boost/math/special_functions/expint.hpp
// Copyright John Maddock 2007. // 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_EXPINT_HPP #define BOOST_MATH_EXPINT_HPP #ifdef _MSC_VER #pragma once #endif #include <boost/math/tools/precision.hpp> #include <boost/math/tools/promotion.hpp> #include <boost/math/tools/fraction.hpp> #include <boost/math/tools/series.hpp> #include <boost/math/policies/error_handling.hpp> #include <boost/math/special_functions/digamma.hpp> #include <boost/math/special_functions/log1p.hpp> #include <boost/math/special_functions/pow.hpp> namespace boost{ namespace math{ template <class T, class Policy> inline typename tools::promote_args<T>::type expint(unsigned n, T z, const Policy& /*pol*/); namespace detail{ template <class T> inline T expint_1_rational(const T& z, const mpl::int_<0>&) { // this function is never actually called BOOST_ASSERT(0); return z; } template <class T> T expint_1_rational(const T& z, const mpl::int_<53>&) { BOOST_MATH_STD_USING T result; if(z <= 1) { // Maximum Deviation Found: 2.006e-18 // Expected Error Term: 2.006e-18 // Max error found at double precision: 2.760e-17 static const T Y = 0.66373538970947265625F; static const T P[6] = { 0.0865197248079397976498L, 0.0320913665303559189999L, -0.245088216639761496153L, -0.0368031736257943745142L, -0.00399167106081113256961L, -0.000111507792921197858394L }; static const T Q[6] = { 1L, 0.37091387659397013215L, 0.056770677104207528384L, 0.00427347600017103698101L, 0.000131049900798434683324L, -0.528611029520217142048e-6L }; result = tools::evaluate_polynomial(P, z) / tools::evaluate_polynomial(Q, z); result += z - log(z) - Y; } else if(z < -boost::math::tools::log_min_value<T>()) { // Maximum Deviation Found (interpolated): 1.444e-17 // Max error found at double precision: 3.119e-17 static const T P[11] = { -0.121013190657725568138e-18L, -0.999999999999998811143L, -43.3058660811817946037L, -724.581482791462469795L, -6046.8250112711035463L, -27182.6254466733970467L, -66598.2652345418633509L, -86273.1567711649528784L, -54844.4587226402067411L, -14751.4895786128450662L, -1185.45720315201027667L }; static const T Q[12] = { 1L, 45.3058660811801465927L, 809.193214954550328455L, 7417.37624454689546708L, 38129.5594484818471461L, 113057.05869159631492L, 192104.047790227984431L, 180329.498380501819718L, 86722.3403467334749201L, 18455.4124737722049515L, 1229.20784182403048905L, -0.776491285282330997549L }; T recip = 1 / z; result = 1 + tools::evaluate_polynomial(P, recip) / tools::evaluate_polynomial(Q, recip); result *= exp(-z) * recip; } else { result = 0; } return result; } template <class T> T expint_1_rational(const T& z, const mpl::int_<64>&) { BOOST_MATH_STD_USING T result; if(z <= 1) { // Maximum Deviation Found: 3.807e-20 // Expected Error Term: 3.807e-20 // Max error found at long double precision: 6.249e-20 static const T Y = 0.66373538970947265625F; static const T P[6] = { 0.0865197248079397956816L, 0.0275114007037026844633L, -0.246594388074877139824L, -0.0237624819878732642231L, -0.00259113319641673986276L, 0.30853660894346057053e-4L }; static const T Q[7] = { 1L, 0.317978365797784100273L, 0.0393622602554758722511L, 0.00204062029115966323229L, 0.732512107100088047854e-5L, -0.202872781770207871975e-5L, 0.52779248094603709945e-7L }; result = tools::evaluate_polynomial(P, z) / tools::evaluate_polynomial(Q, z); result += z - log(z) - Y; } else if(z < -boost::math::tools::log_min_value<T>()) { // Maximum Deviation Found (interpolated): 2.220e-20 // Max error found at long double precision: 1.346e-19 static const T P[14] = { -0.534401189080684443046e-23L, -0.999999999999999999905L, -62.1517806091379402505L, -1568.45688271895145277L, -21015.3431990874009619L, -164333.011755931661949L, -777917.270775426696103L, -2244188.56195255112937L, -3888702.98145335643429L, -3909822.65621952648353L, -2149033.9538897398457L, -584705.537139793925189L, -65815.2605361889477244L, -2038.82870680427258038L }; static const T Q[14] = { 1L, 64.1517806091379399478L, 1690.76044393722763785L, 24035.9534033068949426L, 203679.998633572361706L, 1074661.58459976978285L, 3586552.65020899358773L, 7552186.84989547621411L, 9853333.79353054111434L, 7689642.74550683631258L, 3385553.35146759180739L, 763218.072732396428725L, 73930.2995984054930821L, 2063.86994219629165937L }; T recip = 1 / z; result = 1 + tools::evaluate_polynomial(P, recip) / tools::evaluate_polynomial(Q, recip); result *= exp(-z) * recip; } else { result = 0; } return result; } template <class T> T expint_1_rational(const T& z, const mpl::int_<113>&) { BOOST_MATH_STD_USING T result; if(z <= 1) { // Maximum Deviation Found: 2.477e-35 // Expected Error Term: 2.477e-35 // Max error found at long double precision: 6.810e-35 static const T Y = 0.66373538970947265625F; static const T P[10] = { 0.0865197248079397956434879099175975937L, 0.0369066175910795772830865304506087759L, -0.24272036838415474665971599314725545L, -0.0502166331248948515282379137550178307L, -0.00768384138547489410285101483730424919L, -0.000612574337702109683505224915484717162L, -0.380207107950635046971492617061708534e-4L, -0.136528159460768830763009294683628406e-5L, -0.346839106212658259681029388908658618e-7L, -0.340500302777838063940402160594523429e-9L }; static const T Q[10] = { 1L, 0.426568827778942588160423015589537302L, 0.0841384046470893490592450881447510148L, 0.0100557215850668029618957359471132995L, 0.000799334870474627021737357294799839363L, 0.434452090903862735242423068552687688e-4L, 0.15829674748799079874182885081231252e-5L, 0.354406206738023762100882270033082198e-7L, 0.369373328141051577845488477377890236e-9L, -0.274149801370933606409282434677600112e-12L }; result = tools::evaluate_polynomial(P, z) / tools::evaluate_polynomial(Q, z); result += z - log(z) - Y; } else if(z <= 4) { // Max error in interpolated form: 5.614e-35 // Max error found at long double precision: 7.979e-35 static const T Y = 0.70190334320068359375F; static const T P[17] = { 0.298096656795020369955077350585959794L, 12.9314045995266142913135497455971247L, 226.144334921582637462526628217345501L, 2070.83670924261732722117682067381405L, 10715.1115684330959908244769731347186L, 30728.7876355542048019664777316053311L, 38520.6078609349855436936232610875297L, -27606.0780981527583168728339620565165L, -169026.485055785605958655247592604835L, -254361.919204983608659069868035092282L, -195765.706874132267953259272028679935L, -83352.6826013533205474990119962408675L, -19251.6828496869586415162597993050194L, -2226.64251774578542836725386936102339L, -109.009437301400845902228611986479816L, -1.51492042209561411434644938098833499L }; static const T Q[16] = { 1L, 46.734521442032505570517810766704587L, 908.694714348462269000247450058595655L, 9701.76053033673927362784882748513195L, 63254.2815292641314236625196594947774L, 265115.641285880437335106541757711092L, 732707.841188071900498536533086567735L, 1348514.02492635723327306628712057794L, 1649986.81455283047769673308781585991L, 1326000.828522976970116271208812099L, 683643.09490612171772350481773951341L, 217640.505137263607952365685653352229L, 40288.3467237411710881822569476155485L, 3932.89353979531632559232883283175754L, 169.845369689596739824177412096477219L, 2.17607292280092201170768401876895354L }; T recip = 1 / z; result = Y + tools::evaluate_polynomial(P, recip) / tools::evaluate_polynomial(Q, recip); result *= exp(-z) * recip; } else if(z < -boost::math::tools::log_min_value<T>()) { // Max error in interpolated form: 4.413e-35 // Max error found at long double precision: 8.928e-35 static const T P[19] = { -0.559148411832951463689610809550083986e-40L, -0.999999999999999999999999999999999997L, -166.542326331163836642960118190147367L, -12204.639128796330005065904675153652L, -520807.069767086071806275022036146855L, -14435981.5242137970691490903863125326L, -274574945.737064301247496460758654196L, -3691611582.99810039356254671781473079L, -35622515944.8255047299363690814678763L, -248040014774.502043161750715548451142L, -1243190389769.53458416330946622607913L, -4441730126135.54739052731990368425339L, -11117043181899.7388524310281751971366L, -18976497615396.9717776601813519498961L, -21237496819711.1011661104761906067131L, -14695899122092.5161620333466757812848L, -5737221535080.30569711574295785864903L, -1077042281708.42654526404581272546244L, -68028222642.1941480871395695677675137L }; static const T Q[20] = { 1L, 168.542326331163836642960118190147311L, 12535.7237814586576783518249115343619L, 544891.263372016404143120911148640627L, 15454474.7241010258634446523045237762L, 302495899.896629522673410325891717381L, 4215565948.38886507646911672693270307L, 42552409471.7951815668506556705733344L, 313592377066.753173979584098301610186L, 1688763640223.4541980740597514904542L, 6610992294901.59589748057620192145704L, 18601637235659.6059890851321772682606L, 36944278231087.2571020964163402941583L, 50425858518481.7497071917028793820058L, 45508060902865.0899967797848815980644L, 25649955002765.3817331501988304758142L, 8259575619094.6518520988612711292331L, 1299981487496.12607474362723586264515L, 70242279152.8241187845178443118302693L, -37633302.9409263839042721539363416685L }; T recip = 1 / z; result = 1 + tools::evaluate_polynomial(P, recip) / tools::evaluate_polynomial(Q, recip); result *= exp(-z) * recip; } else { result = 0; } return result; } template <class T> struct expint_fraction { typedef std::pair<T,T> result_type; expint_fraction(unsigned n_, T z_) : b(n_ + z_), i(-1), n(n_){} std::pair<T,T> operator()() { std::pair<T,T> result = std::make_pair(-static_cast<T>((i+1) * (n+i)), b); b += 2; ++i; return result; } private: T b; int i; unsigned n; }; template <class T, class Policy> inline T expint_as_fraction(unsigned n, T z, const Policy& pol) { BOOST_MATH_STD_USING BOOST_MATH_INSTRUMENT_VARIABLE(z) boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); expint_fraction<T> f(n, z); T result = tools::continued_fraction_b( f, boost::math::policies::get_epsilon<T, Policy>(), max_iter); policies::check_series_iterations<T>("boost::math::expint_continued_fraction<%1%>(unsigned,%1%)", max_iter, pol); BOOST_MATH_INSTRUMENT_VARIABLE(result) BOOST_MATH_INSTRUMENT_VARIABLE(max_iter) result = exp(-z) / result; BOOST_MATH_INSTRUMENT_VARIABLE(result) return result; } template <class T> struct expint_series { typedef T result_type; expint_series(unsigned k_, T z_, T x_k_, T denom_, T fact_) : k(k_), z(z_), x_k(x_k_), denom(denom_), fact(fact_){} T operator()() { x_k *= -z; denom += 1; fact *= ++k; return x_k / (denom * fact); } private: unsigned k; T z; T x_k; T denom; T fact; }; template <class T, class Policy> inline T expint_as_series(unsigned n, T z, const Policy& pol) { BOOST_MATH_STD_USING boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); BOOST_MATH_INSTRUMENT_VARIABLE(z) T result = 0; T x_k = -1; T denom = T(1) - n; T fact = 1; unsigned k = 0; for(; k < n - 1;) { result += x_k / (denom * fact); denom += 1; x_k *= -z; fact *= ++k; } BOOST_MATH_INSTRUMENT_VARIABLE(result) result += pow(-z, static_cast<T>(n - 1)) * (boost::math::digamma(static_cast<T>(n)) - log(z)) / fact; BOOST_MATH_INSTRUMENT_VARIABLE(result) expint_series<T> s(k, z, x_k, denom, fact); result = tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, result); policies::check_series_iterations<T>("boost::math::expint_series<%1%>(unsigned,%1%)", max_iter, pol); BOOST_MATH_INSTRUMENT_VARIABLE(result) BOOST_MATH_INSTRUMENT_VARIABLE(max_iter) return result; } template <class T, class Policy, class Tag> T expint_imp(unsigned n, T z, const Policy& pol, const Tag& tag) { BOOST_MATH_STD_USING static const char* function = "boost::math::expint<%1%>(unsigned, %1%)"; if(z < 0) return policies::raise_domain_error<T>(function, "Function requires z >= 0 but got %1%.", z, pol); if(z == 0) return n == 1 ? policies::raise_overflow_error<T>(function, 0, pol) : T(1 / (static_cast<T>(n - 1))); T result; bool f; if(n < 3) { f = z < 0.5; } else { f = z < (static_cast<T>(n - 2) / static_cast<T>(n - 1)); } #ifdef BOOST_MSVC # pragma warning(push) # pragma warning(disable:4127) // conditional expression is constant #endif if(n == 0) result = exp(-z) / z; else if((n == 1) && (Tag::value)) { result = expint_1_rational(z, tag); } else if(f) result = expint_as_series(n, z, pol); else result = expint_as_fraction(n, z, pol); #ifdef BOOST_MSVC # pragma warning(pop) #endif return result; } template <class T> struct expint_i_series { typedef T result_type; expint_i_series(T z_) : k(0), z_k(1), z(z_){} T operator()() { z_k *= z / ++k; return z_k / k; } private: unsigned k; T z_k; T z; }; template <class T, class Policy> T expint_i_as_series(T z, const Policy& pol) { BOOST_MATH_STD_USING T result = log(z); // (log(z) - log(1 / z)) / 2; result += constants::euler<T>(); expint_i_series<T> s(z); boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); result = tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, result); policies::check_series_iterations<T>("boost::math::expint_i_series<%1%>(%1%)", max_iter, pol); return result; } template <class T, class Policy, class Tag> T expint_i_imp(T z, const Policy& pol, const Tag& tag) { static const char* function = "boost::math::expint<%1%>(%1%)"; if(z < 0) return -expint_imp(1, T(-z), pol, tag); if(z == 0) return -policies::raise_overflow_error<T>(function, 0, pol); return expint_i_as_series(z, pol); } template <class T, class Policy> T expint_i_imp(T z, const Policy& pol, const mpl::int_<53>& tag) { BOOST_MATH_STD_USING static const char* function = "boost::math::expint<%1%>(%1%)"; if(z < 0) return -expint_imp(1, -z, pol, tag); if(z == 0) return -policies::raise_overflow_error<T>(function, 0, pol); T result; if(z <= 6) { // Maximum Deviation Found: 2.852e-18 // Expected Error Term: 2.852e-18 // Max Error found at double precision = Poly: 2.636335e-16 Cheb: 4.187027e-16 static const T P[10] = { 2.98677224343598593013L, 0.356343618769377415068L, 0.780836076283730801839L, 0.114670926327032002811L, 0.0499434773576515260534L, 0.00726224593341228159561L, 0.00115478237227804306827L, 0.000116419523609765200999L, 0.798296365679269702435e-5L, 0.2777056254402008721e-6L }; static const T Q[8] = { 1L, -1.17090412365413911947L, 0.62215109846016746276L, -0.195114782069495403315L, 0.0391523431392967238166L, -0.00504800158663705747345L, 0.000389034007436065401822L, -0.138972589601781706598e-4L }; static const T r1 = static_cast<T>(1677624236387711.0L / 4503599627370496.0L); static const T r2 = 0.131401834143860282009280387409357165515556574352422001206362e-16L; static const T r = static_cast<T>(0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392L); T t = (z / 3) - 1; result = tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); t = (z - r1) - r2; result *= t; if(fabs(t) < 0.1) { result += boost::math::log1p(t / r); } else { result += log(z / r); } } else if (z <= 10) { // Maximum Deviation Found: 6.546e-17 // Expected Error Term: 6.546e-17 // Max Error found at double precision = Poly: 6.890169e-17 Cheb: 6.772128e-17 static const T Y = 1.158985137939453125F; static const T P[8] = { 0.00139324086199402804173L, -0.0349921221823888744966L, -0.0264095520754134848538L, -0.00761224003005476438412L, -0.00247496209592143627977L, -0.000374885917942100256775L, -0.554086272024881826253e-4L, -0.396487648924804510056e-5L }; static const T Q[8] = { 1L, 0.744625566823272107711L, 0.329061095011767059236L, 0.100128624977313872323L, 0.0223851099128506347278L, 0.00365334190742316650106L, 0.000402453408512476836472L, 0.263649630720255691787e-4L }; T t = z / 2 - 4; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else if(z <= 20) { // Maximum Deviation Found: 1.843e-17 // Expected Error Term: -1.842e-17 // Max Error found at double precision = Poly: 4.375868e-17 Cheb: 5.860967e-17 static const T Y = 1.0869731903076171875F; static const T P[9] = { -0.00893891094356945667451L, -0.0484607730127134045806L, -0.0652810444222236895772L, -0.0478447572647309671455L, -0.0226059218923777094596L, -0.00720603636917482065907L, -0.00155941947035972031334L, -0.000209750022660200888349L, -0.138652200349182596186e-4L }; static const T Q[9] = { 1L, 1.97017214039061194971L, 1.86232465043073157508L, 1.09601437090337519977L, 0.438873285773088870812L, 0.122537731979686102756L, 0.0233458478275769288159L, 0.00278170769163303669021L, 0.000159150281166108755531L }; T t = z / 5 - 3; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else if(z <= 40) { // Maximum Deviation Found: 5.102e-18 // Expected Error Term: 5.101e-18 // Max Error found at double precision = Poly: 1.441088e-16 Cheb: 1.864792e-16 static const T Y = 1.03937530517578125F; static const T P[9] = { -0.00356165148914447597995L, -0.0229930320357982333406L, -0.0449814350482277917716L, -0.0453759383048193402336L, -0.0272050837209380717069L, -0.00994403059883350813295L, -0.00207592267812291726961L, -0.000192178045857733706044L, -0.113161784705911400295e-9L }; static const T Q[9] = { 1L, 2.84354408840148561131L, 3.6599610090072393012L, 2.75088464344293083595L, 1.2985244073998398643L, 0.383213198510794507409L, 0.0651165455496281337831L, 0.00488071077519227853585L }; T t = z / 10 - 3; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else { // Max Error found at double precision = 3.381886e-17 static const T exp40 = static_cast<T>(2.35385266837019985407899910749034804508871617254555467236651e17L); static const T Y= 1.013065338134765625F; static const T P[6] = { -0.0130653381347656243849L, 0.19029710559486576682L, 94.7365094537197236011L, -2516.35323679844256203L, 18932.0850014925993025L, -38703.1431362056714134L }; static const T Q[7] = { 1L, 61.9733592849439884145L, -2354.56211323420194283L, 22329.1459489893079041L, -70126.245140396567133L, 54738.2833147775537106L, 8297.16296356518409347L }; T t = 1 / z; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); if(z < 41) result *= exp(z) / z; else { // Avoid premature overflow if we can: t = z - 40; if(t > tools::log_max_value<T>()) { result = policies::raise_overflow_error<T>(function, 0, pol); } else { result *= exp(z - 40) / z; if(result > tools::max_value<T>() / exp40) { result = policies::raise_overflow_error<T>(function, 0, pol); } else { result *= exp40; } } } result += z; } return result; } template <class T, class Policy> T expint_i_imp(T z, const Policy& pol, const mpl::int_<64>& tag) { BOOST_MATH_STD_USING static const char* function = "boost::math::expint<%1%>(%1%)"; if(z < 0) return -expint_imp(1, -z, pol, tag); if(z == 0) return -policies::raise_overflow_error<T>(function, 0, pol); T result; if(z <= 6) { // Maximum Deviation Found: 3.883e-21 // Expected Error Term: 3.883e-21 // Max Error found at long double precision = Poly: 3.344801e-19 Cheb: 4.989937e-19 static const T P[11] = { 2.98677224343598593764L, 0.25891613550886736592L, 0.789323584998672832285L, 0.092432587824602399339L, 0.0514236978728625906656L, 0.00658477469745132977921L, 0.00124914538197086254233L, 0.000131429679565472408551L, 0.11293331317982763165e-4L, 0.629499283139417444244e-6L, 0.177833045143692498221e-7L }; static const T Q[9] = { 1L, -1.20352377969742325748L, 0.66707904942606479811L, -0.223014531629140771914L, 0.0493340022262908008636L, -0.00741934273050807310677L, 0.00074353567782087939294L, -0.455861727069603367656e-4L, 0.131515429329812837701e-5L }; static const T r1 = static_cast<T>(1677624236387711.0L / 4503599627370496.0L); static const T r2 = 0.131401834143860282009280387409357165515556574352422001206362e-16L; static const T r = static_cast<T>(0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392L); T t = (z / 3) - 1; result = tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); t = (z - r1) - r2; result *= t; if(fabs(t) < 0.1) { result += boost::math::log1p(t / r); } else { result += log(z / r); } } else if (z <= 10) { // Maximum Deviation Found: 2.622e-21 // Expected Error Term: -2.622e-21 // Max Error found at long double precision = Poly: 1.208328e-20 Cheb: 1.073723e-20 static const T Y = 1.158985137939453125F; static const T P[9] = { 0.00139324086199409049399L, -0.0345238388952337563247L, -0.0382065278072592940767L, -0.0156117003070560727392L, -0.00383276012430495387102L, -0.000697070540945496497992L, -0.877310384591205930343e-4L, -0.623067256376494930067e-5L, -0.377246883283337141444e-6L }; static const T Q[10] = { 1L, 1.08073635708902053767L, 0.553681133533942532909L, 0.176763647137553797451L, 0.0387891748253869928121L, 0.0060603004848394727017L, 0.000670519492939992806051L, 0.4947357050100855646e-4L, 0.204339282037446434827e-5L, 0.146951181174930425744e-7L }; T t = z / 2 - 4; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else if(z <= 20) { // Maximum Deviation Found: 3.220e-20 // Expected Error Term: 3.220e-20 // Max Error found at long double precision = Poly: 7.696841e-20 Cheb: 6.205163e-20 static const T Y = 1.0869731903076171875F; static const T P[10] = { -0.00893891094356946995368L, -0.0487562980088748775943L, -0.0670568657950041926085L, -0.0509577352851442932713L, -0.02551800927409034206L, -0.00892913759760086687083L, -0.00224469630207344379888L, -0.000392477245911296982776L, -0.44424044184395578775e-4L, -0.252788029251437017959e-5L }; static const T Q[10] = { 1L, 2.00323265503572414261L, 1.94688958187256383178L, 1.19733638134417472296L, 0.513137726038353385661L, 0.159135395578007264547L, 0.0358233587351620919881L, 0.0056716655597009417875L, 0.000577048986213535829925L, 0.290976943033493216793e-4L }; T t = z / 5 - 3; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else if(z <= 40) { // Maximum Deviation Found: 2.940e-21 // Expected Error Term: -2.938e-21 // Max Error found at long double precision = Poly: 3.419893e-19 Cheb: 3.359874e-19 static const T Y = 1.03937530517578125F; static const T P[12] = { -0.00356165148914447278177L, -0.0240235006148610849678L, -0.0516699967278057976119L, -0.0586603078706856245674L, -0.0409960120868776180825L, -0.0185485073689590665153L, -0.00537842101034123222417L, -0.000920988084778273760609L, -0.716742618812210980263e-4L, -0.504623302166487346677e-9L, 0.712662196671896837736e-10L, -0.533769629702262072175e-11L }; static const T Q[9] = { 1L, 3.13286733695729715455L, 4.49281223045653491929L, 3.84900294427622911374L, 2.15205199043580378211L, 0.802912186540269232424L, 0.194793170017818925388L, 0.0280128013584653182994L, 0.00182034930799902922549L }; T t = z / 10 - 3; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); BOOST_MATH_INSTRUMENT_VARIABLE(result) result *= exp(z) / z; BOOST_MATH_INSTRUMENT_VARIABLE(result) result += z; BOOST_MATH_INSTRUMENT_VARIABLE(result) } else { // Maximum Deviation Found: 3.536e-20 // Max Error found at long double precision = Poly: 1.310671e-19 Cheb: 8.630943e-11 static const T exp40 = static_cast<T>(2.35385266837019985407899910749034804508871617254555467236651e17L); static const T Y= 1.013065338134765625F; static const T P[9] = { -0.0130653381347656250004L, 0.644487780349757303739L, 143.995670348227433964L, -13918.9322758014173709L, 476260.975133624194484L, -7437102.15135982802122L, 53732298.8764767916542L, -160695051.957997452509L, 137839271.592778020028L }; static const T Q[9] = { 1L, 27.2103343964943718802L, -8785.48528692879413676L, 397530.290000322626766L, -7356441.34957799368252L, 63050914.5343400957524L, -246143779.638307701369L, 384647824.678554961174L, -166288297.874583961493L }; T t = 1 / z; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); if(z < 41) result *= exp(z) / z; else { // Avoid premature overflow if we can: t = z - 40; if(t > tools::log_max_value<T>()) { result = policies::raise_overflow_error<T>(function, 0, pol); } else { result *= exp(z - 40) / z; if(result > tools::max_value<T>() / exp40) { result = policies::raise_overflow_error<T>(function, 0, pol); } else { result *= exp40; } } } result += z; } return result; } template <class T, class Policy> T expint_i_imp(T z, const Policy& pol, const mpl::int_<113>& tag) { BOOST_MATH_STD_USING static const char* function = "boost::math::expint<%1%>(%1%)"; if(z < 0) return -expint_imp(1, -z, pol, tag); if(z == 0) return -policies::raise_overflow_error<T>(function, 0, pol); T result; if(z <= 6) { // Maximum Deviation Found: 1.230e-36 // Expected Error Term: -1.230e-36 // Max Error found at long double precision = Poly: 4.355299e-34 Cheb: 7.512581e-34 static const T P[15] = { 2.98677224343598593765287235997328555L, -0.333256034674702967028780537349334037L, 0.851831522798101228384971644036708463L, -0.0657854833494646206186773614110374948L, 0.0630065662557284456000060708977935073L, -0.00311759191425309373327784154659649232L, 0.00176213568201493949664478471656026771L, -0.491548660404172089488535218163952295e-4L, 0.207764227621061706075562107748176592e-4L, -0.225445398156913584846374273379402765e-6L, 0.996939977231410319761273881672601592e-7L, 0.212546902052178643330520878928100847e-9L, 0.154646053060262871360159325115980023e-9L, 0.143971277122049197323415503594302307e-11L, 0.306243138978114692252817805327426657e-13L }; static const T Q[15] = { 1L, -1.40178870313943798705491944989231793L, 0.943810968269701047641218856758605284L, -0.405026631534345064600850391026113165L, 0.123924153524614086482627660399122762L, -0.0286364505373369439591132549624317707L, 0.00516148845910606985396596845494015963L, -0.000738330799456364820380739850924783649L, 0.843737760991856114061953265870882637e-4L, -0.767957673431982543213661388914587589e-5L, 0.549136847313854595809952100614840031e-6L, -0.299801381513743676764008325949325404e-7L, 0.118419479055346106118129130945423483e-8L, -0.30372295663095470359211949045344607e-10L, 0.382742953753485333207877784720070523e-12L }; static const T r1 = static_cast<T>(1677624236387711.0L / 4503599627370496.0L); static const T r2 = static_cast<T>(266514582277687.0L / 4503599627370496.0L / 4503599627370496.0L); static const T r3 = static_cast<T>(0.283806480836357377069325311780969887585024578164571984232357e-31L); static const T r = static_cast<T>(0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392L); T t = (z / 3) - 1; result = tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); t = ((z - r1) - r2) - r3; result *= t; if(fabs(t) < 0.1) { result += boost::math::log1p(t / r); } else { result += log(z / r); } } else if (z <= 10) { // Maximum Deviation Found: 7.779e-36 // Expected Error Term: -7.779e-36 // Max Error found at long double precision = Poly: 2.576723e-35 Cheb: 1.236001e-34 static const T Y = 1.158985137939453125F; static const T P[15] = { 0.00139324086199409049282472239613554817L, -0.0338173111691991289178779840307998955L, -0.0555972290794371306259684845277620556L, -0.0378677976003456171563136909186202177L, -0.0152221583517528358782902783914356667L, -0.00428283334203873035104248217403126905L, -0.000922782631491644846511553601323435286L, -0.000155513428088853161562660696055496696L, -0.205756580255359882813545261519317096e-4L, -0.220327406578552089820753181821115181e-5L, -0.189483157545587592043421445645377439e-6L, -0.122426571518570587750898968123803867e-7L, -0.635187358949437991465353268374523944e-9L, -0.203015132965870311935118337194860863e-10L, -0.384276705503357655108096065452950822e-12L }; static const T Q[15] = { 1L, 1.58784732785354597996617046880946257L, 1.18550755302279446339364262338114098L, 0.55598993549661368604527040349702836L, 0.184290888380564236919107835030984453L, 0.0459658051803613282360464632326866113L, 0.0089505064268613225167835599456014705L, 0.00139042673882987693424772855926289077L, 0.000174210708041584097450805790176479012L, 0.176324034009707558089086875136647376e-4L, 0.142935845999505649273084545313710581e-5L, 0.907502324487057260675816233312747784e-7L, 0.431044337808893270797934621235918418e-8L, 0.139007266881450521776529705677086902e-9L, 0.234715286125516430792452741830364672e-11L }; T t = z / 2 - 4; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else if(z <= 18) { // Maximum Deviation Found: 1.082e-34 // Expected Error Term: 1.080e-34 // Max Error found at long double precision = Poly: 1.958294e-34 Cheb: 2.472261e-34 static const T Y = 1.091579437255859375F; static const T P[17] = { -0.00685089599550151282724924894258520532L, -0.0443313550253580053324487059748497467L, -0.071538561252424027443296958795814874L, -0.0622923153354102682285444067843300583L, -0.0361631270264607478205393775461208794L, -0.0153192826839624850298106509601033261L, -0.00496967904961260031539602977748408242L, -0.00126989079663425780800919171538920589L, -0.000258933143097125199914724875206326698L, -0.422110326689204794443002330541441956e-4L, -0.546004547590412661451073996127115221e-5L, -0.546775260262202177131068692199272241e-6L, -0.404157632825805803833379568956559215e-7L, -0.200612596196561323832327013027419284e-8L, -0.502538501472133913417609379765434153e-10L, -0.326283053716799774936661568391296584e-13L, 0.869226483473172853557775877908693647e-15L }; static const T Q[15] = { 1L, 2.23227220874479061894038229141871087L, 2.40221000361027971895657505660959863L, 1.65476320985936174728238416007084214L, 0.816828602963895720369875535001248227L, 0.306337922909446903672123418670921066L, 0.0902400121654409267774593230720600752L, 0.0212708882169429206498765100993228086L, 0.00404442626252467471957713495828165491L, 0.0006195601618842253612635241404054589L, 0.755930932686543009521454653994321843e-4L, 0.716004532773778954193609582677482803e-5L, 0.500881663076471627699290821742924233e-6L, 0.233593219218823384508105943657387644e-7L, 0.554900353169148897444104962034267682e-9L }; T t = z / 4 - 3.5; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else if(z <= 26) { // Maximum Deviation Found: 3.163e-35 // Expected Error Term: 3.163e-35 // Max Error found at long double precision = Poly: 4.158110e-35 Cheb: 5.385532e-35 static const T Y = 1.051731109619140625F; static const T P[14] = { -0.00144552494420652573815404828020593565L, -0.0126747451594545338365684731262912741L, -0.01757394877502366717526779263438073L, -0.0126838952395506921945756139424722588L, -0.0060045057928894974954756789352443522L, -0.00205349237147226126653803455793107903L, -0.000532606040579654887676082220195624207L, -0.000107344687098019891474772069139014662L, -0.169536802705805811859089949943435152e-4L, -0.20863311729206543881826553010120078e-5L, -0.195670358542116256713560296776654385e-6L, -0.133291168587253145439184028259772437e-7L, -0.595500337089495614285777067722823397e-9L, -0.133141358866324100955927979606981328e-10L }; static const T Q[14] = { 1L, 1.72490783907582654629537013560044682L, 1.44524329516800613088375685659759765L, 0.778241785539308257585068744978050181L, 0.300520486589206605184097270225725584L, 0.0879346899691339661394537806057953957L, 0.0200802415843802892793583043470125006L, 0.00362842049172586254520256100538273214L, 0.000519731362862955132062751246769469957L, 0.584092147914050999895178697392282665e-4L, 0.501851497707855358002773398333542337e-5L, 0.313085677467921096644895738538865537e-6L, 0.127552010539733113371132321521204458e-7L, 0.25737310826983451144405899970774587e-9L }; T t = z / 4 - 5.5; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); BOOST_MATH_INSTRUMENT_VARIABLE(result) result *= exp(z) / z; BOOST_MATH_INSTRUMENT_VARIABLE(result) result += z; BOOST_MATH_INSTRUMENT_VARIABLE(result) } else if(z <= 42) { // Maximum Deviation Found: 7.972e-36 // Expected Error Term: 7.962e-36 // Max Error found at long double precision = Poly: 1.711721e-34 Cheb: 3.100018e-34 static const T Y = 1.032726287841796875F; static const T P[15] = { -0.00141056919297307534690895009969373233L, -0.0123384175302540291339020257071411437L, -0.0298127270706864057791526083667396115L, -0.0390686759471630584626293670260768098L, -0.0338226792912607409822059922949035589L, -0.0211659736179834946452561197559654582L, -0.0100428887460879377373158821400070313L, -0.00370717396015165148484022792801682932L, -0.0010768667551001624764329000496561659L, -0.000246127328761027039347584096573123531L, -0.437318110527818613580613051861991198e-4L, -0.587532682329299591501065482317771497e-5L, -0.565697065670893984610852937110819467e-6L, -0.350233957364028523971768887437839573e-7L, -0.105428907085424234504608142258423505e-8L }; static const T Q[16] = { 1L, 3.17261315255467581204685605414005525L, 4.85267952971640525245338392887217426L, 4.74341914912439861451492872946725151L, 3.31108463283559911602405970817931801L, 1.74657006336994649386607925179848899L, 0.718255607416072737965933040353653244L, 0.234037553177354542791975767960643864L, 0.0607470145906491602476833515412605389L, 0.0125048143774226921434854172947548724L, 0.00201034366420433762935768458656609163L, 0.000244823338417452367656368849303165721L, 0.213511655166983177960471085462540807e-4L, 0.119323998465870686327170541547982932e-5L, 0.322153582559488797803027773591727565e-7L, -0.161635525318683508633792845159942312e-16L }; T t = z / 8 - 4.25; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); BOOST_MATH_INSTRUMENT_VARIABLE(result) result *= exp(z) / z; BOOST_MATH_INSTRUMENT_VARIABLE(result) result += z; BOOST_MATH_INSTRUMENT_VARIABLE(result) } else if(z <= 56) { // Maximum Deviation Found: 4.469e-36 // Expected Error Term: 4.468e-36 // Max Error found at long double precision = Poly: 1.288958e-35 Cheb: 2.304586e-35 static const T Y = 1.0216197967529296875F; static const T P[12] = { -0.000322999116096627043476023926572650045L, -0.00385606067447365187909164609294113346L, -0.00686514524727568176735949971985244415L, -0.00606260649593050194602676772589601799L, -0.00334382362017147544335054575436194357L, -0.00126108534260253075708625583630318043L, -0.000337881489347846058951220431209276776L, -0.648480902304640018785370650254018022e-4L, -0.87652644082970492211455290209092766e-5L, -0.794712243338068631557849449519994144e-6L, -0.434084023639508143975983454830954835e-7L, -0.107839681938752337160494412638656696e-8L }; static const T Q[12] = { 1L, 2.09913805456661084097134805151524958L, 2.07041755535439919593503171320431849L, 1.26406517226052371320416108604874734L, 0.529689923703770353961553223973435569L, 0.159578150879536711042269658656115746L, 0.0351720877642000691155202082629857131L, 0.00565313621289648752407123620997063122L, 0.000646920278540515480093843570291218295L, 0.499904084850091676776993523323213591e-4L, 0.233740058688179614344680531486267142e-5L, 0.498800627828842754845418576305379469e-7L }; T t = z / 7 - 7; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); BOOST_MATH_INSTRUMENT_VARIABLE(result) result *= exp(z) / z; BOOST_MATH_INSTRUMENT_VARIABLE(result) result += z; BOOST_MATH_INSTRUMENT_VARIABLE(result) } else if(z <= 84) { // Maximum Deviation Found: 5.588e-35 // Expected Error Term: -5.566e-35 // Max Error found at long double precision = Poly: 9.976345e-35 Cheb: 8.358865e-35 static const T Y = 1.015148162841796875F; static const T P[11] = { -0.000435714784725086961464589957142615216L, -0.00432114324353830636009453048419094314L, -0.0100740363285526177522819204820582424L, -0.0116744115827059174392383504427640362L, -0.00816145387784261141360062395898644652L, -0.00371380272673500791322744465394211508L, -0.00112958263488611536502153195005736563L, -0.000228316462389404645183269923754256664L, -0.29462181955852860250359064291292577e-4L, -0.21972450610957417963227028788460299e-5L, -0.720558173805289167524715527536874694e-7L }; static const T Q[11] = { 1L, 2.95918362458402597039366979529287095L, 3.96472247520659077944638411856748924L, 3.15563251550528513747923714884142131L, 1.64674612007093983894215359287448334L, 0.58695020129846594405856226787156424L, 0.144358385319329396231755457772362793L, 0.024146911506411684815134916238348063L, 0.0026257132337460784266874572001650153L, 0.000167479843750859222348869769094711093L, 0.475673638665358075556452220192497036e-5L }; T t = z / 14 - 5; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); BOOST_MATH_INSTRUMENT_VARIABLE(result) result *= exp(z) / z; BOOST_MATH_INSTRUMENT_VARIABLE(result) result += z; BOOST_MATH_INSTRUMENT_VARIABLE(result) } else if(z <= 210) { // Maximum Deviation Found: 4.448e-36 // Expected Error Term: 4.445e-36 // Max Error found at long double precision = Poly: 2.058532e-35 Cheb: 2.165465e-27 static const T Y= 1.00849151611328125F; static const T P[9] = { -0.0084915161132812500000001440233607358L, 1.84479378737716028341394223076147872L, -130.431146923726715674081563022115568L, 4336.26945491571504885214176203512015L, -76279.0031974974730095170437591004177L, 729577.956271997673695191455111727774L, -3661928.69330208734947103004900349266L, 8570600.041606912735872059184527855L, -6758379.93672362080947905580906028645L }; static const T Q[10] = { 1L, -99.4868026047611434569541483506091713L, 3879.67753690517114249705089803055473L, -76495.82413252517165830203774900806L, 820773.726408311894342553758526282667L, -4803087.64956923577571031564909646579L, 14521246.227703545012713173740895477L, -19762752.0196769712258527849159393044L, 8354144.67882768405803322344185185517L, 355076.853106511136734454134915432571L }; T t = 1 / z; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); result *= exp(z) / z; result += z; } else // z > 210 { // Maximum Deviation Found: 3.963e-37 // Expected Error Term: 3.963e-37 // Max Error found at long double precision = Poly: 1.248049e-36 Cheb: 2.843486e-29 static const T exp40 = static_cast<T>(2.35385266837019985407899910749034804508871617254555467236651e17L); static const T Y= 1.00252532958984375F; static const T P[8] = { -0.00252532958984375000000000000000000085L, 1.16591386866059087390621952073890359L, -67.8483431314018462417456828499277579L, 1567.68688154683822956359536287575892L, -17335.4683325819116482498725687644986L, 93632.6567462673524739954389166550069L, -225025.189335919133214440347510936787L, 175864.614717440010942804684741336853L }; static const T Q[9] = { 1L, -65.6998869881600212224652719706425129L, 1642.73850032324014781607859416890077L, -19937.2610222467322481947237312818575L, 124136.267326632742667972126625064538L, -384614.251466704550678760562965502293L, 523355.035910385688578278384032026998L, -217809.552260834025885677791936351294L, -8555.81719551123640677261226549550872L }; T t = 1 / z; result = Y + tools::evaluate_polynomial(P, t) / tools::evaluate_polynomial(Q, t); if(z < 41) result *= exp(z) / z; else { // Avoid premature overflow if we can: t = z - 40; if(t > tools::log_max_value<T>()) { result = policies::raise_overflow_error<T>(function, 0, pol); } else { result *= exp(z - 40) / z; if(result > tools::max_value<T>() / exp40) { result = policies::raise_overflow_error<T>(function, 0, pol); } else { result *= exp40; } } } result += z; } return result; } template <class T, class Policy> inline typename tools::promote_args<T>::type expint_forwarder(T z, const Policy& /*pol*/, mpl::true_ const&) { 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; 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; return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::expint_i_imp( static_cast<value_type>(z), forwarding_policy(), tag_type()), "boost::math::expint<%1%>(%1%)"); } template <class T> inline typename tools::promote_args<T>::type expint_forwarder(unsigned n, T z, const mpl::false_&) { return boost::math::expint(n, z, policies::policy<>()); } } // namespace detail template <class T, class Policy> inline typename tools::promote_args<T>::type expint(unsigned n, 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; 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; return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::expint_imp( n, static_cast<value_type>(z), forwarding_policy(), tag_type()), "boost::math::expint<%1%>(unsigned, %1%)"); } template <class T, class U> inline typename detail::expint_result<T, U>::type expint(T const z, U const u) { typedef typename policies::is_policy<U>::type tag_type; return detail::expint_forwarder(z, u, tag_type()); } template <class T> inline typename tools::promote_args<T>::type expint(T z) { return expint(z, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_EXPINT_HPP