boost/math/special_functions/detail/bessel_k1.hpp
// Copyright (c) 2006 Xiaogang Zhang // Copyright (c) 2017 John Maddock // 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_BESSEL_K1_HPP #define BOOST_MATH_BESSEL_K1_HPP #ifdef _MSC_VER #pragma once #pragma warning(push) #pragma warning(disable:4702) // Unreachable code (release mode only warning) #endif #include <boost/math/tools/rational.hpp> #include <boost/math/tools/big_constant.hpp> #include <boost/math/policies/error_handling.hpp> #include <boost/math/tools/assert.hpp> #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128) // // This is the only way we can avoid // warning: non-standard suffix on floating constant [-Wpedantic] // when building with -Wall -pedantic. Neither __extension__ // nor #pragma diagnostic ignored work :( // #pragma GCC system_header #endif // Modified Bessel function of the second kind of order zero // minimax rational approximations on intervals, see // Russon and Blair, Chalk River Report AECL-3461, 1969, // as revised by Pavel Holoborodko in "Rational Approximations // for the Modified Bessel Function of the Second Kind - K0(x) // for Computations with Double Precision", see // http://www.advanpix.com/2016/01/05/rational-approximations-for-the-modified-bessel-function-of-the-second-kind-k1-for-computations-with-double-precision/ // // The actual coefficients used are our own derivation (by JM) // since we extend to both greater and lesser precision than the // references above. We can also improve performance WRT to // Holoborodko without loss of precision. namespace boost { namespace math { namespace detail{ template <typename T> T bessel_k1(const T&); template <class T, class tag> struct bessel_k1_initializer { struct init { init() { do_init(tag()); } static void do_init(const std::integral_constant<int, 113>&) { bessel_k1(T(0.5)); bessel_k1(T(2)); bessel_k1(T(6)); } static void do_init(const std::integral_constant<int, 64>&) { bessel_k1(T(0.5)); bessel_k1(T(6)); } template <class U> static void do_init(const U&) {} void force_instantiate()const {} }; static const init initializer; static void force_instantiate() { initializer.force_instantiate(); } }; template <class T, class tag> const typename bessel_k1_initializer<T, tag>::init bessel_k1_initializer<T, tag>::initializer; template <typename T, int N> inline T bessel_k1_imp(const T&, const std::integral_constant<int, N>&) { BOOST_MATH_ASSERT(0); return 0; } template <typename T> T bessel_k1_imp(const T& x, const std::integral_constant<int, 24>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found: 3.090e-12 // Expected Error Term : -3.053e-12 // Maximum Relative Change in Control Points : 4.927e-02 // Max Error found at float precision = Poly : 7.918347e-10 static const T Y = 8.695471287e-02f; static const T P[] = { -3.621379531e-03f, 7.131781976e-03f, -1.535278300e-05f }; static const T Q[] = { 1.000000000e+00f, -5.173102701e-02f, 9.203530671e-04f }; T a = x * x / 4; a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2; // Maximum Deviation Found: 3.556e-08 // Expected Error Term : -3.541e-08 // Maximum Relative Change in Control Points : 8.203e-02 static const T P2[] = { -3.079657469e-01f, -8.537108913e-02f, -4.640275408e-03f, -1.156442414e-04f }; return tools::evaluate_polynomial(P2, T(x * x)) * x + 1 / x + log(x) * a; } else { // Maximum Deviation Found: 3.369e-08 // Expected Error Term : -3.227e-08 // Maximum Relative Change in Control Points : 9.917e-02 // Max Error found at float precision = Poly : 6.084411e-08 static const T Y = 1.450342178f; static const T P[] = { -1.970280088e-01f, 2.188747807e-02f, 7.270394756e-01f, 2.490678196e-01f }; static const T Q[] = { 1.000000000e+00f, 2.274292882e+00f, 9.904984851e-01f, 4.585534549e-02f }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k1_imp(const T& x, const std::integral_constant<int, 53>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found: 1.922e-17 // Expected Error Term : 1.921e-17 // Maximum Relative Change in Control Points : 5.287e-03 // Max Error found at double precision = Poly : 2.004747e-17 static const T Y = 8.69547128677368164e-02f; static const T P[] = { -3.62137953440350228e-03, 7.11842087490330300e-03, 1.00302560256614306e-05, 1.77231085381040811e-06 }; static const T Q[] = { 1.00000000000000000e+00, -4.80414794429043831e-02, 9.85972641934416525e-04, -8.91196859397070326e-06 }; T a = x * x / 4; a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2; // Maximum Deviation Found: 4.053e-17 // Expected Error Term : -4.053e-17 // Maximum Relative Change in Control Points : 3.103e-04 // Max Error found at double precision = Poly : 1.246698e-16 static const T P2[] = { -3.07965757829206184e-01, -7.80929703673074907e-02, -2.70619343754051620e-03, -2.49549522229072008e-05 }; static const T Q2[] = { 1.00000000000000000e+00, -2.36316836412163098e-02, 2.64524577525962719e-04, -1.49749618004162787e-06 }; return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a; } else { // Maximum Deviation Found: 8.883e-17 // Expected Error Term : -1.641e-17 // Maximum Relative Change in Control Points : 2.786e-01 // Max Error found at double precision = Poly : 1.258798e-16 static const T Y = 1.45034217834472656f; static const T P[] = { -1.97028041029226295e-01, -2.32408961548087617e+00, -7.98269784507699938e+00, -2.39968410774221632e+00, 3.28314043780858713e+01, 5.67713761158496058e+01, 3.30907788466509823e+01, 6.62582288933739787e+00, 3.08851840645286691e-01 }; static const T Q[] = { 1.00000000000000000e+00, 1.41811409298826118e+01, 7.35979466317556420e+01, 1.77821793937080859e+02, 2.11014501598705982e+02, 1.19425262951064454e+02, 2.88448064302447607e+01, 2.27912927104139732e+00, 2.50358186953478678e-02 }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k1_imp(const T& x, const std::integral_constant<int, 64>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found: 5.549e-23 // Expected Error Term : -5.548e-23 // Maximum Relative Change in Control Points : 2.002e-03 // Max Error found at float80 precision = Poly : 9.352785e-22 static const T Y = 8.695471286773681640625e-02f; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -3.621379534403483072861e-03), BOOST_MATH_BIG_CONSTANT(T, 64, 7.102135866103952705932e-03), BOOST_MATH_BIG_CONSTANT(T, 64, 4.167545240236717601167e-05), BOOST_MATH_BIG_CONSTANT(T, 64, 2.537484002571894870830e-06), BOOST_MATH_BIG_CONSTANT(T, 64, 6.603228256820000135990e-09) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 64, -4.354457194045068370363e-02), BOOST_MATH_BIG_CONSTANT(T, 64, 8.709137201220209072820e-04), BOOST_MATH_BIG_CONSTANT(T, 64, -9.676151796359590545143e-06), BOOST_MATH_BIG_CONSTANT(T, 64, 5.162715192766245311659e-08) }; T a = x * x / 4; a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2; // Maximum Deviation Found: 1.995e-23 // Expected Error Term : 1.995e-23 // Maximum Relative Change in Control Points : 8.174e-04 // Max Error found at float80 precision = Poly : 4.137325e-20 static const T P2[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -3.079657578292062244054e-01), BOOST_MATH_BIG_CONSTANT(T, 64, -7.963049154965966503231e-02), BOOST_MATH_BIG_CONSTANT(T, 64, -3.103277523735639924895e-03), BOOST_MATH_BIG_CONSTANT(T, 64, -4.023052834702215699504e-05), BOOST_MATH_BIG_CONSTANT(T, 64, -1.719459155018493821839e-07) }; static const T Q2[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 64, -1.863917670410152669768e-02), BOOST_MATH_BIG_CONSTANT(T, 64, 1.699367098849735298090e-04), BOOST_MATH_BIG_CONSTANT(T, 64, -9.309358790546076298429e-07), BOOST_MATH_BIG_CONSTANT(T, 64, 2.708893480271612711933e-09) }; return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a; } else { // Maximum Deviation Found: 9.785e-20 // Expected Error Term : -3.302e-21 // Maximum Relative Change in Control Points : 3.432e-01 // Max Error found at float80 precision = Poly : 1.083755e-19 static const T Y = 1.450342178344726562500e+00f; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 64, -1.970280410292263112917e-01), BOOST_MATH_BIG_CONSTANT(T, 64, -4.058564803062959169322e+00), BOOST_MATH_BIG_CONSTANT(T, 64, -3.036658174194917777473e+01), BOOST_MATH_BIG_CONSTANT(T, 64, -9.576825392332820142173e+01), BOOST_MATH_BIG_CONSTANT(T, 64, -6.706969489248020941949e+01), BOOST_MATH_BIG_CONSTANT(T, 64, 3.264572499406168221382e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 8.584972047303151034100e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 8.422082733280017909550e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 3.738005441471368178383e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 7.016938390144121276609e+01), BOOST_MATH_BIG_CONSTANT(T, 64, 4.319614662598089438939e+00), BOOST_MATH_BIG_CONSTANT(T, 64, 3.710715864316521856193e-02) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 64, 2.298433045824439052398e+01), BOOST_MATH_BIG_CONSTANT(T, 64, 2.082047745067709230037e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 9.662367854250262046592e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 2.504148628460454004686e+03), BOOST_MATH_BIG_CONSTANT(T, 64, 3.712730364911389908905e+03), BOOST_MATH_BIG_CONSTANT(T, 64, 3.108002081150068641112e+03), BOOST_MATH_BIG_CONSTANT(T, 64, 1.400149940532448553143e+03), BOOST_MATH_BIG_CONSTANT(T, 64, 3.083303048095846226299e+02), BOOST_MATH_BIG_CONSTANT(T, 64, 2.748706060530351833346e+01), BOOST_MATH_BIG_CONSTANT(T, 64, 6.321900849331506946977e-01), }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k1_imp(const T& x, const std::integral_constant<int, 113>&) { BOOST_MATH_STD_USING if(x <= 1) { // Maximum Deviation Found: 7.120e-35 // Expected Error Term : -7.119e-35 // Maximum Relative Change in Control Points : 1.207e-03 // Max Error found at float128 precision = Poly : 7.143688e-35 static const T Y = 8.695471286773681640625000000000000000e-02f; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 113, -3.621379534403483072916666666666595475e-03), BOOST_MATH_BIG_CONSTANT(T, 113, 7.074117676930975433219826471336547627e-03), BOOST_MATH_BIG_CONSTANT(T, 113, 9.631337631362776369069668419033041661e-05), BOOST_MATH_BIG_CONSTANT(T, 113, 3.468935967870048731821071646104412775e-06), BOOST_MATH_BIG_CONSTANT(T, 113, 2.956705020559599861444492614737168261e-08), BOOST_MATH_BIG_CONSTANT(T, 113, 2.347140307321161346703214099534250263e-10), BOOST_MATH_BIG_CONSTANT(T, 113, 5.569608494081482873946791086435679661e-13) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 113, -3.580768910152105375615558920428350204e-02), BOOST_MATH_BIG_CONSTANT(T, 113, 6.197467671701485365363068445534557369e-04), BOOST_MATH_BIG_CONSTANT(T, 113, -6.707466533308630411966030561446666237e-06), BOOST_MATH_BIG_CONSTANT(T, 113, 4.846687802282250112624373388491123527e-08), BOOST_MATH_BIG_CONSTANT(T, 113, -2.248493131151981569517383040323900343e-10), BOOST_MATH_BIG_CONSTANT(T, 113, 5.319279786372775264555728921709381080e-13) }; T a = x * x / 4; a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2; // Maximum Deviation Found: 4.473e-37 // Expected Error Term : 4.473e-37 // Maximum Relative Change in Control Points : 8.550e-04 // Max Error found at float128 precision = Poly : 8.167701e-35 static const T P2[] = { BOOST_MATH_BIG_CONSTANT(T, 113, -3.079657578292062244053600156878870690e-01), BOOST_MATH_BIG_CONSTANT(T, 113, -8.133183745732467770755578848987414875e-02), BOOST_MATH_BIG_CONSTANT(T, 113, -3.548968792764174773125420229299431951e-03), BOOST_MATH_BIG_CONSTANT(T, 113, -5.886125468718182876076972186152445490e-05), BOOST_MATH_BIG_CONSTANT(T, 113, -4.506712111733707245745396404449639865e-07), BOOST_MATH_BIG_CONSTANT(T, 113, -1.632502325880313239698965376754406011e-09), BOOST_MATH_BIG_CONSTANT(T, 113, -2.311973065898784812266544485665624227e-12) }; static const T Q2[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 113, -1.311471216733781016657962995723287450e-02), BOOST_MATH_BIG_CONSTANT(T, 113, 8.571876054797365417068164018709472969e-05), BOOST_MATH_BIG_CONSTANT(T, 113, -3.630181215268238731442496851497901293e-07), BOOST_MATH_BIG_CONSTANT(T, 113, 1.070176111227805048604885986867484807e-09), BOOST_MATH_BIG_CONSTANT(T, 113, -2.129046580769872602793220056461084761e-12), BOOST_MATH_BIG_CONSTANT(T, 113, 2.294906469421390890762001971790074432e-15) }; return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a; } else if(x < 4) { // Max error in interpolated form: 5.307e-37 // Max Error found at float128 precision = Poly: 7.087862e-35 static const T Y = 1.5023040771484375f; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 113, -2.489899398329369710528254347931380044e-01), BOOST_MATH_BIG_CONSTANT(T, 113, -6.819080211203854781858815596508456873e+00), BOOST_MATH_BIG_CONSTANT(T, 113, -7.599915699069767382647695624952723034e+01), BOOST_MATH_BIG_CONSTANT(T, 113, -4.450211910821295507926582231071300718e+02), BOOST_MATH_BIG_CONSTANT(T, 113, -1.451374687870925175794150513723956533e+03), BOOST_MATH_BIG_CONSTANT(T, 113, -2.405805746895098802803503988539098226e+03), BOOST_MATH_BIG_CONSTANT(T, 113, -5.638808326778389656403861103277220518e+02), BOOST_MATH_BIG_CONSTANT(T, 113, 5.513958744081268456191778822780865708e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 1.121301640926540743072258116122834804e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 1.080094900175649541266613109971296190e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 5.896531083639613332407534434915552429e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 1.856602122319645694042555107114028437e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 3.237121918853145421414003823957537419e+02), BOOST_MATH_BIG_CONSTANT(T, 113, 2.842072954561323076230238664623893504e+01), BOOST_MATH_BIG_CONSTANT(T, 113, 1.039705646510167437971862966128055524e+00), BOOST_MATH_BIG_CONSTANT(T, 113, 1.008418100718254816100425022904039530e-02) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 113, 2.927456835239137986889227412815459529e+01), BOOST_MATH_BIG_CONSTANT(T, 113, 3.598985593265577043711382994516531273e+02), BOOST_MATH_BIG_CONSTANT(T, 113, 2.449897377085510281395819892689690579e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 1.025555887684561913263090023158085327e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 2.774140447181062463181892531100679195e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 4.962055507843204417243602332246120418e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 5.908269326976180183216954452196772931e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 4.655160454422016855911700790722577942e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 2.383586885019548163464418964577684608e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 7.679920375586960324298491662159976419e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 1.478586421028842906987799049804565008e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 1.565384974896746094224942654383537090e+02), BOOST_MATH_BIG_CONSTANT(T, 113, 7.902617937084010911005732488607114511e+00), BOOST_MATH_BIG_CONSTANT(T, 113, 1.429293010387921526110949911029094926e-01), BOOST_MATH_BIG_CONSTANT(T, 113, 3.880342607911083143560111853491047663e-04) }; return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); } else { // Maximum Deviation Found: 4.359e-37 // Expected Error Term : -6.565e-40 // Maximum Relative Change in Control Points : 1.880e-01 // Max Error found at float128 precision = Poly : 2.943572e-35 static const T Y = 1.308816909790039062500000000000000000f; static const T P[] = { BOOST_MATH_BIG_CONSTANT(T, 113, -5.550277247453881129211735759447737350e-02), BOOST_MATH_BIG_CONSTANT(T, 113, -3.485883080219574328217554864956175929e+00), BOOST_MATH_BIG_CONSTANT(T, 113, -8.903760658131484239300875153154881958e+01), BOOST_MATH_BIG_CONSTANT(T, 113, -1.144813672213626237418235110712293337e+03), BOOST_MATH_BIG_CONSTANT(T, 113, -6.498400501156131446691826557494158173e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 1.573531831870363502604119835922166116e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 5.417416550054632009958262596048841154e+05), BOOST_MATH_BIG_CONSTANT(T, 113, 4.271266450613557412825896604269130661e+06), BOOST_MATH_BIG_CONSTANT(T, 113, 1.898386013314389952534433455681107783e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 5.353798784656436259250791761023512750e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 9.839619195427352438957774052763490067e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 1.169246368651532232388152442538005637e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 8.696368884166831199967845883371116431e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 3.810226630422736458064005843327500169e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 8.854996610560406127438950635716757614e+06), BOOST_MATH_BIG_CONSTANT(T, 113, 8.981057433937398731355768088809437625e+05), BOOST_MATH_BIG_CONSTANT(T, 113, 2.519440069856232098711793483639792952e+04) }; static const T Q[] = { BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00), BOOST_MATH_BIG_CONSTANT(T, 113, 7.127348248283623146544565916604103560e+01), BOOST_MATH_BIG_CONSTANT(T, 113, 2.205092684176906740104488180754982065e+03), BOOST_MATH_BIG_CONSTANT(T, 113, 3.911249195069050636298346469740075758e+04), BOOST_MATH_BIG_CONSTANT(T, 113, 4.426103406579046249654548481377792614e+05), BOOST_MATH_BIG_CONSTANT(T, 113, 3.365861555422488771286500241966208541e+06), BOOST_MATH_BIG_CONSTANT(T, 113, 1.765377714160383676864913709252529840e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 6.453822726931857253365138260720815246e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 1.643207885048369990391975749439783892e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 2.882540678243694621895816336640877878e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 3.410120808992380266174106812005338148e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 2.628138016559335882019310900426773027e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 1.250794693811010646965360198541047961e+08), BOOST_MATH_BIG_CONSTANT(T, 113, 3.378723408195485594610593014072950078e+07), BOOST_MATH_BIG_CONSTANT(T, 113, 4.488253856312453816451380319061865560e+06), BOOST_MATH_BIG_CONSTANT(T, 113, 2.202167197882689873967723350537104582e+05), BOOST_MATH_BIG_CONSTANT(T, 113, 1.673233230356966539460728211412989843e+03) }; if(x < tools::log_max_value<T>()) return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x)); else { T ex = exp(-x / 2); return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex; } } } template <typename T> T bessel_k1_imp(const T& x, const std::integral_constant<int, 0>&) { if(boost::math::tools::digits<T>() <= 24) return bessel_k1_imp(x, std::integral_constant<int, 24>()); else if(boost::math::tools::digits<T>() <= 53) return bessel_k1_imp(x, std::integral_constant<int, 53>()); else if(boost::math::tools::digits<T>() <= 64) return bessel_k1_imp(x, std::integral_constant<int, 64>()); else if(boost::math::tools::digits<T>() <= 113) return bessel_k1_imp(x, std::integral_constant<int, 113>()); BOOST_MATH_ASSERT(0); return 0; } template <typename T> inline T bessel_k1(const T& x) { typedef std::integral_constant<int, ((std::numeric_limits<T>::digits == 0) || (std::numeric_limits<T>::radix != 2)) ? 0 : std::numeric_limits<T>::digits <= 24 ? 24 : std::numeric_limits<T>::digits <= 53 ? 53 : std::numeric_limits<T>::digits <= 64 ? 64 : std::numeric_limits<T>::digits <= 113 ? 113 : -1 > tag_type; bessel_k1_initializer<T, tag_type>::force_instantiate(); return bessel_k1_imp(x, tag_type()); } }}} // namespaces #ifdef _MSC_VER #pragma warning(pop) #endif #endif // BOOST_MATH_BESSEL_K1_HPP