boost/math/special_functions/zeta.hpp
// Copyright John Maddock 2007, 2014. // 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_ZETA_HPP #define BOOST_MATH_ZETA_HPP #ifdef _MSC_VER #pragma once #endif #include <boost/math/special_functions/math_fwd.hpp> #include <boost/math/tools/precision.hpp> #include <boost/math/tools/series.hpp> #include <boost/math/tools/big_constant.hpp> #include <boost/math/policies/error_handling.hpp> #include <boost/math/special_functions/gamma.hpp> #include <boost/math/special_functions/factorials.hpp> #include <boost/math/special_functions/sin_pi.hpp> namespace boost{ namespace math{ namespace detail{ #if 0 // // This code is commented out because we have a better more rapidly converging series // now. Retained for future reference and in case the new code causes any issues down the line.... // template <class T, class Policy> struct zeta_series_cache_size { // // Work how large to make our cache size when evaluating the series // evaluation: normally this is just large enough for the series // to have converged, but for arbitrary precision types we need a // really large cache to achieve reasonable precision in a reasonable // time. This is important when constructing rational approximations // to zeta for example. // typedef typename boost::math::policies::precision<T,Policy>::type precision_type; typedef typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<0> >, mpl::int_<5000>, typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<64> >, mpl::int_<70>, typename mpl::if_< mpl::less_equal<precision_type, mpl::int_<113> >, mpl::int_<100>, mpl::int_<5000> >::type >::type >::type type; }; template <class T, class Policy> T zeta_series_imp(T s, T sc, const Policy&) { // // Series evaluation from: // Havil, J. Gamma: Exploring Euler's Constant. // Princeton, NJ: Princeton University Press, 2003. // // See also http://mathworld.wolfram.com/RiemannZetaFunction.html // BOOST_MATH_STD_USING T sum = 0; T mult = 0.5; T change; typedef typename zeta_series_cache_size<T,Policy>::type cache_size; T powers[cache_size::value] = { 0, }; unsigned n = 0; do{ T binom = -static_cast<T>(n); T nested_sum = 1; if(n < sizeof(powers) / sizeof(powers[0])) powers[n] = pow(static_cast<T>(n + 1), -s); for(unsigned k = 1; k <= n; ++k) { T p; if(k < sizeof(powers) / sizeof(powers[0])) { p = powers[k]; //p = pow(k + 1, -s); } else p = pow(static_cast<T>(k + 1), -s); nested_sum += binom * p; binom *= (k - static_cast<T>(n)) / (k + 1); } change = mult * nested_sum; sum += change; mult /= 2; ++n; }while(fabs(change / sum) > tools::epsilon<T>()); return sum * 1 / -boost::math::powm1(T(2), sc); } // // Classical p-series: // template <class T> struct zeta_series2 { typedef T result_type; zeta_series2(T _s) : s(-_s), k(1){} T operator()() { BOOST_MATH_STD_USING return pow(static_cast<T>(k++), s); } private: T s; unsigned k; }; template <class T, class Policy> inline T zeta_series2_imp(T s, const Policy& pol) { boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();; zeta_series2<T> f(s); T result = tools::sum_series( f, policies::get_epsilon<T, Policy>(), max_iter); policies::check_series_iterations<T>("boost::math::zeta_series2<%1%>(%1%)", max_iter, pol); return result; } #endif template <class T, class Policy> T zeta_polynomial_series(T s, T sc, Policy const &) { // // This is algorithm 3 from: // // "An Efficient Algorithm for the Riemann Zeta Function", P. Borwein, // Canadian Mathematical Society, Conference Proceedings. // See: http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P155.pdf // BOOST_MATH_STD_USING int n = itrunc(T(log(boost::math::tools::epsilon<T>()) / -2)); T sum = 0; T two_n = ldexp(T(1), n); int ej_sign = 1; for(int j = 0; j < n; ++j) { sum += ej_sign * -two_n / pow(T(j + 1), s); ej_sign = -ej_sign; } T ej_sum = 1; T ej_term = 1; for(int j = n; j <= 2 * n - 1; ++j) { sum += ej_sign * (ej_sum - two_n) / pow(T(j + 1), s); ej_sign = -ej_sign; ej_term *= 2 * n - j; ej_term /= j - n + 1; ej_sum += ej_term; } return -sum / (two_n * (-powm1(T(2), sc))); } template <class T, class Policy> T zeta_imp_prec(T s, T sc, const Policy& pol, const mpl::int_<0>&) { BOOST_MATH_STD_USING T result; if(s >= policies::digits<T, Policy>()) return 1; result = zeta_polynomial_series(s, sc, pol); #if 0 // Old code archived for future reference: // // Only use power series if it will converge in 100 // iterations or less: the more iterations it consumes // the slower convergence becomes so we have to be very // careful in it's usage. // if (s > -log(tools::epsilon<T>()) / 4.5) result = detail::zeta_series2_imp(s, pol); else result = detail::zeta_series_imp(s, sc, pol); #endif return result; } template <class T, class Policy> inline T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<53>&) { BOOST_MATH_STD_USING T result; if(s < 1) { // Rational Approximation // Maximum Deviation Found: 2.020e-18 // Expected Error Term: -2.020e-18 // Max error found at double precision: 3.994987e-17 static const T P[6] = { static_cast<T>(0.24339294433593750202L), static_cast<T>(-0.49092470516353571651L), static_cast<T>(0.0557616214776046784287L), static_cast<T>(-0.00320912498879085894856L), static_cast<T>(0.000451534528645796438704L), static_cast<T>(-0.933241270357061460782e-5L), }; static const T Q[6] = { static_cast<T>(1L), static_cast<T>(-0.279960334310344432495L), static_cast<T>(0.0419676223309986037706L), static_cast<T>(-0.00413421406552171059003L), static_cast<T>(0.00024978985622317935355L), static_cast<T>(-0.101855788418564031874e-4L), }; result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc); result -= 1.2433929443359375F; result += (sc); result /= (sc); } else if(s <= 2) { // Maximum Deviation Found: 9.007e-20 // Expected Error Term: 9.007e-20 static const T P[6] = { static_cast<T>(0.577215664901532860516L), static_cast<T>(0.243210646940107164097L), static_cast<T>(0.0417364673988216497593L), static_cast<T>(0.00390252087072843288378L), static_cast<T>(0.000249606367151877175456L), static_cast<T>(0.110108440976732897969e-4L), }; static const T Q[6] = { static_cast<T>(1.0), static_cast<T>(0.295201277126631761737L), static_cast<T>(0.043460910607305495864L), static_cast<T>(0.00434930582085826330659L), static_cast<T>(0.000255784226140488490982L), static_cast<T>(0.10991819782396112081e-4L), }; result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc)); result += 1 / (-sc); } else if(s <= 4) { // Maximum Deviation Found: 5.946e-22 // Expected Error Term: -5.946e-22 static const float Y = 0.6986598968505859375; static const T P[6] = { static_cast<T>(-0.0537258300023595030676L), static_cast<T>(0.0445163473292365591906L), static_cast<T>(0.0128677673534519952905L), static_cast<T>(0.00097541770457391752726L), static_cast<T>(0.769875101573654070925e-4L), static_cast<T>(0.328032510000383084155e-5L), }; static const T Q[7] = { 1.0f, static_cast<T>(0.33383194553034051422L), static_cast<T>(0.0487798431291407621462L), static_cast<T>(0.00479039708573558490716L), static_cast<T>(0.000270776703956336357707L), static_cast<T>(0.106951867532057341359e-4L), static_cast<T>(0.236276623974978646399e-7L), }; result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2)); result += Y + 1 / (-sc); } else if(s <= 7) { // Maximum Deviation Found: 2.955e-17 // Expected Error Term: 2.955e-17 // Max error found at double precision: 2.009135e-16 static const T P[6] = { static_cast<T>(-2.49710190602259410021L), static_cast<T>(-2.60013301809475665334L), static_cast<T>(-0.939260435377109939261L), static_cast<T>(-0.138448617995741530935L), static_cast<T>(-0.00701721240549802377623L), static_cast<T>(-0.229257310594893932383e-4L), }; static const T Q[9] = { 1.0f, static_cast<T>(0.706039025937745133628L), static_cast<T>(0.15739599649558626358L), static_cast<T>(0.0106117950976845084417L), static_cast<T>(-0.36910273311764618902e-4L), static_cast<T>(0.493409563927590008943e-5L), static_cast<T>(-0.234055487025287216506e-6L), static_cast<T>(0.718833729365459760664e-8L), static_cast<T>(-0.1129200113474947419e-9L), }; result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4)); result = 1 + exp(result); } else if(s < 15) { // Maximum Deviation Found: 7.117e-16 // Expected Error Term: 7.117e-16 // Max error found at double precision: 9.387771e-16 static const T P[7] = { static_cast<T>(-4.78558028495135619286L), static_cast<T>(-1.89197364881972536382L), static_cast<T>(-0.211407134874412820099L), static_cast<T>(-0.000189204758260076688518L), static_cast<T>(0.00115140923889178742086L), static_cast<T>(0.639949204213164496988e-4L), static_cast<T>(0.139348932445324888343e-5L), }; static const T Q[9] = { 1.0f, static_cast<T>(0.244345337378188557777L), static_cast<T>(0.00873370754492288653669L), static_cast<T>(-0.00117592765334434471562L), static_cast<T>(-0.743743682899933180415e-4L), static_cast<T>(-0.21750464515767984778e-5L), static_cast<T>(0.471001264003076486547e-8L), static_cast<T>(-0.833378440625385520576e-10L), static_cast<T>(0.699841545204845636531e-12L), }; result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7)); result = 1 + exp(result); } else if(s < 36) { // Max error in interpolated form: 1.668e-17 // Max error found at long double precision: 1.669714e-17 static const T P[8] = { static_cast<T>(-10.3948950573308896825L), static_cast<T>(-2.85827219671106697179L), static_cast<T>(-0.347728266539245787271L), static_cast<T>(-0.0251156064655346341766L), static_cast<T>(-0.00119459173416968685689L), static_cast<T>(-0.382529323507967522614e-4L), static_cast<T>(-0.785523633796723466968e-6L), static_cast<T>(-0.821465709095465524192e-8L), }; static const T Q[10] = { 1.0f, static_cast<T>(0.208196333572671890965L), static_cast<T>(0.0195687657317205033485L), static_cast<T>(0.00111079638102485921877L), static_cast<T>(0.408507746266039256231e-4L), static_cast<T>(0.955561123065693483991e-6L), static_cast<T>(0.118507153474022900583e-7L), static_cast<T>(0.222609483627352615142e-14L), }; result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15)); result = 1 + exp(result); } else if(s < 56) { result = 1 + pow(T(2), -s); } else { result = 1; } return result; } template <class T, class Policy> T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<64>&) { BOOST_MATH_STD_USING T result; if(s < 1) { // Rational Approximation // Maximum Deviation Found: 3.099e-20 // Expected Error Term: 3.099e-20 // Max error found at long double precision: 5.890498e-20 static const T P[6] = { BOOST_MATH_BIG_CONSTANT(T, 64, 0.243392944335937499969), BOOST_MATH_BIG_CONSTANT(T, 64, -0.496837806864865688082), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0680008039723709987107), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00511620413006619942112), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000455369899250053003335), BOOST_MATH_BIG_CONSTANT(T, 64, -0.279496685273033761927e-4), }; static const T Q[7] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, -0.30425480068225790522), BOOST_MATH_BIG_CONSTANT(T, 64, 0.050052748580371598736), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00519355671064700627862), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000360623385771198350257), BOOST_MATH_BIG_CONSTANT(T, 64, -0.159600883054550987633e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.339770279812410586032e-6), }; result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc); result -= 1.2433929443359375F; result += (sc); result /= (sc); } else if(s <= 2) { // Maximum Deviation Found: 1.059e-21 // Expected Error Term: 1.059e-21 // Max error found at long double precision: 1.626303e-19 static const T P[6] = { BOOST_MATH_BIG_CONSTANT(T, 64, 0.577215664901532860605), BOOST_MATH_BIG_CONSTANT(T, 64, 0.222537368917162139445), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0356286324033215682729), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00304465292366350081446), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000178102511649069421904), BOOST_MATH_BIG_CONSTANT(T, 64, 0.700867470265983665042e-5), }; static const T Q[7] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 0.259385759149531030085), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0373974962106091316854), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00332735159183332820617), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000188690420706998606469), BOOST_MATH_BIG_CONSTANT(T, 64, 0.635994377921861930071e-5), BOOST_MATH_BIG_CONSTANT(T, 64, 0.226583954978371199405e-7), }; result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc)); result += 1 / (-sc); } else if(s <= 4) { // Maximum Deviation Found: 5.946e-22 // Expected Error Term: -5.946e-22 static const float Y = 0.6986598968505859375; static const T P[7] = { BOOST_MATH_BIG_CONSTANT(T, 64, -0.053725830002359501027), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0470551187571475844778), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0101339410415759517471), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00100240326666092854528), BOOST_MATH_BIG_CONSTANT(T, 64, 0.685027119098122814867e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.390972820219765942117e-5), BOOST_MATH_BIG_CONSTANT(T, 64, 0.540319769113543934483e-7), }; static const T Q[8] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 0.286577739726542730421), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0447355811517733225843), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00430125107610252363302), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000284956969089786662045), BOOST_MATH_BIG_CONSTANT(T, 64, 0.116188101609848411329e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.278090318191657278204e-6), BOOST_MATH_BIG_CONSTANT(T, 64, -0.19683620233222028478e-8), }; result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2)); result += Y + 1 / (-sc); } else if(s <= 7) { // Max error found at long double precision: 8.132216e-19 static const T P[8] = { BOOST_MATH_BIG_CONSTANT(T, 64, -2.49710190602259407065), BOOST_MATH_BIG_CONSTANT(T, 64, -3.36664913245960625334), BOOST_MATH_BIG_CONSTANT(T, 64, -1.77180020623777595452), BOOST_MATH_BIG_CONSTANT(T, 64, -0.464717885249654313933), BOOST_MATH_BIG_CONSTANT(T, 64, -0.0643694921293579472583), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00464265386202805715487), BOOST_MATH_BIG_CONSTANT(T, 64, -0.000165556579779704340166), BOOST_MATH_BIG_CONSTANT(T, 64, -0.252884970740994069582e-5), }; static const T Q[9] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 1.01300131390690459085), BOOST_MATH_BIG_CONSTANT(T, 64, 0.387898115758643503827), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0695071490045701135188), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00586908595251442839291), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000217752974064612188616), BOOST_MATH_BIG_CONSTANT(T, 64, 0.397626583349419011731e-5), BOOST_MATH_BIG_CONSTANT(T, 64, -0.927884739284359700764e-8), BOOST_MATH_BIG_CONSTANT(T, 64, 0.119810501805618894381e-9), }; result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4)); result = 1 + exp(result); } else if(s < 15) { // Max error in interpolated form: 1.133e-18 // Max error found at long double precision: 2.183198e-18 static const T P[9] = { BOOST_MATH_BIG_CONSTANT(T, 64, -4.78558028495135548083), BOOST_MATH_BIG_CONSTANT(T, 64, -3.23873322238609358947), BOOST_MATH_BIG_CONSTANT(T, 64, -0.892338582881021799922), BOOST_MATH_BIG_CONSTANT(T, 64, -0.131326296217965913809), BOOST_MATH_BIG_CONSTANT(T, 64, -0.0115651591773783712996), BOOST_MATH_BIG_CONSTANT(T, 64, -0.000657728968362695775205), BOOST_MATH_BIG_CONSTANT(T, 64, -0.252051328129449973047e-4), BOOST_MATH_BIG_CONSTANT(T, 64, -0.626503445372641798925e-6), BOOST_MATH_BIG_CONSTANT(T, 64, -0.815696314790853893484e-8), }; static const T Q[9] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 0.525765665400123515036), BOOST_MATH_BIG_CONSTANT(T, 64, 0.10852641753657122787), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0115669945375362045249), BOOST_MATH_BIG_CONSTANT(T, 64, 0.000732896513858274091966), BOOST_MATH_BIG_CONSTANT(T, 64, 0.30683952282420248448e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.819649214609633126119e-6), BOOST_MATH_BIG_CONSTANT(T, 64, 0.117957556472335968146e-7), BOOST_MATH_BIG_CONSTANT(T, 64, -0.193432300973017671137e-12), }; result = tools::evaluate_polynomial(P, T(s - 7)) / tools::evaluate_polynomial(Q, T(s - 7)); result = 1 + exp(result); } else if(s < 42) { // Max error in interpolated form: 1.668e-17 // Max error found at long double precision: 1.669714e-17 static const T P[9] = { BOOST_MATH_BIG_CONSTANT(T, 64, -10.3948950573308861781), BOOST_MATH_BIG_CONSTANT(T, 64, -2.82646012777913950108), BOOST_MATH_BIG_CONSTANT(T, 64, -0.342144362739570333665), BOOST_MATH_BIG_CONSTANT(T, 64, -0.0249285145498722647472), BOOST_MATH_BIG_CONSTANT(T, 64, -0.00122493108848097114118), BOOST_MATH_BIG_CONSTANT(T, 64, -0.423055371192592850196e-4), BOOST_MATH_BIG_CONSTANT(T, 64, -0.1025215577185967488e-5), BOOST_MATH_BIG_CONSTANT(T, 64, -0.165096762663509467061e-7), BOOST_MATH_BIG_CONSTANT(T, 64, -0.145392555873022044329e-9), }; static const T Q[10] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.0), BOOST_MATH_BIG_CONSTANT(T, 64, 0.205135978585281988052), BOOST_MATH_BIG_CONSTANT(T, 64, 0.0192359357875879453602), BOOST_MATH_BIG_CONSTANT(T, 64, 0.00111496452029715514119), BOOST_MATH_BIG_CONSTANT(T, 64, 0.434928449016693986857e-4), BOOST_MATH_BIG_CONSTANT(T, 64, 0.116911068726610725891e-5), BOOST_MATH_BIG_CONSTANT(T, 64, 0.206704342290235237475e-7), BOOST_MATH_BIG_CONSTANT(T, 64, 0.209772836100827647474e-9), BOOST_MATH_BIG_CONSTANT(T, 64, -0.939798249922234703384e-16), BOOST_MATH_BIG_CONSTANT(T, 64, 0.264584017421245080294e-18), }; result = tools::evaluate_polynomial(P, T(s - 15)) / tools::evaluate_polynomial(Q, T(s - 15)); result = 1 + exp(result); } else if(s < 63) { result = 1 + pow(T(2), -s); } else { result = 1; } return result; } template <class T, class Policy> T zeta_imp_prec(T s, T sc, const Policy&, const mpl::int_<113>&) { BOOST_MATH_STD_USING T result; if(s < 1) { // Rational Approximation // Maximum Deviation Found: 9.493e-37 // Expected Error Term: 9.492e-37 // Max error found at long double precision: 7.281332e-31 static const T P[10] = { BOOST_MATH_BIG_CONSTANT(T, 113, -1.0), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0353008629988648122808504280990313668), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0107795651204927743049369868548706909), BOOST_MATH_BIG_CONSTANT(T, 113, 0.000523961870530500751114866884685172975), BOOST_MATH_BIG_CONSTANT(T, 113, -0.661805838304910731947595897966487515e-4), BOOST_MATH_BIG_CONSTANT(T, 113, -0.658932670403818558510656304189164638e-5), BOOST_MATH_BIG_CONSTANT(T, 113, -0.103437265642266106533814021041010453e-6), BOOST_MATH_BIG_CONSTANT(T, 113, 0.116818787212666457105375746642927737e-7), BOOST_MATH_BIG_CONSTANT(T, 113, 0.660690993901506912123512551294239036e-9), BOOST_MATH_BIG_CONSTANT(T, 113, 0.113103113698388531428914333768142527e-10), }; static const T Q[11] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, -0.387483472099602327112637481818565459), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0802265315091063135271497708694776875), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0110727276164171919280036408995078164), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00112552716946286252000434849173787243), BOOST_MATH_BIG_CONSTANT(T, 113, -0.874554160748626916455655180296834352e-4), BOOST_MATH_BIG_CONSTANT(T, 113, 0.530097847491828379568636739662278322e-5), BOOST_MATH_BIG_CONSTANT(T, 113, -0.248461553590496154705565904497247452e-6), BOOST_MATH_BIG_CONSTANT(T, 113, 0.881834921354014787309644951507523899e-8), BOOST_MATH_BIG_CONSTANT(T, 113, -0.217062446168217797598596496310953025e-9), BOOST_MATH_BIG_CONSTANT(T, 113, 0.315823200002384492377987848307151168e-11), }; result = tools::evaluate_polynomial(P, sc) / tools::evaluate_polynomial(Q, sc); result += (sc); result /= (sc); } else if(s <= 2) { // Maximum Deviation Found: 1.616e-37 // Expected Error Term: -1.615e-37 static const T P[10] = { BOOST_MATH_BIG_CONSTANT(T, 113, 0.577215664901532860606512090082402431), BOOST_MATH_BIG_CONSTANT(T, 113, 0.255597968739771510415479842335906308), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0494056503552807274142218876983542205), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00551372778611700965268920983472292325), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00043667616723970574871427830895192731), BOOST_MATH_BIG_CONSTANT(T, 113, 0.268562259154821957743669387915239528e-4), BOOST_MATH_BIG_CONSTANT(T, 113, 0.109249633923016310141743084480436612e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.273895554345300227466534378753023924e-7), BOOST_MATH_BIG_CONSTANT(T, 113, 0.583103205551702720149237384027795038e-9), BOOST_MATH_BIG_CONSTANT(T, 113, -0.835774625259919268768735944711219256e-11), }; static const T Q[11] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, 0.316661751179735502065583176348292881), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0540401806533507064453851182728635272), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00598621274107420237785899476374043797), BOOST_MATH_BIG_CONSTANT(T, 113, 0.000474907812321704156213038740142079615), BOOST_MATH_BIG_CONSTANT(T, 113, 0.272125421722314389581695715835862418e-4), BOOST_MATH_BIG_CONSTANT(T, 113, 0.112649552156479800925522445229212933e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.301838975502992622733000078063330461e-7), BOOST_MATH_BIG_CONSTANT(T, 113, 0.422960728687211282539769943184270106e-9), BOOST_MATH_BIG_CONSTANT(T, 113, -0.377105263588822468076813329270698909e-11), BOOST_MATH_BIG_CONSTANT(T, 113, -0.581926559304525152432462127383600681e-13), }; result = tools::evaluate_polynomial(P, T(-sc)) / tools::evaluate_polynomial(Q, T(-sc)); result += 1 / (-sc); } else if(s <= 4) { // Maximum Deviation Found: 1.891e-36 // Expected Error Term: -1.891e-36 // Max error found: 2.171527e-35 static const float Y = 0.6986598968505859375; static const T P[11] = { BOOST_MATH_BIG_CONSTANT(T, 113, -0.0537258300023595010275848333539748089), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0429086930802630159457448174466342553), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0136148228754303412510213395034056857), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00190231601036042925183751238033763915), BOOST_MATH_BIG_CONSTANT(T, 113, 0.000186880390916311438818302549192456581), BOOST_MATH_BIG_CONSTANT(T, 113, 0.145347370745893262394287982691323657e-4), BOOST_MATH_BIG_CONSTANT(T, 113, 0.805843276446813106414036600485884885e-6), BOOST_MATH_BIG_CONSTANT(T, 113, 0.340818159286739137503297172091882574e-7), BOOST_MATH_BIG_CONSTANT(T, 113, 0.115762357488748996526167305116837246e-8), BOOST_MATH_BIG_CONSTANT(T, 113, 0.231904754577648077579913403645767214e-10), BOOST_MATH_BIG_CONSTANT(T, 113, 0.340169592866058506675897646629036044e-12), }; static const T Q[12] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, 0.363755247765087100018556983050520554), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0696581979014242539385695131258321598), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00882208914484611029571547753782014817), BOOST_MATH_BIG_CONSTANT(T, 113, 0.000815405623261946661762236085660996718), BOOST_MATH_BIG_CONSTANT(T, 113, 0.571366167062457197282642344940445452e-4), BOOST_MATH_BIG_CONSTANT(T, 113, 0.309278269271853502353954062051797838e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.12822982083479010834070516053794262e-6), BOOST_MATH_BIG_CONSTANT(T, 113, 0.397876357325018976733953479182110033e-8), BOOST_MATH_BIG_CONSTANT(T, 113, 0.8484432107648683277598472295289279e-10), BOOST_MATH_BIG_CONSTANT(T, 113, 0.105677416606909614301995218444080615e-11), BOOST_MATH_BIG_CONSTANT(T, 113, 0.547223964564003701979951154093005354e-15), }; result = tools::evaluate_polynomial(P, T(s - 2)) / tools::evaluate_polynomial(Q, T(s - 2)); result += Y + 1 / (-sc); } else if(s <= 6) { // Max error in interpolated form: 1.510e-37 // Max error found at long double precision: 2.769266e-34 static const T Y = 3.28348541259765625F; static const T P[13] = { BOOST_MATH_BIG_CONSTANT(T, 113, 0.786383506575062179339611614117697622), BOOST_MATH_BIG_CONSTANT(T, 113, 0.495766593395271370974685959652073976), BOOST_MATH_BIG_CONSTANT(T, 113, -0.409116737851754766422360889037532228), BOOST_MATH_BIG_CONSTANT(T, 113, -0.57340744006238263817895456842655987), BOOST_MATH_BIG_CONSTANT(T, 113, -0.280479899797421910694892949057963111), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0753148409447590257157585696212649869), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0122934003684672788499099362823748632), BOOST_MATH_BIG_CONSTANT(T, 113, -0.00126148398446193639247961370266962927), BOOST_MATH_BIG_CONSTANT(T, 113, -0.828465038179772939844657040917364896e-4), BOOST_MATH_BIG_CONSTANT(T, 113, -0.361008916706050977143208468690645684e-5), BOOST_MATH_BIG_CONSTANT(T, 113, -0.109879825497910544424797771195928112e-6), BOOST_MATH_BIG_CONSTANT(T, 113, -0.214539416789686920918063075528797059e-8), BOOST_MATH_BIG_CONSTANT(T, 113, -0.15090220092460596872172844424267351e-10), }; static const T Q[14] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, 1.69490865837142338462982225731926485), BOOST_MATH_BIG_CONSTANT(T, 113, 1.22697696630994080733321401255942464), BOOST_MATH_BIG_CONSTANT(T, 113, 0.495409420862526540074366618006341533), BOOST_MATH_BIG_CONSTANT(T, 113, 0.122368084916843823462872905024259633), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0191412993625268971656513890888208623), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00191401538628980617753082598351559642), BOOST_MATH_BIG_CONSTANT(T, 113, 0.000123318142456272424148930280876444459), BOOST_MATH_BIG_CONSTANT(T, 113, 0.531945488232526067889835342277595709e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.161843184071894368337068779669116236e-6), BOOST_MATH_BIG_CONSTANT(T, 113, 0.305796079600152506743828859577462778e-8), BOOST_MATH_BIG_CONSTANT(T, 113, 0.233582592298450202680170811044408894e-10), BOOST_MATH_BIG_CONSTANT(T, 113, -0.275363878344548055574209713637734269e-13), BOOST_MATH_BIG_CONSTANT(T, 113, 0.221564186807357535475441900517843892e-15), }; result = tools::evaluate_polynomial(P, T(s - 4)) / tools::evaluate_polynomial(Q, T(s - 4)); result -= Y; result = 1 + exp(result); } else if(s < 10) { // Max error in interpolated form: 1.999e-34 // Max error found at long double precision: 2.156186e-33 static const T P[13] = { BOOST_MATH_BIG_CONSTANT(T, 113, -4.0545627381873738086704293881227365), BOOST_MATH_BIG_CONSTANT(T, 113, -4.70088348734699134347906176097717782), BOOST_MATH_BIG_CONSTANT(T, 113, -2.36921550900925512951976617607678789), BOOST_MATH_BIG_CONSTANT(T, 113, -0.684322583796369508367726293719322866), BOOST_MATH_BIG_CONSTANT(T, 113, -0.126026534540165129870721937592996324), BOOST_MATH_BIG_CONSTANT(T, 113, -0.015636903921778316147260572008619549), BOOST_MATH_BIG_CONSTANT(T, 113, -0.00135442294754728549644376325814460807), BOOST_MATH_BIG_CONSTANT(T, 113, -0.842793965853572134365031384646117061e-4), BOOST_MATH_BIG_CONSTANT(T, 113, -0.385602133791111663372015460784978351e-5), BOOST_MATH_BIG_CONSTANT(T, 113, -0.130458500394692067189883214401478539e-6), BOOST_MATH_BIG_CONSTANT(T, 113, -0.315861074947230418778143153383660035e-8), BOOST_MATH_BIG_CONSTANT(T, 113, -0.500334720512030826996373077844707164e-10), BOOST_MATH_BIG_CONSTANT(T, 113, -0.420204769185233365849253969097184005e-12), }; static const T Q[14] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, 0.97663511666410096104783358493318814), BOOST_MATH_BIG_CONSTANT(T, 113, 0.40878780231201806504987368939673249), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0963890666609396058945084107597727252), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0142207619090854604824116070866614505), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00139010220902667918476773423995750877), BOOST_MATH_BIG_CONSTANT(T, 113, 0.940669540194694997889636696089994734e-4), BOOST_MATH_BIG_CONSTANT(T, 113, 0.458220848507517004399292480807026602e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.16345521617741789012782420625435495e-6), BOOST_MATH_BIG_CONSTANT(T, 113, 0.414007452533083304371566316901024114e-8), BOOST_MATH_BIG_CONSTANT(T, 113, 0.68701473543366328016953742622661377e-10), BOOST_MATH_BIG_CONSTANT(T, 113, 0.603461891080716585087883971886075863e-12), BOOST_MATH_BIG_CONSTANT(T, 113, 0.294670713571839023181857795866134957e-16), BOOST_MATH_BIG_CONSTANT(T, 113, -0.147003914536437243143096875069813451e-18), }; result = tools::evaluate_polynomial(P, T(s - 6)) / tools::evaluate_polynomial(Q, T(s - 6)); result = 1 + exp(result); } else if(s < 17) { // Max error in interpolated form: 1.641e-32 // Max error found at long double precision: 1.696121e-32 static const T P[13] = { BOOST_MATH_BIG_CONSTANT(T, 113, -6.91319491921722925920883787894829678), BOOST_MATH_BIG_CONSTANT(T, 113, -3.65491257639481960248690596951049048), BOOST_MATH_BIG_CONSTANT(T, 113, -0.813557553449954526442644544105257881), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0994317301685870959473658713841138083), BOOST_MATH_BIG_CONSTANT(T, 113, -0.00726896610245676520248617014211734906), BOOST_MATH_BIG_CONSTANT(T, 113, -0.000317253318715075854811266230916762929), BOOST_MATH_BIG_CONSTANT(T, 113, -0.66851422826636750855184211580127133e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.879464154730985406003332577806849971e-7), BOOST_MATH_BIG_CONSTANT(T, 113, 0.113838903158254250631678791998294628e-7), BOOST_MATH_BIG_CONSTANT(T, 113, 0.379184410304927316385211327537817583e-9), BOOST_MATH_BIG_CONSTANT(T, 113, 0.612992858643904887150527613446403867e-11), BOOST_MATH_BIG_CONSTANT(T, 113, 0.347873737198164757035457841688594788e-13), BOOST_MATH_BIG_CONSTANT(T, 113, -0.289187187441625868404494665572279364e-15), }; static const T Q[14] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, 0.427310044448071818775721584949868806), BOOST_MATH_BIG_CONSTANT(T, 113, 0.074602514873055756201435421385243062), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00688651562174480772901425121653945942), BOOST_MATH_BIG_CONSTANT(T, 113, 0.000360174847635115036351323894321880445), BOOST_MATH_BIG_CONSTANT(T, 113, 0.973556847713307543918865405758248777e-5), BOOST_MATH_BIG_CONSTANT(T, 113, -0.853455848314516117964634714780874197e-8), BOOST_MATH_BIG_CONSTANT(T, 113, -0.118203513654855112421673192194622826e-7), BOOST_MATH_BIG_CONSTANT(T, 113, -0.462521662511754117095006543363328159e-9), BOOST_MATH_BIG_CONSTANT(T, 113, -0.834212591919475633107355719369463143e-11), BOOST_MATH_BIG_CONSTANT(T, 113, -0.5354594751002702935740220218582929e-13), BOOST_MATH_BIG_CONSTANT(T, 113, 0.406451690742991192964889603000756203e-15), BOOST_MATH_BIG_CONSTANT(T, 113, 0.887948682401000153828241615760146728e-19), BOOST_MATH_BIG_CONSTANT(T, 113, -0.34980761098820347103967203948619072e-21), }; result = tools::evaluate_polynomial(P, T(s - 10)) / tools::evaluate_polynomial(Q, T(s - 10)); result = 1 + exp(result); } else if(s < 30) { // Max error in interpolated form: 1.563e-31 // Max error found at long double precision: 1.562725e-31 static const T P[13] = { BOOST_MATH_BIG_CONSTANT(T, 113, -11.7824798233959252791987402769438322), BOOST_MATH_BIG_CONSTANT(T, 113, -4.36131215284987731928174218354118102), BOOST_MATH_BIG_CONSTANT(T, 113, -0.732260980060982349410898496846972204), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0744985185694913074484248803015717388), BOOST_MATH_BIG_CONSTANT(T, 113, -0.00517228281320594683022294996292250527), BOOST_MATH_BIG_CONSTANT(T, 113, -0.000260897206152101522569969046299309939), BOOST_MATH_BIG_CONSTANT(T, 113, -0.989553462123121764865178453128769948e-5), BOOST_MATH_BIG_CONSTANT(T, 113, -0.286916799741891410827712096608826167e-6), BOOST_MATH_BIG_CONSTANT(T, 113, -0.637262477796046963617949532211619729e-8), BOOST_MATH_BIG_CONSTANT(T, 113, -0.106796831465628373325491288787760494e-9), BOOST_MATH_BIG_CONSTANT(T, 113, -0.129343095511091870860498356205376823e-11), BOOST_MATH_BIG_CONSTANT(T, 113, -0.102397936697965977221267881716672084e-13), BOOST_MATH_BIG_CONSTANT(T, 113, -0.402663128248642002351627980255756363e-16), }; static const T Q[14] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, 0.311288325355705609096155335186466508), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0438318468940415543546769437752132748), BOOST_MATH_BIG_CONSTANT(T, 113, 0.00374396349183199548610264222242269536), BOOST_MATH_BIG_CONSTANT(T, 113, 0.000218707451200585197339671707189281302), BOOST_MATH_BIG_CONSTANT(T, 113, 0.927578767487930747532953583797351219e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.294145760625753561951137473484889639e-6), BOOST_MATH_BIG_CONSTANT(T, 113, 0.704618586690874460082739479535985395e-8), BOOST_MATH_BIG_CONSTANT(T, 113, 0.126333332872897336219649130062221257e-9), BOOST_MATH_BIG_CONSTANT(T, 113, 0.16317315713773503718315435769352765e-11), BOOST_MATH_BIG_CONSTANT(T, 113, 0.137846712823719515148344938160275695e-13), BOOST_MATH_BIG_CONSTANT(T, 113, 0.580975420554224366450994232723910583e-16), BOOST_MATH_BIG_CONSTANT(T, 113, -0.291354445847552426900293580511392459e-22), BOOST_MATH_BIG_CONSTANT(T, 113, 0.73614324724785855925025452085443636e-25), }; result = tools::evaluate_polynomial(P, T(s - 17)) / tools::evaluate_polynomial(Q, T(s - 17)); result = 1 + exp(result); } else if(s < 74) { // Max error in interpolated form: 2.311e-27 // Max error found at long double precision: 2.297544e-27 static const T P[14] = { BOOST_MATH_BIG_CONSTANT(T, 113, -20.7944102007844314586649688802236072), BOOST_MATH_BIG_CONSTANT(T, 113, -4.95759941987499442499908748130192187), BOOST_MATH_BIG_CONSTANT(T, 113, -0.563290752832461751889194629200298688), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0406197001137935911912457120706122877), BOOST_MATH_BIG_CONSTANT(T, 113, -0.0020846534789473022216888863613422293), BOOST_MATH_BIG_CONSTANT(T, 113, -0.808095978462109173749395599401375667e-4), BOOST_MATH_BIG_CONSTANT(T, 113, -0.244706022206249301640890603610060959e-5), BOOST_MATH_BIG_CONSTANT(T, 113, -0.589477682919645930544382616501666572e-7), BOOST_MATH_BIG_CONSTANT(T, 113, -0.113699573675553496343617442433027672e-8), BOOST_MATH_BIG_CONSTANT(T, 113, -0.174767860183598149649901223128011828e-10), BOOST_MATH_BIG_CONSTANT(T, 113, -0.210051620306761367764549971980026474e-12), BOOST_MATH_BIG_CONSTANT(T, 113, -0.189187969537370950337212675466400599e-14), BOOST_MATH_BIG_CONSTANT(T, 113, -0.116313253429564048145641663778121898e-16), BOOST_MATH_BIG_CONSTANT(T, 113, -0.376708747782400769427057630528578187e-19), }; static const T Q[16] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.0), BOOST_MATH_BIG_CONSTANT(T, 113, 0.205076752981410805177554569784219717), BOOST_MATH_BIG_CONSTANT(T, 113, 0.0202526722696670378999575738524540269), BOOST_MATH_BIG_CONSTANT(T, 113, 0.001278305290005994980069466658219057), BOOST_MATH_BIG_CONSTANT(T, 113, 0.576404779858501791742255670403304787e-4), BOOST_MATH_BIG_CONSTANT(T, 113, 0.196477049872253010859712483984252067e-5), BOOST_MATH_BIG_CONSTANT(T, 113, 0.521863830500876189501054079974475762e-7), BOOST_MATH_BIG_CONSTANT(T, 113, 0.109524209196868135198775445228552059e-8), BOOST_MATH_BIG_CONSTANT(T, 113, 0.181698713448644481083966260949267825e-10), BOOST_MATH_BIG_CONSTANT(T, 113, 0.234793316975091282090312036524695562e-12), BOOST_MATH_BIG_CONSTANT(T, 113, 0.227490441461460571047545264251399048e-14), BOOST_MATH_BIG_CONSTANT(T, 113, 0.151500292036937400913870642638520668e-16), BOOST_MATH_BIG_CONSTANT(T, 113, 0.543475775154780935815530649335936121e-19), BOOST_MATH_BIG_CONSTANT(T, 113, 0.241647013434111434636554455083309352e-28), BOOST_MATH_BIG_CONSTANT(T, 113, -0.557103423021951053707162364713587374e-31), BOOST_MATH_BIG_CONSTANT(T, 113, 0.618708773442584843384712258199645166e-34), }; result = tools::evaluate_polynomial(P, T(s - 30)) / tools::evaluate_polynomial(Q, T(s - 30)); result = 1 + exp(result); } else if(s < 117) { result = 1 + pow(T(2), -s); } else { result = 1; } return result; } template <class T, class Policy> T zeta_imp_odd_integer(int s, const T&, const Policy&, const mpl::true_&) { static const T results[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.2020569031595942853997381615114500), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0369277551433699263313654864570342), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0083492773819228268397975498497968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0020083928260822144178527692324121), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0004941886041194645587022825264699), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0001227133475784891467518365263574), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000305882363070204935517285106451), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000076371976378997622736002935630), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000019082127165539389256569577951), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000004769329867878064631167196044), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000001192199259653110730677887189), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000298035035146522801860637051), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000074507117898354294919810042), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000018626597235130490064039099), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000004656629065033784072989233), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000001164155017270051977592974), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000291038504449709968692943), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000072759598350574810145209), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000018189896503070659475848), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000004547473783042154026799), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000001136868407680227849349), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000284217097688930185546), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000071054273952108527129), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000017763568435791203275), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000004440892103143813364), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000001110223025141066134), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000277555756213612417), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000069388939045441537), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000017347234760475766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000004336808690020650), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000001084202172494241), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000271050543122347), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000067762635780452), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000016940658945098), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000004235164736273), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000001058791184068), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000264697796017), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000066174449004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000016543612251), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000004135903063), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000001033975766), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000258493941), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000064623485), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000016155871), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000004038968), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000001009742), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000252435), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000063109), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000015777), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000003944), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000986), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000247), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000062), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000015), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000004), BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000001), }; return s > 113 ? 1 : results[(s - 3) / 2]; } template <class T, class Policy> T zeta_imp_odd_integer(int s, const T& sc, const Policy& pol, const mpl::false_&) { static BOOST_MATH_THREAD_LOCAL bool is_init = false; static BOOST_MATH_THREAD_LOCAL T results[50] = {}; static BOOST_MATH_THREAD_LOCAL int digits = tools::digits<T>(); int current_digits = tools::digits<T>(); if(digits != current_digits) { // Oh my precision has changed... is_init = false; } if(!is_init) { is_init = true; digits = current_digits; for(unsigned k = 0; k < sizeof(results) / sizeof(results[0]); ++k) { T arg = k * 2 + 3; T c_arg = 1 - arg; results[k] = zeta_polynomial_series(arg, c_arg, pol); } } unsigned index = (s - 3) / 2; return index >= sizeof(results) / sizeof(results[0]) ? zeta_polynomial_series(T(s), sc, pol): results[index]; } template <class T, class Policy, class Tag> T zeta_imp(T s, T sc, const Policy& pol, const Tag& tag) { BOOST_MATH_STD_USING static const char* function = "boost::math::zeta<%1%>"; if(sc == 0) return policies::raise_pole_error<T>( function, "Evaluation of zeta function at pole %1%", s, pol); T result; // // Trivial case: // if(s > policies::digits<T, Policy>()) return 1; // // Start by seeing if we have a simple closed form: // if(floor(s) == s) { #ifndef BOOST_NO_EXCEPTIONS // Without exceptions we expect itrunc to return INT_MAX on overflow // and we fall through anyway. try { #endif int v = itrunc(s); if(v == s) { if(v < 0) { if(((-v) & 1) == 0) return 0; int n = (-v + 1) / 2; if(n <= (int)boost::math::max_bernoulli_b2n<T>::value) return T((-v & 1) ? -1 : 1) * boost::math::unchecked_bernoulli_b2n<T>(n) / (1 - v); } else if((v & 1) == 0) { if(((v / 2) <= (int)boost::math::max_bernoulli_b2n<T>::value) && (v <= (int)boost::math::max_factorial<T>::value)) return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) * boost::math::unchecked_bernoulli_b2n<T>(v / 2) / boost::math::unchecked_factorial<T>(v); return T(((v / 2 - 1) & 1) ? -1 : 1) * ldexp(T(1), v - 1) * pow(constants::pi<T, Policy>(), v) * boost::math::bernoulli_b2n<T>(v / 2) / boost::math::factorial<T>(v); } else return zeta_imp_odd_integer(v, sc, pol, mpl::bool_<(Tag::value <= 113) && Tag::value>()); } #ifndef BOOST_NO_EXCEPTIONS } catch(const boost::math::rounding_error&){} // Just fall through, s is too large to round catch(const std::overflow_error&){} #endif } if(fabs(s) < tools::root_epsilon<T>()) { result = -0.5f - constants::log_root_two_pi<T, Policy>() * s; } else if(s < 0) { std::swap(s, sc); if(floor(sc/2) == sc/2) result = 0; else { if(s > max_factorial<T>::value) { T mult = boost::math::sin_pi(0.5f * sc, pol) * 2 * zeta_imp(s, sc, pol, tag); result = boost::math::lgamma(s, pol); result -= s * log(2 * constants::pi<T>()); if(result > tools::log_max_value<T>()) return sign(mult) * policies::raise_overflow_error<T>(function, 0, pol); result = exp(result); if(tools::max_value<T>() / fabs(mult) < result) return boost::math::sign(mult) * policies::raise_overflow_error<T>(function, 0, pol); result *= mult; } else { result = boost::math::sin_pi(0.5f * sc, pol) * 2 * pow(2 * constants::pi<T>(), -s) * boost::math::tgamma(s, pol) * zeta_imp(s, sc, pol, tag); } } } else { result = zeta_imp_prec(s, sc, pol, tag); } return result; } template <class T, class Policy, class tag> struct zeta_initializer { struct init { init() { do_init(tag()); } static void do_init(const mpl::int_<0>&){ boost::math::zeta(static_cast<T>(5), Policy()); } static void do_init(const mpl::int_<53>&){ boost::math::zeta(static_cast<T>(5), Policy()); } static void do_init(const mpl::int_<64>&) { boost::math::zeta(static_cast<T>(0.5), Policy()); boost::math::zeta(static_cast<T>(1.5), Policy()); boost::math::zeta(static_cast<T>(3.5), Policy()); boost::math::zeta(static_cast<T>(6.5), Policy()); boost::math::zeta(static_cast<T>(14.5), Policy()); boost::math::zeta(static_cast<T>(40.5), Policy()); boost::math::zeta(static_cast<T>(5), Policy()); } static void do_init(const mpl::int_<113>&) { boost::math::zeta(static_cast<T>(0.5), Policy()); boost::math::zeta(static_cast<T>(1.5), Policy()); boost::math::zeta(static_cast<T>(3.5), Policy()); boost::math::zeta(static_cast<T>(5.5), Policy()); boost::math::zeta(static_cast<T>(9.5), Policy()); boost::math::zeta(static_cast<T>(16.5), Policy()); boost::math::zeta(static_cast<T>(25.5), Policy()); boost::math::zeta(static_cast<T>(70.5), Policy()); boost::math::zeta(static_cast<T>(5), Policy()); } void force_instantiate()const{} }; static const init initializer; static void force_instantiate() { initializer.force_instantiate(); } }; template <class T, class Policy, class tag> const typename zeta_initializer<T, Policy, tag>::init zeta_initializer<T, Policy, tag>::initializer; } // detail template <class T, class Policy> inline typename tools::promote_args<T>::type zeta(T s, const Policy&) { 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; //typedef mpl::int_<0> tag_type; detail::zeta_initializer<value_type, forwarding_policy, tag_type>::force_instantiate(); return policies::checked_narrowing_cast<result_type, forwarding_policy>(detail::zeta_imp( static_cast<value_type>(s), static_cast<value_type>(1 - static_cast<value_type>(s)), forwarding_policy(), tag_type()), "boost::math::zeta<%1%>(%1%)"); } template <class T> inline typename tools::promote_args<T>::type zeta(T s) { return zeta(s, policies::policy<>()); } }} // namespaces #endif // BOOST_MATH_ZETA_HPP