mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-15 16:42:12 -05:00
143 lines
4.5 KiB
Plaintext
143 lines
4.5 KiB
Plaintext
[section:error_function Error Functions]
|
|
|
|
[h4 Synopsis]
|
|
|
|
``
|
|
#include <boost/math/special_functions/erf.hpp>
|
|
``
|
|
|
|
namespace boost{ namespace math{
|
|
|
|
template <class T>
|
|
``__sf_result`` erf(T z);
|
|
|
|
template <class T, class ``__Policy``>
|
|
``__sf_result`` erf(T z, const ``__Policy``&);
|
|
|
|
template <class T>
|
|
``__sf_result`` erfc(T z);
|
|
|
|
template <class T, class ``__Policy``>
|
|
``__sf_result`` erfc(T z, const ``__Policy``&);
|
|
|
|
}} // namespaces
|
|
|
|
The return type of these functions is computed using the __arg_promotion_rules:
|
|
the return type is `double` if T is an integer type, and T otherwise.
|
|
|
|
[optional_policy]
|
|
|
|
[h4 Description]
|
|
|
|
template <class T>
|
|
``__sf_result`` erf(T z);
|
|
|
|
template <class T, class ``__Policy``>
|
|
``__sf_result`` erf(T z, const ``__Policy``&);
|
|
|
|
Returns the [@http://en.wikipedia.org/wiki/Error_function error function]
|
|
[@http://functions.wolfram.com/GammaBetaErf/Erf/ erf] of z:
|
|
|
|
[equation erf1]
|
|
|
|
[graph erf]
|
|
|
|
template <class T>
|
|
``__sf_result`` erfc(T z);
|
|
|
|
template <class T, class ``__Policy``>
|
|
``__sf_result`` erfc(T z, const ``__Policy``&);
|
|
|
|
Returns the complement of the [@http://functions.wolfram.com/GammaBetaErf/Erfc/ error function] of z:
|
|
|
|
[equation erf2]
|
|
|
|
[graph erfc]
|
|
|
|
[h4 Accuracy]
|
|
|
|
The following table shows the peak errors (in units of epsilon)
|
|
found on various platforms with various floating point types,
|
|
along with comparisons to the __gsl, __glibc, __hpc and __cephes libraries.
|
|
Unless otherwise specified any floating point type that is narrower
|
|
than the one shown will have __zero_error.
|
|
|
|
[table_erf]
|
|
|
|
[table_erfc]
|
|
|
|
[h4 Testing]
|
|
|
|
The tests for these functions come in two parts:
|
|
basic sanity checks use spot values calculated using
|
|
[@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf Mathworld's online evaluator],
|
|
while accuracy checks use high-precision test values calculated at 1000-bit precision with
|
|
[@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation.
|
|
Note that the generic and type-specific
|
|
versions of these functions use differing implementations internally, so this
|
|
gives us reasonably independent test data. Using our test data to test other
|
|
"known good" implementations also provides an additional sanity check.
|
|
|
|
[h4 Implementation]
|
|
|
|
All versions of these functions first use the usual reflection formulas
|
|
to make their arguments positive:
|
|
|
|
erf(-z) = 1 - erf(z);
|
|
|
|
erfc(-z) = 2 - erfc(z); // preferred when -z < -0.5
|
|
|
|
erfc(-z) = 1 + erf(z); // preferred when -0.5 <= -z < 0
|
|
|
|
The generic versions of these functions are implemented in terms of
|
|
the incomplete gamma function.
|
|
|
|
When the significand (mantissa) size is recognised
|
|
(currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double)
|
|
then a series of rational approximations [jm_rationals] are used.
|
|
|
|
For `z <= 0.5` then a rational approximation to erf is used, based on the
|
|
observation that erf is an odd function and therefore erf is calculated using:
|
|
|
|
erf(z) = z * (C + R(z*z));
|
|
|
|
where the rational approximation R(z*z) is optimised for absolute error:
|
|
as long as its absolute error is small enough compared to the constant C, then any
|
|
round-off error incurred during the computation of R(z*z) will effectively
|
|
disappear from the result. As a result the error for erf and erfc in this
|
|
region is very low: the last bit is incorrect in only a very small number of
|
|
cases.
|
|
|
|
For `z > 0.5` we observe that over a small interval \[a, b) then:
|
|
|
|
erfc(z) * exp(z*z) * z ~ c
|
|
|
|
for some constant c.
|
|
|
|
Therefore for `z > 0.5` we calculate erfc using:
|
|
|
|
erfc(z) = exp(-z*z) * (C + R(z - B)) / z;
|
|
|
|
Again R(z - B) is optimised for absolute error, and the constant `C` is
|
|
the average of `erfc(z) * exp(z*z) * z` taken at the endpoints of the range.
|
|
Once again, as long as the absolute error in R(z - B) is small
|
|
compared to `c` then `c + R(z - B)` will be correctly rounded, and the error
|
|
in the result will depend only on the accuracy of the exp function. In practice,
|
|
in all but a very small number of cases, the error is confined to the last bit
|
|
of the result. The constant `B` is chosen so that the left hand end of the range
|
|
of the rational approximation is 0.
|
|
|
|
For large `z` over a range \[a, +[infin]\] the above approximation is modified to:
|
|
|
|
erfc(z) = exp(-z*z) * (C + R(1 / z)) / z;
|
|
|
|
[endsect]
|
|
[/ :error_function The Error Functions]
|
|
|
|
[/
|
|
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).
|
|
]
|