[template perf[name value] [value]]
[template para[text] '''<para>'''[text]'''</para>''']

[section:perf Performance]

[section:perf_over Performance Overview]
[performance_overview]
[endsect]

[section:interp Interpreting these Results]

In all of the following tables, the best performing
result in each row, is assigned a relative value of "1" and shown
in bold, so a score of "2" means ['"twice as slow as the best 
performing result".]  Actual timings in seconds per function call
are also shown in parenthesis.  

Result were obtained on a system
with an Intel 2.8GHz Pentium 4 processor with 2Gb of RAM and running
either Windows XP or Mandriva Linux.  

[caution As usual
with performance results these should be taken with a large pinch
of salt: relative performance is known to shift quite a bit depending
upon the architecture of the particular test system used.  Further
more, our performance results were obtained using our own test data:
these test values are designed to provide good coverage of our code and test
all the appropriate corner cases.  They do not necessarily represent
"typical" usage: whatever that may be!
]

[note Since these tests were run, most compilers have improved their code optimisation,
and processor speeds have improved too, so these results are known to be out of date.]

[endsect]

[section:getting_best Getting the Best Performance from this Library]

By far the most important thing you can do when using this library
is turn on your compiler's optimisation options.  As the following 
table shows the penalty for using the library in debug mode can be 
quite large.  

[table  Performance Comparison of Release and Debug Settings
[[Function]
      [Microsoft Visual C++ 8.0
      
      Debug Settings: /Od /ZI
      ]
         [Microsoft Visual C++ 8.0
   
         Release settings: /Ox /arch:SSE2
         ]]

[[__erf][[perf msvc-debug-erf..[para 16.65][para (1.028e-006s)]]][[perf msvc-erf..[para *1.00*][para (1.483e-007s)]]]]
[[__erf_inv][[perf msvc-debug-erf_inv..[para 19.28][para (1.215e-006s)]]][[perf msvc-erf_inv..[para *1.00*][para (4.888e-007s)]]]]
[[__ibeta and __ibetac][[perf msvc-debug-ibeta..[para 8.32][para (1.540e-005s)]]][[perf msvc-ibeta..[para *1.00*][para (1.852e-006s)]]]]
[[__ibeta_inv and __ibetac_inv][[perf msvc-debug-ibeta_inv..[para 10.25][para (7.492e-005s)]]][[perf msvc-ibeta_inv..[para *1.00*][para (7.311e-006s)]]]]
[[__ibeta_inva, __ibetac_inva, __ibeta_invb and __ibetac_invb][[perf msvc-debug-ibeta_invab..[para 8.57][para (2.441e-004s)]]][[perf msvc-ibeta_invab..[para *1.00*][para (2.847e-005s)]]]]
[[__gamma_p and __gamma_q][[perf msvc-debug-igamma..[para 10.98][para (1.044e-005s)]]][[perf msvc-igamma..[para *1.00*][para (9.504e-007s)]]]]
[[__gamma_p_inv and __gamma_q_inv][[perf msvc-debug-igamma_inv..[para 10.25][para (3.721e-005s)]]][[perf msvc-igamma_inv..[para *1.00*][para (3.631e-006s)]]]]
[[__gamma_p_inva and __gamma_q_inva][[perf msvc-debug-igamma_inva..[para 11.26][para (1.124e-004s)]]][[perf msvc-igamma_inva..[para *1.00*][para (9.982e-006s)]]]]
]

[endsect]

[section:comp_compilers Comparing Compilers]

After a good choice of build settings the next most important thing 
you can do, is choose your compiler
- and the standard C library it sits on top of - very carefully.  GCC-3.x
in particular has been found to be particularly bad at inlining code, 
and performing the kinds of high level transformations that good C++ performance
demands (thankfully GCC-4.x is somewhat better in this respect).

[table  Performance Comparison of Various Windows Compilers
[[Function]
   [Intel C++ 10.0
   
   ( /Ox /Qipo /QxN )
   ]
      [Microsoft Visual C++ 8.0
      
      ( /Ox /arch:SSE2 )
      ]
      [Cygwin G++ 3.4
      
      ( /O3 )
      ]]
[[__erf][[perf intel-erf..[para *1.00*][para (4.118e-008s)]]][[perf msvc-erf..[para *1.00*][para (1.483e-007s)]]][[perf gcc-erf..[para 3.24][para (1.336e-007s)]]]]
[[__erf_inv][[perf intel-erf_inv..[para *1.00*][para (4.439e-008s)]]][[perf msvc-erf_inv..[para *1.00*][para (4.888e-007s)]]][[perf gcc-erf_inv..[para 7.88][para (3.500e-007s)]]]]
[[__ibeta and __ibetac][[perf intel-ibeta..[para *1.00*][para (1.631e-006s)]]][[perf msvc-ibeta..[para 1.14][para (1.852e-006s)]]][[perf gcc-ibeta..[para 3.05][para (4.975e-006s)]]]]
[[__ibeta_inv and __ibetac_inv][[perf intel-ibeta_inv..[para *1.00*][para (6.133e-006s)]]][[perf msvc-ibeta_inv..[para 1.19][para (7.311e-006s)]]][[perf gcc-ibeta_inv..[para 2.60][para (1.597e-005s)]]]]
[[__ibeta_inva, __ibetac_inva, __ibeta_invb and __ibetac_invb][[perf intel-ibeta_invab..[para *1.00*][para (2.453e-005s)]]][[perf msvc-ibeta_invab..[para 1.16][para (2.847e-005s)]]][[perf gcc-ibeta_invab..[para 2.83][para (6.947e-005s)]]]]
[[__gamma_p and __gamma_q][[perf intel-igamma..[para *1.00*][para (6.735e-007s)]]][[perf msvc-igamma..[para 1.41][para (9.504e-007s)]]][[perf gcc-igamma..[para 2.78][para (1.872e-006s)]]]]
[[__gamma_p_inv and __gamma_q_inv][[perf intel-igamma_inv..[para *1.00*][para (2.637e-006s)]]][[perf msvc-igamma_inv..[para 1.38][para (3.631e-006s)]]][[perf gcc-igamma_inv..[para 3.31][para (8.736e-006s)]]]]
[[__gamma_p_inva and __gamma_q_inva][[perf intel-igamma_inva..[para *1.00*][para (7.716e-006s)]]][[perf msvc-igamma_inva..[para 1.29][para (9.982e-006s)]]][[perf gcc-igamma_inva..[para 2.56][para (1.974e-005s)]]]]
]

[endsect]

[section:tuning Performance Tuning Macros]

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.  

[table
[[Macro][Meaning]]
[[BOOST_MATH_POLY_METHOD]
   [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.]]
[[BOOST_MATH_RATIONAL_METHOD]
   [Determines how symmetrical rational functions are evaluated: mostly
   this only effects how the Lanczos approximation is evaluated, and how
   the `evaluate_rational` function behaves.  Define to one
   of the values 0, 1, 2 or 3: see below for the meaning of these values.
   ]]
[[BOOST_MATH_MAX_POLY_ORDER]
   [The maximum order of polynomial or rational function that will
   be evaluated by a method other than 0 (a simple "for" loop).
   ]]
[[BOOST_MATH_INT_TABLE_TYPE(RT, IT)]
   [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:
   
      #define BOOST_MATH_INT_TABLE_TYPE(RT, IT) RT
   
   to boost/math/tools/user.hpp, otherwise the default of:
   
      #define BOOST_MATH_INT_TABLE_TYPE(RT, IT) IT
   
   Set in boost/math/config.hpp is fine, and may well result in smaller
   code.
   ]]
]

The values to which `BOOST_MATH_POLY_METHOD` and `BOOST_MATH_RATIONAL_METHOD`
may be set are as follows:

[table
[[Value][Effect]]
[[0][The polynomial or rational function is evaluated using Horner's
      method, and a simple for-loop.  
      
      Note that if the order of the polynomial
      or rational function is a runtime parameter, or the order is
      greater than the value of `BOOST_MATH_MAX_POLY_ORDER`, then
      this method is always used, irrespective of the value
      of `BOOST_MATH_POLY_METHOD` or `BOOST_MATH_RATIONAL_METHOD`.]]
[[1][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 `BOOST_MATH_MAX_POLY_ORDER`. ]]
[[2][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 `BOOST_MATH_MAX_POLY_ORDER`.]]
[[3][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 `BOOST_MATH_MAX_POLY_ORDER`.]]
]

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.

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.

[table A Comparison of Polynomial Evaluation Methods
[[Compiler/platform][Method 0][Method 1][Method 2][Method 3]]
[[Microsoft C++ 9.0, Polynomial evaluation] [[perf msvc-Polynomial-method-0..[para 1.26][para (7.421e-008s)]]][[perf msvc-Polynomial-method-1..[para 1.22][para (7.226e-008s)]]][[perf msvc-Polynomial-method-2..[para *1.00*][para (5.901e-008s)]]][[perf msvc-Polynomial-method-3..[para 1.04][para (6.115e-008s)]]]]
[[Microsoft C++ 9.0, Rational evaluation] [[perf msvc-Rational-method-0..[para *1.00*][para (1.008e-007s)]]][[perf msvc-Rational-method-1..[para *1.00*][para (1.008e-007s)]]][[perf msvc-Rational-method-2..[para 1.43][para (1.445e-007s)]]][[perf msvc-Rational-method-3..[para 1.40][para (1.409e-007s)]]]]
[[Intel C++ 11.1 (Windows), Polynomial evaluation] [[perf intel-Polynomial-method-0..[para 1.18][para (6.517e-008s)]]][[perf intel-Polynomial-method-1..[para 1.18][para (6.505e-008s)]]][[perf intel-Polynomial-method-2..[para *1.00*][para (5.516e-008s)]]][[perf intel-Polynomial-method-3..[para *1.00*][para (5.516e-008s)]]]]
[[Intel C++ 11.1 (Windows), Rational evaluation] [[perf intel-Rational-method-0..[para *1.00*][para (8.947e-008s)]]][[perf intel-Rational-method-1..[para 1.02][para (9.130e-008s)]]][[perf intel-Rational-method-2..[para 1.49][para (1.333e-007s)]]][[perf intel-Rational-method-3..[para 1.04][para (9.325e-008s)]]]]
[[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-1.13 and Cephes]

All the results were measured on a 2.0GHz Intel T5800 Core 2 Duo, 4Gb RAM, Windows Vista
machine, with the test program compiled with Microsoft Visual C++ 2009 using the /Ox option.

[table
[[Function][Boost][GSL-1.9][Cephes]]
[[__cbrt]
   [[perf msvc-cbrt..[para *1.00*][para (4.873e-007s)]]]
   [N\/A]
   [[perf msvc-cbrt-cephes..[para *1.00*][para (6.699e-007s)]]]]
[[__log1p]
   [[perf msvc-log1p..[para *1.00*][para (1.664e-007s)]]]
   [[perf msvc-log1p-gsl..[para *1.00*][para (2.677e-007s)]]]
   [[perf msvc-log1p-cephes..[para *1.00*][para (1.189e-007s)]]]]
[[__expm1]
   [[perf msvc-expm1..[para *1.00*][para (8.760e-008s)]]]
   [[perf msvc-expm1-gsl..[para *1.00*][para (1.248e-007s)]]]
   [[perf msvc-expm1-cephes..[para *1.00*][para (8.169e-008s)]]]]
[[__tgamma]
   [[perf msvc-gamma..[para 1.80][para (2.997e-007s)]]]
   [[perf msvc-gamma-gsl..[para 1.54][para (2.569e-007s)]]]
   [[perf msvc-gamma-cephes..[para *1.00*][para (1.666e-007s)]]]]
[[__lgamma]
   [[perf msvc-lgamma..[para 2.20][para (3.045e-007s)]]]
   [[perf msvc-lgamma-gsl..[para 4.14][para (5.713e-007s)]]]
   [[perf msvc-lgamma-cephes..[para *1.00*][para (1.381e-007s)]]]]
[[__erf and __erfc]
   [[perf msvc-erf..[para *1.00*][para (1.483e-007s)]]]
   [[perf msvc-erf-gsl..[para *1.00*][para (7.052e-007s)]]]
   [[perf msvc-erf-cephes..[para *1.00*][para (1.722e-007s)]]]]
[[__gamma_p and __gamma_q]
   [[perf msvc-igamma..[para *1.00*][para (6.182e-007s)]]]
   [[perf msvc-igamma-gsl..[para 3.57][para (2.209e-006s)]]]
   [[perf msvc-igamma-cephes..[para 4.29][para (2.651e-006s)]]]]
[[__gamma_p_inv and __gamma_q_inv]
   [[perf msvc-igamma_inv..[para *1.00*][para (1.943e-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 (1.670e-006s)]]]
   [[perf msvc-ibeta-cephes..[para 1.16][para (1.935e-006s)]]]
   [[perf msvc-ibeta-cephes..[para 1.16][para (1.935e-006s)]]]]
[[__ibeta_inv and __ibetac_inv]
   [[perf msvc-ibeta_inv..[para *1.00*][para (6.075e-006s)]]]
   [N\/A]
   [[perf msvc-ibeta_inv-cephes..[para 2.45][para (1.489e-005s)]]]]
[[__cyl_bessel_j]
   [[perf msvc-cyl_bessel_j..[para 17.89[footnote The performance here is dominated by a few cases where the parameters grow very large:
   faster asymptotic expansions are available, but are of limited (or even frankly terrible) precision.  The same issue
   effects all of our Bessel function implementations, but doesn't necessarily show in the current performance data.
   More investigation is needed here.]][para (4.248e-005s)]]]
   [[perf msvc-cyl_bessel_j-gsl..[para *1.00*][para (5.214e-006s)]]]
   [[perf msvc-cyl_bessel_j-cephes..[para *1.00*][para (2.374e-006s)]]]]
[[__cyl_bessel_i]
   [[perf msvc-cyl_bessel_i..[para *1.00*][para (5.924e-006s)]]]
   [[perf msvc-cyl_bessel_i-gsl..[para *1.00*][para (4.487e-006s)]]]
   [[perf msvc-cyl_bessel_i-cephes..[para *1.00*][para (4.823e-006s)]]]]
[[__cyl_bessel_k]
   [[perf msvc-cyl_bessel_k..[para *1.00*][para (2.783e-006s)]]]
   [[perf msvc-cyl_bessel_k-gsl..[para *1.00*][para (3.927e-006s)]]]
   [N\/A]]
[[__cyl_neumann]
   [[perf msvc-cyl_neumann..[para *1.00*][para (4.465e-006s)]]]
   [[perf msvc-cyl_neumann-gsl..[para *1.00*][para (1.230e-005s)]]]
   [[perf msvc-cyl_neumann-cephes..[para *1.00*][para (4.977e-006s)]]]]
]

[heading Comparison to the R and DCDFLIB Statistical Libraries on Windows]

All the results were measured on a 2.0GHz Intel T5800 Core 2 Duo, 4Gb RAM, Windows Vista
machine, with the test program compiled with Microsoft Visual C++ 2009, and
R-2.9.2 compiled in "standalone mode" with MinGW-4.3 
(R-2.9.2 appears not to be buildable with Visual C++).

[table A Comparison to the R Statistical Library on Windows XP
[[Statistical Function][Boost][R][DCDFLIB]]
[[__beta_distrib CDF][[perf msvc-dist-beta-cdf..[para 1.08][para (1.385e-006s)]]][[perf msvc-dist-beta-R-cdf..[para *1.00*][para (1.278e-006s)]]][[perf msvc-dist-beta-dcd-cdf..[para 1.06][para (1.349e-006s)]]]]
[[__beta_distrib Quantile][[perf msvc-dist-beta-quantile..[para *1.00*][para (4.975e-006s)]]][[perf msvc-dist-beta-R-quantile..[para 67.66[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 (3.366e-004s)]]][[perf msvc-dist-beta-dcd-quantile..[para 4.23][para (2.103e-005s)]]]]
[[__binomial_distrib CDF][[perf msvc-dist-binomial-cdf..[para 1.06][para (4.503e-007s)]]][[perf msvc-dist-binom-R-cdf..[para 1.81][para (7.680e-007s)]]][[perf msvc-dist-binomial-dcd-cdf..[para *1.00*][para (4.239e-007s)]]]]
[[__binomial_distrib Quantile][[perf msvc-dist-binomial-quantile..[para *1.00*][para (3.254e-006s)]]][[perf msvc-dist-binom-R-quantile..[para 1.15][para (3.746e-006s)]]][[perf msvc-dist-binomial-dcd-quantile..[para 7.25][para (2.358e-005s)]]]]
[[__cauchy_distrib CDF][[perf msvc-dist-cauchy-cdf..[para *1.00*][para (1.134e-007s)]]][[perf msvc-dist-cauchy-R-cdf..[para 1.08][para (1.227e-007s)]]][NA]]
[[__cauchy_distrib Quantile][[perf msvc-dist-cauchy-quantile..[para *1.00*][para (1.203e-007s)]]][[perf msvc-dist-cauchy-R-quantile..[para *1.00*][para (1.203e-007s)]]][NA]]
[[__chi_squared_distrib CDF][[perf msvc-dist-chi_squared-cdf..[para 1.21][para (5.021e-007s)]]][[perf msvc-dist-chisq-R-cdf..[para 2.83][para (1.176e-006s)]]][[perf msvc-dist-chi_squared-dcd-cdf..[para *1.00*][para (4.155e-007s)]]]]
[[__chi_squared_distrib Quantile][[perf msvc-dist-chi_squared-quantile..[para *1.00*][para (1.930e-006s)]]][[perf msvc-dist-chisq-R-quantile..[para 2.72][para (5.243e-006s)]]][[perf msvc-dist-chi_squared-dcd-quantile..[para 5.73][para (1.106e-005s)]]]]
[[__exp_distrib CDF][[perf msvc-dist-exponential-cdf..[para *1.00*][para (3.798e-008s)]]][[perf msvc-dist-exp-R-cdf..[para 5.89][para (2.236e-007s)]]][NA]]
[[__exp_distrib Quantile][[perf msvc-dist-exponential-quantile..[para 1.41][para (9.006e-008s)]]][[perf msvc-dist-exp-R-quantile..[para *1.00*][para (6.380e-008s)]]][NA]]
[[__F_distrib CDF][[perf msvc-dist-fisher_f-cdf..[para *1.00*][para (9.556e-007s)]]][[perf msvc-dist-f-R-cdf..[para 1.34][para (1.283e-006s)]]][[perf msvc-dist-f-dcd-cdf..[para 1.24][para (1.183e-006s)]]]]
[[__F_distrib Quantile][[perf msvc-dist-fisher_f-quantile..[para *1.00*][para (6.987e-006s)]]][[perf msvc-dist-f-R-quantile..[para 1.33][para (9.325e-006s)]]][[perf msvc-dist-f-dcd-quantile..[para 3.16][para (2.205e-005s)]]]]
[[__gamma_distrib CDF][[perf msvc-dist-gamma-cdf..[para 1.52][para (6.240e-007s)]]][[perf msvc-dist-gamma-R-cdf..[para 3.11][para (1.279e-006s)]]][[perf msvc-dist-gam-dcd-cdf..[para *1.00*][para (4.111e-007s)]]]]
[[__gamma_distrib Quantile][[perf msvc-dist-gamma-quantile..[para 1.24][para (2.179e-006s)]]][[perf msvc-dist-gamma-R-quantile..[para 6.25][para (1.102e-005s)]]][[perf msvc-dist-gam-dcd-quantile..[para *1.00*][para (1.764e-006s)]]]] 

[[__hypergeometric_distrib CDF][[perf msvc-dist-hypergeometric-cdf..[para 3.60[footnote This result is somewhat misleading: for small values of the parameters there is  virtually no difference between the two libraries, but for large values the Boost implementation is /much/ slower, albeit with much improved precision.]][para (5.987e-007s)]]][[perf msvc-dist-hypergeo-R-cdf..[para *1.00*][para (1.665e-007s)]]][NA]]
[[__hypergeometric_distrib Quantile][[perf msvc-dist-hypergeometric-quantile..[para *1.00*][para (5.684e-007s)]]][[perf msvc-dist-hypergeo-R-quantile..[para 3.53][para (2.004e-006s)]]][NA]] 
[[__logistic_distrib CDF][[perf msvc-dist-logistic-cdf..[para *1.00*][para (1.714e-007s)]]][[perf msvc-dist-logis-R-cdf..[para 5.24][para (8.984e-007s)]]][NA]]
[[__logistic_distrib Quantile][[perf msvc-dist-logistic-quantile..[para 1.02][para (2.084e-007s)]]][[perf msvc-dist-logis-R-quantile..[para *1.00*][para (2.043e-007s)]]][NA]] 

[[__lognormal_distrib CDF][[perf msvc-dist-lognormal-cdf..[para *1.00*][para (3.579e-007s)]]][[perf msvc-dist-lnorm-R-cdf..[para 1.49][para (5.332e-007s)]]][NA]]
[[__lognormal_distrib Quantile][[perf msvc-dist-lognormal-quantile..[para *1.00*][para (9.622e-007s)]]][[perf msvc-dist-lnorm-R-quantile..[para 1.57][para (1.507e-006s)]]][NA]]
[[__negative_binomial_distrib CDF][[perf msvc-dist-negative_binomial-cdf..[para *1.00*][para (6.227e-007s)]]][[perf msvc-dist-nbinom-R-cdf..[para 2.25][para (1.403e-006s)]]][[perf msvc-dist-negative_binomial-dcd-cdf..[para 2.21][para (1.378e-006s)]]]]
[[__negative_binomial_distrib Quantile][[perf msvc-dist-negative_binomial-quantile..[para *1.00*][para (8.594e-006s)]]][[perf msvc-dist-nbinom-R-quantile..[para 43.43[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 (3.732e-004s)]]][[perf msvc-dist-negative_binomial-dcd-quantile..[para 3.48][para (2.994e-005s)]]]]

[[__non_central_chi_squared_distrib CDF][[perf msvc-dist-non_central_chi_squared-cdf..[para 2.16][para (3.926e-006s)]]][[perf msvc-dist-nchisq-R-cdf..[para 79.93][para (1.450e-004s)]]][[perf msvc-dist-non_central_chi_squared-dcd-cdf..[para *1.00*][para (1.814e-006s)]]]]
[[__non_central_chi_squared_distrib Quantile][[perf msvc-dist-non_central_chi_squared-quantile..[para 5.00][para (3.393e-004s)]]][[perf msvc-dist-nchisq-R-quantile..[para 393.90[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 (2.673e-002s)]]][[perf msvc-dist-non_central_chi_squared-dcd-quantile..[para *1.00*][para (6.786e-005s)]]]]
[[__non_central_F_distrib CDF][[perf msvc-dist-non_central_f-cdf..[para 1.59][para (1.128e-005s)]]][[perf msvc-dist-nf-R-cdf..[para *1.00*][para (7.087e-006s)]]][[perf msvc-dist-fnc-cdf..[para *1.00*][para (4.274e-006s)]]]]
[[__non_central_F_distrib Quantile][[perf msvc-dist-non_central_f-quantile..[para *1.00*][para (4.750e-004s)]]][[perf msvc-dist-nf-R-quantile..[para 1.62][para (7.681e-004s)]]][[perf msvc-dist-ncf-quantile..[para *1.00*][para (4.274e-006s)]]]]
[[__non_central_t_distrib CDF][[perf msvc-dist-non_central_t-cdf..[para 3.41][para (1.852e-005s)]]][[perf msvc-dist-nt-R-cdf..[para *1.00*][para (5.436e-006s)]]][NA]]
[[__non_central_t_distrib Quantile][[perf msvc-dist-non_central_t-quantile..[para 1.31][para (5.768e-004s)]]][[perf msvc-dist-nt-R-quantile..[para *1.00*[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 (4.411e-004s)]]][NA]]

[[__normal_distrib CDF][[perf msvc-dist-normal-cdf..[para *1.00*][para (8.373e-008s)]]][[perf msvc-dist-norm-R-cdf..[para 1.68][para (1.409e-007s)]]][[perf msvc-dist-nor-dcd-cdf..[para 6.01][para (5.029e-007s)]]]]
[[__normal_distrib Quantile][[perf msvc-dist-normal-quantile..[para 1.29][para (1.521e-007s)]]][[perf msvc-dist-norm-R-quantile..[para *1.00*][para (1.182e-007s)]]][[perf msvc-dist-nor-dcd-quantile..[para 10.85][para (1.283e-006s)]]]]
[[__poisson_distrib CDF][[perf msvc-dist-poisson-cdf..[para 1.18][para (5.193e-007s)]]][[perf msvc-dist-pois-R-cdf..[para 2.98][para (1.314e-006s)]]][[perf msvc-dist-poi-dcd-cdf..[para *1.00*][para (4.410e-007s)]]]]
[[__poisson_distrib][[perf msvc-dist-poisson-quantile..[para *1.00*][para (1.203e-006s)]]][[perf msvc-dist-pois-R-quantile..[para 2.20][para (2.642e-006s)]]][[perf msvc-dist-poi-dcd-quantile..[para 7.86][para (9.457e-006s)]]]]
[[__students_t_distrib CDF][[perf msvc-dist-students_t-cdf..[para *1.00*][para (8.655e-007s)]]][[perf msvc-dist-t-R-cdf..[para 1.06][para (9.166e-007s)]]][[perf msvc-dist-t-dcd-cdf..[para 1.04][para (8.999e-007s)]]]]
[[__students_t_distrib Quantile][[perf msvc-dist-students_t-quantile..[para *1.00*][para (2.294e-006s)]]][[perf msvc-dist-t-R-quantile..[para 1.36][para (3.131e-006s)]]][[perf msvc-dist-t-dcd-quantile..[para 4.82][para (1.106e-005s)]]]]
[[__weibull_distrib CDF][[perf msvc-dist-weibull-cdf..[para *1.00*][para (1.865e-007s)]]][[perf msvc-dist-weibull-R-cdf..[para 2.33][para (4.341e-007s)]]][NA]]
[[__weibull_distrib Quantile][[perf msvc-dist-weibull-quantile..[para *1.00*][para (3.608e-007s)]]][[perf msvc-dist-weibull-R-quantile..[para 1.22][para (4.410e-007s)]]][NA]]
]

[heading Comparison to the R Statistical Library on Linux]

All the results were measured on a 2.0GHz Intel T5800 Core 2 Duo, 4Gb RAM, Ubuntu Linux 9
machine, with the test program and R-2.9.2 compiled with GNU G++ 4.3.3 using -O3 -DNDEBUG=1.

[table A Comparison to the R Statistical Library on Linux
[[Statistical Function][Boost][R][DCDFLIB]]
[[__beta_distrib CDF][[perf gcc-4_3_2-dist-beta-cdf..[para 2.09][para (3.189e-006s)]]][[perf gcc-4_3_2-dist-beta-R-cdf..[para *1.00*][para (1.526e-006s)]]][[perf gcc-4_3_2-dist-beta-dcd-cdf..[para 1.19][para (1.822e-006s)]]]]
[[__beta_distrib Quantile][[perf gcc-4_3_2-dist-beta-quantile..[para *1.00*][para (1.185e-005s)]]][[perf gcc-4_3_2-dist-beta-R-quantile..[para 30.51[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 (3.616e-004s)]]][[perf gcc-4_3_2-dist-beta-dcd-quantile..[para 2.52][para (2.989e-005s)]]]]
[[__binomial_distrib CDF][[perf gcc-4_3_2-dist-binomial-cdf..[para 4.41][para (9.175e-007s)]]][[perf gcc-4_3_2-dist-binom-R-cdf..[para 3.59][para (7.476e-007s)]]][[perf gcc-4_3_2-dist-binomial-dcd-cdf..[para *1.00*][para (2.081e-007s)]]]]
[[__binomial_distrib Quantile][[perf gcc-4_3_2-dist-binomial-quantile..[para 1.57][para (6.925e-006s)]]][[perf gcc-4_3_2-dist-binom-R-quantile..[para *1.00*][para (4.407e-006s)]]][[perf gcc-4_3_2-dist-binomial-dcd-quantile..[para 7.43][para (3.274e-005s)]]]]
[[__cauchy_distrib CDF][[perf gcc-4_3_2-dist-cauchy-cdf..[para *1.00*][para (1.594e-007s)]]][[perf gcc-4_3_2-dist-cauchy-R-cdf..[para 1.04][para (1.654e-007s)]]][NA]]
[[__cauchy_distrib Quantile][[perf gcc-4_3_2-dist-cauchy-quantile..[para 1.21][para (1.752e-007s)]]][[perf gcc-4_3_2-dist-cauchy-R-quantile..[para *1.00*][para (1.448e-007s)]]][NA]]
[[__chi_squared_distrib CDF][[perf gcc-4_3_2-dist-chi_squared-cdf..[para 2.61][para (1.376e-006s)]]][[perf gcc-4_3_2-dist-chisq-R-cdf..[para 2.36][para (1.243e-006s)]]][[perf gcc-4_3_2-dist-chi_squared-dcd-cdf..[para *1.00*][para (5.270e-007s)]]]]
[[__chi_squared_distrib Quantile][[perf gcc-4_3_2-dist-chi_squared-quantile..[para *1.00*][para (4.252e-006s)]]][[perf gcc-4_3_2-dist-chisq-R-quantile..[para 1.34][para (5.700e-006s)]]][[perf gcc-4_3_2-dist-chi_squared-dcd-quantile..[para 3.47][para (1.477e-005s)]]]]
[[__exp_distrib CDF][[perf gcc-4_3_2-dist-exponential-cdf..[para *1.00*][para (1.342e-007s)]]][[perf gcc-4_3_2-dist-exp-R-cdf..[para 1.25][para (1.677e-007s)]]][NA]]
[[__exp_distrib Quantile][[perf gcc-4_3_2-dist-exponential-quantile..[para *1.00*][para (8.827e-008s)]]][[perf gcc-4_3_2-dist-exp-R-quantile..[para 1.07][para (9.470e-008s)]]][NA]]
[[__F_distrib CDF][[perf gcc-4_3_2-dist-fisher_f-cdf..[para 1.62][para (2.324e-006s)]]][[perf gcc-4_3_2-dist-f-R-cdf..[para 1.19][para (1.711e-006s)]]][[perf gcc-4_3_2-dist-f-dcd-cdf..[para *1.00*][para (1.437e-006s)]]]]
[[__F_distrib Quantile][[perf gcc-4_3_2-dist-fisher_f-quantile..[para 1.53][para (1.577e-005s)]]][[perf gcc-4_3_2-dist-f-R-quantile..[para *1.00*][para (1.033e-005s)]]][[perf gcc-4_3_2-dist-f-dcd-quantile..[para 2.63][para (2.719e-005s)]]]]
[[__gamma_distrib CDF][[perf gcc-4_3_2-dist-gamma-cdf..[para 3.18][para (1.582e-006s)]]][[perf gcc-4_3_2-dist-gamma-R-cdf..[para 2.63][para (1.309e-006s)]]][[perf gcc-4_3_2-dist-gam-dcd-cdf..[para *1.00*][para (4.980e-007s)]]]]
[[__gamma_distrib Quantile][[perf gcc-4_3_2-dist-gamma-quantile..[para 2.19][para (4.770e-006s)]]][[perf gcc-4_3_2-dist-gamma-R-quantile..[para 6.94][para (1.513e-005s)]]][[perf gcc-4_3_2-dist-gam-dcd-quantile..[para *1.00*][para (2.179e-006s)]]]] 

[[__hypergeometric_distrib CDF][[perf gcc-4_3_2-dist-hypergeometric-cdf..[para 2.20[footnote This result is somewhat misleading: for small values of the parameters there is  virtually no difference between the two libraries, but for large values the Boost implementation is /much/ slower, albeit with much improved precision.]][para (3.522e-007s)]]][[perf gcc-4_3_2-dist-hypergeo-R-cdf..[para *1.00*][para (1.601e-007s)]]][NA]]
[[__hypergeometric_distrib Quantile][[perf gcc-4_3_2-dist-hypergeometric-quantile..[para *1.00*][para (8.279e-007s)]]][[perf gcc-4_3_2-dist-hypergeo-R-quantile..[para 2.57][para (2.125e-006s)]]][NA]] 
[[__logistic_distrib CDF][[perf gcc-4_3_2-dist-logistic-cdf..[para *1.00*][para (9.398e-008s)]]][[perf gcc-4_3_2-dist-logis-R-cdf..[para 2.75][para (2.588e-007s)]]][NA]]
[[__logistic_distrib Quantile][[perf gcc-4_3_2-dist-logistic-quantile..[para *1.00*][para (9.893e-008s)]]][[perf gcc-4_3_2-dist-logis-R-quantile..[para 1.30][para (1.285e-007s)]]][NA]] 

[[__lognormal_distrib CDF][[perf gcc-4_3_2-dist-lognormal-cdf..[para *1.00*][para (1.831e-007s)]]][[perf gcc-4_3_2-dist-lnorm-R-cdf..[para 1.39][para (2.539e-007s)]]][NA]]
[[__lognormal_distrib Quantile][[perf gcc-4_3_2-dist-lognormal-quantile..[para 1.10][para (5.551e-007s)]]][[perf gcc-4_3_2-dist-lnorm-R-quantile..[para *1.00*][para (5.037e-007s)]]][NA]]
[[__negative_binomial_distrib CDF][[perf gcc-4_3_2-dist-negative_binomial-cdf..[para 1.08][para (1.563e-006s)]]][[perf gcc-4_3_2-dist-nbinom-R-cdf..[para *1.00*][para (1.444e-006s)]]][[perf gcc-4_3_2-dist-negative_binomial-dcd-cdf..[para *1.00*][para (1.444e-006s)]]]]
[[__negative_binomial_distrib Quantile][[perf gcc-4_3_2-dist-negative_binomial-quantile..[para *1.00*][para (1.700e-005s)]]][[perf gcc-4_3_2-dist-nbinom-R-quantile..[para 25.92[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 (4.407e-004s)]]][[perf gcc-4_3_2-dist-negative_binomial-dcd-quantile..[para 1.93][para (3.274e-005s)]]]]

[[__non_central_chi_squared_distrib CDF][[perf gcc-4_3_2-dist-non_central_chi_squared-cdf..[para 5.06][para (2.841e-005s)]]][[perf gcc-4_3_2-dist-nchisq-R-cdf..[para 25.01][para (1.405e-004s)]]][[perf gcc-4_3_2-dist-non_central_chi_squared-dcd-cdf..[para *1.00*][para (5.617e-006s)]]]]
[[__non_central_chi_squared_distrib Quantile][[perf gcc-4_3_2-dist-non_central_chi_squared-quantile..[para 8.47][para (1.879e-003s)]]][[perf gcc-4_3_2-dist-nchisq-R-quantile..[para 144.91[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 (3.214e-002s)]]][[perf gcc-4_3_2-dist-non_central_chi_squared-dcd-quantile..[para *1.00*][para (2.218e-004s)]]]]
[[__non_central_F_distrib CDF][[perf gcc-4_3_2-dist-non_central_f-cdf..[para 10.33][para (5.868e-005s)]]][[perf gcc-4_3_2-dist-nf-R-cdf..[para 1.42][para (8.058e-006s)]]][[perf gcc-4_3_2-dist-fnc-dcd-cdf..[para *1.00*][para (5.682e-006s)]]]]
[[__non_central_F_distrib Quantile][[perf gcc-4_3_2-dist-non_central_f-quantile..[para 5.64][para (7.869e-004s)]]][[perf gcc-4_3_2-dist-nf-R-quantile..[para 6.63][para (9.256e-004s)]]][[perf gcc-4_3_2-dist-fnc-dcd-quantile..[para *1.00*][para (1.396e-004s)]]]]
[[__non_central_t_distrib CDF][[perf gcc-4_3_2-dist-non_central_t-cdf..[para 4.91][para (3.357e-005s)]]][[perf gcc-4_3_2-dist-nt-R-cdf..[para *1.00*][para (6.844e-006s)]]][NA]]
[[__non_central_t_distrib Quantile][[perf gcc-4_3_2-dist-non_central_t-quantile..[para 1.57][para (9.265e-004s)]]][[perf gcc-4_3_2-dist-nt-R-quantile..[para *1.00*[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 (5.916e-004s)]]][NA]]

[[__normal_distrib CDF][[perf gcc-4_3_2-dist-normal-cdf..[para *1.00*][para (1.074e-007s)]]][[perf gcc-4_3_2-dist-norm-R-cdf..[para 1.16][para (1.245e-007s)]]][[perf gcc-4_3_2-dist-nor-dcd-cdf..[para 5.36][para (5.762e-007s)]]]]
[[__normal_distrib Quantile][[perf gcc-4_3_2-dist-normal-quantile..[para 1.28][para (1.902e-007s)]]][[perf gcc-4_3_2-dist-norm-R-quantile..[para *1.00*][para (1.490e-007s)]]][[perf gcc-4_3_2-dist-nor-dcd-quantile..[para 10.35][para (1.542e-006s)]]]]
[[__poisson_distrib CDF][[perf gcc-4_3_2-dist-poisson-cdf..[para 2.43][para (1.198e-006s)]]][[perf gcc-4_3_2-dist-pois-R-cdf..[para 2.25][para (1.110e-006s)]]][[perf gcc-4_3_2-dist-poi-dcd-cdf..[para *1.00*][para (4.937e-007s)]]]]
[[__poisson_distrib][[perf gcc-4_3_2-dist-poisson-quantile..[para 1.11][para (3.032e-006s)]]][[perf gcc-4_3_2-dist-pois-R-quantile..[para *1.00*][para (2.724e-006s)]]][[perf gcc-4_3_2-dist-poi-dcd-quantile..[para 4.07][para (1.110e-005s)]]]]
[[__students_t_distrib CDF][[perf gcc-4_3_2-dist-students_t-cdf..[para 2.17][para (2.020e-006s)]]][[perf gcc-4_3_2-dist-t-R-cdf..[para *1.00*][para (9.321e-007s)]]][[perf gcc-4_3_2-dist-t-dcd-cdf..[para 1.10][para (1.021e-006s)]]]]
[[__students_t_distrib Quantile][[perf gcc-4_3_2-dist-students_t-quantile..[para 1.18][para (3.972e-006s)]]][[perf gcc-4_3_2-dist-t-R-quantile..[para *1.00*][para (3.364e-006s)]]][[perf gcc-4_3_2-dist-t-dcd-quantile..[para 3.89][para (1.308e-005s)]]]]
[[__weibull_distrib CDF][[perf gcc-4_3_2-dist-weibull-cdf..[para *1.00*][para (3.662e-007s)]]][[perf gcc-4_3_2-dist-weibull-R-cdf..[para 1.04][para (3.808e-007s)]]][NA]]
[[__weibull_distrib Quantile][[perf gcc-4_3_2-dist-weibull-quantile..[para *1.00*][para (4.112e-007s)]]][[perf gcc-4_3_2-dist-weibull-R-quantile..[para 1.05][para (4.317e-007s)]]][NA]]
]

[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).
]