| // (C) Copyright John Maddock 2006. | |
| // Use, modification and distribution are subject to 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) | |
| #ifndef BOOST_MATH_SPECIAL_ERF_HPP | |
| #define BOOST_MATH_SPECIAL_ERF_HPP | |
| #ifdef _MSC_VER | |
| #pragma once | |
| #endif | |
| #include <boost/math/special_functions/math_fwd.hpp> | |
| #include <boost/math/tools/config.hpp> | |
| #include <boost/math/special_functions/gamma.hpp> | |
| #include <boost/math/tools/roots.hpp> | |
| #include <boost/math/policies/error_handling.hpp> | |
| namespace boost{ namespace math{ | |
| namespace detail | |
| { | |
| // | |
| // Asymptotic series for large z: | |
| // | |
| template <class T> | |
| struct erf_asympt_series_t | |
| { | |
| erf_asympt_series_t(T z) : xx(2 * -z * z), tk(1) | |
| { | |
| BOOST_MATH_STD_USING | |
| result = -exp(-z * z) / sqrt(boost::math::constants::pi<T>()); | |
| result /= z; | |
| } | |
| typedef T result_type; | |
| T operator()() | |
| { | |
| BOOST_MATH_STD_USING | |
| T r = result; | |
| result *= tk / xx; | |
| tk += 2; | |
| if( fabs(r) < fabs(result)) | |
| result = 0; | |
| return r; | |
| } | |
| private: | |
| T result; | |
| T xx; | |
| int tk; | |
| }; | |
| // | |
| // How large z has to be in order to ensure that the series converges: | |
| // | |
| template <class T> | |
| inline float erf_asymptotic_limit_N(const T&) | |
| { | |
| return (std::numeric_limits<float>::max)(); | |
| } | |
| inline float erf_asymptotic_limit_N(const mpl::int_<24>&) | |
| { | |
| return 2.8F; | |
| } | |
| inline float erf_asymptotic_limit_N(const mpl::int_<53>&) | |
| { | |
| return 4.3F; | |
| } | |
| inline float erf_asymptotic_limit_N(const mpl::int_<64>&) | |
| { | |
| return 4.8F; | |
| } | |
| inline float erf_asymptotic_limit_N(const mpl::int_<106>&) | |
| { | |
| return 6.5F; | |
| } | |
| inline float erf_asymptotic_limit_N(const mpl::int_<113>&) | |
| { | |
| return 6.8F; | |
| } | |
| template <class T, class Policy> | |
| inline T erf_asymptotic_limit() | |
| { | |
| typedef typename policies::precision<T, Policy>::type precision_type; | |
| typedef typename mpl::if_< | |
| mpl::less_equal<precision_type, mpl::int_<24> >, | |
| typename mpl::if_< | |
| mpl::less_equal<precision_type, mpl::int_<0> >, | |
| mpl::int_<0>, | |
| mpl::int_<24> | |
| >::type, | |
| typename mpl::if_< | |
| mpl::less_equal<precision_type, mpl::int_<53> >, | |
| mpl::int_<53>, | |
| typename mpl::if_< | |
| mpl::less_equal<precision_type, mpl::int_<64> >, | |
| mpl::int_<64>, | |
| typename mpl::if_< | |
| mpl::less_equal<precision_type, mpl::int_<106> >, | |
| mpl::int_<106>, | |
| typename mpl::if_< | |
| mpl::less_equal<precision_type, mpl::int_<113> >, | |
| mpl::int_<113>, | |
| mpl::int_<0> | |
| >::type | |
| >::type | |
| >::type | |
| >::type | |
| >::type tag_type; | |
| return erf_asymptotic_limit_N(tag_type()); | |
| } | |
| template <class T, class Policy, class Tag> | |
| T erf_imp(T z, bool invert, const Policy& pol, const Tag& t) | |
| { | |
| BOOST_MATH_STD_USING | |
| BOOST_MATH_INSTRUMENT_CODE("Generic erf_imp called"); | |
| if(z < 0) | |
| { | |
| if(!invert) | |
| return -erf_imp(T(-z), invert, pol, t); | |
| else | |
| return 1 + erf_imp(T(-z), false, pol, t); | |
| } | |
| T result; | |
| if(!invert && (z > detail::erf_asymptotic_limit<T, Policy>())) | |
| { | |
| detail::erf_asympt_series_t<T> s(z); | |
| boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); | |
| result = boost::math::tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, 1); | |
| policies::check_series_iterations("boost::math::erf<%1%>(%1%, %1%)", max_iter, pol); | |
| } | |
| else | |
| { | |
| T x = z * z; | |
| if(x < 0.6) | |
| { | |
| // Compute P: | |
| result = z * exp(-x); | |
| result /= sqrt(boost::math::constants::pi<T>()); | |
| if(result != 0) | |
| result *= 2 * detail::lower_gamma_series(T(0.5f), x, pol); | |
| } | |
| else if(x < 1.1f) | |
| { | |
| // Compute Q: | |
| invert = !invert; | |
| result = tgamma_small_upper_part(T(0.5f), x, pol); | |
| result /= sqrt(boost::math::constants::pi<T>()); | |
| } | |
| else | |
| { | |
| // Compute Q: | |
| invert = !invert; | |
| result = z * exp(-x); | |
| result /= sqrt(boost::math::constants::pi<T>()); | |
| result *= upper_gamma_fraction(T(0.5f), x, policies::get_epsilon<T, Policy>()); | |
| } | |
| } | |
| if(invert) | |
| result = 1 - result; | |
| return result; | |
| } | |
| template <class T, class Policy> | |
| T erf_imp(T z, bool invert, const Policy& pol, const mpl::int_<53>& t) | |
| { | |
| BOOST_MATH_STD_USING | |
| BOOST_MATH_INSTRUMENT_CODE("53-bit precision erf_imp called"); | |
| if(z < 0) | |
| { | |
| if(!invert) | |
| return -erf_imp(-z, invert, pol, t); | |
| else if(z < -0.5) | |
| return 2 - erf_imp(-z, invert, pol, t); | |
| else | |
| return 1 + erf_imp(-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 < 1e-10) | |
| { | |
| if(z == 0) | |
| { | |
| result = T(0); | |
| } | |
| else | |
| { | |
| result = static_cast<T>(z * 1.125f + z * 0.003379167095512573896158903121545171688L); | |
| } | |
| } | |
| else | |
| { | |
| // Maximum Deviation Found: 1.561e-17 | |
| // Expected Error Term: 1.561e-17 | |
| // Maximum Relative Change in Control Points: 1.155e-04 | |
| // Max Error found at double precision = 2.961182e-17 | |
| static const T Y = 1.044948577880859375f; | |
| static const T P[] = { | |
| 0.0834305892146531832907L, | |
| -0.338165134459360935041L, | |
| -0.0509990735146777432841L, | |
| -0.00772758345802133288487L, | |
| -0.000322780120964605683831L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 0.455004033050794024546L, | |
| 0.0875222600142252549554L, | |
| 0.00858571925074406212772L, | |
| 0.000370900071787748000569L, | |
| }; | |
| T zz = z * z; | |
| result = z * (Y + tools::evaluate_polynomial(P, zz) / tools::evaluate_polynomial(Q, zz)); | |
| } | |
| } | |
| else if(invert ? (z < 28) : (z < 5.8f)) | |
| { | |
| // | |
| // We'll be calculating erfc: | |
| // | |
| invert = !invert; | |
| if(z < 1.5f) | |
| { | |
| // Maximum Deviation Found: 3.702e-17 | |
| // Expected Error Term: 3.702e-17 | |
| // Maximum Relative Change in Control Points: 2.845e-04 | |
| // Max Error found at double precision = 4.841816e-17 | |
| static const T Y = 0.405935764312744140625f; | |
| static const T P[] = { | |
| -0.098090592216281240205L, | |
| 0.178114665841120341155L, | |
| 0.191003695796775433986L, | |
| 0.0888900368967884466578L, | |
| 0.0195049001251218801359L, | |
| 0.00180424538297014223957L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.84759070983002217845L, | |
| 1.42628004845511324508L, | |
| 0.578052804889902404909L, | |
| 0.12385097467900864233L, | |
| 0.0113385233577001411017L, | |
| 0.337511472483094676155e-5L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 0.5) / tools::evaluate_polynomial(Q, z - 0.5); | |
| result *= exp(-z * z) / z; | |
| } | |
| else if(z < 2.5f) | |
| { | |
| // Max Error found at double precision = 6.599585e-18 | |
| // Maximum Deviation Found: 3.909e-18 | |
| // Expected Error Term: 3.909e-18 | |
| // Maximum Relative Change in Control Points: 9.886e-05 | |
| static const T Y = 0.50672817230224609375f; | |
| static const T P[] = { | |
| -0.0243500476207698441272L, | |
| 0.0386540375035707201728L, | |
| 0.04394818964209516296L, | |
| 0.0175679436311802092299L, | |
| 0.00323962406290842133584L, | |
| 0.000235839115596880717416L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.53991494948552447182L, | |
| 0.982403709157920235114L, | |
| 0.325732924782444448493L, | |
| 0.0563921837420478160373L, | |
| 0.00410369723978904575884L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 1.5) / tools::evaluate_polynomial(Q, z - 1.5); | |
| result *= exp(-z * z) / z; | |
| } | |
| else if(z < 4.5f) | |
| { | |
| // Maximum Deviation Found: 1.512e-17 | |
| // Expected Error Term: 1.512e-17 | |
| // Maximum Relative Change in Control Points: 2.222e-04 | |
| // Max Error found at double precision = 2.062515e-17 | |
| static const T Y = 0.5405750274658203125f; | |
| static const T P[] = { | |
| 0.00295276716530971662634L, | |
| 0.0137384425896355332126L, | |
| 0.00840807615555585383007L, | |
| 0.00212825620914618649141L, | |
| 0.000250269961544794627958L, | |
| 0.113212406648847561139e-4L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.04217814166938418171L, | |
| 0.442597659481563127003L, | |
| 0.0958492726301061423444L, | |
| 0.0105982906484876531489L, | |
| 0.000479411269521714493907L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 3.5) / tools::evaluate_polynomial(Q, z - 3.5); | |
| 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[] = { | |
| 0.00628057170626964891937L, | |
| 0.0175389834052493308818L, | |
| -0.212652252872804219852L, | |
| -0.687717681153649930619L, | |
| -2.5518551727311523996L, | |
| -3.22729451764143718517L, | |
| -2.8175401114513378771L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 2.79257750980575282228L, | |
| 11.0567237927800161565L, | |
| 15.930646027911794143L, | |
| 22.9367376522880577224L, | |
| 13.5064170191802889145L, | |
| 5.48409182238641741584L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, 1 / z) / tools::evaluate_polynomial(Q, 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 L>T erf_imp(T z, bool invert, const L& 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(-z, invert, pol, t); | |
| else if(z < -0.5) | |
| return 2 - erf_imp(-z, invert, pol, t); | |
| else | |
| return 1 + erf_imp(-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) | |
| { | |
| result = z * 1.125 + z * 0.003379167095512573896158903121545171688L; | |
| } | |
| 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[] = { | |
| 0.0834305892146531988966L, | |
| -0.338097283075565413695L, | |
| -0.0509602734406067204596L, | |
| -0.00904906346158537794396L, | |
| -0.000489468651464798669181L, | |
| -0.200305626366151877759e-4L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 0.455817300515875172439L, | |
| 0.0916537354356241792007L, | |
| 0.0102722652675910031202L, | |
| 0.000650511752687851548735L, | |
| 0.189532519105655496778e-4L, | |
| }; | |
| result = z * (Y + tools::evaluate_polynomial(P, z * z) / tools::evaluate_polynomial(Q, 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[] = { | |
| -0.0980905922162812031672L, | |
| 0.159989089922969141329L, | |
| 0.222359821619935712378L, | |
| 0.127303921703577362312L, | |
| 0.0384057530342762400273L, | |
| 0.00628431160851156719325L, | |
| 0.000441266654514391746428L, | |
| 0.266689068336295642561e-7L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 2.03237474985469469291L, | |
| 1.78355454954969405222L, | |
| 0.867940326293760578231L, | |
| 0.248025606990021698392L, | |
| 0.0396649631833002269861L, | |
| 0.00279220237309449026796L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 0.5f) / tools::evaluate_polynomial(Q, 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[] = { | |
| -0.024350047620769840217L, | |
| 0.0343522687935671451309L, | |
| 0.0505420824305544949541L, | |
| 0.0257479325917757388209L, | |
| 0.00669349844190354356118L, | |
| 0.00090807914416099524444L, | |
| 0.515917266698050027934e-4L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.71657861671930336344L, | |
| 1.26409634824280366218L, | |
| 0.512371437838969015941L, | |
| 0.120902623051120950935L, | |
| 0.0158027197831887485261L, | |
| 0.000897871370778031611439L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 1.5f) / tools::evaluate_polynomial(Q, 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[] = { | |
| 0.0029527671653097284033L, | |
| 0.0141853245895495604051L, | |
| 0.0104959584626432293901L, | |
| 0.00343963795976100077626L, | |
| 0.00059065441194877637899L, | |
| 0.523435380636174008685e-4L, | |
| 0.189896043050331257262e-5L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.19352160185285642574L, | |
| 0.603256964363454392857L, | |
| 0.165411142458540585835L, | |
| 0.0259729870946203166468L, | |
| 0.00221657568292893699158L, | |
| 0.804149464190309799804e-4L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 3.5f) / tools::evaluate_polynomial(Q, 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[] = { | |
| 0.00593438793008050214106L, | |
| 0.0280666231009089713937L, | |
| -0.141597835204583050043L, | |
| -0.978088201154300548842L, | |
| -5.47351527796012049443L, | |
| -13.8677304660245326627L, | |
| -27.1274948720539821722L, | |
| -29.2545152747009461519L, | |
| -16.8865774499799676937L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 4.72948911186645394541L, | |
| 23.6750543147695749212L, | |
| 60.0021517335693186785L, | |
| 131.766251645149522868L, | |
| 178.167924971283482513L, | |
| 182.499390505915222699L, | |
| 104.365251479578577989L, | |
| 30.8365511891224291717L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, 1 / z) / tools::evaluate_polynomial(Q, 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 L>T erf_imp(T z, bool invert, const L& 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(-z, invert, pol, t); | |
| else if(z < -0.5) | |
| return 2 - erf_imp(-z, invert, pol, t); | |
| else | |
| return 1 + erf_imp(-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) | |
| { | |
| result = z * 1.125 + z * 0.003379167095512573896158903121545171688L; | |
| } | |
| 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[] = { | |
| 0.0442269454158250738961589031215451778L, | |
| -0.35549265736002144875335323556961233L, | |
| -0.0582179564566667896225454670863270393L, | |
| -0.0112694696904802304229950538453123925L, | |
| -0.000805730648981801146251825329609079099L, | |
| -0.566304966591936566229702842075966273e-4L, | |
| -0.169655010425186987820201021510002265e-5L, | |
| -0.344448249920445916714548295433198544e-7L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 0.466542092785657604666906909196052522L, | |
| 0.100005087012526447295176964142107611L, | |
| 0.0128341535890117646540050072234142603L, | |
| 0.00107150448466867929159660677016658186L, | |
| 0.586168368028999183607733369248338474e-4L, | |
| 0.196230608502104324965623171516808796e-5L, | |
| 0.313388521582925207734229967907890146e-7L, | |
| }; | |
| result = z * (Y + tools::evaluate_polynomial(P, z * z) / tools::evaluate_polynomial(Q, 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[] = { | |
| -0.0640320213544647969396032886581290455L, | |
| 0.200769874440155895637857443946706731L, | |
| 0.378447199873537170666487408805779826L, | |
| 0.30521399466465939450398642044975127L, | |
| 0.146890026406815277906781824723458196L, | |
| 0.0464837937749539978247589252732769567L, | |
| 0.00987895759019540115099100165904822903L, | |
| 0.00137507575429025512038051025154301132L, | |
| 0.0001144764551085935580772512359680516L, | |
| 0.436544865032836914773944382339900079e-5L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 2.47651182872457465043733800302427977L, | |
| 2.78706486002517996428836400245547955L, | |
| 1.87295924621659627926365005293130693L, | |
| 0.829375825174365625428280908787261065L, | |
| 0.251334771307848291593780143950311514L, | |
| 0.0522110268876176186719436765734722473L, | |
| 0.00718332151250963182233267040106902368L, | |
| 0.000595279058621482041084986219276392459L, | |
| 0.226988669466501655990637599399326874e-4L, | |
| 0.270666232259029102353426738909226413e-10L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 0.5f) / tools::evaluate_polynomial(Q, 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[] = { | |
| -0.0289965858925328393392496555094848345L, | |
| 0.0868181194868601184627743162571779226L, | |
| 0.169373435121178901746317404936356745L, | |
| 0.13350446515949251201104889028133486L, | |
| 0.0617447837290183627136837688446313313L, | |
| 0.0185618495228251406703152962489700468L, | |
| 0.00371949406491883508764162050169531013L, | |
| 0.000485121708792921297742105775823900772L, | |
| 0.376494706741453489892108068231400061e-4L, | |
| 0.133166058052466262415271732172490045e-5L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 2.32970330146503867261275580968135126L, | |
| 2.46325715420422771961250513514928746L, | |
| 1.55307882560757679068505047390857842L, | |
| 0.644274289865972449441174485441409076L, | |
| 0.182609091063258208068606847453955649L, | |
| 0.0354171651271241474946129665801606795L, | |
| 0.00454060370165285246451879969534083997L, | |
| 0.000349871943711566546821198612518656486L, | |
| 0.123749319840299552925421880481085392e-4L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 1.0f) / tools::evaluate_polynomial(Q, 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[] = { | |
| -0.0201233630504573402185161184151016606L, | |
| 0.0331864357574860196516686996302305002L, | |
| 0.0716562720864787193337475444413405461L, | |
| 0.0545835322082103985114927569724880658L, | |
| 0.0236692635189696678976549720784989593L, | |
| 0.00656970902163248872837262539337601845L, | |
| 0.00120282643299089441390490459256235021L, | |
| 0.000142123229065182650020762792081622986L, | |
| 0.991531438367015135346716277792989347e-5L, | |
| 0.312857043762117596999398067153076051e-6L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 2.13506082409097783827103424943508554L, | |
| 2.06399257267556230937723190496806215L, | |
| 1.18678481279932541314830499880691109L, | |
| 0.447733186643051752513538142316799562L, | |
| 0.11505680005657879437196953047542148L, | |
| 0.020163993632192726170219663831914034L, | |
| 0.00232708971840141388847728782209730585L, | |
| 0.000160733201627963528519726484608224112L, | |
| 0.507158721790721802724402992033269266e-5L, | |
| 0.18647774409821470950544212696270639e-12L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 1.5f) / tools::evaluate_polynomial(Q, 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[] = { | |
| -0.00902152521745813634562524098263360074L, | |
| 0.0145207142776691539346923710537580927L, | |
| 0.0301681239582193983824211995978678571L, | |
| 0.0215548540823305814379020678660434461L, | |
| 0.00864683476267958365678294164340749949L, | |
| 0.00219693096885585491739823283511049902L, | |
| 0.000364961639163319762492184502159894371L, | |
| 0.388174251026723752769264051548703059e-4L, | |
| 0.241918026931789436000532513553594321e-5L, | |
| 0.676586625472423508158937481943649258e-7L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.93669171363907292305550231764920001L, | |
| 1.69468476144051356810672506101377494L, | |
| 0.880023580986436640372794392579985511L, | |
| 0.299099106711315090710836273697708402L, | |
| 0.0690593962363545715997445583603382337L, | |
| 0.0108427016361318921960863149875360222L, | |
| 0.00111747247208044534520499324234317695L, | |
| 0.686843205749767250666787987163701209e-4L, | |
| 0.192093541425429248675532015101904262e-5L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 2.25f) / tools::evaluate_polynomial(Q, 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[] = { | |
| -0.0033703486408887424921155540591370375L, | |
| 0.0104948043110005245215286678898115811L, | |
| 0.0148530118504000311502310457390417795L, | |
| 0.00816693029245443090102738825536188916L, | |
| 0.00249716579989140882491939681805594585L, | |
| 0.0004655591010047353023978045800916647L, | |
| 0.531129557920045295895085236636025323e-4L, | |
| 0.343526765122727069515775194111741049e-5L, | |
| 0.971120407556888763695313774578711839e-7L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.59911256167540354915906501335919317L, | |
| 1.136006830764025173864831382946934L, | |
| 0.468565867990030871678574840738423023L, | |
| 0.122821824954470343413956476900662236L, | |
| 0.0209670914950115943338996513330141633L, | |
| 0.00227845718243186165620199012883547257L, | |
| 0.000144243326443913171313947613547085553L, | |
| 0.407763415954267700941230249989140046e-5L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 3.0f) / tools::evaluate_polynomial(Q, 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[] = { | |
| 0.00118142849742309772151454518093813615L, | |
| 0.0072201822885703318172366893469382745L, | |
| 0.0078782276276860110721875733778481505L, | |
| 0.00418229166204362376187593976656261146L, | |
| 0.00134198400587769200074194304298642705L, | |
| 0.000283210387078004063264777611497435572L, | |
| 0.405687064094911866569295610914844928e-4L, | |
| 0.39348283801568113807887364414008292e-5L, | |
| 0.248798540917787001526976889284624449e-6L, | |
| 0.929502490223452372919607105387474751e-8L, | |
| 0.156161469668275442569286723236274457e-9L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.52955245103668419479878456656709381L, | |
| 1.06263944820093830054635017117417064L, | |
| 0.441684612681607364321013134378316463L, | |
| 0.121665258426166960049773715928906382L, | |
| 0.0232134512374747691424978642874321434L, | |
| 0.00310778180686296328582860464875562636L, | |
| 0.000288361770756174705123674838640161693L, | |
| 0.177529187194133944622193191942300132e-4L, | |
| 0.655068544833064069223029299070876623e-6L, | |
| 0.11005507545746069573608988651927452e-7L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 4.5f) / tools::evaluate_polynomial(Q, 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[] = { | |
| 0.000293236907400849056269309713064107674L, | |
| 0.00225110719535060642692275221961480162L, | |
| 0.00190984458121502831421717207849429799L, | |
| 0.000747757733460111743833929141001680706L, | |
| 0.000170663175280949889583158597373928096L, | |
| 0.246441188958013822253071608197514058e-4L, | |
| 0.229818000860544644974205957895688106e-5L, | |
| 0.134886977703388748488480980637704864e-6L, | |
| 0.454764611880548962757125070106650958e-8L, | |
| 0.673002744115866600294723141176820155e-10L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.12843690320861239631195353379313367L, | |
| 0.569900657061622955362493442186537259L, | |
| 0.169094404206844928112348730277514273L, | |
| 0.0324887449084220415058158657252147063L, | |
| 0.00419252877436825753042680842608219552L, | |
| 0.00036344133176118603523976748563178578L, | |
| 0.204123895931375107397698245752850347e-4L, | |
| 0.674128352521481412232785122943508729e-6L, | |
| 0.997637501418963696542159244436245077e-8L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z - 6.5f) / tools::evaluate_polynomial(Q, 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[] = { | |
| 0.000282420728751494363613829834891390121L, | |
| 0.00175387065018002823433704079355125161L, | |
| 0.0021344978564889819420775336322920375L, | |
| 0.00124151356560137532655039683963075661L, | |
| 0.000423600733566948018555157026862139644L, | |
| 0.914030340865175237133613697319509698e-4L, | |
| 0.126999927156823363353809747017945494e-4L, | |
| 0.110610959842869849776179749369376402e-5L, | |
| 0.55075079477173482096725348704634529e-7L, | |
| 0.119735694018906705225870691331543806e-8L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 1.69889613396167354566098060039549882L, | |
| 1.28824647372749624464956031163282674L, | |
| 0.572297795434934493541628008224078717L, | |
| 0.164157697425571712377043857240773164L, | |
| 0.0315311145224594430281219516531649562L, | |
| 0.00405588922155632380812945849777127458L, | |
| 0.000336929033691445666232029762868642417L, | |
| 0.164033049810404773469413526427932109e-4L, | |
| 0.356615210500531410114914617294694857e-6L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, z / 2 - 4.75f) / tools::evaluate_polynomial(Q, 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[] = { | |
| 0.000920922048732849448079451574171836943L, | |
| 0.00321439044532288750501700028748922439L, | |
| -0.250455263029390118657884864261823431L, | |
| -0.906807635364090342031792404764598142L, | |
| -8.92233572835991735876688745989985565L, | |
| -21.7797433494422564811782116907878495L, | |
| -91.1451915251976354349734589601171659L, | |
| -144.1279109655993927069052125017673L, | |
| -313.845076581796338665519022313775589L, | |
| -273.11378811923343424081101235736475L, | |
| -271.651566205951067025696102600443452L, | |
| -60.0530577077238079968843307523245547L, | |
| }; | |
| static const T Q[] = { | |
| 1L, | |
| 3.49040448075464744191022350947892036L, | |
| 34.3563592467165971295915749548313227L, | |
| 84.4993232033879023178285731843850461L, | |
| 376.005865281206894120659401340373818L, | |
| 629.95369438888946233003926191755125L, | |
| 1568.35771983533158591604513304269098L, | |
| 1646.02452040831961063640827116581021L, | |
| 2299.96860633240298708910425594484895L, | |
| 1222.73204392037452750381340219906374L, | |
| 799.359797306084372350264298361110448L, | |
| 72.7415265778588087243442792401576737L, | |
| }; | |
| result = Y + tools::evaluate_polynomial(P, 1 / z) / tools::evaluate_polynomial(Q, 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 L>T erf_imp(T z, bool invert, const L& l, const mpl::int_<113>& t) | |
| } // 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()); | |
| 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()); | |
| 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()); | |
| 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 | |