performance.qbk

Boost provides free peer-reviewed portable C++ source libraries. We emphasize libraries that work

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.

[table A Comparison of Polynomial Evaluation Methods
[[Compiler/platform][Method 0][Method 1][Method 2][Method 3]]
[[Microsoft C++ 8.0, Polynomial evaluation] [[perf msvc-Polynomial-method-0..[para 1.34][para (1.161e-007s)]]][[perf msvc-Polynomial-method-1..[para 1.13][para (9.777e-008s)]]][[perf msvc-Polynomial-method-2..[para 1.07][para (9.289e-008s)]]][[perf msvc-Polynomial-method-3..[para *1.00*][para (8.678e-008s)]]]]
[[Microsoft C++ 8.0, Rational evaluation] [[perf msvc-Rational-method-0..[para *1.00*][para (1.443e-007s)]]][[perf msvc-Rational-method-1..[para 1.03][para (1.492e-007s)]]][[perf msvc-Rational-method-2..[para 1.20][para (1.736e-007s)]]][[perf msvc-Rational-method-3..[para 1.07][para (1.540e-007s)]]]]
[[Intel C++ 10.0 (Windows), Polynomial evaluation] [[perf intel-Polynomial-method-0..[para 1.03][para (7.702e-008s)]]][[perf intel-Polynomial-method-1..[para 1.03][para (7.702e-008s)]]][[perf intel-Polynomial-method-2..[para *1.00*][para (7.446e-008s)]]][[perf intel-Polynomial-method-3..[para 1.03][para (7.690e-008s)]]]]
[[Intel C++ 10.0 (Windows), Rational evaluation] [[perf intel-Rational-method-0..[para *1.00*][para (1.245e-007s)]]][[perf intel-Rational-method-1..[para *1.00*][para (1.245e-007s)]]][[perf intel-Rational-method-2..[para 1.18][para (1.465e-007s)]]][[perf intel-Rational-method-3..[para 1.06][para (1.318e-007s)]]]]
[[GNU G++ 4.2 (Linux), Polynomial evaluation] [[perf gcc-4_2-ld-Polynomial-method-0..[para 1.61][para (1.220e-007s)]]][[perf gcc-4_2-ld-Polynomial-method-1..[para 1.68][para (1.269e-007s)]]][[perf gcc-4_2-ld-Polynomial-method-2..[para 1.23][para (9.275e-008s)]]][[perf gcc-4_2-ld-Polynomial-method-3..[para *1.00*][para (7.566e-008s)]]]]
[[GNU G++ 4.2 (Linux), Rational evaluation] [[perf gcc-4_2-ld-Rational-method-0..[para 1.26][para (1.660e-007s)]]][[perf gcc-4_2-ld-Rational-method-1..[para 1.33][para (1.758e-007s)]]][[perf gcc-4_2-ld-Rational-method-2..[para *1.00*][para (1.318e-007s)]]][[perf gcc-4_2-ld-Rational-method-3..[para 1.15][para (1.513e-007s)]]]]
[[Intel C++ 10.0 (Linux), Polynomial evaluation] [[perf intel-linux-Polynomial-method-0..[para 1.15][para (9.154e-008s)]]][[perf intel-linux-Polynomial-method-1..[para 1.15][para (9.154e-008s)]]][[perf intel-linux-Polynomial-method-2..[para *1.00*][para (7.934e-008s)]]][[perf intel-linux-Polynomial-method-3..[para *1.00*][para (7.934e-008s)]]]]
[[Intel C++ 10.0 (Linux), Rational evaluation] [[perf intel-linux-Rational-method-0..[para *1.00*][para (1.245e-007s)]]][[perf intel-linux-Rational-method-1..[para *1.00*][para (1.245e-007s)]]][[perf intel-linux-Rational-method-2..[para 1.35][para (1.684e-007s)]]][[perf intel-linux-Rational-method-3..[para 1.04][para (1.294e-007s)]]]]
]

There is one final performance tuning option that is available as a compile time
[link math_toolkit.policy policy].  Normally when evaluating functions at `double`
precision, these are actually evaluated at `long double` precision internally:
this helps to ensure that as close to full `double` precision as possible is 
achieved, but may slow down execution in some environments.  The defaults for
this policy can be changed by 
[link math_toolkit.policy.pol_ref.policy_defaults 
defining the macro `BOOST_MATH_PROMOTE_DOUBLE_POLICY`]
to `false`, or 
[link math_toolkit.policy.pol_ref.internal_promotion 
by specifying a specific policy] when calling the special
functions or distributions.  See also the 
[link math_toolkit.policy.pol_tutorial policy tutorial].

[table  Performance Comparison with and Without Internal Promotion to long double
[[Function]
   [GCC 4.2 , Linux
   
   (with internal promotion of double to long double).
   ]
      [GCC 4.2, Linux
      
      (without promotion of double).
      ]
]
[[__erf][[perf gcc-4_2-ld-erf..[para 1.48][para (1.387e-007s)]]][[perf gcc-4_2-erf..[para *1.00*][para (9.377e-008s)]]]]
[[__erf_inv][[perf gcc-4_2-ld-erf_inv..[para 1.11][para (4.009e-007s)]]][[perf gcc-4_2-erf_inv..[para *1.00*][para (3.598e-007s)]]]]
[[__ibeta and __ibetac][[perf gcc-4_2-ld-ibeta..[para 1.29][para (5.354e-006s)]]][[perf gcc-4_2-ibeta..[para *1.00*][para (4.137e-006s)]]]]
[[__ibeta_inv and __ibetac_inv][[perf gcc-4_2-ld-ibeta_inv..[para 1.44][para (2.220e-005s)]]][[perf gcc-4_2-ibeta_inv..[para *1.00*][para (1.538e-005s)]]]]
[[__ibeta_inva, __ibetac_inva, __ibeta_invb and __ibetac_invb][[perf gcc-4_2-ld-ibeta_invab..[para 1.25][para (7.009e-005s)]]][[perf gcc-4_2-ibeta_invab..[para *1.00*][para (5.607e-005s)]]]]
[[__gamma_p and __gamma_q][[perf gcc-4_2-ld-igamma..[para 1.26][para (3.116e-006s)]]][[perf gcc-4_2-igamma..[para *1.00*][para (2.464e-006s)]]]]
[[__gamma_p_inv and __gamma_q_inv][[perf gcc-4_2-ld-igamma_inv..[para 1.27][para (1.178e-005s)]]][[perf gcc-4_2-igamma_inv..[para *1.00*][para (9.291e-006s)]]]]
[[__gamma_p_inva and __gamma_q_inva][[perf gcc-4_2-ld-igamma_inva..[para 1.20][para (2.765e-005s)]]][[perf gcc-4_2-igamma_inva..[para *1.00*][para (2.311e-005s)]]]]
]

[endsect]
[section:comparisons Comparisons to Other Open Source Libraries]

We've run our performance tests both for our own code, and against other
open source implementations of the same functions.  The results are 
presented below to give you a rough idea of how they all compare.

[caution
You should exercise extreme caution when interpreting
these results, relative performance may vary by platform, the tests use
data that gives good code coverage of /our/ code, but which may skew the 
results towards the corner cases.  Finally, remember that different 
libraries make different choices with regard to performance verses 
numerical stability.
]

[heading Comparison to GSL-1.9 and Cephes]

All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Windows XP
machine, with all the libraries compiled with Microsoft Visual C++ 2005 using 
the `/Ox /arch:SSE2` options.

[table
[[Function][Boost][GSL-1.9][Cephes]]
[[__tgamma][[perf msvc-gamma..[para 1.50][para (2.566e-007s)]]][[perf msvc-gamma-gsl..[para 1.54][para (2.627e-007s)]]][[perf msvc-gamma-cephes..[para *1.00*][para (1.709e-007s)]]]]
[[__lgamma][[perf msvc-lgamma..[para 1.73][para (2.688e-007s)]]][[perf msvc-lgamma-gsl..[para 3.61][para (5.621e-007s)]]][[perf msvc-lgamma-cephes..[para *1.00*][para (1.556e-007s)]]]]
[[__gamma_p and __gamma_q][[perf msvc-igamma..[para *1.00*][para (9.504e-007s)]]][[perf msvc-igamma-gsl..[para 2.15][para (2.042e-006s)]]][[perf msvc-igamma-cephes..[para 2.57][para (2.439e-006s)]]]]
[[__gamma_p_inv and __gamma_q_inv][[perf msvc-igamma_inv..[para *1.00*][para (3.631e-006s)]]][N\/A][+INF [footnote Cephes gets stuck in an infinite loop while trying to execute our test cases.]]]
[[__ibeta and __ibetac][[perf msvc-ibeta..[para *1.00*][para (1.852e-006s)]]][[perf msvc-ibeta-cephes..[para 1.07][para (1.974e-006s)]]][[perf msvc-ibeta-cephes..[para 1.07][para (1.974e-006s)]]]]
[[__ibeta_inv and __ibetac_inv][[perf msvc-ibeta_inv..[para *1.00*][para (7.311e-006s)]]][N\/A][[perf msvc-ibeta_inv-cephes..[para 2.24][para (1.637e-005s)]]]]
]

[heading Comparison to the R Statistical Library on Windows]

All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Windows XP
machine, with the test program compiled with Microsoft Visual C++ 2005, and
R-2.5.0 compiled in "standalone mode" with MinGW-3.4 
(R-2.5.0 appears not to be buildable with Visual C++).

[table A Comparison to the R Statistical Library on Windows XP
[[Statistical Function][Boost][R]]
[[__beta_distrib CDF][[perf msvc-dist-beta-cdf..[para 1.20][para (1.916e-006s)]]][[perf msvc-dist-beta-R-cdf..[para *1.00*][para (1.597e-006s)]]]]
[[__beta_distrib Quantile][[perf msvc-dist-beta-quantile..[para *1.00*][para (6.570e-006s)]]][[perf msvc-dist-beta-R-quantile..[para 74.66[footnote There are a small number of our test cases where the R library fails to converge on a result: these tend to dominate the performance result.]][para (4.905e-004s)]]]]
[[__binomial_distrib CDF][[perf msvc-dist-binomial-cdf..[para *1.00*][para (5.276e-007s)]]][[perf msvc-dist-binom-R-cdf..[para 2.45][para (1.293e-006s)]]]]
[[__binomial_distrib Quantile][[perf msvc-dist-binomial-quantile..[para *1.00*][para (4.013e-006s)]]][[perf msvc-dist-binom-R-quantile..[para 1.32][para (5.280e-006s)]]]]
[[__cauchy_distrib CDF][[perf msvc-dist-cauchy-cdf..[para *1.00*][para (1.231e-007s)]]][[perf msvc-dist-cauchy-R-cdf..[para 1.28][para (1.576e-007s)]]]]
[[__cauchy_distrib Quantile][[perf msvc-dist-cauchy-quantile..[para *1.00*][para (1.498e-007s)]]][[perf msvc-dist-cauchy-R-quantile..[para *1.00*][para (1.498e-007s)]]]]
[[__chi_squared_distrib CDF][[perf msvc-dist-chi_squared-cdf..[para *1.00*][para (7.889e-007s)]]][[perf msvc-dist-chisq-R-cdf..[para 2.48][para (1.955e-006s)]]]]
[[__chi_squared_distrib Quantile][[perf msvc-dist-chi_squared-quantile..[para *1.00*][para (4.303e-006s)]]][[perf msvc-dist-chisq-R-quantile..[para 1.61][para (6.925e-006s)]]]]
[[__exp_distrib CDF][[perf msvc-dist-exponential-cdf..[para *1.00*][para (1.955e-007s)]]][[perf msvc-dist-exp-R-cdf..[para 1.97][para (3.844e-007s)]]]]
[[__exp_distrib Quantile][[perf msvc-dist-exponential-quantile..[para 1.07][para (1.206e-007s)]]][[perf msvc-dist-exp-R-quantile..[para *1.00*][para (1.126e-007s)]]]]
[[__F_distrib CDF][[perf msvc-dist-fisher_f-cdf..[para *1.00*][para (1.309e-006s)]]][[perf msvc-dist-f-R-cdf..[para 2.12][para (2.780e-006s)]]]]
[[__F_distrib Quantile][[perf msvc-dist-fisher_f-quantile..[para *1.00*][para (7.204e-006s)]]][[perf msvc-dist-f-R-quantile..[para 1.78][para (1.280e-005s)]]]]
[[__gamma_distrib CDF][[perf msvc-dist-gamma-cdf..[para *1.00*][para (1.076e-006s)]]][[perf msvc-dist-gamma-R-cdf..[para 2.07][para (2.227e-006s)]]]]
[[__gamma_distrib Quantile][[perf msvc-dist-gamma-quantile..[para *1.00*][para (5.189e-006s)]]][[perf msvc-dist-gamma-R-quantile..[para 1.14][para (5.937e-006s)]]]] 
[[__lognormal_distrib CDF][[perf msvc-dist-lognormal-cdf..[para *1.00*][para (2.078e-007s)]]][[perf msvc-dist-lnorm-R-cdf..[para 1.41][para (2.930e-007s)]]]]
[[__lognormal_distrib Quantile][[perf msvc-dist-lognormal-quantile..[para *1.00*][para (6.692e-007s)]]][[perf msvc-dist-lnorm-R-quantile..[para 1.63][para (1.090e-006s)]]]]
[[__negative_binomial_distrib CDF][[perf msvc-dist-negative_binomial-cdf..[para *1.00*][para (9.005e-007s)]]][[perf msvc-dist-nbinom-R-cdf..[para 2.42][para (2.178e-006s)]]]]
[[__negative_binomial_distrib Quantile][[perf msvc-dist-negative_binomial-quantile..[para *1.00*][para (9.601e-006s)]]][[perf msvc-dist-nbinom-R-quantile..[para 53.59[footnote The R library appears to use a linear-search strategy, that can perform very badly in a small number of pathological cases, but may or may not be more efficient in "typical" cases]][para (5.145e-004s)]]]]
[[__normal_distrib CDF][[perf msvc-dist-normal-cdf..[para *1.00*][para (5.926e-008s)]]][[perf msvc-dist-norm-R-cdf..[para 3.01][para (1.785e-007s)]]]]
[[__normal_distrib Quantile][[perf msvc-dist-normal-quantile..[para *1.00*][para (1.248e-007s)]]][[perf msvc-dist-norm-R-quantile..[para 1.05][para (1.311e-007s)]]]]
[[__poisson_distrib CDF][[perf msvc-dist-poisson-cdf..[para *1.00*][para (8.999e-007s)]]][[perf msvc-dist-pois-R-cdf..[para 2.42][para (2.175e-006s)]]]]
[[__poisson_distrib][[perf msvc-dist-poisson-quantile..[para *1.00*][para (1.853e-006s)]]][[perf msvc-dist-pois-R-quantile..[para 2.17][para (4.014e-006s)]]]]
[[__students_t_distrib CDF][[perf msvc-dist-students_t-cdf..[para *1.00*][para (1.223e-006s)]]][[perf msvc-dist-t-R-cdf..[para 1.13][para (1.376e-006s)]]]]
[[__students_t_distrib Quantile][[perf msvc-dist-students_t-quantile..[para *1.00*][para (2.570e-006s)]]][[perf msvc-dist-t-R-quantile..[para 1.04][para (2.668e-006s)]]]]
[[__weibull_distrib CDF][[perf msvc-dist-weibull-cdf..[para *1.00*][para (4.741e-007s)]]][[perf msvc-dist-weibull-R-cdf..[para 1.46][para (6.943e-007s)]]]]
[[__weibull_distrib Quantile][[perf msvc-dist-weibull-quantile..[para *1.00*][para (7.926e-007s)]]][[perf msvc-dist-weibull-R-quantile..[para 1.08][para (8.542e-007s)]]]]
]

[heading Comparison to the R Statistical Library on Linux]

All the results were measured on a 2.8GHz Intel Pentium 4, 2Gb RAM, Mandriva Linux
machine, with the test program and R-2.5.0 compiled with GNU G++ 4.2.0.

[table A Comparison to the R Statistical Library on Linux
[[Statistical Function][Boost][R]]
[[__beta_distrib CDF][[perf gcc-4_2-dist-beta-cdf..[para 1.71][para (3.508e-006s)]]][[perf gcc-4_2-dist-beta-R-cdf..[para *1.00*][para (2.050e-006s)]]]]
[[__beta_distrib Quantile][[perf gcc-4_2-dist-beta-quantile..[para *1.00*][para (1.294e-005s)]]][[perf gcc-4_2-dist-beta-R-quantile..[para 44.06[footnote There are a small number of our test cases where the R library fails to converge on a result: these tend to dominate the performance result.]][para (5.701e-004s)]]]]
[[__binomial_distrib CDF][[perf gcc-4_2-dist-binomial-cdf..[para 1.22][para (1.342e-006s)]]][[perf gcc-4_2-dist-binom-R-cdf..[para *1.00*][para (1.104e-006s)]]]]
[[__binomial_distrib Quantile][[perf gcc-4_2-dist-binomial-quantile..[para 1.36][para (7.083e-006s)]]][[perf gcc-4_2-dist-binom-R-quantile..[para *1.00*][para (5.194e-006s)]]]]
[[__cauchy_distrib CDF][[perf gcc-4_2-dist-cauchy-cdf..[para *1.00*][para (1.372e-007s)]]][[perf gcc-4_2-dist-cauchy-R-cdf..[para 1.47][para (2.017e-007s)]]]]
[[__cauchy_distrib Quantile][[perf gcc-4_2-dist-cauchy-quantile..[para *1.00*][para (1.542e-007s)]]][[perf gcc-4_2-dist-cauchy-R-quantile..[para 1.14][para (1.752e-007s)]]]]
[[__chi_squared_distrib CDF][[perf gcc-4_2-dist-chi_squared-cdf..[para 1.04][para (1.820e-006s)]]][[perf gcc-4_2-dist-chisq-R-cdf..[para *1.00*][para (1.753e-006s)]]]]
[[__chi_squared_distrib Quantile][[perf gcc-4_2-dist-chi_squared-quantile..[para 1.39][para (9.345e-006s)]]][[perf gcc-4_2-dist-chisq-R-quantile..[para *1.00*][para (6.728e-006s)]]]]
[[__exp_distrib CDF][[perf gcc-4_2-dist-exponential-cdf..[para *1.00*][para (2.195e-007s)]]][[perf gcc-4_2-dist-exp-R-cdf..[para 1.17][para (2.561e-007s)]]]]
[[__exp_distrib Quantile][[perf gcc-4_2-dist-exponential-quantile..[para *1.00*][para (1.123e-007s)]]][[perf gcc-4_2-dist-exp-R-quantile..[para 1.03][para (1.155e-007s)]]]]
[[__F_distrib CDF][[perf gcc-4_2-dist-fisher_f-cdf..[para *1.00*][para (2.744e-006s)]]][[perf gcc-4_2-dist-f-R-cdf..[para 1.08][para (2.970e-006s)]]]]
[[__F_distrib Quantile][[perf gcc-4_2-dist-fisher_f-quantile..[para 1.14][para (1.550e-005s)]]][[perf gcc-4_2-dist-f-R-quantile..[para *1.00*][para (1.359e-005s)]]]]
[[__gamma_distrib CDF][[perf gcc-4_2-dist-gamma-cdf..[para 1.29][para (2.578e-006s)]]][[perf gcc-4_2-dist-gamma-R-cdf..[para *1.00*][para (1.992e-006s)]]]]
[[__gamma_distrib Quantile][[perf gcc-4_2-dist-gamma-quantile..[para 1.77][para (1.020e-005s)]]][[perf gcc-4_2-dist-gamma-R-quantile..[para *1.00*][para (5.757e-006s)]]]] 
[[__lognormal_distrib CDF][[perf gcc-4_2-dist-lognormal-cdf..[para *1.00*][para (1.782e-007s)]]][[perf gcc-4_2-dist-lnorm-R-cdf..[para 2.00][para (3.564e-007s)]]]]
[[__lognormal_distrib Quantile][[perf gcc-4_2-dist-lognormal-quantile..[para *1.00*][para (7.093e-007s)]]][[perf gcc-4_2-dist-lnorm-R-quantile..[para 1.07][para (7.607e-007s)]]]]
[[__negative_binomial_distrib CDF][[perf gcc-4_2-dist-negative_binomial-cdf..[para 1.03][para (2.209e-006s)]]][[perf gcc-4_2-dist-nbinom-R-cdf..[para *1.00*][para (2.141e-006s)]]]]
[[__negative_binomial_distrib Quantile][[perf gcc-4_2-dist-negative_binomial-quantile..[para *1.00*][para (1.826e-005s)]]][[perf gcc-4_2-dist-nbinom-R-quantile..[para 30.07[footnote The R library appears to use a linear-search strategy, that can perform very badly in a small number of pathological cases, but may or may not be more efficient in "typical" cases]][para (5.490e-004s)]]]]
[[__normal_distrib CDF][[perf gcc-4_2-dist-normal-cdf..[para *1.00*][para (8.542e-008s)]]][[perf gcc-4_2-dist-norm-R-cdf..[para 2.09][para (1.782e-007s)]]]]
[[__normal_distrib Quantile][[perf gcc-4_2-dist-normal-quantile..[para *1.00*][para (1.362e-007s)]]][[perf gcc-4_2-dist-norm-R-quantile..[para 1.26][para (1.722e-007s)]]]]
[[__poisson_distrib CDF][[perf gcc-4_2-dist-poisson-cdf..[para 1.10][para (1.953e-006s)]]][[perf gcc-4_2-dist-pois-R-cdf..[para *1.00*][para (1.775e-006s)]]]]
[[__poisson_distrib][[perf gcc-4_2-dist-poisson-quantile..[para 1.12][para (4.214e-006s)]]][[perf gcc-4_2-dist-pois-R-quantile..[para *1.00*][para (3.752e-006s)]]]]
[[__students_t_distrib CDF][[perf gcc-4_2-dist-students_t-cdf..[para 1.55][para (2.441e-006s)]]][[perf gcc-4_2-dist-t-R-cdf..[para *1.00*][para (1.576e-006s)]]]]
[[__students_t_distrib Quantile][[perf gcc-4_2-dist-students_t-quantile..[para 1.33][para (3.972e-006s)]]][[perf gcc-4_2-dist-t-R-quantile..[para *1.00*][para (2.990e-006s)]]]]
[[__weibull_distrib CDF][[perf gcc-4_2-dist-weibull-cdf..[para *1.00*][para (6.640e-007s)]]][[perf gcc-4_2-dist-weibull-R-cdf..[para 1.06][para (7.031e-007s)]]]]
[[__weibull_distrib Quantile][[perf gcc-4_2-dist-weibull-quantile..[para *1.00*][para (7.504e-007s)]]][[perf gcc-4_2-dist-weibull-R-quantile..[para 1.03][para (7.710e-007s)]]]]
]

[endsect]

[section:perf_test_app The Performance Test Application]

Under ['boost-path]\/libs\/math\/performance you will find a 
(fairly rudimentary) performance test application for this library.

To run this application yourself, build the all the .cpp files in
['boost-path]\/libs\/math\/performance into an application using
your usual release-build settings.  Run the application with --help
to see a full list of options, or with --all to test everything 
(which takes quite a while), or with --tune to test the 
[link math_toolkit.perf.tuning available performance tuning options].

If you want to use this application to test the effect of changing
any of the __policy_section, then you will need to build and run it twice: 
once with the default __policy_section, and then a second time with the 
__policy_section you want to test set as the default.

[endsect]

[endsect]

[/ 
  Copyright 2006 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]