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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"> |
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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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