<th>
<p>
Method 1<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 1<br> (Integer Coefficients)
</p>
</th>
<th>
<p>
Method 2<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 2<br> (Integer Coefficients)
</p>
</th>
<th>
<p>
Method 3<br> (Double Coefficients)
</p>
</th>
<th>
<p>
Method 3<br> (Integer Coefficients)
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Order 2
</p>
</td>
<td>
<p>
<span class="grey">-</span>
</p>
</td>
<td>
<p>
<span class="grey">-</span>
</p>
</td>
<td>
<p>
<span class="red">2.12<br> (89ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.95<br> (82ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 3
</p>
</td>
<td>
<p>
<span class="red">2.10<br> (88ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.10<br> (88ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.05<br> (86ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.10<br> (88ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (44ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 4
</p>
</td>
<td>
<p>
<span class="red">2.12<br> (89ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.21<br> (93ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.98<br> (83ns)</span>
</p>
</td>
<td>
<p>
<span class="red">2.10<br> (88ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (43ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (43ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.02<br> (43ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (42ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 5
</p>
</td>
<td>
<p>
<span class="green">1.07<br> (90ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.15<br> (97ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (91ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (84ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (122ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.46<br> (123ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (122ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (122ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 6
</p>
</td>
<td>
<p>
<span class="green">1.16<br> (102ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.58<br> (139ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (88ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.03<br> (91ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.44<br> (127ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.44<br> (127ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.41<br> (124ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.38<br> (121ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 7
</p>
</td>
<td>
<p>
<span class="blue">1.29<br> (121ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.44<br> (135ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.01<br> (95ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (94ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.38<br> (130ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.36<br> (128ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.33<br> (125ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.36<br> (128ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 8
</p>
</td>
<td>
<p>
<span class="blue">1.33<br> (134ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.52<br> (154ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (101ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (109ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.38<br> (139ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.31<br> (132ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.39<br> (140ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.37<br> (138ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 9
</p>
</td>
<td>
<p>
<span class="green">1.18<br> (141ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.45<br> (172ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (119ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.08<br> (128ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.13<br> (135ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.26<br> (150ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.26<br> (150ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.27<br> (151ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 10
</p>
</td>
<td>
<p>
<span class="blue">1.29<br> (180ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.28<br> (178ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (146ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (139ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.06<br> (147ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.06<br> (147ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.18<br> (164ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.17<br> (163ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 11
</p>
</td>
<td>
<p>
<span class="blue">1.28<br> (187ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.28<br> (187ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.06<br> (155ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.05<br> (154ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.03<br> (151ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (146ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.19<br> (174ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.47<br> (215ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 12
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (197ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.38<br> (223ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.04<br> (168ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.04<br> (169ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (162ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.04<br> (169ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (198ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.52<br> (246ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 13
</p>
</td>
<td>
<p>
<span class="blue">1.23<br> (209ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.29<br> (220ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.15<br> (196ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.10<br> (187ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (170ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.15<br> (196ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (208ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.61<br> (273ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 14
</p>
</td>
<td>
<p>
<span class="blue">1.28<br> (242ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.39<br> (262ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.15<br> (218ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.14<br> (216ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (189ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.01<br> (191ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.49<br> (282ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.53<br> (290ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 15
</p>
</td>
<td>
<p>
<span class="blue">1.28<br> (260ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.34<br> (273ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.12<br> (227ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.15<br> (233ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (203ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (203ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.38<br> (280ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.47<br> (298ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 16
</p>
</td>
<td>
<p>
<span class="blue">1.35<br> (288ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.40<br> (300ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (261ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.18<br> (252ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (214ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.23<br> (264ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.43<br> (305ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.52<br> (325ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 17
</p>
</td>
<td>
<p>
<span class="green">1.16<br> (259ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.47<br> (328ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.15<br> (256ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.35<br> (302ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (223ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (273ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.50<br> (334ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.52<br> (339ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 18
</p>
</td>
<td>
<p>
<span class="green">1.10<br> (273ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.46<br> (363ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.10<br> (273ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.75<br> (434ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (248ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.30<br> (322ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.41<br> (349ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.46<br> (363ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 19
</p>
</td>
<td>
<p>
<span class="blue">1.26<br> (330ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.35<br> (352ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.24<br> (324ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.33<br> (348ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (261ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (319ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.44<br> (377ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.46<br> (381ns)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Order 20
</p>
</td>
<td>
<p>
<span class="blue">1.24<br> (330ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.60<br> (427ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.22<br> (327ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.56<br> (416ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.00<br> (267ns)</span>
</p>
</td>
<td>
<p>
<span class="green">1.19<br> (317ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.57<br> (418ns)</span>
</p>
</td>
<td>
<p>
<span class="blue">1.56<br> (416ns)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
[table_Polynomial_Method_Comparison_with_GNU_C_version_5_1_0_on_linux]
</p>
<p>
[table_Rational_Method_Comparison_with_GNU_C_version_5_1_0_on_linux]
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 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, 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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