math/special_functions/expint.hpp

0001 //  Copyright John Maddock 2007.
0002 //  Use, modification and distribution are subject to the
0003 //  Boost Software License, Version 1.0. (See accompanying file
0004 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
0005
0006 #ifndef BOOST_MATH_EXPINT_HPP
0007 #define BOOST_MATH_EXPINT_HPP
0008
0009 #ifdef _MSC_VER
0010 #pragma once
0011 #pragma warning(push)
0012 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
0013 #endif
0014
0015 #include <boost/math/tools/precision.hpp>
0016 #include <boost/math/tools/promotion.hpp>
0017 #include <boost/math/tools/fraction.hpp>
0018 #include <boost/math/tools/series.hpp>
0019 #include <boost/math/policies/error_handling.hpp>
0020 #include <boost/math/special_functions/math_fwd.hpp>
0021 #include <boost/math/special_functions/digamma.hpp>
0022 #include <boost/math/special_functions/log1p.hpp>
0023 #include <boost/math/special_functions/pow.hpp>
0024
0025 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
0026 //
0027 // This is the only way we can avoid
0028 // warning: non-standard suffix on floating constant [-Wpedantic]
0029 // when building with -Wall -pedantic.  Neither __extension__
0030 // nor #pragma diagnostic ignored work :(
0031 //
0032 #pragma GCC system_header
0033 #endif
0034
0035 namespace boost{ namespace math{
0036
0037 template <class T, class Policy>
0038 inline typename tools::promote_args<T>::type
0039    expint(unsigned n, T z, const Policy& /*pol*/);
0040
0041 namespace detail{
0042
0043 template <class T>
0044 inline T expint_1_rational(const T& z, const std::integral_constant<int, 0>&)
0045 {
0046    // this function is never actually called
0047    BOOST_MATH_ASSERT(0);
0048    return z;
0049 }
0050
0051 template <class T>
0052 T expint_1_rational(const T& z, const std::integral_constant<int, 53>&)
0053 {
0054    BOOST_MATH_STD_USING
0055    T result;
0056    if(z <= 1)
0057    {
0058       // Maximum Deviation Found:                     2.006e-18
0059       // Expected Error Term:                         2.006e-18
0060       // Max error found at double precision:         2.760e-17
0061       static const T Y = 0.66373538970947265625F;
0062       static const T P[6] = {
0063          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0865197248079397976498),
0064          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0320913665303559189999),
0065          BOOST_MATH_BIG_CONSTANT(T, 53, -0.245088216639761496153),
0066          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0368031736257943745142),
0067          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00399167106081113256961),
0068          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000111507792921197858394)
0069       };
0070       static const T Q[6] = {
0071          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
0072          BOOST_MATH_BIG_CONSTANT(T, 53, 0.37091387659397013215),
0073          BOOST_MATH_BIG_CONSTANT(T, 53, 0.056770677104207528384),
0074          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00427347600017103698101),
0075          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000131049900798434683324),
0076          BOOST_MATH_BIG_CONSTANT(T, 53, -0.528611029520217142048e-6)
0077       };
0078       result = tools::evaluate_polynomial(P, z)
0079          / tools::evaluate_polynomial(Q, z);
0080       result += z - log(z) - Y;
0081    }
0082    else if(z < -boost::math::tools::log_min_value<T>())
0083    {
0084       // Maximum Deviation Found (interpolated):      1.444e-17
0085       // Max error found at double precision:         3.119e-17
0086       static const T P[11] = {
0087          BOOST_MATH_BIG_CONSTANT(T, 53, -0.121013190657725568138e-18),
0088          BOOST_MATH_BIG_CONSTANT(T, 53, -0.999999999999998811143),
0089          BOOST_MATH_BIG_CONSTANT(T, 53, -43.3058660811817946037),
0090          BOOST_MATH_BIG_CONSTANT(T, 53, -724.581482791462469795),
0091          BOOST_MATH_BIG_CONSTANT(T, 53, -6046.8250112711035463),
0092          BOOST_MATH_BIG_CONSTANT(T, 53, -27182.6254466733970467),
0093          BOOST_MATH_BIG_CONSTANT(T, 53, -66598.2652345418633509),
0094          BOOST_MATH_BIG_CONSTANT(T, 53, -86273.1567711649528784),
0095          BOOST_MATH_BIG_CONSTANT(T, 53, -54844.4587226402067411),
0096          BOOST_MATH_BIG_CONSTANT(T, 53, -14751.4895786128450662),
0097          BOOST_MATH_BIG_CONSTANT(T, 53, -1185.45720315201027667)
0098       };
0099       static const T Q[12] = {
0100          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
0101          BOOST_MATH_BIG_CONSTANT(T, 53, 45.3058660811801465927),
0102          BOOST_MATH_BIG_CONSTANT(T, 53, 809.193214954550328455),
0103          BOOST_MATH_BIG_CONSTANT(T, 53, 7417.37624454689546708),
0104          BOOST_MATH_BIG_CONSTANT(T, 53, 38129.5594484818471461),
0105          BOOST_MATH_BIG_CONSTANT(T, 53, 113057.05869159631492),
0106          BOOST_MATH_BIG_CONSTANT(T, 53, 192104.047790227984431),
0107          BOOST_MATH_BIG_CONSTANT(T, 53, 180329.498380501819718),
0108          BOOST_MATH_BIG_CONSTANT(T, 53, 86722.3403467334749201),
0109          BOOST_MATH_BIG_CONSTANT(T, 53, 18455.4124737722049515),
0110          BOOST_MATH_BIG_CONSTANT(T, 53, 1229.20784182403048905),
0111          BOOST_MATH_BIG_CONSTANT(T, 53, -0.776491285282330997549)
0112       };
0113       T recip = 1 / z;
0114       result = 1 + tools::evaluate_polynomial(P, recip)
0115          / tools::evaluate_polynomial(Q, recip);
0116       result *= exp(-z) * recip;
0117    }
0118    else
0119    {
0120       result = 0;
0121    }
0122    return result;
0123 }
0124
0125 template <class T>
0126 T expint_1_rational(const T& z, const std::integral_constant<int, 64>&)
0127 {
0128    BOOST_MATH_STD_USING
0129    T result;
0130    if(z <= 1)
0131    {
0132       // Maximum Deviation Found:                     3.807e-20
0133       // Expected Error Term:                         3.807e-20
0134       // Max error found at long double precision:    6.249e-20
0135
0136       static const T Y = 0.66373538970947265625F;
0137       static const T P[6] = {
0138          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0865197248079397956816),
0139          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0275114007037026844633),
0140          BOOST_MATH_BIG_CONSTANT(T, 64, -0.246594388074877139824),
0141          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0237624819878732642231),
0142          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00259113319641673986276),
0143          BOOST_MATH_BIG_CONSTANT(T, 64, 0.30853660894346057053e-4)
0144       };
0145       static const T Q[7] = {
0146          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
0147          BOOST_MATH_BIG_CONSTANT(T, 64, 0.317978365797784100273),
0148          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0393622602554758722511),
0149          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00204062029115966323229),
0150          BOOST_MATH_BIG_CONSTANT(T, 64, 0.732512107100088047854e-5),
0151          BOOST_MATH_BIG_CONSTANT(T, 64, -0.202872781770207871975e-5),
0152          BOOST_MATH_BIG_CONSTANT(T, 64, 0.52779248094603709945e-7)
0153       };
0154       result = tools::evaluate_polynomial(P, z)
0155          / tools::evaluate_polynomial(Q, z);
0156       result += z - log(z) - Y;
0157    }
0158    else if(z < -boost::math::tools::log_min_value<T>())
0159    {
0160       // Maximum Deviation Found (interpolated):     2.220e-20
0161       // Max error found at long double precision:   1.346e-19
0162       static const T P[14] = {
0163          BOOST_MATH_BIG_CONSTANT(T, 64, -0.534401189080684443046e-23),
0164          BOOST_MATH_BIG_CONSTANT(T, 64, -0.999999999999999999905),
0165          BOOST_MATH_BIG_CONSTANT(T, 64, -62.1517806091379402505),
0166          BOOST_MATH_BIG_CONSTANT(T, 64, -1568.45688271895145277),
0167          BOOST_MATH_BIG_CONSTANT(T, 64, -21015.3431990874009619),
0168          BOOST_MATH_BIG_CONSTANT(T, 64, -164333.011755931661949),
0169          BOOST_MATH_BIG_CONSTANT(T, 64, -777917.270775426696103),
0170          BOOST_MATH_BIG_CONSTANT(T, 64, -2244188.56195255112937),
0171          BOOST_MATH_BIG_CONSTANT(T, 64, -3888702.98145335643429),
0172          BOOST_MATH_BIG_CONSTANT(T, 64, -3909822.65621952648353),
0173          BOOST_MATH_BIG_CONSTANT(T, 64, -2149033.9538897398457),
0174          BOOST_MATH_BIG_CONSTANT(T, 64, -584705.537139793925189),
0175          BOOST_MATH_BIG_CONSTANT(T, 64, -65815.2605361889477244),
0176          BOOST_MATH_BIG_CONSTANT(T, 64, -2038.82870680427258038)
0177       };
0178       static const T Q[14] = {
0179          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
0180          BOOST_MATH_BIG_CONSTANT(T, 64, 64.1517806091379399478),
0181          BOOST_MATH_BIG_CONSTANT(T, 64, 1690.76044393722763785),
0182          BOOST_MATH_BIG_CONSTANT(T, 64, 24035.9534033068949426),
0183          BOOST_MATH_BIG_CONSTANT(T, 64, 203679.998633572361706),
0184          BOOST_MATH_BIG_CONSTANT(T, 64, 1074661.58459976978285),
0185          BOOST_MATH_BIG_CONSTANT(T, 64, 3586552.65020899358773),
0186          BOOST_MATH_BIG_CONSTANT(T, 64, 7552186.84989547621411),
0187          BOOST_MATH_BIG_CONSTANT(T, 64, 9853333.79353054111434),
0188          BOOST_MATH_BIG_CONSTANT(T, 64, 7689642.74550683631258),
0189          BOOST_MATH_BIG_CONSTANT(T, 64, 3385553.35146759180739),
0190          BOOST_MATH_BIG_CONSTANT(T, 64, 763218.072732396428725),
0191          BOOST_MATH_BIG_CONSTANT(T, 64, 73930.2995984054930821),
0192          BOOST_MATH_BIG_CONSTANT(T, 64, 2063.86994219629165937)
0193       };
0194       T recip = 1 / z;
0195       result = 1 + tools::evaluate_polynomial(P, recip)
0196          / tools::evaluate_polynomial(Q, recip);
0197       result *= exp(-z) * recip;
0198    }
0199    else
0200    {
0201       result = 0;
0202    }
0203    return result;
0204 }
0205
0206 template <class T>
0207 T expint_1_rational(const T& z, const std::integral_constant<int, 113>&)
0208 {
0209    BOOST_MATH_STD_USING
0210    T result;
0211    if(z <= 1)
0212    {
0213       // Maximum Deviation Found:                     2.477e-35
0214       // Expected Error Term:                         2.477e-35
0215       // Max error found at long double precision:    6.810e-35
0216
0217       static const T Y = 0.66373538970947265625F;
0218       static const T P[10] = {
0219          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0865197248079397956434879099175975937),
0220          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0369066175910795772830865304506087759),
0221          BOOST_MATH_BIG_CONSTANT(T, 113, -0.24272036838415474665971599314725545),
0222          BOOST_MATH_BIG_CONSTANT(T, 113, -0.0502166331248948515282379137550178307),
0223          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00768384138547489410285101483730424919),
0224          BOOST_MATH_BIG_CONSTANT(T, 113, -0.000612574337702109683505224915484717162),
0225          BOOST_MATH_BIG_CONSTANT(T, 113, -0.380207107950635046971492617061708534e-4),
0226          BOOST_MATH_BIG_CONSTANT(T, 113, -0.136528159460768830763009294683628406e-5),
0227          BOOST_MATH_BIG_CONSTANT(T, 113, -0.346839106212658259681029388908658618e-7),
0228          BOOST_MATH_BIG_CONSTANT(T, 113, -0.340500302777838063940402160594523429e-9)
0229       };
0230       static const T Q[10] = {
0231          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
0232          BOOST_MATH_BIG_CONSTANT(T, 113, 0.426568827778942588160423015589537302),
0233          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0841384046470893490592450881447510148),
0234          BOOST_MATH_BIG_CONSTANT(T, 113, 0.0100557215850668029618957359471132995),
0235          BOOST_MATH_BIG_CONSTANT(T, 113, 0.000799334870474627021737357294799839363),
0236          BOOST_MATH_BIG_CONSTANT(T, 113, 0.434452090903862735242423068552687688e-4),
0237          BOOST_MATH_BIG_CONSTANT(T, 113, 0.15829674748799079874182885081231252e-5),
0238          BOOST_MATH_BIG_CONSTANT(T, 113, 0.354406206738023762100882270033082198e-7),
0239          BOOST_MATH_BIG_CONSTANT(T, 113, 0.369373328141051577845488477377890236e-9),
0240          BOOST_MATH_BIG_CONSTANT(T, 113, -0.274149801370933606409282434677600112e-12)
0241       };
0242       result = tools::evaluate_polynomial(P, z)
0243          / tools::evaluate_polynomial(Q, z);
0244       result += z - log(z) - Y;
0245    }
0246    else if(z <= 4)
0247    {
0248       // Max error in interpolated form:             5.614e-35
0249       // Max error found at long double precision:   7.979e-35
0250
0251       static const T Y = 0.70190334320068359375F;
0252
0253       static const T P[16] = {
0254          BOOST_MATH_BIG_CONSTANT(T, 113, 0.298096656795020369955077350585959794),
0255          BOOST_MATH_BIG_CONSTANT(T, 113, 12.9314045995266142913135497455971247),
0256          BOOST_MATH_BIG_CONSTANT(T, 113, 226.144334921582637462526628217345501),
0257          BOOST_MATH_BIG_CONSTANT(T, 113, 2070.83670924261732722117682067381405),
0258          BOOST_MATH_BIG_CONSTANT(T, 113, 10715.1115684330959908244769731347186),
0259          BOOST_MATH_BIG_CONSTANT(T, 113, 30728.7876355542048019664777316053311),
0260          BOOST_MATH_BIG_CONSTANT(T, 113, 38520.6078609349855436936232610875297),
0261          BOOST_MATH_BIG_CONSTANT(T, 113, -27606.0780981527583168728339620565165),
0262          BOOST_MATH_BIG_CONSTANT(T, 113, -169026.485055785605958655247592604835),
0263          BOOST_MATH_BIG_CONSTANT(T, 113, -254361.919204983608659069868035092282),
0264          BOOST_MATH_BIG_CONSTANT(T, 113, -195765.706874132267953259272028679935),
0265          BOOST_MATH_BIG_CONSTANT(T, 113, -83352.6826013533205474990119962408675),
0266          BOOST_MATH_BIG_CONSTANT(T, 113, -19251.6828496869586415162597993050194),
0267          BOOST_MATH_BIG_CONSTANT(T, 113, -2226.64251774578542836725386936102339),
0268          BOOST_MATH_BIG_CONSTANT(T, 113, -109.009437301400845902228611986479816),
0269          BOOST_MATH_BIG_CONSTANT(T, 113, -1.51492042209561411434644938098833499)
0270       };
0271       static const T Q[16] = {
0272          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
0273          BOOST_MATH_BIG_CONSTANT(T, 113, 46.734521442032505570517810766704587),
0274          BOOST_MATH_BIG_CONSTANT(T, 113, 908.694714348462269000247450058595655),
0275          BOOST_MATH_BIG_CONSTANT(T, 113, 9701.76053033673927362784882748513195),
0276          BOOST_MATH_BIG_CONSTANT(T, 113, 63254.2815292641314236625196594947774),
0277          BOOST_MATH_BIG_CONSTANT(T, 113, 265115.641285880437335106541757711092),
0278          BOOST_MATH_BIG_CONSTANT(T, 113, 732707.841188071900498536533086567735),
0279          BOOST_MATH_BIG_CONSTANT(T, 113, 1348514.02492635723327306628712057794),
0280          BOOST_MATH_BIG_CONSTANT(T, 113, 1649986.81455283047769673308781585991),
0281          BOOST_MATH_BIG_CONSTANT(T, 113, 1326000.828522976970116271208812099),
0282          BOOST_MATH_BIG_CONSTANT(T, 113, 683643.09490612171772350481773951341),
0283          BOOST_MATH_BIG_CONSTANT(T, 113, 217640.505137263607952365685653352229),
0284          BOOST_MATH_BIG_CONSTANT(T, 113, 40288.3467237411710881822569476155485),
0285          BOOST_MATH_BIG_CONSTANT(T, 113, 3932.89353979531632559232883283175754),
0286          BOOST_MATH_BIG_CONSTANT(T, 113, 169.845369689596739824177412096477219),
0287          BOOST_MATH_BIG_CONSTANT(T, 113, 2.17607292280092201170768401876895354)
0288       };
0289       T recip = 1 / z;
0290       result = Y + tools::evaluate_polynomial(P, recip)
0291          / tools::evaluate_polynomial(Q, recip);
0292       result *= exp(-z) * recip;
0293    }
0294    else if(z < -boost::math::tools::log_min_value<T>())
0295    {
0296       // Max error in interpolated form:             4.413e-35
0297       // Max error found at long double precision:   8.928e-35
0298
0299       static const T P[19] = {
0300          BOOST_MATH_BIG_CONSTANT(T, 113, -0.559148411832951463689610809550083986e-40),
0301          BOOST_MATH_BIG_CONSTANT(T, 113, -0.999999999999999999999999999999999997),
0302          BOOST_MATH_BIG_CONSTANT(T, 113, -166.542326331163836642960118190147367),
0303          BOOST_MATH_BIG_CONSTANT(T, 113, -12204.639128796330005065904675153652),
0304          BOOST_MATH_BIG_CONSTANT(T, 113, -520807.069767086071806275022036146855),
0305          BOOST_MATH_BIG_CONSTANT(T, 113, -14435981.5242137970691490903863125326),
0306          BOOST_MATH_BIG_CONSTANT(T, 113, -274574945.737064301247496460758654196),
0307          BOOST_MATH_BIG_CONSTANT(T, 113, -3691611582.99810039356254671781473079),
0308          BOOST_MATH_BIG_CONSTANT(T, 113, -35622515944.8255047299363690814678763),
0309          BOOST_MATH_BIG_CONSTANT(T, 113, -248040014774.502043161750715548451142),
0310          BOOST_MATH_BIG_CONSTANT(T, 113, -1243190389769.53458416330946622607913),
0311          BOOST_MATH_BIG_CONSTANT(T, 113, -4441730126135.54739052731990368425339),
0312          BOOST_MATH_BIG_CONSTANT(T, 113, -11117043181899.7388524310281751971366),
0313          BOOST_MATH_BIG_CONSTANT(T, 113, -18976497615396.9717776601813519498961),
0314          BOOST_MATH_BIG_CONSTANT(T, 113, -21237496819711.1011661104761906067131),
0315          BOOST_MATH_BIG_CONSTANT(T, 113, -14695899122092.5161620333466757812848),
0316          BOOST_MATH_BIG_CONSTANT(T, 113, -5737221535080.30569711574295785864903),
0317          BOOST_MATH_BIG_CONSTANT(T, 113, -1077042281708.42654526404581272546244),
0318          BOOST_MATH_BIG_CONSTANT(T, 113, -68028222642.1941480871395695677675137)
0319       };
0320       static const T Q[20] = {
0321          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
0322          BOOST_MATH_BIG_CONSTANT(T, 113, 168.542326331163836642960118190147311),
0323          BOOST_MATH_BIG_CONSTANT(T, 113, 12535.7237814586576783518249115343619),
0324          BOOST_MATH_BIG_CONSTANT(T, 113, 544891.263372016404143120911148640627),
0325          BOOST_MATH_BIG_CONSTANT(T, 113, 15454474.7241010258634446523045237762),
0326          BOOST_MATH_BIG_CONSTANT(T, 113, 302495899.896629522673410325891717381),
0327          BOOST_MATH_BIG_CONSTANT(T, 113, 4215565948.38886507646911672693270307),
0328          BOOST_MATH_BIG_CONSTANT(T, 113, 42552409471.7951815668506556705733344),
0329          BOOST_MATH_BIG_CONSTANT(T, 113, 313592377066.753173979584098301610186),
0330          BOOST_MATH_BIG_CONSTANT(T, 113, 1688763640223.4541980740597514904542),
0331          BOOST_MATH_BIG_CONSTANT(T, 113, 6610992294901.59589748057620192145704),
0332          BOOST_MATH_BIG_CONSTANT(T, 113, 18601637235659.6059890851321772682606),
0333          BOOST_MATH_BIG_CONSTANT(T, 113, 36944278231087.2571020964163402941583),
0334          BOOST_MATH_BIG_CONSTANT(T, 113, 50425858518481.7497071917028793820058),
0335          BOOST_MATH_BIG_CONSTANT(T, 113, 45508060902865.0899967797848815980644),
0336          BOOST_MATH_BIG_CONSTANT(T, 113, 25649955002765.3817331501988304758142),
0337          BOOST_MATH_BIG_CONSTANT(T, 113, 8259575619094.6518520988612711292331),
0338          BOOST_MATH_BIG_CONSTANT(T, 113, 1299981487496.12607474362723586264515),
0339          BOOST_MATH_BIG_CONSTANT(T, 113, 70242279152.8241187845178443118302693),
0340          BOOST_MATH_BIG_CONSTANT(T, 113, -37633302.9409263839042721539363416685)
0341       };
0342       T recip = 1 / z;
0343       result = 1 + tools::evaluate_polynomial(P, recip)
0344          / tools::evaluate_polynomial(Q, recip);
0345       result *= exp(-z) * recip;
0346    }
0347    else
0348    {
0349       result = 0;
0350    }
0351    return result;
0352 }
0353
0354 template <class T>
0355 struct expint_fraction
0356 {
0357    typedef std::pair<T,T> result_type;
0358    expint_fraction(unsigned n_, T z_) : b(n_ + z_), i(-1), n(n_){}
0359    std::pair<T,T> operator()()
0360    {
0361       std::pair<T,T> result = std::make_pair(-static_cast<T>((i+1) * (n+i)), b);
0362       b += 2;
0363       ++i;
0364       return result;
0365    }
0366 private:
0367    T b;
0368    int i;
0369    unsigned n;
0370 };
0371
0372 template <class T, class Policy>
0373 inline T expint_as_fraction(unsigned n, T z, const Policy& pol)
0374 {
0375    BOOST_MATH_STD_USING
0376    BOOST_MATH_INSTRUMENT_VARIABLE(z)
0377    std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
0378    expint_fraction<T> f(n, z);
0379    T result = tools::continued_fraction_b(
0380       f,
0381       boost::math::policies::get_epsilon<T, Policy>(),
0382       max_iter);
0383    policies::check_series_iterations<T>("boost::math::expint_continued_fraction<%1%>(unsigned,%1%)", max_iter, pol);
0384    BOOST_MATH_INSTRUMENT_VARIABLE(result)
0385    BOOST_MATH_INSTRUMENT_VARIABLE(max_iter)
0386    result = exp(-z) / result;
0387    BOOST_MATH_INSTRUMENT_VARIABLE(result)
0388    return result;
0389 }
0390
0391 template <class T>
0392 struct expint_series
0393 {
0394    typedef T result_type;
0395    expint_series(unsigned k_, T z_, T x_k_, T denom_, T fact_)
0396       : k(k_), z(z_), x_k(x_k_), denom(denom_), fact(fact_){}
0397    T operator()()
0398    {
0399       x_k *= -z;
0400       denom += 1;
0401       fact *= ++k;
0402       return x_k / (denom * fact);
0403    }
0404 private:
0405    unsigned k;
0406    T z;
0407    T x_k;
0408    T denom;
0409    T fact;
0410 };
0411
0412 template <class T, class Policy>
0413 inline T expint_as_series(unsigned n, T z, const Policy& pol)
0414 {
0415    BOOST_MATH_STD_USING
0416    std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
0417
0418    BOOST_MATH_INSTRUMENT_VARIABLE(z)
0419
0420    T result = 0;
0421    T x_k = -1;
0422    T denom = T(1) - n;
0423    T fact = 1;
0424    unsigned k = 0;
0425    for(; k < n - 1;)
0426    {
0427       result += x_k / (denom * fact);
0428       denom += 1;
0429       x_k *= -z;
0430       fact *= ++k;
0431    }
0432    BOOST_MATH_INSTRUMENT_VARIABLE(result)
0433    result += pow(-z, static_cast<T>(n - 1))
0434       * (boost::math::digamma(static_cast<T>(n), pol) - log(z)) / fact;
0435    BOOST_MATH_INSTRUMENT_VARIABLE(result)
0436
0437    expint_series<T> s(k, z, x_k, denom, fact);
0438    result = tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, result);
0439    policies::check_series_iterations<T>("boost::math::expint_series<%1%>(unsigned,%1%)", max_iter, pol);
0440    BOOST_MATH_INSTRUMENT_VARIABLE(result)
0441    BOOST_MATH_INSTRUMENT_VARIABLE(max_iter)
0442    return result;
0443 }
0444
0445 template <class T, class Policy, class Tag>
0446 T expint_imp(unsigned n, T z, const Policy& pol, const Tag& tag)
0447 {
0448    BOOST_MATH_STD_USING
0449    static const char* function = "boost::math::expint<%1%>(unsigned, %1%)";
0450    if(z < 0)
0451       return policies::raise_domain_error<T>(function, "Function requires z >= 0 but got %1%.", z, pol);
0452    if(z == 0)
0453       return n == 1 ? policies::raise_overflow_error<T>(function, nullptr, pol) : T(1 / (static_cast<T>(n - 1)));
0454
0455    T result;
0456
0457    bool f;
0458    if(n < 3)
0459    {
0460       f = z < T(0.5);
0461    }
0462    else
0463    {
0464       f = z < (static_cast<T>(n - 2) / static_cast<T>(n - 1));
0465    }
0466 #ifdef _MSC_VER
0467 #  pragma warning(push)
0468 #  pragma warning(disable:4127) // conditional expression is constant
0469 #endif
0470    if(n == 0)
0471       result = exp(-z) / z;
0472    else if((n == 1) && (Tag::value))
0473    {
0474       result = expint_1_rational(z, tag);
0475    }
0476    else if(f)
0477       result = expint_as_series(n, z, pol);
0478    else
0479       result = expint_as_fraction(n, z, pol);
0480 #ifdef _MSC_VER
0481 #  pragma warning(pop)
0482 #endif
0483
0484    return result;
0485 }
0486
0487 template <class T>
0488 struct expint_i_series
0489 {
0490    typedef T result_type;
0491    expint_i_series(T z_) : k(0), z_k(1), z(z_){}
0492    T operator()()
0493    {
0494       z_k *= z / ++k;
0495       return z_k / k;
0496    }
0497 private:
0498    unsigned k;
0499    T z_k;
0500    T z;
0501 };
0502
0503 template <class T, class Policy>
0504 T expint_i_as_series(T z, const Policy& pol)
0505 {
0506    BOOST_MATH_STD_USING
0507    T result = log(z); // (log(z) - log(1 / z)) / 2;
0508    result += constants::euler<T>();
0509    expint_i_series<T> s(z);
0510    std::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
0511    result = tools::sum_series(s, policies::get_epsilon<T, Policy>(), max_iter, result);
0512    policies::check_series_iterations<T>("boost::math::expint_i_series<%1%>(%1%)", max_iter, pol);
0513    return result;
0514 }
0515
0516 template <class T, class Policy, class Tag>
0517 T expint_i_imp(T z, const Policy& pol, const Tag& tag)
0518 {
0519    static const char* function = "boost::math::expint<%1%>(%1%)";
0520    if(z < 0)
0521       return -expint_imp(1, T(-z), pol, tag);
0522    if(z == 0)
0523       return -policies::raise_overflow_error<T>(function, nullptr, pol);
0524    return expint_i_as_series(z, pol);
0525 }
0526
0527 template <class T, class Policy>
0528 T expint_i_imp(T z, const Policy& pol, const std::integral_constant<int, 53>& tag)
0529 {
0530    BOOST_MATH_STD_USING
0531    static const char* function = "boost::math::expint<%1%>(%1%)";
0532    if(z < 0)
0533       return -expint_imp(1, T(-z), pol, tag);
0534    if(z == 0)
0535       return -policies::raise_overflow_error<T>(function, nullptr, pol);
0536
0537    T result;
0538
0539    if(z <= 6)
0540    {
0541       // Maximum Deviation Found:                     2.852e-18
0542       // Expected Error Term:                         2.852e-18
0543       // Max Error found at double precision =        Poly: 2.636335e-16   Cheb: 4.187027e-16
0544       static const T P[10] = {
0545          BOOST_MATH_BIG_CONSTANT(T, 53, 2.98677224343598593013),
0546          BOOST_MATH_BIG_CONSTANT(T, 53, 0.356343618769377415068),
0547          BOOST_MATH_BIG_CONSTANT(T, 53, 0.780836076283730801839),
0548          BOOST_MATH_BIG_CONSTANT(T, 53, 0.114670926327032002811),
0549          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0499434773576515260534),
0550          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00726224593341228159561),
0551          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00115478237227804306827),
0552          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000116419523609765200999),
0553          BOOST_MATH_BIG_CONSTANT(T, 53, 0.798296365679269702435e-5),
0554          BOOST_MATH_BIG_CONSTANT(T, 53, 0.2777056254402008721e-6)
0555       };
0556       static const T Q[8] = {
0557          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
0558          BOOST_MATH_BIG_CONSTANT(T, 53, -1.17090412365413911947),
0559          BOOST_MATH_BIG_CONSTANT(T, 53, 0.62215109846016746276),
0560          BOOST_MATH_BIG_CONSTANT(T, 53, -0.195114782069495403315),
0561          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0391523431392967238166),
0562          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00504800158663705747345),
0563          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000389034007436065401822),
0564          BOOST_MATH_BIG_CONSTANT(T, 53, -0.138972589601781706598e-4)
0565       };
0566
0567       static const T c1 = BOOST_MATH_BIG_CONSTANT(T, 53, 1677624236387711.0);
0568       static const T c2 = BOOST_MATH_BIG_CONSTANT(T, 53, 4503599627370496.0);
0569       static const T r1 = static_cast<T>(c1 / c2);
0570       static const T r2 = BOOST_MATH_BIG_CONSTANT(T, 53, 0.131401834143860282009280387409357165515556574352422001206362e-16);
0571       static const T r = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 53, 0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392));
0572       T t = (z / 3) - 1;
0573       result = tools::evaluate_polynomial(P, t)
0574          / tools::evaluate_polynomial(Q, t);
0575       t = (z - r1) - r2;
0576       result *= t;
0577       if(fabs(t) < T(0.1))
0578       {
0579          result += boost::math::log1p(t / r, pol);
0580       }
0581       else
0582       {
0583          result += log(z / r);
0584       }
0585    }
0586    else if (z <= 10)
0587    {
0588       // Maximum Deviation Found:                     6.546e-17
0589       // Expected Error Term:                         6.546e-17
0590       // Max Error found at double precision =        Poly: 6.890169e-17   Cheb: 6.772128e-17
0591       static const T Y = 1.158985137939453125F;
0592       static const T P[8] = {
0593          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00139324086199402804173),
0594          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0349921221823888744966),
0595          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0264095520754134848538),
0596          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00761224003005476438412),
0597          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00247496209592143627977),
0598          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000374885917942100256775),
0599          BOOST_MATH_BIG_CONSTANT(T, 53, -0.554086272024881826253e-4),
0600          BOOST_MATH_BIG_CONSTANT(T, 53, -0.396487648924804510056e-5)
0601       };
0602       static const T Q[8] = {
0603          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
0604          BOOST_MATH_BIG_CONSTANT(T, 53, 0.744625566823272107711),
0605          BOOST_MATH_BIG_CONSTANT(T, 53, 0.329061095011767059236),
0606          BOOST_MATH_BIG_CONSTANT(T, 53, 0.100128624977313872323),
0607          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0223851099128506347278),
0608          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00365334190742316650106),
0609          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000402453408512476836472),
0610          BOOST_MATH_BIG_CONSTANT(T, 53, 0.263649630720255691787e-4)
0611       };
0612       T t = z / 2 - 4;
0613       result = Y + tools::evaluate_polynomial(P, t)
0614          / tools::evaluate_polynomial(Q, t);
0615       result *= exp(z) / z;
0616       result += z;
0617    }
0618    else if(z <= 20)
0619    {
0620       // Maximum Deviation Found:                     1.843e-17
0621       // Expected Error Term:                         -1.842e-17
0622       // Max Error found at double precision =        Poly: 4.375868e-17   Cheb: 5.860967e-17
0623
0624       static const T Y = 1.0869731903076171875F;
0625       static const T P[9] = {
0626          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00893891094356945667451),
0627          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0484607730127134045806),
0628          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0652810444222236895772),
0629          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0478447572647309671455),
0630          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0226059218923777094596),
0631          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00720603636917482065907),
0632          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00155941947035972031334),
0633          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000209750022660200888349),
0634          BOOST_MATH_BIG_CONSTANT(T, 53, -0.138652200349182596186e-4)
0635       };
0636       static const T Q[9] = {
0637          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
0638          BOOST_MATH_BIG_CONSTANT(T, 53, 1.97017214039061194971),
0639          BOOST_MATH_BIG_CONSTANT(T, 53, 1.86232465043073157508),
0640          BOOST_MATH_BIG_CONSTANT(T, 53, 1.09601437090337519977),
0641          BOOST_MATH_BIG_CONSTANT(T, 53, 0.438873285773088870812),
0642          BOOST_MATH_BIG_CONSTANT(T, 53, 0.122537731979686102756),
0643          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0233458478275769288159),
0644          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00278170769163303669021),
0645          BOOST_MATH_BIG_CONSTANT(T, 53, 0.000159150281166108755531)
0646       };
0647       T t = z / 5 - 3;
0648       result = Y + tools::evaluate_polynomial(P, t)
0649          / tools::evaluate_polynomial(Q, t);
0650       result *= exp(z) / z;
0651       result += z;
0652    }
0653    else if(z <= 40)
0654    {
0655       // Maximum Deviation Found:                     5.102e-18
0656       // Expected Error Term:                         5.101e-18
0657       // Max Error found at double precision =        Poly: 1.441088e-16   Cheb: 1.864792e-16
0658
0659
0660       static const T Y = 1.03937530517578125F;
0661       static const T P[9] = {
0662          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00356165148914447597995),
0663          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0229930320357982333406),
0664          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0449814350482277917716),
0665          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0453759383048193402336),
0666          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0272050837209380717069),
0667          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00994403059883350813295),
0668          BOOST_MATH_BIG_CONSTANT(T, 53, -0.00207592267812291726961),
0669          BOOST_MATH_BIG_CONSTANT(T, 53, -0.000192178045857733706044),
0670          BOOST_MATH_BIG_CONSTANT(T, 53, -0.113161784705911400295e-9)
0671       };
0672       static const T Q[9] = {
0673          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
0674          BOOST_MATH_BIG_CONSTANT(T, 53, 2.84354408840148561131),
0675          BOOST_MATH_BIG_CONSTANT(T, 53, 3.6599610090072393012),
0676          BOOST_MATH_BIG_CONSTANT(T, 53, 2.75088464344293083595),
0677          BOOST_MATH_BIG_CONSTANT(T, 53, 1.2985244073998398643),
0678          BOOST_MATH_BIG_CONSTANT(T, 53, 0.383213198510794507409),
0679          BOOST_MATH_BIG_CONSTANT(T, 53, 0.0651165455496281337831),
0680          BOOST_MATH_BIG_CONSTANT(T, 53, 0.00488071077519227853585)
0681       };
0682       T t = z / 10 - 3;
0683       result = Y + tools::evaluate_polynomial(P, t)
0684          / tools::evaluate_polynomial(Q, t);
0685       result *= exp(z) / z;
0686       result += z;
0687    }
0688    else
0689    {
0690       // Max Error found at double precision =        3.381886e-17
0691       static const T exp40 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 53, 2.35385266837019985407899910749034804508871617254555467236651e17));
0692       static const T Y= 1.013065338134765625F;
0693       static const T P[6] = {
0694          BOOST_MATH_BIG_CONSTANT(T, 53, -0.0130653381347656243849),
0695          BOOST_MATH_BIG_CONSTANT(T, 53, 0.19029710559486576682),
0696          BOOST_MATH_BIG_CONSTANT(T, 53, 94.7365094537197236011),
0697          BOOST_MATH_BIG_CONSTANT(T, 53, -2516.35323679844256203),
0698          BOOST_MATH_BIG_CONSTANT(T, 53, 18932.0850014925993025),
0699          BOOST_MATH_BIG_CONSTANT(T, 53, -38703.1431362056714134)
0700       };
0701       static const T Q[7] = {
0702          BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
0703          BOOST_MATH_BIG_CONSTANT(T, 53, 61.9733592849439884145),
0704          BOOST_MATH_BIG_CONSTANT(T, 53, -2354.56211323420194283),
0705          BOOST_MATH_BIG_CONSTANT(T, 53, 22329.1459489893079041),
0706          BOOST_MATH_BIG_CONSTANT(T, 53, -70126.245140396567133),
0707          BOOST_MATH_BIG_CONSTANT(T, 53, 54738.2833147775537106),
0708          BOOST_MATH_BIG_CONSTANT(T, 53, 8297.16296356518409347)
0709       };
0710       T t = 1 / z;
0711       result = Y + tools::evaluate_polynomial(P, t)
0712          / tools::evaluate_polynomial(Q, t);
0713       if(z < 41)
0714          result *= exp(z) / z;
0715       else
0716       {
0717          // Avoid premature overflow if we can:
0718          t = z - 40;
0719          if(t > tools::log_max_value<T>())
0720          {
0721             result = policies::raise_overflow_error<T>(function, nullptr, pol);
0722          }
0723          else
0724          {
0725             result *= exp(z - 40) / z;
0726             if(result > tools::max_value<T>() / exp40)
0727             {
0728                result = policies::raise_overflow_error<T>(function, nullptr, pol);
0729             }
0730             else
0731             {
0732                result *= exp40;
0733             }
0734          }
0735       }
0736       result += z;
0737    }
0738    return result;
0739 }
0740
0741 template <class T, class Policy>
0742 T expint_i_imp(T z, const Policy& pol, const std::integral_constant<int, 64>& tag)
0743 {
0744    BOOST_MATH_STD_USING
0745    static const char* function = "boost::math::expint<%1%>(%1%)";
0746    if(z < 0)
0747       return -expint_imp(1, T(-z), pol, tag);
0748    if(z == 0)
0749       return -policies::raise_overflow_error<T>(function, nullptr, pol);
0750
0751    T result;
0752
0753    if(z <= 6)
0754    {
0755       // Maximum Deviation Found:                     3.883e-21
0756       // Expected Error Term:                         3.883e-21
0757       // Max Error found at long double precision =   Poly: 3.344801e-19   Cheb: 4.989937e-19
0758
0759       static const T P[11] = {
0760          BOOST_MATH_BIG_CONSTANT(T, 64, 2.98677224343598593764),
0761          BOOST_MATH_BIG_CONSTANT(T, 64, 0.25891613550886736592),
0762          BOOST_MATH_BIG_CONSTANT(T, 64, 0.789323584998672832285),
0763          BOOST_MATH_BIG_CONSTANT(T, 64, 0.092432587824602399339),
0764          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0514236978728625906656),
0765          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00658477469745132977921),
0766          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00124914538197086254233),
0767          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000131429679565472408551),
0768          BOOST_MATH_BIG_CONSTANT(T, 64, 0.11293331317982763165e-4),
0769          BOOST_MATH_BIG_CONSTANT(T, 64, 0.629499283139417444244e-6),
0770          BOOST_MATH_BIG_CONSTANT(T, 64, 0.177833045143692498221e-7)
0771       };
0772       static const T Q[9] = {
0773          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
0774          BOOST_MATH_BIG_CONSTANT(T, 64, -1.20352377969742325748),
0775          BOOST_MATH_BIG_CONSTANT(T, 64, 0.66707904942606479811),
0776          BOOST_MATH_BIG_CONSTANT(T, 64, -0.223014531629140771914),
0777          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0493340022262908008636),
0778          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00741934273050807310677),
0779          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00074353567782087939294),
0780          BOOST_MATH_BIG_CONSTANT(T, 64, -0.455861727069603367656e-4),
0781          BOOST_MATH_BIG_CONSTANT(T, 64, 0.131515429329812837701e-5)
0782       };
0783
0784       static const T c1 = BOOST_MATH_BIG_CONSTANT(T, 64, 1677624236387711.0);
0785       static const T c2 = BOOST_MATH_BIG_CONSTANT(T, 64, 4503599627370496.0);
0786       static const T r1 = c1 / c2;
0787       static const T r2 = BOOST_MATH_BIG_CONSTANT(T, 64, 0.131401834143860282009280387409357165515556574352422001206362e-16);
0788       static const T r = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392));
0789       T t = (z / 3) - 1;
0790       result = tools::evaluate_polynomial(P, t)
0791          / tools::evaluate_polynomial(Q, t);
0792       t = (z - r1) - r2;
0793       result *= t;
0794       if(fabs(t) < T(0.1))
0795       {
0796          result += boost::math::log1p(t / r, pol);
0797       }
0798       else
0799       {
0800          result += log(z / r);
0801       }
0802    }
0803    else if (z <= 10)
0804    {
0805       // Maximum Deviation Found:                     2.622e-21
0806       // Expected Error Term:                         -2.622e-21
0807       // Max Error found at long double precision =   Poly: 1.208328e-20   Cheb: 1.073723e-20
0808
0809       static const T Y = 1.158985137939453125F;
0810       static const T P[9] = {
0811          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00139324086199409049399),
0812          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0345238388952337563247),
0813          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0382065278072592940767),
0814          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0156117003070560727392),
0815          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00383276012430495387102),
0816          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000697070540945496497992),
0817          BOOST_MATH_BIG_CONSTANT(T, 64, -0.877310384591205930343e-4),
0818          BOOST_MATH_BIG_CONSTANT(T, 64, -0.623067256376494930067e-5),
0819          BOOST_MATH_BIG_CONSTANT(T, 64, -0.377246883283337141444e-6)
0820       };
0821       static const T Q[10] = {
0822          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
0823          BOOST_MATH_BIG_CONSTANT(T, 64, 1.08073635708902053767),
0824          BOOST_MATH_BIG_CONSTANT(T, 64, 0.553681133533942532909),
0825          BOOST_MATH_BIG_CONSTANT(T, 64, 0.176763647137553797451),
0826          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0387891748253869928121),
0827          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0060603004848394727017),
0828          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000670519492939992806051),
0829          BOOST_MATH_BIG_CONSTANT(T, 64, 0.4947357050100855646e-4),
0830          BOOST_MATH_BIG_CONSTANT(T, 64, 0.204339282037446434827e-5),
0831          BOOST_MATH_BIG_CONSTANT(T, 64, 0.146951181174930425744e-7)
0832       };
0833       T t = z / 2 - 4;
0834       result = Y + tools::evaluate_polynomial(P, t)
0835          / tools::evaluate_polynomial(Q, t);
0836       result *= exp(z) / z;
0837       result += z;
0838    }
0839    else if(z <= 20)
0840    {
0841       // Maximum Deviation Found:                     3.220e-20
0842       // Expected Error Term:                         3.220e-20
0843       // Max Error found at long double precision =   Poly: 7.696841e-20   Cheb: 6.205163e-20
0844
0845
0846       static const T Y = 1.0869731903076171875F;
0847       static const T P[10] = {
0848          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00893891094356946995368),
0849          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0487562980088748775943),
0850          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0670568657950041926085),
0851          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0509577352851442932713),
0852          BOOST_MATH_BIG_CONSTANT(T, 64, -0.02551800927409034206),
0853          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00892913759760086687083),
0854          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00224469630207344379888),
0855          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000392477245911296982776),
0856          BOOST_MATH_BIG_CONSTANT(T, 64, -0.44424044184395578775e-4),
0857          BOOST_MATH_BIG_CONSTANT(T, 64, -0.252788029251437017959e-5)
0858       };
0859       static const T Q[10] = {
0860          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
0861          BOOST_MATH_BIG_CONSTANT(T, 64, 2.00323265503572414261),
0862          BOOST_MATH_BIG_CONSTANT(T, 64, 1.94688958187256383178),
0863          BOOST_MATH_BIG_CONSTANT(T, 64, 1.19733638134417472296),
0864          BOOST_MATH_BIG_CONSTANT(T, 64, 0.513137726038353385661),
0865          BOOST_MATH_BIG_CONSTANT(T, 64, 0.159135395578007264547),
0866          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0358233587351620919881),
0867          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0056716655597009417875),
0868          BOOST_MATH_BIG_CONSTANT(T, 64, 0.000577048986213535829925),
0869          BOOST_MATH_BIG_CONSTANT(T, 64, 0.290976943033493216793e-4)
0870       };
0871       T t = z / 5 - 3;
0872       result = Y + tools::evaluate_polynomial(P, t)
0873          / tools::evaluate_polynomial(Q, t);
0874       result *= exp(z) / z;
0875       result += z;
0876    }
0877    else if(z <= 40)
0878    {
0879       // Maximum Deviation Found:                     2.940e-21
0880       // Expected Error Term:                         -2.938e-21
0881       // Max Error found at long double precision =   Poly: 3.419893e-19   Cheb: 3.359874e-19
0882
0883       static const T Y = 1.03937530517578125F;
0884       static const T P[12] = {
0885          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00356165148914447278177),
0886          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0240235006148610849678),
0887          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0516699967278057976119),
0888          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0586603078706856245674),
0889          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0409960120868776180825),
0890          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0185485073689590665153),
0891          BOOST_MATH_BIG_CONSTANT(T, 64, -0.00537842101034123222417),
0892          BOOST_MATH_BIG_CONSTANT(T, 64, -0.000920988084778273760609),
0893          BOOST_MATH_BIG_CONSTANT(T, 64, -0.716742618812210980263e-4),
0894          BOOST_MATH_BIG_CONSTANT(T, 64, -0.504623302166487346677e-9),
0895          BOOST_MATH_BIG_CONSTANT(T, 64, 0.712662196671896837736e-10),
0896          BOOST_MATH_BIG_CONSTANT(T, 64, -0.533769629702262072175e-11)
0897       };
0898       static const T Q[9] = {
0899          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
0900          BOOST_MATH_BIG_CONSTANT(T, 64, 3.13286733695729715455),
0901          BOOST_MATH_BIG_CONSTANT(T, 64, 4.49281223045653491929),
0902          BOOST_MATH_BIG_CONSTANT(T, 64, 3.84900294427622911374),
0903          BOOST_MATH_BIG_CONSTANT(T, 64, 2.15205199043580378211),
0904          BOOST_MATH_BIG_CONSTANT(T, 64, 0.802912186540269232424),
0905          BOOST_MATH_BIG_CONSTANT(T, 64, 0.194793170017818925388),
0906          BOOST_MATH_BIG_CONSTANT(T, 64, 0.0280128013584653182994),
0907          BOOST_MATH_BIG_CONSTANT(T, 64, 0.00182034930799902922549)
0908       };
0909       T t = z / 10 - 3;
0910       result = Y + tools::evaluate_polynomial(P, t)
0911          / tools::evaluate_polynomial(Q, t);
0912       BOOST_MATH_INSTRUMENT_VARIABLE(result)
0913       result *= exp(z) / z;
0914       BOOST_MATH_INSTRUMENT_VARIABLE(result)
0915       result += z;
0916       BOOST_MATH_INSTRUMENT_VARIABLE(result)
0917    }
0918    else
0919    {
0920       // Maximum Deviation Found:                     3.536e-20
0921       // Max Error found at long double precision =   Poly: 1.310671e-19   Cheb: 8.630943e-11
0922
0923       static const T exp40 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.35385266837019985407899910749034804508871617254555467236651e17));
0924       static const T Y= 1.013065338134765625F;
0925       static const T P[9] = {
0926          BOOST_MATH_BIG_CONSTANT(T, 64, -0.0130653381347656250004),
0927          BOOST_MATH_BIG_CONSTANT(T, 64, 0.644487780349757303739),
0928          BOOST_MATH_BIG_CONSTANT(T, 64, 143.995670348227433964),
0929          BOOST_MATH_BIG_CONSTANT(T, 64, -13918.9322758014173709),
0930          BOOST_MATH_BIG_CONSTANT(T, 64, 476260.975133624194484),
0931          BOOST_MATH_BIG_CONSTANT(T, 64, -7437102.15135982802122),
0932          BOOST_MATH_BIG_CONSTANT(T, 64, 53732298.8764767916542),
0933          BOOST_MATH_BIG_CONSTANT(T, 64, -160695051.957997452509),
0934          BOOST_MATH_BIG_CONSTANT(T, 64, 137839271.592778020028)
0935       };
0936       static const T Q[9] = {
0937          BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
0938          BOOST_MATH_BIG_CONSTANT(T, 64, 27.2103343964943718802),
0939          BOOST_MATH_BIG_CONSTANT(T, 64, -8785.48528692879413676),
0940          BOOST_MATH_BIG_CONSTANT(T, 64, 397530.290000322626766),
0941          BOOST_MATH_BIG_CONSTANT(T, 64, -7356441.34957799368252),
0942          BOOST_MATH_BIG_CONSTANT(T, 64, 63050914.5343400957524),
0943          BOOST_MATH_BIG_CONSTANT(T, 64, -246143779.638307701369),
0944          BOOST_MATH_BIG_CONSTANT(T, 64, 384647824.678554961174),
0945          BOOST_MATH_BIG_CONSTANT(T, 64, -166288297.874583961493)
0946       };
0947       T t = 1 / z;
0948       result = Y + tools::evaluate_polynomial(P, t)
0949          / tools::evaluate_polynomial(Q, t);
0950       if(z < 41)
0951          result *= exp(z) / z;
0952       else
0953       {
0954          // Avoid premature overflow if we can:
0955          t = z - 40;
0956          if(t > tools::log_max_value<T>())
0957          {
0958             result = policies::raise_overflow_error<T>(function, nullptr, pol);
0959          }
0960          else
0961          {
0962             result *= exp(z - 40) / z;
0963             if(result > tools::max_value<T>() / exp40)
0964             {
0965                result = policies::raise_overflow_error<T>(function, nullptr, pol);
0966             }
0967             else
0968             {
0969                result *= exp40;
0970             }
0971          }
0972       }
0973       result += z;
0974    }
0975    return result;
0976 }
0977
0978 template <class T, class Policy>
0979 void expint_i_imp_113a(T& result, const T& z, const Policy& pol)
0980 {
0981    BOOST_MATH_STD_USING
0982    // Maximum Deviation Found:                     1.230e-36
0983    // Expected Error Term:                         -1.230e-36
0984    // Max Error found at long double precision =   Poly: 4.355299e-34   Cheb: 7.512581e-34
0985
0986
0987    static const T P[15] = {
0988       BOOST_MATH_BIG_CONSTANT(T, 113, 2.98677224343598593765287235997328555),
0989       BOOST_MATH_BIG_CONSTANT(T, 113, -0.333256034674702967028780537349334037),
0990       BOOST_MATH_BIG_CONSTANT(T, 113, 0.851831522798101228384971644036708463),
0991       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0657854833494646206186773614110374948),
0992       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0630065662557284456000060708977935073),
0993       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00311759191425309373327784154659649232),
0994       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00176213568201493949664478471656026771),
0995       BOOST_MATH_BIG_CONSTANT(T, 113, -0.491548660404172089488535218163952295e-4),
0996       BOOST_MATH_BIG_CONSTANT(T, 113, 0.207764227621061706075562107748176592e-4),
0997       BOOST_MATH_BIG_CONSTANT(T, 113, -0.225445398156913584846374273379402765e-6),
0998       BOOST_MATH_BIG_CONSTANT(T, 113, 0.996939977231410319761273881672601592e-7),
0999       BOOST_MATH_BIG_CONSTANT(T, 113, 0.212546902052178643330520878928100847e-9),
1000       BOOST_MATH_BIG_CONSTANT(T, 113, 0.154646053060262871360159325115980023e-9),
1001       BOOST_MATH_BIG_CONSTANT(T, 113, 0.143971277122049197323415503594302307e-11),
1002       BOOST_MATH_BIG_CONSTANT(T, 113, 0.306243138978114692252817805327426657e-13)
1003    };
1004    static const T Q[15] = {
1005       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1006       BOOST_MATH_BIG_CONSTANT(T, 113, -1.40178870313943798705491944989231793),
1007       BOOST_MATH_BIG_CONSTANT(T, 113, 0.943810968269701047641218856758605284),
1008       BOOST_MATH_BIG_CONSTANT(T, 113, -0.405026631534345064600850391026113165),
1009       BOOST_MATH_BIG_CONSTANT(T, 113, 0.123924153524614086482627660399122762),
1010       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0286364505373369439591132549624317707),
1011       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00516148845910606985396596845494015963),
1012       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000738330799456364820380739850924783649),
1013       BOOST_MATH_BIG_CONSTANT(T, 113, 0.843737760991856114061953265870882637e-4),
1014       BOOST_MATH_BIG_CONSTANT(T, 113, -0.767957673431982543213661388914587589e-5),
1015       BOOST_MATH_BIG_CONSTANT(T, 113, 0.549136847313854595809952100614840031e-6),
1016       BOOST_MATH_BIG_CONSTANT(T, 113, -0.299801381513743676764008325949325404e-7),
1017       BOOST_MATH_BIG_CONSTANT(T, 113, 0.118419479055346106118129130945423483e-8),
1018       BOOST_MATH_BIG_CONSTANT(T, 113, -0.30372295663095470359211949045344607e-10),
1019       BOOST_MATH_BIG_CONSTANT(T, 113, 0.382742953753485333207877784720070523e-12)
1020    };
1021
1022    static const T c1 = BOOST_MATH_BIG_CONSTANT(T, 113, 1677624236387711.0);
1023    static const T c2 = BOOST_MATH_BIG_CONSTANT(T, 113, 4503599627370496.0);
1024    static const T c3 = BOOST_MATH_BIG_CONSTANT(T, 113, 266514582277687.0);
1025    static const T c4 = BOOST_MATH_BIG_CONSTANT(T, 113, 4503599627370496.0);
1026    static const T c5 = BOOST_MATH_BIG_CONSTANT(T, 113, 4503599627370496.0);
1027    static const T r1 = c1 / c2;
1028    static const T r2 = c3 / c4 / c5;
1029    static const T r3 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.283806480836357377069325311780969887585024578164571984232357e-31));
1030    static const T r = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.372507410781366634461991866580119133535689497771654051555657435242200120636201854384926049951548942392));
1031    T t = (z / 3) - 1;
1032    result = tools::evaluate_polynomial(P, t)
1033       / tools::evaluate_polynomial(Q, t);
1034    t = ((z - r1) - r2) - r3;
1035    result *= t;
1036    if(fabs(t) < 0.1)
1037    {
1038       result += boost::math::log1p(t / r, pol);
1039    }
1040    else
1041    {
1042       result += log(z / r);
1043    }
1044 }
1045
1046 template <class T>
1047 void expint_i_113b(T& result, const T& z)
1048 {
1049    BOOST_MATH_STD_USING
1050    // Maximum Deviation Found:                     7.779e-36
1051    // Expected Error Term:                         -7.779e-36
1052    // Max Error found at long double precision =   Poly: 2.576723e-35   Cheb: 1.236001e-34
1053
1054    static const T Y = 1.158985137939453125F;
1055    static const T P[15] = {
1056       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139324086199409049282472239613554817),
1057       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0338173111691991289178779840307998955),
1058       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0555972290794371306259684845277620556),
1059       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0378677976003456171563136909186202177),
1060       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0152221583517528358782902783914356667),
1061       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00428283334203873035104248217403126905),
1062       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000922782631491644846511553601323435286),
1063       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000155513428088853161562660696055496696),
1064       BOOST_MATH_BIG_CONSTANT(T, 113, -0.205756580255359882813545261519317096e-4),
1065       BOOST_MATH_BIG_CONSTANT(T, 113, -0.220327406578552089820753181821115181e-5),
1066       BOOST_MATH_BIG_CONSTANT(T, 113, -0.189483157545587592043421445645377439e-6),
1067       BOOST_MATH_BIG_CONSTANT(T, 113, -0.122426571518570587750898968123803867e-7),
1068       BOOST_MATH_BIG_CONSTANT(T, 113, -0.635187358949437991465353268374523944e-9),
1069       BOOST_MATH_BIG_CONSTANT(T, 113, -0.203015132965870311935118337194860863e-10),
1070       BOOST_MATH_BIG_CONSTANT(T, 113, -0.384276705503357655108096065452950822e-12)
1071    };
1072    static const T Q[15] = {
1073       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1074       BOOST_MATH_BIG_CONSTANT(T, 113, 1.58784732785354597996617046880946257),
1075       BOOST_MATH_BIG_CONSTANT(T, 113, 1.18550755302279446339364262338114098),
1076       BOOST_MATH_BIG_CONSTANT(T, 113, 0.55598993549661368604527040349702836),
1077       BOOST_MATH_BIG_CONSTANT(T, 113, 0.184290888380564236919107835030984453),
1078       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0459658051803613282360464632326866113),
1079       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0089505064268613225167835599456014705),
1080       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139042673882987693424772855926289077),
1081       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000174210708041584097450805790176479012),
1082       BOOST_MATH_BIG_CONSTANT(T, 113, 0.176324034009707558089086875136647376e-4),
1083       BOOST_MATH_BIG_CONSTANT(T, 113, 0.142935845999505649273084545313710581e-5),
1084       BOOST_MATH_BIG_CONSTANT(T, 113, 0.907502324487057260675816233312747784e-7),
1085       BOOST_MATH_BIG_CONSTANT(T, 113, 0.431044337808893270797934621235918418e-8),
1086       BOOST_MATH_BIG_CONSTANT(T, 113, 0.139007266881450521776529705677086902e-9),
1087       BOOST_MATH_BIG_CONSTANT(T, 113, 0.234715286125516430792452741830364672e-11)
1088    };
1089    T t = z / 2 - 4;
1090    result = Y + tools::evaluate_polynomial(P, t)
1091       / tools::evaluate_polynomial(Q, t);
1092    result *= exp(z) / z;
1093    result += z;
1094 }
1095
1096 template <class T>
1097 void expint_i_113c(T& result, const T& z)
1098 {
1099    BOOST_MATH_STD_USING
1100    // Maximum Deviation Found:                     1.082e-34
1101    // Expected Error Term:                         1.080e-34
1102    // Max Error found at long double precision =   Poly: 1.958294e-34   Cheb: 2.472261e-34
1103
1104
1105    static const T Y = 1.091579437255859375F;
1106    static const T P[17] = {
1107       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00685089599550151282724924894258520532),
1108       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0443313550253580053324487059748497467),
1109       BOOST_MATH_BIG_CONSTANT(T, 113, -0.071538561252424027443296958795814874),
1110       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0622923153354102682285444067843300583),
1111       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0361631270264607478205393775461208794),
1112       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0153192826839624850298106509601033261),
1113       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00496967904961260031539602977748408242),
1114       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126989079663425780800919171538920589),
1115       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000258933143097125199914724875206326698),
1116       BOOST_MATH_BIG_CONSTANT(T, 113, -0.422110326689204794443002330541441956e-4),
1117       BOOST_MATH_BIG_CONSTANT(T, 113, -0.546004547590412661451073996127115221e-5),
1118       BOOST_MATH_BIG_CONSTANT(T, 113, -0.546775260262202177131068692199272241e-6),
1119       BOOST_MATH_BIG_CONSTANT(T, 113, -0.404157632825805803833379568956559215e-7),
1120       BOOST_MATH_BIG_CONSTANT(T, 113, -0.200612596196561323832327013027419284e-8),
1121       BOOST_MATH_BIG_CONSTANT(T, 113, -0.502538501472133913417609379765434153e-10),
1122       BOOST_MATH_BIG_CONSTANT(T, 113, -0.326283053716799774936661568391296584e-13),
1123       BOOST_MATH_BIG_CONSTANT(T, 113, 0.869226483473172853557775877908693647e-15)
1124    };
1125    static const T Q[15] = {
1126       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1127       BOOST_MATH_BIG_CONSTANT(T, 113, 2.23227220874479061894038229141871087),
1128       BOOST_MATH_BIG_CONSTANT(T, 113, 2.40221000361027971895657505660959863),
1129       BOOST_MATH_BIG_CONSTANT(T, 113, 1.65476320985936174728238416007084214),
1130       BOOST_MATH_BIG_CONSTANT(T, 113, 0.816828602963895720369875535001248227),
1131       BOOST_MATH_BIG_CONSTANT(T, 113, 0.306337922909446903672123418670921066),
1132       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0902400121654409267774593230720600752),
1133       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0212708882169429206498765100993228086),
1134       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00404442626252467471957713495828165491),
1135       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0006195601618842253612635241404054589),
1136       BOOST_MATH_BIG_CONSTANT(T, 113, 0.755930932686543009521454653994321843e-4),
1137       BOOST_MATH_BIG_CONSTANT(T, 113, 0.716004532773778954193609582677482803e-5),
1138       BOOST_MATH_BIG_CONSTANT(T, 113, 0.500881663076471627699290821742924233e-6),
1139       BOOST_MATH_BIG_CONSTANT(T, 113, 0.233593219218823384508105943657387644e-7),
1140       BOOST_MATH_BIG_CONSTANT(T, 113, 0.554900353169148897444104962034267682e-9)
1141    };
1142    T t = z / 4 - 3.5;
1143    result = Y + tools::evaluate_polynomial(P, t)
1144       / tools::evaluate_polynomial(Q, t);
1145    result *= exp(z) / z;
1146    result += z;
1147 }
1148
1149 template <class T>
1150 void expint_i_113d(T& result, const T& z)
1151 {
1152    BOOST_MATH_STD_USING
1153    // Maximum Deviation Found:                     3.163e-35
1154    // Expected Error Term:                         3.163e-35
1155    // Max Error found at long double precision =   Poly: 4.158110e-35   Cheb: 5.385532e-35
1156
1157    static const T Y = 1.051731109619140625F;
1158    static const T P[14] = {
1159       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00144552494420652573815404828020593565),
1160       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0126747451594545338365684731262912741),
1161       BOOST_MATH_BIG_CONSTANT(T, 113, -0.01757394877502366717526779263438073),
1162       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0126838952395506921945756139424722588),
1163       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0060045057928894974954756789352443522),
1164       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00205349237147226126653803455793107903),
1165       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000532606040579654887676082220195624207),
1166       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000107344687098019891474772069139014662),
1167       BOOST_MATH_BIG_CONSTANT(T, 113, -0.169536802705805811859089949943435152e-4),
1168       BOOST_MATH_BIG_CONSTANT(T, 113, -0.20863311729206543881826553010120078e-5),
1169       BOOST_MATH_BIG_CONSTANT(T, 113, -0.195670358542116256713560296776654385e-6),
1170       BOOST_MATH_BIG_CONSTANT(T, 113, -0.133291168587253145439184028259772437e-7),
1171       BOOST_MATH_BIG_CONSTANT(T, 113, -0.595500337089495614285777067722823397e-9),
1172       BOOST_MATH_BIG_CONSTANT(T, 113, -0.133141358866324100955927979606981328e-10)
1173    };
1174    static const T Q[14] = {
1175       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1176       BOOST_MATH_BIG_CONSTANT(T, 113, 1.72490783907582654629537013560044682),
1177       BOOST_MATH_BIG_CONSTANT(T, 113, 1.44524329516800613088375685659759765),
1178       BOOST_MATH_BIG_CONSTANT(T, 113, 0.778241785539308257585068744978050181),
1179       BOOST_MATH_BIG_CONSTANT(T, 113, 0.300520486589206605184097270225725584),
1180       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0879346899691339661394537806057953957),
1181       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0200802415843802892793583043470125006),
1182       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00362842049172586254520256100538273214),
1183       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000519731362862955132062751246769469957),
1184       BOOST_MATH_BIG_CONSTANT(T, 113, 0.584092147914050999895178697392282665e-4),
1185       BOOST_MATH_BIG_CONSTANT(T, 113, 0.501851497707855358002773398333542337e-5),
1186       BOOST_MATH_BIG_CONSTANT(T, 113, 0.313085677467921096644895738538865537e-6),
1187       BOOST_MATH_BIG_CONSTANT(T, 113, 0.127552010539733113371132321521204458e-7),
1188       BOOST_MATH_BIG_CONSTANT(T, 113, 0.25737310826983451144405899970774587e-9)
1189    };
1190    T t = z / 4 - 5.5;
1191    result = Y + tools::evaluate_polynomial(P, t)
1192       / tools::evaluate_polynomial(Q, t);
1193    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1194    result *= exp(z) / z;
1195    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1196    result += z;
1197    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1198 }
1199
1200 template <class T>
1201 void expint_i_113e(T& result, const T& z)
1202 {
1203    BOOST_MATH_STD_USING
1204    // Maximum Deviation Found:                     7.972e-36
1205    // Expected Error Term:                         7.962e-36
1206    // Max Error found at long double precision =   Poly: 1.711721e-34   Cheb: 3.100018e-34
1207
1208    static const T Y = 1.032726287841796875F;
1209    static const T P[15] = {
1210       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00141056919297307534690895009969373233),
1211       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0123384175302540291339020257071411437),
1212       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0298127270706864057791526083667396115),
1213       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0390686759471630584626293670260768098),
1214       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0338226792912607409822059922949035589),
1215       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0211659736179834946452561197559654582),
1216       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0100428887460879377373158821400070313),
1217       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00370717396015165148484022792801682932),
1218       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0010768667551001624764329000496561659),
1219       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000246127328761027039347584096573123531),
1220       BOOST_MATH_BIG_CONSTANT(T, 113, -0.437318110527818613580613051861991198e-4),
1221       BOOST_MATH_BIG_CONSTANT(T, 113, -0.587532682329299591501065482317771497e-5),
1222       BOOST_MATH_BIG_CONSTANT(T, 113, -0.565697065670893984610852937110819467e-6),
1223       BOOST_MATH_BIG_CONSTANT(T, 113, -0.350233957364028523971768887437839573e-7),
1224       BOOST_MATH_BIG_CONSTANT(T, 113, -0.105428907085424234504608142258423505e-8)
1225    };
1226    static const T Q[16] = {
1227       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1228       BOOST_MATH_BIG_CONSTANT(T, 113, 3.17261315255467581204685605414005525),
1229       BOOST_MATH_BIG_CONSTANT(T, 113, 4.85267952971640525245338392887217426),
1230       BOOST_MATH_BIG_CONSTANT(T, 113, 4.74341914912439861451492872946725151),
1231       BOOST_MATH_BIG_CONSTANT(T, 113, 3.31108463283559911602405970817931801),
1232       BOOST_MATH_BIG_CONSTANT(T, 113, 1.74657006336994649386607925179848899),
1233       BOOST_MATH_BIG_CONSTANT(T, 113, 0.718255607416072737965933040353653244),
1234       BOOST_MATH_BIG_CONSTANT(T, 113, 0.234037553177354542791975767960643864),
1235       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0607470145906491602476833515412605389),
1236       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0125048143774226921434854172947548724),
1237       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00201034366420433762935768458656609163),
1238       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000244823338417452367656368849303165721),
1239       BOOST_MATH_BIG_CONSTANT(T, 113, 0.213511655166983177960471085462540807e-4),
1240       BOOST_MATH_BIG_CONSTANT(T, 113, 0.119323998465870686327170541547982932e-5),
1241       BOOST_MATH_BIG_CONSTANT(T, 113, 0.322153582559488797803027773591727565e-7),
1242       BOOST_MATH_BIG_CONSTANT(T, 113, -0.161635525318683508633792845159942312e-16)
1243    };
1244    T t = z / 8 - 4.25;
1245    result = Y + tools::evaluate_polynomial(P, t)
1246       / tools::evaluate_polynomial(Q, t);
1247    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1248    result *= exp(z) / z;
1249    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1250    result += z;
1251    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1252 }
1253
1254 template <class T>
1255 void expint_i_113f(T& result, const T& z)
1256 {
1257    BOOST_MATH_STD_USING
1258    // Maximum Deviation Found:                     4.469e-36
1259    // Expected Error Term:                         4.468e-36
1260    // Max Error found at long double precision =   Poly: 1.288958e-35   Cheb: 2.304586e-35
1261
1262    static const T Y = 1.0216197967529296875F;
1263    static const T P[12] = {
1264       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000322999116096627043476023926572650045),
1265       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00385606067447365187909164609294113346),
1266       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00686514524727568176735949971985244415),
1267       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00606260649593050194602676772589601799),
1268       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00334382362017147544335054575436194357),
1269       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126108534260253075708625583630318043),
1270       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000337881489347846058951220431209276776),
1271       BOOST_MATH_BIG_CONSTANT(T, 113, -0.648480902304640018785370650254018022e-4),
1272       BOOST_MATH_BIG_CONSTANT(T, 113, -0.87652644082970492211455290209092766e-5),
1273       BOOST_MATH_BIG_CONSTANT(T, 113, -0.794712243338068631557849449519994144e-6),
1274       BOOST_MATH_BIG_CONSTANT(T, 113, -0.434084023639508143975983454830954835e-7),
1275       BOOST_MATH_BIG_CONSTANT(T, 113, -0.107839681938752337160494412638656696e-8)
1276    };
1277    static const T Q[12] = {
1278       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1279       BOOST_MATH_BIG_CONSTANT(T, 113, 2.09913805456661084097134805151524958),
1280       BOOST_MATH_BIG_CONSTANT(T, 113, 2.07041755535439919593503171320431849),
1281       BOOST_MATH_BIG_CONSTANT(T, 113, 1.26406517226052371320416108604874734),
1282       BOOST_MATH_BIG_CONSTANT(T, 113, 0.529689923703770353961553223973435569),
1283       BOOST_MATH_BIG_CONSTANT(T, 113, 0.159578150879536711042269658656115746),
1284       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0351720877642000691155202082629857131),
1285       BOOST_MATH_BIG_CONSTANT(T, 113, 0.00565313621289648752407123620997063122),
1286       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000646920278540515480093843570291218295),
1287       BOOST_MATH_BIG_CONSTANT(T, 113, 0.499904084850091676776993523323213591e-4),
1288       BOOST_MATH_BIG_CONSTANT(T, 113, 0.233740058688179614344680531486267142e-5),
1289       BOOST_MATH_BIG_CONSTANT(T, 113, 0.498800627828842754845418576305379469e-7)
1290    };
1291    T t = z / 7 - 7;
1292    result = Y + tools::evaluate_polynomial(P, t)
1293       / tools::evaluate_polynomial(Q, t);
1294    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1295    result *= exp(z) / z;
1296    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1297    result += z;
1298    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1299 }
1300
1301 template <class T>
1302 void expint_i_113g(T& result, const T& z)
1303 {
1304    BOOST_MATH_STD_USING
1305    // Maximum Deviation Found:                     5.588e-35
1306    // Expected Error Term:                         -5.566e-35
1307    // Max Error found at long double precision =   Poly: 9.976345e-35   Cheb: 8.358865e-35
1308
1309    static const T Y = 1.015148162841796875F;
1310    static const T P[11] = {
1311       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000435714784725086961464589957142615216),
1312       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00432114324353830636009453048419094314),
1313       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0100740363285526177522819204820582424),
1314       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0116744115827059174392383504427640362),
1315       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00816145387784261141360062395898644652),
1316       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00371380272673500791322744465394211508),
1317       BOOST_MATH_BIG_CONSTANT(T, 113, -0.00112958263488611536502153195005736563),
1318       BOOST_MATH_BIG_CONSTANT(T, 113, -0.000228316462389404645183269923754256664),
1319       BOOST_MATH_BIG_CONSTANT(T, 113, -0.29462181955852860250359064291292577e-4),
1320       BOOST_MATH_BIG_CONSTANT(T, 113, -0.21972450610957417963227028788460299e-5),
1321       BOOST_MATH_BIG_CONSTANT(T, 113, -0.720558173805289167524715527536874694e-7)
1322    };
1323    static const T Q[11] = {
1324       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1325       BOOST_MATH_BIG_CONSTANT(T, 113, 2.95918362458402597039366979529287095),
1326       BOOST_MATH_BIG_CONSTANT(T, 113, 3.96472247520659077944638411856748924),
1327       BOOST_MATH_BIG_CONSTANT(T, 113, 3.15563251550528513747923714884142131),
1328       BOOST_MATH_BIG_CONSTANT(T, 113, 1.64674612007093983894215359287448334),
1329       BOOST_MATH_BIG_CONSTANT(T, 113, 0.58695020129846594405856226787156424),
1330       BOOST_MATH_BIG_CONSTANT(T, 113, 0.144358385319329396231755457772362793),
1331       BOOST_MATH_BIG_CONSTANT(T, 113, 0.024146911506411684815134916238348063),
1332       BOOST_MATH_BIG_CONSTANT(T, 113, 0.0026257132337460784266874572001650153),
1333       BOOST_MATH_BIG_CONSTANT(T, 113, 0.000167479843750859222348869769094711093),
1334       BOOST_MATH_BIG_CONSTANT(T, 113, 0.475673638665358075556452220192497036e-5)
1335    };
1336    T t = z / 14 - 5;
1337    result = Y + tools::evaluate_polynomial(P, t)
1338       / tools::evaluate_polynomial(Q, t);
1339    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1340    result *= exp(z) / z;
1341    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1342    result += z;
1343    BOOST_MATH_INSTRUMENT_VARIABLE(result)
1344 }
1345
1346 template <class T>
1347 void expint_i_113h(T& result, const T& z)
1348 {
1349    BOOST_MATH_STD_USING
1350    // Maximum Deviation Found:                     4.448e-36
1351    // Expected Error Term:                         4.445e-36
1352    // Max Error found at long double precision =   Poly: 2.058532e-35   Cheb: 2.165465e-27
1353
1354    static const T Y= 1.00849151611328125F;
1355    static const T P[9] = {
1356       BOOST_MATH_BIG_CONSTANT(T, 113, -0.0084915161132812500000001440233607358),
1357       BOOST_MATH_BIG_CONSTANT(T, 113, 1.84479378737716028341394223076147872),
1358       BOOST_MATH_BIG_CONSTANT(T, 113, -130.431146923726715674081563022115568),
1359       BOOST_MATH_BIG_CONSTANT(T, 113, 4336.26945491571504885214176203512015),
1360       BOOST_MATH_BIG_CONSTANT(T, 113, -76279.0031974974730095170437591004177),
1361       BOOST_MATH_BIG_CONSTANT(T, 113, 729577.956271997673695191455111727774),
1362       BOOST_MATH_BIG_CONSTANT(T, 113, -3661928.69330208734947103004900349266),
1363       BOOST_MATH_BIG_CONSTANT(T, 113, 8570600.041606912735872059184527855),
1364       BOOST_MATH_BIG_CONSTANT(T, 113, -6758379.93672362080947905580906028645)
1365    };
1366    static const T Q[10] = {
1367       BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1368       BOOST_MATH_BIG_CONSTANT(T, 113, -99.4868026047611434569541483506091713),
1369       BOOST_MATH_BIG_CONSTANT(T, 113, 3879.67753690517114249705089803055473),
1370       BOOST_MATH_BIG_CONSTANT(T, 113, -76495.82413252517165830203774900806),
1371       BOOST_MATH_BIG_CONSTANT(T, 113, 820773.726408311894342553758526282667),
1372       BOOST_MATH_BIG_CONSTANT(T, 113, -4803087.64956923577571031564909646579),
1373       BOOST_MATH_BIG_CONSTANT(T, 113, 14521246.227703545012713173740895477),
1374       BOOST_MATH_BIG_CONSTANT(T, 113, -19762752.0196769712258527849159393044),
1375       BOOST_MATH_BIG_CONSTANT(T, 113, 8354144.67882768405803322344185185517),
1376       BOOST_MATH_BIG_CONSTANT(T, 113, 355076.853106511136734454134915432571)
1377    };
1378    T t = 1 / z;
1379    result = Y + tools::evaluate_polynomial(P, t)
1380       / tools::evaluate_polynomial(Q, t);
1381    result *= exp(z) / z;
1382    result += z;
1383 }
1384
1385 template <class T, class Policy>
1386 T expint_i_imp(T z, const Policy& pol, const std::integral_constant<int, 113>& tag)
1387 {
1388    BOOST_MATH_STD_USING
1389    static const char* function = "boost::math::expint<%1%>(%1%)";
1390    if(z < 0)
1391       return -expint_imp(1, T(-z), pol, tag);
1392    if(z == 0)
1393       return -policies::raise_overflow_error<T>(function, nullptr, pol);
1394
1395    T result;
1396
1397    if(z <= 6)
1398    {
1399       expint_i_imp_113a(result, z, pol);
1400    }
1401    else if (z <= 10)
1402    {
1403       expint_i_113b(result, z);
1404    }
1405    else if(z <= 18)
1406    {
1407       expint_i_113c(result, z);
1408    }
1409    else if(z <= 26)
1410    {
1411       expint_i_113d(result, z);
1412    }
1413    else if(z <= 42)
1414    {
1415       expint_i_113e(result, z);
1416    }
1417    else if(z <= 56)
1418    {
1419       expint_i_113f(result, z);
1420    }
1421    else if(z <= 84)
1422    {
1423       expint_i_113g(result, z);
1424    }
1425    else if(z <= 210)
1426    {
1427       expint_i_113h(result, z);
1428    }
1429    else // z > 210
1430    {
1431       // Maximum Deviation Found:                     3.963e-37
1432       // Expected Error Term:                         3.963e-37
1433       // Max Error found at long double precision =   Poly: 1.248049e-36   Cheb: 2.843486e-29
1434
1435       static const T exp40 = static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2.35385266837019985407899910749034804508871617254555467236651e17));
1436       static const T Y= 1.00252532958984375F;
1437       static const T P[8] = {
1438          BOOST_MATH_BIG_CONSTANT(T, 113, -0.00252532958984375000000000000000000085),
1439          BOOST_MATH_BIG_CONSTANT(T, 113, 1.16591386866059087390621952073890359),
1440          BOOST_MATH_BIG_CONSTANT(T, 113, -67.8483431314018462417456828499277579),
1441          BOOST_MATH_BIG_CONSTANT(T, 113, 1567.68688154683822956359536287575892),
1442          BOOST_MATH_BIG_CONSTANT(T, 113, -17335.4683325819116482498725687644986),
1443          BOOST_MATH_BIG_CONSTANT(T, 113, 93632.6567462673524739954389166550069),
1444          BOOST_MATH_BIG_CONSTANT(T, 113, -225025.189335919133214440347510936787),
1445          BOOST_MATH_BIG_CONSTANT(T, 113, 175864.614717440010942804684741336853)
1446       };
1447       static const T Q[9] = {
1448          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
1449          BOOST_MATH_BIG_CONSTANT(T, 113, -65.6998869881600212224652719706425129),
1450          BOOST_MATH_BIG_CONSTANT(T, 113, 1642.73850032324014781607859416890077),
1451          BOOST_MATH_BIG_CONSTANT(T, 113, -19937.2610222467322481947237312818575),
1452          BOOST_MATH_BIG_CONSTANT(T, 113, 124136.267326632742667972126625064538),
1453          BOOST_MATH_BIG_CONSTANT(T, 113, -384614.251466704550678760562965502293),
1454          BOOST_MATH_BIG_CONSTANT(T, 113, 523355.035910385688578278384032026998),
1455          BOOST_MATH_BIG_CONSTANT(T, 113, -217809.552260834025885677791936351294),
1456          BOOST_MATH_BIG_CONSTANT(T, 113, -8555.81719551123640677261226549550872)
1457       };
1458       T t = 1 / z;
1459       result = Y + tools::evaluate_polynomial(P, t)
1460          / tools::evaluate_polynomial(Q, t);
1461       if(z < 41)
1462          result *= exp(z) / z;
1463       else
1464       {
1465          // Avoid premature overflow if we can:
1466          t = z - 40;
1467          if(t > tools::log_max_value<T>())
1468          {
1469             result = policies::raise_overflow_error<T>(function, nullptr, pol);
1470          }
1471          else
1472          {
1473             result *= exp(z - 40) / z;
1474             if(result > tools::max_value<T>() / exp40)
1475             {
1476                result = policies::raise_overflow_error<T>(function, nullptr, pol);
1477             }
1478             else
1479             {
1480                result *= exp40;
1481             }
1482          }
1483       }
1484       result += z;
1485    }
1486    return result;
1487 }
1488
1489 template <class T, class Policy, class tag>
1490 struct expint_i_initializer
1491 {
1492    struct init
1493    {
1494       init()
1495       {
1496          do_init(tag());
1497       }
1498       static void do_init(const std::integral_constant<int, 0>&){}
1499       static void do_init(const std::integral_constant<int, 53>&)
1500       {
1501          boost::math::expint(T(5), Policy());
1502          boost::math::expint(T(7), Policy());
1503          boost::math::expint(T(18), Policy());
1504          boost::math::expint(T(38), Policy());
1505          boost::math::expint(T(45), Policy());
1506       }
1507       static void do_init(const std::integral_constant<int, 64>&)
1508       {
1509          boost::math::expint(T(5), Policy());
1510          boost::math::expint(T(7), Policy());
1511          boost::math::expint(T(18), Policy());
1512          boost::math::expint(T(38), Policy());
1513          boost::math::expint(T(45), Policy());
1514       }
1515       static void do_init(const std::integral_constant<int, 113>&)
1516       {
1517          boost::math::expint(T(5), Policy());
1518          boost::math::expint(T(7), Policy());
1519          boost::math::expint(T(17), Policy());
1520          boost::math::expint(T(25), Policy());
1521          boost::math::expint(T(40), Policy());
1522          boost::math::expint(T(50), Policy());
1523          boost::math::expint(T(80), Policy());
1524          boost::math::expint(T(200), Policy());
1525          boost::math::expint(T(220), Policy());
1526       }
1527       void force_instantiate()const{}
1528    };
1529    static const init initializer;
1530    static void force_instantiate()
1531    {
1532       initializer.force_instantiate();
1533    }
1534 };
1535
1536 template <class T, class Policy, class tag>
1537 const typename expint_i_initializer<T, Policy, tag>::init expint_i_initializer<T, Policy, tag>::initializer;
1538
1539 template <class T, class Policy, class tag>
1540 struct expint_1_initializer
1541 {
1542    struct init
1543    {
1544       init()
1545       {
1546          do_init(tag());
1547       }
1548       static void do_init(const std::integral_constant<int, 0>&){}
1549       static void do_init(const std::integral_constant<int, 53>&)
1550       {
1551          boost::math::expint(1, T(0.5), Policy());
1552          boost::math::expint(1, T(2), Policy());
1553       }
1554       static void do_init(const std::integral_constant<int, 64>&)
1555       {
1556          boost::math::expint(1, T(0.5), Policy());
1557          boost::math::expint(1, T(2), Policy());
1558       }
1559       static void do_init(const std::integral_constant<int, 113>&)
1560       {
1561          boost::math::expint(1, T(0.5), Policy());
1562          boost::math::expint(1, T(2), Policy());
1563          boost::math::expint(1, T(6), Policy());
1564       }
1565       void force_instantiate()const{}
1566    };
1567    static const init initializer;
1568    static void force_instantiate()
1569    {
1570       initializer.force_instantiate();
1571    }
1572 };
1573
1574 template <class T, class Policy, class tag>
1575 const typename expint_1_initializer<T, Policy, tag>::init expint_1_initializer<T, Policy, tag>::initializer;
1576
1577 template <class T, class Policy>
1578 inline typename tools::promote_args<T>::type
1579    expint_forwarder(T z, const Policy& /*pol*/, std::true_type const&)
1580 {
1581    typedef typename tools::promote_args<T>::type result_type;
1582    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1583    typedef typename policies::precision<result_type, Policy>::type precision_type;
1584    typedef typename policies::normalise<
1585       Policy,
1586       policies::promote_float<false>,
1587       policies::promote_double<false>,
1588       policies::discrete_quantile<>,
1589       policies::assert_undefined<> >::type forwarding_policy;
1590    typedef std::integral_constant<int,
1591       precision_type::value <= 0 ? 0 :
1592       precision_type::value <= 53 ? 53 :
1593       precision_type::value <= 64 ? 64 :
1594       precision_type::value <= 113 ? 113 : 0
1595    > tag_type;
1596
1597    expint_i_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1598
1599    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::expint_i_imp(
1600       static_cast<value_type>(z),
1601       forwarding_policy(),
1602       tag_type()), "boost::math::expint<%1%>(%1%)");
1603 }
1604
1605 template <class T>
1606 inline typename tools::promote_args<T>::type
1607 expint_forwarder(unsigned n, T z, const std::false_type&)
1608 {
1609    return boost::math::expint(n, z, policies::policy<>());
1610 }
1611
1612 } // namespace detail
1613
1614 template <class T, class Policy>
1615 inline typename tools::promote_args<T>::type
1616    expint(unsigned n, T z, const Policy& /*pol*/)
1617 {
1618    typedef typename tools::promote_args<T>::type result_type;
1619    typedef typename policies::evaluation<result_type, Policy>::type value_type;
1620    typedef typename policies::precision<result_type, Policy>::type precision_type;
1621    typedef typename policies::normalise<
1622       Policy,
1623       policies::promote_float<false>,
1624       policies::promote_double<false>,
1625       policies::discrete_quantile<>,
1626       policies::assert_undefined<> >::type forwarding_policy;
1627    typedef std::integral_constant<int,
1628       precision_type::value <= 0 ? 0 :
1629       precision_type::value <= 53 ? 53 :
1630       precision_type::value <= 64 ? 64 :
1631       precision_type::value <= 113 ? 113 : 0
1632    > tag_type;
1633
1634    detail::expint_1_initializer<value_type, forwarding_policy, tag_type>::force_instantiate();
1635
1636    return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::expint_imp(
1637       n,
1638       static_cast<value_type>(z),
1639       forwarding_policy(),
1640       tag_type()), "boost::math::expint<%1%>(unsigned, %1%)");
1641 }
1642
1643 template <class T, class U>
1644 inline typename detail::expint_result<T, U>::type
1645    expint(T const z, U const u)
1646 {
1647    typedef typename policies::is_policy<U>::type tag_type;
1648    return detail::expint_forwarder(z, u, tag_type());
1649 }
1650
1651 template <class T>
1652 inline typename tools::promote_args<T>::type
1653    expint(T z)
1654 {
1655    return expint(z, policies::policy<>());
1656 }
1657
1658 }} // namespaces
1659
1660 #ifdef _MSC_VER
1661 #pragma warning(pop)
1662 #endif
1663
1664 #endif // BOOST_MATH_EXPINT_HPP
1665
1666