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- </div>
- <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>
- <div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem">
- define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
- </li></ol></div>
- <p>
- to boost/math/tools/user.hpp, otherwise the default of:
- </p>
- <div class="orderedlist"><ol class="orderedlist" type="1"><li class="listitem">
- define BOOST_MATH_INT_TABLE_TYPE(RT, IT) IT
- </li></ol></div>
- <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 21.3. 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 21.4. 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>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
- </p>
- </div></td>
- </tr></table>
- <hr>
- <div class="spirit-nav">
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