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1156 lines
53 KiB
C++
1156 lines
53 KiB
C++
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// (C) Copyright John Maddock 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SPECIAL_ERF_HPP
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#define BOOST_MATH_SPECIAL_ERF_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/math_fwd.hpp>
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#include <boost/math/tools/config.hpp>
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#include <boost/math/special_functions/gamma.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/math/policies/error_handling.hpp>
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#include <boost/math/tools/big_constant.hpp>
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namespace boost{ namespace math{
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namespace detail
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{
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//
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// Asymptotic series for large z:
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//
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template <class T>
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struct erf_asympt_series_t
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{
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erf_asympt_series_t(T z) : xx(2 * -z * z), tk(1)
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{
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BOOST_MATH_STD_USING
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result = -exp(-z * z) / sqrt(boost::math::constants::pi<T>());
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result /= z;
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}
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typedef T result_type;
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T operator()()
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{
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BOOST_MATH_STD_USING
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T r = result;
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result *= tk / xx;
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tk += 2;
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if( fabs(r) < fabs(result))
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result = 0;
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return r;
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}
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private:
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T result;
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T xx;
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int tk;
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};
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//
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// How large z has to be in order to ensure that the series converges:
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//
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template <class T>
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inline float erf_asymptotic_limit_N(const T&)
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{
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return (std::numeric_limits<float>::max)();
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}
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inline float erf_asymptotic_limit_N(const mpl::int_<24>&)
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{
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return 2.8F;
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}
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inline float erf_asymptotic_limit_N(const mpl::int_<53>&)
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{
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return 4.3F;
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}
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inline float erf_asymptotic_limit_N(const mpl::int_<64>&)
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{
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return 4.8F;
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}
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inline float erf_asymptotic_limit_N(const mpl::int_<106>&)
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{
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return 6.5F;
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}
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inline float erf_asymptotic_limit_N(const mpl::int_<113>&)
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{
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return 6.8F;
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}
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template <class T, class Policy>
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inline T erf_asymptotic_limit()
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{
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typedef typename policies::precision<T, Policy>::type precision_type;
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typedef typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<24> >,
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typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<0> >,
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mpl::int_<0>,
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mpl::int_<24>
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>::type,
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typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<53> >,
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mpl::int_<53>,
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typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<64> >,
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mpl::int_<64>,
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typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<106> >,
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mpl::int_<106>,
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typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<113> >,
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mpl::int_<113>,
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mpl::int_<0>
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>::type
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>::type
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>::type
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>::type
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>::type tag_type;
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return erf_asymptotic_limit_N(tag_type());
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}
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template <class T, class Policy, class Tag>
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T erf_imp(T z, bool invert, const Policy& pol, const Tag& t)
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{
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BOOST_MATH_STD_USING
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BOOST_MATH_INSTRUMENT_CODE("Generic erf_imp called");
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if(z < 0)
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{
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if(!invert)
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return -erf_imp(T(-z), invert, pol, t);
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else
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return 1 + erf_imp(T(-z), false, pol, t);
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}
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T result;
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if(!invert && (z > detail::erf_asymptotic_limit<T, Policy>()))
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{
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detail::erf_asympt_series_t<T> s(z);
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boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
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result = boost::math::tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, 1);
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policies::check_series_iterations<T>("boost::math::erf<%1%>(%1%, %1%)", max_iter, pol);
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}
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else
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{
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T x = z * z;
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if(x < 0.6)
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{
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// Compute P:
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result = z * exp(-x);
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result /= sqrt(boost::math::constants::pi<T>());
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if(result != 0)
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result *= 2 * detail::lower_gamma_series(T(0.5f), x, pol);
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}
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else if(x < 1.1f)
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{
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// Compute Q:
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invert = !invert;
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result = tgamma_small_upper_part(T(0.5f), x, pol);
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result /= sqrt(boost::math::constants::pi<T>());
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}
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else
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{
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// Compute Q:
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invert = !invert;
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result = z * exp(-x);
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result /= sqrt(boost::math::constants::pi<T>());
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result *= upper_gamma_fraction(T(0.5f), x, policies::get_epsilon<T, Policy>());
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}
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}
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if(invert)
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result = 1 - result;
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return result;
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}
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template <class T, class Policy>
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T erf_imp(T z, bool invert, const Policy& pol, const mpl::int_<53>& t)
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{
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BOOST_MATH_STD_USING
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BOOST_MATH_INSTRUMENT_CODE("53-bit precision erf_imp called");
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if(z < 0)
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{
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if(!invert)
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return -erf_imp(T(-z), invert, pol, t);
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else if(z < -0.5)
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return 2 - erf_imp(T(-z), invert, pol, t);
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else
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return 1 + erf_imp(T(-z), false, pol, t);
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}
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T result;
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//
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// Big bunch of selection statements now to pick
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// which implementation to use,
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// try to put most likely options first:
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//
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if(z < 0.5)
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{
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//
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// We're going to calculate erf:
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//
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if(z < 1e-10)
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{
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if(z == 0)
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{
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result = T(0);
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}
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else
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{
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static const T c = BOOST_MATH_BIG_CONSTANT(T, 53, 0.003379167095512573896158903121545171688);
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result = static_cast<T>(z * 1.125f + z * c);
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}
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}
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else
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{
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// Maximum Deviation Found: 1.561e-17
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// Expected Error Term: 1.561e-17
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// Maximum Relative Change in Control Points: 1.155e-04
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// Max Error found at double precision = 2.961182e-17
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static const T Y = 1.044948577880859375f;
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static const T P[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0834305892146531832907),
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BOOST_MATH_BIG_CONSTANT(T, 53, -0.338165134459360935041),
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BOOST_MATH_BIG_CONSTANT(T, 53, -0.0509990735146777432841),
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BOOST_MATH_BIG_CONSTANT(T, 53, -0.00772758345802133288487),
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BOOST_MATH_BIG_CONSTANT(T, 53, -0.000322780120964605683831),
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};
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static const T Q[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.455004033050794024546),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0875222600142252549554),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.00858571925074406212772),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.000370900071787748000569),
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};
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T zz = z * z;
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result = z * (Y + tools::evaluate_polynomial(P, zz) / tools::evaluate_polynomial(Q, zz));
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}
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}
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else if(invert ? (z < 28) : (z < 5.8f))
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{
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//
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// We'll be calculating erfc:
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//
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invert = !invert;
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if(z < 1.5f)
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{
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// Maximum Deviation Found: 3.702e-17
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// Expected Error Term: 3.702e-17
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// Maximum Relative Change in Control Points: 2.845e-04
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// Max Error found at double precision = 4.841816e-17
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static const T Y = 0.405935764312744140625f;
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static const T P[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, -0.098090592216281240205),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.178114665841120341155),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.191003695796775433986),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0888900368967884466578),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0195049001251218801359),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.00180424538297014223957),
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};
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static const T Q[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.84759070983002217845),
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.42628004845511324508),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.578052804889902404909),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.12385097467900864233),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0113385233577001411017),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.337511472483094676155e-5),
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};
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BOOST_MATH_INSTRUMENT_VARIABLE(Y);
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BOOST_MATH_INSTRUMENT_VARIABLE(P[0]);
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BOOST_MATH_INSTRUMENT_VARIABLE(Q[0]);
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BOOST_MATH_INSTRUMENT_VARIABLE(z);
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result = Y + tools::evaluate_polynomial(P, T(z - 0.5)) / tools::evaluate_polynomial(Q, T(z - 0.5));
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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result *= exp(-z * z) / z;
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BOOST_MATH_INSTRUMENT_VARIABLE(result);
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}
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else if(z < 2.5f)
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{
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// Max Error found at double precision = 6.599585e-18
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// Maximum Deviation Found: 3.909e-18
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// Expected Error Term: 3.909e-18
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// Maximum Relative Change in Control Points: 9.886e-05
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static const T Y = 0.50672817230224609375f;
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static const T P[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, -0.0243500476207698441272),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0386540375035707201728),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.04394818964209516296),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175679436311802092299),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.00323962406290842133584),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.000235839115596880717416),
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};
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static const T Q[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.53991494948552447182),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.982403709157920235114),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.325732924782444448493),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0563921837420478160373),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.00410369723978904575884),
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};
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result = Y + tools::evaluate_polynomial(P, T(z - 1.5)) / tools::evaluate_polynomial(Q, T(z - 1.5));
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result *= exp(-z * z) / z;
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}
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else if(z < 4.5f)
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{
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// Maximum Deviation Found: 1.512e-17
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// Expected Error Term: 1.512e-17
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// Maximum Relative Change in Control Points: 2.222e-04
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// Max Error found at double precision = 2.062515e-17
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static const T Y = 0.5405750274658203125f;
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static const T P[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.00295276716530971662634),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0137384425896355332126),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.00840807615555585383007),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.00212825620914618649141),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.000250269961544794627958),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.113212406648847561139e-4),
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};
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static const T Q[] = {
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
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BOOST_MATH_BIG_CONSTANT(T, 53, 1.04217814166938418171),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.442597659481563127003),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0958492726301061423444),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.0105982906484876531489),
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BOOST_MATH_BIG_CONSTANT(T, 53, 0.000479411269521714493907),
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};
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result = Y + tools::evaluate_polynomial(P, T(z - 3.5)) / tools::evaluate_polynomial(Q, T(z - 3.5));
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|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
// Max Error found at double precision = 2.997958e-17
|
||
|
|
// Maximum Deviation Found: 2.860e-17
|
||
|
|
// Expected Error Term: 2.859e-17
|
||
|
|
// Maximum Relative Change in Control Points: 1.357e-05
|
||
|
|
static const T Y = 0.5579090118408203125f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 0.00628057170626964891937),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 0.0175389834052493308818),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, -0.212652252872804219852),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, -0.687717681153649930619),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, -2.5518551727311523996),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, -3.22729451764143718517),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, -2.8175401114513378771),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 2.79257750980575282228),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 11.0567237927800161565),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 15.930646027911794143),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 22.9367376522880577224),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 13.5064170191802889145),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 53, 5.48409182238641741584),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// Any value of z larger than 28 will underflow to zero:
|
||
|
|
//
|
||
|
|
result = 0;
|
||
|
|
invert = !invert;
|
||
|
|
}
|
||
|
|
|
||
|
|
if(invert)
|
||
|
|
{
|
||
|
|
result = 1 - result;
|
||
|
|
}
|
||
|
|
|
||
|
|
return result;
|
||
|
|
} // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const mpl::int_<53>& t)
|
||
|
|
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
T erf_imp(T z, bool invert, const Policy& pol, const mpl::int_<64>& t)
|
||
|
|
{
|
||
|
|
BOOST_MATH_STD_USING
|
||
|
|
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("64-bit precision erf_imp called");
|
||
|
|
|
||
|
|
if(z < 0)
|
||
|
|
{
|
||
|
|
if(!invert)
|
||
|
|
return -erf_imp(T(-z), invert, pol, t);
|
||
|
|
else if(z < -0.5)
|
||
|
|
return 2 - erf_imp(T(-z), invert, pol, t);
|
||
|
|
else
|
||
|
|
return 1 + erf_imp(T(-z), false, pol, t);
|
||
|
|
}
|
||
|
|
|
||
|
|
T result;
|
||
|
|
|
||
|
|
//
|
||
|
|
// Big bunch of selection statements now to pick which
|
||
|
|
// implementation to use, try to put most likely options
|
||
|
|
// first:
|
||
|
|
//
|
||
|
|
if(z < 0.5)
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// We're going to calculate erf:
|
||
|
|
//
|
||
|
|
if(z == 0)
|
||
|
|
{
|
||
|
|
result = 0;
|
||
|
|
}
|
||
|
|
else if(z < 1e-10)
|
||
|
|
{
|
||
|
|
static const T c = BOOST_MATH_BIG_CONSTANT(T, 64, 0.003379167095512573896158903121545171688);
|
||
|
|
result = z * 1.125 + z * c;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
// Max Error found at long double precision = 1.623299e-20
|
||
|
|
// Maximum Deviation Found: 4.326e-22
|
||
|
|
// Expected Error Term: -4.326e-22
|
||
|
|
// Maximum Relative Change in Control Points: 1.474e-04
|
||
|
|
static const T Y = 1.044948577880859375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0834305892146531988966),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.338097283075565413695),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0509602734406067204596),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.00904906346158537794396),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.000489468651464798669181),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.200305626366151877759e-4),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.455817300515875172439),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0916537354356241792007),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0102722652675910031202),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000650511752687851548735),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.189532519105655496778e-4),
|
||
|
|
};
|
||
|
|
result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else if(invert ? (z < 110) : (z < 6.4f))
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// We'll be calculating erfc:
|
||
|
|
//
|
||
|
|
invert = !invert;
|
||
|
|
if(z < 1.5)
|
||
|
|
{
|
||
|
|
// Max Error found at long double precision = 3.239590e-20
|
||
|
|
// Maximum Deviation Found: 2.241e-20
|
||
|
|
// Expected Error Term: -2.241e-20
|
||
|
|
// Maximum Relative Change in Control Points: 5.110e-03
|
||
|
|
static const T Y = 0.405935764312744140625f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.0980905922162812031672),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.159989089922969141329),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.222359821619935712378),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.127303921703577362312),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0384057530342762400273),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00628431160851156719325),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000441266654514391746428),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.266689068336295642561e-7),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 2.03237474985469469291),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.78355454954969405222),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.867940326293760578231),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.248025606990021698392),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0396649631833002269861),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00279220237309449026796),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 2.5)
|
||
|
|
{
|
||
|
|
// Max Error found at long double precision = 3.686211e-21
|
||
|
|
// Maximum Deviation Found: 1.495e-21
|
||
|
|
// Expected Error Term: -1.494e-21
|
||
|
|
// Maximum Relative Change in Control Points: 1.793e-04
|
||
|
|
static const T Y = 0.50672817230224609375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.024350047620769840217),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0343522687935671451309),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0505420824305544949541),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0257479325917757388209),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00669349844190354356118),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00090807914416099524444),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.515917266698050027934e-4),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.71657861671930336344),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.26409634824280366218),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.512371437838969015941),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.120902623051120950935),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0158027197831887485261),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.000897871370778031611439),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 4.5)
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 1.107e-20
|
||
|
|
// Expected Error Term: -1.106e-20
|
||
|
|
// Maximum Relative Change in Control Points: 1.709e-04
|
||
|
|
// Max Error found at long double precision = 1.446908e-20
|
||
|
|
static const T Y = 0.5405750274658203125f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0029527671653097284033),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0141853245895495604051),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0104959584626432293901),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00343963795976100077626),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00059065441194877637899),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.523435380636174008685e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.189896043050331257262e-5),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.19352160185285642574),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.603256964363454392857),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.165411142458540585835),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0259729870946203166468),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00221657568292893699158),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.804149464190309799804e-4),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 3.5f)) / tools::evaluate_polynomial(Q, T(z - 3.5f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
// Max Error found at long double precision = 7.961166e-21
|
||
|
|
// Maximum Deviation Found: 6.677e-21
|
||
|
|
// Expected Error Term: 6.676e-21
|
||
|
|
// Maximum Relative Change in Control Points: 2.319e-05
|
||
|
|
static const T Y = 0.55825519561767578125f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.00593438793008050214106),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 0.0280666231009089713937),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.141597835204583050043),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -0.978088201154300548842),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -5.47351527796012049443),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -13.8677304660245326627),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -27.1274948720539821722),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -29.2545152747009461519),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, -16.8865774499799676937),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 4.72948911186645394541),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 23.6750543147695749212),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 60.0021517335693186785),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 131.766251645149522868),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 178.167924971283482513),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 182.499390505915222699),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 104.365251479578577989),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 64, 30.8365511891224291717),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// Any value of z larger than 110 will underflow to zero:
|
||
|
|
//
|
||
|
|
result = 0;
|
||
|
|
invert = !invert;
|
||
|
|
}
|
||
|
|
|
||
|
|
if(invert)
|
||
|
|
{
|
||
|
|
result = 1 - result;
|
||
|
|
}
|
||
|
|
|
||
|
|
return result;
|
||
|
|
} // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const mpl::int_<64>& t)
|
||
|
|
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
T erf_imp(T z, bool invert, const Policy& pol, const mpl::int_<113>& t)
|
||
|
|
{
|
||
|
|
BOOST_MATH_STD_USING
|
||
|
|
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("113-bit precision erf_imp called");
|
||
|
|
|
||
|
|
if(z < 0)
|
||
|
|
{
|
||
|
|
if(!invert)
|
||
|
|
return -erf_imp(T(-z), invert, pol, t);
|
||
|
|
else if(z < -0.5)
|
||
|
|
return 2 - erf_imp(T(-z), invert, pol, t);
|
||
|
|
else
|
||
|
|
return 1 + erf_imp(T(-z), false, pol, t);
|
||
|
|
}
|
||
|
|
|
||
|
|
T result;
|
||
|
|
|
||
|
|
//
|
||
|
|
// Big bunch of selection statements now to pick which
|
||
|
|
// implementation to use, try to put most likely options
|
||
|
|
// first:
|
||
|
|
//
|
||
|
|
if(z < 0.5)
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// We're going to calculate erf:
|
||
|
|
//
|
||
|
|
if(z == 0)
|
||
|
|
{
|
||
|
|
result = 0;
|
||
|
|
}
|
||
|
|
else if(z < 1e-20)
|
||
|
|
{
|
||
|
|
static const T c = BOOST_MATH_BIG_CONSTANT(T, 113, 0.003379167095512573896158903121545171688);
|
||
|
|
result = z * 1.125 + z * c;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
// Max Error found at long double precision = 2.342380e-35
|
||
|
|
// Maximum Deviation Found: 6.124e-36
|
||
|
|
// Expected Error Term: -6.124e-36
|
||
|
|
// Maximum Relative Change in Control Points: 3.492e-10
|
||
|
|
static const T Y = 1.0841522216796875f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0442269454158250738961589031215451778),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.35549265736002144875335323556961233),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0582179564566667896225454670863270393),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0112694696904802304229950538453123925),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.000805730648981801146251825329609079099),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.566304966591936566229702842075966273e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.169655010425186987820201021510002265e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.344448249920445916714548295433198544e-7),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.466542092785657604666906909196052522),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.100005087012526447295176964142107611),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0128341535890117646540050072234142603),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00107150448466867929159660677016658186),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.586168368028999183607733369248338474e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.196230608502104324965623171516808796e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.313388521582925207734229967907890146e-7),
|
||
|
|
};
|
||
|
|
result = z * (Y + tools::evaluate_polynomial(P, T(z * z)) / tools::evaluate_polynomial(Q, T(z * z)));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else if(invert ? (z < 110) : (z < 8.65f))
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// We'll be calculating erfc:
|
||
|
|
//
|
||
|
|
invert = !invert;
|
||
|
|
if(z < 1)
|
||
|
|
{
|
||
|
|
// Max Error found at long double precision = 3.246278e-35
|
||
|
|
// Maximum Deviation Found: 1.388e-35
|
||
|
|
// Expected Error Term: 1.387e-35
|
||
|
|
// Maximum Relative Change in Control Points: 6.127e-05
|
||
|
|
static const T Y = 0.371877193450927734375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0640320213544647969396032886581290455),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.200769874440155895637857443946706731),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.378447199873537170666487408805779826),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.30521399466465939450398642044975127),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.146890026406815277906781824723458196),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0464837937749539978247589252732769567),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00987895759019540115099100165904822903),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00137507575429025512038051025154301132),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0001144764551085935580772512359680516),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.436544865032836914773944382339900079e-5),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2.47651182872457465043733800302427977),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2.78706486002517996428836400245547955),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.87295924621659627926365005293130693),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.829375825174365625428280908787261065),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.251334771307848291593780143950311514),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0522110268876176186719436765734722473),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00718332151250963182233267040106902368),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000595279058621482041084986219276392459),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.226988669466501655990637599399326874e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.270666232259029102353426738909226413e-10),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 0.5f)) / tools::evaluate_polynomial(Q, T(z - 0.5f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 1.5)
|
||
|
|
{
|
||
|
|
// Max Error found at long double precision = 2.215785e-35
|
||
|
|
// Maximum Deviation Found: 1.539e-35
|
||
|
|
// Expected Error Term: 1.538e-35
|
||
|
|
// Maximum Relative Change in Control Points: 6.104e-05
|
||
|
|
static const T Y = 0.45658016204833984375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0289965858925328393392496555094848345),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0868181194868601184627743162571779226),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.169373435121178901746317404936356745),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.13350446515949251201104889028133486),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0617447837290183627136837688446313313),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0185618495228251406703152962489700468),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00371949406491883508764162050169531013),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000485121708792921297742105775823900772),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.376494706741453489892108068231400061e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.133166058052466262415271732172490045e-5),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2.32970330146503867261275580968135126),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2.46325715420422771961250513514928746),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.55307882560757679068505047390857842),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.644274289865972449441174485441409076),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.182609091063258208068606847453955649),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0354171651271241474946129665801606795),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00454060370165285246451879969534083997),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000349871943711566546821198612518656486),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.123749319840299552925421880481085392e-4),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 1.0f)) / tools::evaluate_polynomial(Q, T(z - 1.0f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 2.25)
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 1.418e-35
|
||
|
|
// Expected Error Term: 1.418e-35
|
||
|
|
// Maximum Relative Change in Control Points: 1.316e-04
|
||
|
|
// Max Error found at long double precision = 1.998462e-35
|
||
|
|
static const T Y = 0.50250148773193359375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0201233630504573402185161184151016606),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0331864357574860196516686996302305002),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0716562720864787193337475444413405461),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0545835322082103985114927569724880658),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0236692635189696678976549720784989593),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00656970902163248872837262539337601845),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00120282643299089441390490459256235021),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000142123229065182650020762792081622986),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.991531438367015135346716277792989347e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.312857043762117596999398067153076051e-6),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2.13506082409097783827103424943508554),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2.06399257267556230937723190496806215),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.18678481279932541314830499880691109),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.447733186643051752513538142316799562),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.11505680005657879437196953047542148),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.020163993632192726170219663831914034),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00232708971840141388847728782209730585),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000160733201627963528519726484608224112),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.507158721790721802724402992033269266e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.18647774409821470950544212696270639e-12),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 1.5f)) / tools::evaluate_polynomial(Q, T(z - 1.5f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if (z < 3)
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 3.575e-36
|
||
|
|
// Expected Error Term: 3.575e-36
|
||
|
|
// Maximum Relative Change in Control Points: 7.103e-05
|
||
|
|
// Max Error found at long double precision = 5.794737e-36
|
||
|
|
static const T Y = 0.52896785736083984375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.00902152521745813634562524098263360074),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0145207142776691539346923710537580927),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0301681239582193983824211995978678571),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0215548540823305814379020678660434461),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00864683476267958365678294164340749949),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00219693096885585491739823283511049902),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000364961639163319762492184502159894371),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.388174251026723752769264051548703059e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.241918026931789436000532513553594321e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.676586625472423508158937481943649258e-7),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.93669171363907292305550231764920001),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.69468476144051356810672506101377494),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.880023580986436640372794392579985511),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.299099106711315090710836273697708402),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0690593962363545715997445583603382337),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0108427016361318921960863149875360222),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00111747247208044534520499324234317695),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.686843205749767250666787987163701209e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.192093541425429248675532015101904262e-5),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 2.25f)) / tools::evaluate_polynomial(Q, T(z - 2.25f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 3.5)
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 8.126e-37
|
||
|
|
// Expected Error Term: -8.126e-37
|
||
|
|
// Maximum Relative Change in Control Points: 1.363e-04
|
||
|
|
// Max Error found at long double precision = 1.747062e-36
|
||
|
|
static const T Y = 0.54037380218505859375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.0033703486408887424921155540591370375),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0104948043110005245215286678898115811),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0148530118504000311502310457390417795),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00816693029245443090102738825536188916),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00249716579989140882491939681805594585),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0004655591010047353023978045800916647),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.531129557920045295895085236636025323e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.343526765122727069515775194111741049e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.971120407556888763695313774578711839e-7),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.59911256167540354915906501335919317),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.136006830764025173864831382946934),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.468565867990030871678574840738423023),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.122821824954470343413956476900662236),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0209670914950115943338996513330141633),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00227845718243186165620199012883547257),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000144243326443913171313947613547085553),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.407763415954267700941230249989140046e-5),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 3.0f)) / tools::evaluate_polynomial(Q, T(z - 3.0f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 5.5)
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 5.804e-36
|
||
|
|
// Expected Error Term: -5.803e-36
|
||
|
|
// Maximum Relative Change in Control Points: 2.475e-05
|
||
|
|
// Max Error found at long double precision = 1.349545e-35
|
||
|
|
static const T Y = 0.55000019073486328125f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00118142849742309772151454518093813615),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0072201822885703318172366893469382745),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0078782276276860110721875733778481505),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00418229166204362376187593976656261146),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00134198400587769200074194304298642705),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000283210387078004063264777611497435572),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.405687064094911866569295610914844928e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.39348283801568113807887364414008292e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.248798540917787001526976889284624449e-6),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.929502490223452372919607105387474751e-8),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.156161469668275442569286723236274457e-9),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.52955245103668419479878456656709381),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.06263944820093830054635017117417064),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.441684612681607364321013134378316463),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.121665258426166960049773715928906382),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0232134512374747691424978642874321434),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00310778180686296328582860464875562636),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000288361770756174705123674838640161693),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.177529187194133944622193191942300132e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.655068544833064069223029299070876623e-6),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.11005507545746069573608988651927452e-7),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 4.5f)) / tools::evaluate_polynomial(Q, T(z - 4.5f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 7.5)
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 1.007e-36
|
||
|
|
// Expected Error Term: 1.007e-36
|
||
|
|
// Maximum Relative Change in Control Points: 1.027e-03
|
||
|
|
// Max Error found at long double precision = 2.646420e-36
|
||
|
|
static const T Y = 0.5574436187744140625f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000293236907400849056269309713064107674),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00225110719535060642692275221961480162),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190984458121502831421717207849429799),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000747757733460111743833929141001680706),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000170663175280949889583158597373928096),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.246441188958013822253071608197514058e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.229818000860544644974205957895688106e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.134886977703388748488480980637704864e-6),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.454764611880548962757125070106650958e-8),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.673002744115866600294723141176820155e-10),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.12843690320861239631195353379313367),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.569900657061622955362493442186537259),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.169094404206844928112348730277514273),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0324887449084220415058158657252147063),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00419252877436825753042680842608219552),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00036344133176118603523976748563178578),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.204123895931375107397698245752850347e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.674128352521481412232785122943508729e-6),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.997637501418963696542159244436245077e-8),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z - 6.5f)) / tools::evaluate_polynomial(Q, T(z - 6.5f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else if(z < 11.5)
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 8.380e-36
|
||
|
|
// Expected Error Term: 8.380e-36
|
||
|
|
// Maximum Relative Change in Control Points: 2.632e-06
|
||
|
|
// Max Error found at long double precision = 9.849522e-36
|
||
|
|
static const T Y = 0.56083202362060546875f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000282420728751494363613829834891390121),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00175387065018002823433704079355125161),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0021344978564889819420775336322920375),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00124151356560137532655039683963075661),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000423600733566948018555157026862139644),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.914030340865175237133613697319509698e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.126999927156823363353809747017945494e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.110610959842869849776179749369376402e-5),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.55075079477173482096725348704634529e-7),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.119735694018906705225870691331543806e-8),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.69889613396167354566098060039549882),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.28824647372749624464956031163282674),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.572297795434934493541628008224078717),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.164157697425571712377043857240773164),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.0315311145224594430281219516531649562),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00405588922155632380812945849777127458),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000336929033691445666232029762868642417),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.164033049810404773469413526427932109e-4),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.356615210500531410114914617294694857e-6),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(z / 2 - 4.75f)) / tools::evaluate_polynomial(Q, T(z / 2 - 4.75f));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
// Maximum Deviation Found: 1.132e-35
|
||
|
|
// Expected Error Term: -1.132e-35
|
||
|
|
// Maximum Relative Change in Control Points: 4.674e-04
|
||
|
|
// Max Error found at long double precision = 1.162590e-35
|
||
|
|
static const T Y = 0.5632686614990234375f;
|
||
|
|
static const T P[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.000920922048732849448079451574171836943),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 0.00321439044532288750501700028748922439),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.250455263029390118657884864261823431),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -0.906807635364090342031792404764598142),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -8.92233572835991735876688745989985565),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -21.7797433494422564811782116907878495),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -91.1451915251976354349734589601171659),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -144.1279109655993927069052125017673),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -313.845076581796338665519022313775589),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -273.11378811923343424081101235736475),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -271.651566205951067025696102600443452),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, -60.0530577077238079968843307523245547),
|
||
|
|
};
|
||
|
|
static const T Q[] = {
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 3.49040448075464744191022350947892036),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 34.3563592467165971295915749548313227),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 84.4993232033879023178285731843850461),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 376.005865281206894120659401340373818),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 629.95369438888946233003926191755125),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1568.35771983533158591604513304269098),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1646.02452040831961063640827116581021),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 2299.96860633240298708910425594484895),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 1222.73204392037452750381340219906374),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 799.359797306084372350264298361110448),
|
||
|
|
BOOST_MATH_BIG_CONSTANT(T, 113, 72.7415265778588087243442792401576737),
|
||
|
|
};
|
||
|
|
result = Y + tools::evaluate_polynomial(P, T(1 / z)) / tools::evaluate_polynomial(Q, T(1 / z));
|
||
|
|
result *= exp(-z * z) / z;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
//
|
||
|
|
// Any value of z larger than 110 will underflow to zero:
|
||
|
|
//
|
||
|
|
result = 0;
|
||
|
|
invert = !invert;
|
||
|
|
}
|
||
|
|
|
||
|
|
if(invert)
|
||
|
|
{
|
||
|
|
result = 1 - result;
|
||
|
|
}
|
||
|
|
|
||
|
|
return result;
|
||
|
|
} // template <class T, class Lanczos>T erf_imp(T z, bool invert, const Lanczos& l, const mpl::int_<113>& t)
|
||
|
|
|
||
|
|
template <class T, class Policy, class tag>
|
||
|
|
struct erf_initializer
|
||
|
|
{
|
||
|
|
struct init
|
||
|
|
{
|
||
|
|
init()
|
||
|
|
{
|
||
|
|
do_init(tag());
|
||
|
|
}
|
||
|
|
static void do_init(const mpl::int_<0>&){}
|
||
|
|
static void do_init(const mpl::int_<53>&)
|
||
|
|
{
|
||
|
|
boost::math::erf(static_cast<T>(1e-12), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(0.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(1.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(2.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(4.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(5.25), Policy());
|
||
|
|
}
|
||
|
|
static void do_init(const mpl::int_<64>&)
|
||
|
|
{
|
||
|
|
boost::math::erf(static_cast<T>(1e-12), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(0.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(1.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(2.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(4.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(5.25), Policy());
|
||
|
|
}
|
||
|
|
static void do_init(const mpl::int_<113>&)
|
||
|
|
{
|
||
|
|
boost::math::erf(static_cast<T>(1e-22), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(0.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(1.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(2.125), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(2.75), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(3.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(5.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(7.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(11.25), Policy());
|
||
|
|
boost::math::erf(static_cast<T>(12.5), Policy());
|
||
|
|
}
|
||
|
|
void force_instantiate()const{}
|
||
|
|
};
|
||
|
|
static const init initializer;
|
||
|
|
static void force_instantiate()
|
||
|
|
{
|
||
|
|
initializer.force_instantiate();
|
||
|
|
}
|
||
|
|
};
|
||
|
|
|
||
|
|
template <class T, class Policy, class tag>
|
||
|
|
const typename erf_initializer<T, Policy, tag>::init erf_initializer<T, Policy, tag>::initializer;
|
||
|
|
|
||
|
|
} // namespace detail
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline typename tools::promote_args<T>::type erf(T z, const Policy& /* pol */)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T>::type result_type;
|
||
|
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
||
|
|
typedef typename policies::precision<result_type, Policy>::type precision_type;
|
||
|
|
typedef typename policies::normalise<
|
||
|
|
Policy,
|
||
|
|
policies::promote_float<false>,
|
||
|
|
policies::promote_double<false>,
|
||
|
|
policies::discrete_quantile<>,
|
||
|
|
policies::assert_undefined<> >::type forwarding_policy;
|
||
|
|
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
|
||
|
|
|
||
|
|
typedef typename mpl::if_<
|
||
|
|
mpl::less_equal<precision_type, mpl::int_<0> >,
|
||
|
|
mpl::int_<0>,
|
||
|
|
typename mpl::if_<
|
||
|
|
mpl::less_equal<precision_type, mpl::int_<53> >,
|
||
|
|
mpl::int_<53>, // double
|
||
|
|
typename mpl::if_<
|
||
|
|
mpl::less_equal<precision_type, mpl::int_<64> >,
|
||
|
|
mpl::int_<64>, // 80-bit long double
|
||
|
|
typename mpl::if_<
|
||
|
|
mpl::less_equal<precision_type, mpl::int_<113> >,
|
||
|
|
mpl::int_<113>, // 128-bit long double
|
||
|
|
mpl::int_<0> // too many bits, use generic version.
|
||
|
|
>::type
|
||
|
|
>::type
|
||
|
|
>::type
|
||
|
|
>::type tag_type;
|
||
|
|
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
|
||
|
|
|
||
|
|
detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
|
||
|
|
|
||
|
|
return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
|
||
|
|
static_cast<value_type>(z),
|
||
|
|
false,
|
||
|
|
forwarding_policy(),
|
||
|
|
tag_type()), "boost::math::erf<%1%>(%1%, %1%)");
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, class Policy>
|
||
|
|
inline typename tools::promote_args<T>::type erfc(T z, const Policy& /* pol */)
|
||
|
|
{
|
||
|
|
typedef typename tools::promote_args<T>::type result_type;
|
||
|
|
typedef typename policies::evaluation<result_type, Policy>::type value_type;
|
||
|
|
typedef typename policies::precision<result_type, Policy>::type precision_type;
|
||
|
|
typedef typename policies::normalise<
|
||
|
|
Policy,
|
||
|
|
policies::promote_float<false>,
|
||
|
|
policies::promote_double<false>,
|
||
|
|
policies::discrete_quantile<>,
|
||
|
|
policies::assert_undefined<> >::type forwarding_policy;
|
||
|
|
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("result_type = " << typeid(result_type).name());
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("value_type = " << typeid(value_type).name());
|
||
|
|
BOOST_MATH_INSTRUMENT_CODE("precision_type = " << typeid(precision_type).name());
|
||
|
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typedef typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<0> >,
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mpl::int_<0>,
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typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<53> >,
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mpl::int_<53>, // double
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typename mpl::if_<
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mpl::less_equal<precision_type, mpl::int_<64> >,
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||
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mpl::int_<64>, // 80-bit long double
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||
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typename mpl::if_<
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||
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mpl::less_equal<precision_type, mpl::int_<113> >,
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||
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mpl::int_<113>, // 128-bit long double
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||
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mpl::int_<0> // too many bits, use generic version.
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>::type
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||
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>::type
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||
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|
>::type
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||
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|
>::type tag_type;
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||
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||
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|
BOOST_MATH_INSTRUMENT_CODE("tag_type = " << typeid(tag_type).name());
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||
|
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|
||
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|
detail::erf_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); // Force constants to be initialized before main
|
||
|
|
|
||
|
|
return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::erf_imp(
|
||
|
|
static_cast<value_type>(z),
|
||
|
|
true,
|
||
|
|
forwarding_policy(),
|
||
|
|
tag_type()), "boost::math::erfc<%1%>(%1%, %1%)");
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
inline typename tools::promote_args<T>::type erf(T z)
|
||
|
|
{
|
||
|
|
return boost::math::erf(z, policies::policy<>());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
inline typename tools::promote_args<T>::type erfc(T z)
|
||
|
|
{
|
||
|
|
return boost::math::erfc(z, policies::policy<>());
|
||
|
|
}
|
||
|
|
|
||
|
|
} // namespace math
|
||
|
|
} // namespace boost
|
||
|
|
|
||
|
|
#include <boost/math/special_functions/detail/erf_inv.hpp>
|
||
|
|
|
||
|
|
#endif // BOOST_MATH_SPECIAL_ERF_HPP
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|