special_functions/detail/bessel_k1.hpp

0001 //  Copyright (c) 2006 Xiaogang Zhang
0002 //  Copyright (c) 2017 John Maddock
0003 //  Use, modification and distribution are subject to the
0004 //  Boost Software License, Version 1.0. (See accompanying file
0005 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
0006
0007 #ifndef BOOST_MATH_BESSEL_K1_HPP
0008 #define BOOST_MATH_BESSEL_K1_HPP
0009
0010 #ifdef _MSC_VER
0011 #pragma once
0012 #pragma warning(push)
0013 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
0014 #endif
0015
0016 #include <boost/math/tools/rational.hpp>
0017 #include <boost/math/tools/big_constant.hpp>
0018 #include <boost/math/policies/error_handling.hpp>
0019 #include <boost/math/tools/assert.hpp>
0020
0021 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
0022 //
0023 // This is the only way we can avoid
0024 // warning: non-standard suffix on floating constant [-Wpedantic]
0025 // when building with -Wall -pedantic.  Neither __extension__
0026 // nor #pragma diagnostic ignored work :(
0027 //
0028 #pragma GCC system_header
0029 #endif
0030
0031 // Modified Bessel function of the second kind of order zero
0032 // minimax rational approximations on intervals, see
0033 // Russon and Blair, Chalk River Report AECL-3461, 1969,
0034 // as revised by Pavel Holoborodko in "Rational Approximations
0035 // for the Modified Bessel Function of the Second Kind - K0(x)
0036 // for Computations with Double Precision", see
0037 // http://www.advanpix.com/2016/01/05/rational-approximations-for-the-modified-bessel-function-of-the-second-kind-k1-for-computations-with-double-precision/
0038 //
0039 // The actual coefficients used are our own derivation (by JM)
0040 // since we extend to both greater and lesser precision than the
0041 // references above.  We can also improve performance WRT to
0042 // Holoborodko without loss of precision.
0043
0044 namespace boost { namespace math { namespace detail{
0045
0046    template <typename T>
0047    T bessel_k1(const T&);
0048
0049    template <class T, class tag>
0050    struct bessel_k1_initializer
0051    {
0052       struct init
0053       {
0054          init()
0055          {
0056             do_init(tag());
0057          }
0058          static void do_init(const std::integral_constant<int, 113>&)
0059          {
0060             bessel_k1(T(0.5));
0061             bessel_k1(T(2));
0062             bessel_k1(T(6));
0063          }
0064          static void do_init(const std::integral_constant<int, 64>&)
0065          {
0066             bessel_k1(T(0.5));
0067             bessel_k1(T(6));
0068          }
0069          template <class U>
0070          static void do_init(const U&) {}
0071          void force_instantiate()const {}
0072       };
0073       static const init initializer;
0074       static void force_instantiate()
0075       {
0076          initializer.force_instantiate();
0077       }
0078    };
0079
0080    template <class T, class tag>
0081    const typename bessel_k1_initializer<T, tag>::init bessel_k1_initializer<T, tag>::initializer;
0082
0083
0084    template <typename T, int N>
0085    inline T bessel_k1_imp(const T&, const std::integral_constant<int, N>&)
0086    {
0087       BOOST_MATH_ASSERT(0);
0088       return 0;
0089    }
0090
0091    template <typename T>
0092    T bessel_k1_imp(const T& x, const std::integral_constant<int, 24>&)
0093    {
0094       BOOST_MATH_STD_USING
0095       if(x <= 1)
0096       {
0097          // Maximum Deviation Found:                     3.090e-12
0098          // Expected Error Term : -3.053e-12
0099          // Maximum Relative Change in Control Points : 4.927e-02
0100          // Max Error found at float precision = Poly : 7.918347e-10
0101          static const T Y = 8.695471287e-02f;
0102          static const T P[] =
0103          {
0104             -3.621379531e-03f,
0105             7.131781976e-03f,
0106             -1.535278300e-05f
0107          };
0108          static const T Q[] =
0109          {
0110             1.000000000e+00f,
0111             -5.173102701e-02f,
0112             9.203530671e-04f
0113          };
0114
0115          T a = x * x / 4;
0116          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
0117
0118          // Maximum Deviation Found:                     3.556e-08
0119          // Expected Error Term : -3.541e-08
0120          // Maximum Relative Change in Control Points : 8.203e-02
0121          static const T P2[] =
0122          {
0123             -3.079657469e-01f,
0124             -8.537108913e-02f,
0125             -4.640275408e-03f,
0126             -1.156442414e-04f
0127          };
0128
0129          return tools::evaluate_polynomial(P2, T(x * x)) * x + 1 / x + log(x) * a;
0130       }
0131       else
0132       {
0133          // Maximum Deviation Found:                     3.369e-08
0134          // Expected Error Term : -3.227e-08
0135          // Maximum Relative Change in Control Points : 9.917e-02
0136          // Max Error found at float precision = Poly : 6.084411e-08
0137          static const T Y = 1.450342178f;
0138          static const T P[] =
0139          {
0140             -1.970280088e-01f,
0141             2.188747807e-02f,
0142             7.270394756e-01f,
0143             2.490678196e-01f
0144          };
0145          static const T Q[] =
0146          {
0147             1.000000000e+00f,
0148             2.274292882e+00f,
0149             9.904984851e-01f,
0150             4.585534549e-02f
0151          };
0152          if(x < tools::log_max_value<T>())
0153             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
0154          else
0155          {
0156             T ex = exp(-x / 2);
0157             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
0158          }
0159       }
0160    }
0161
0162    template <typename T>
0163    T bessel_k1_imp(const T& x, const std::integral_constant<int, 53>&)
0164    {
0165       BOOST_MATH_STD_USING
0166       if(x <= 1)
0167       {
0168          // Maximum Deviation Found:                     1.922e-17
0169          // Expected Error Term : 1.921e-17
0170          // Maximum Relative Change in Control Points : 5.287e-03
0171          // Max Error found at double precision = Poly : 2.004747e-17
0172          static const T Y = 8.69547128677368164e-02f;
0173          static const T P[] =
0174          {
0175             -3.62137953440350228e-03,
0176             7.11842087490330300e-03,
0177             1.00302560256614306e-05,
0178             1.77231085381040811e-06
0179          };
0180          static const T Q[] =
0181          {
0182             1.00000000000000000e+00,
0183             -4.80414794429043831e-02,
0184             9.85972641934416525e-04,
0185             -8.91196859397070326e-06
0186          };
0187
0188          T a = x * x / 4;
0189          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
0190
0191          // Maximum Deviation Found:                     4.053e-17
0192          // Expected Error Term : -4.053e-17
0193          // Maximum Relative Change in Control Points : 3.103e-04
0194          // Max Error found at double precision = Poly : 1.246698e-16
0195
0196          static const T P2[] =
0197          {
0198             -3.07965757829206184e-01,
0199             -7.80929703673074907e-02,
0200             -2.70619343754051620e-03,
0201             -2.49549522229072008e-05
0202          };
0203          static const T Q2[] =
0204          {
0205             1.00000000000000000e+00,
0206             -2.36316836412163098e-02,
0207             2.64524577525962719e-04,
0208             -1.49749618004162787e-06
0209          };
0210
0211          return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a;
0212       }
0213       else
0214       {
0215          // Maximum Deviation Found:                     8.883e-17
0216          // Expected Error Term : -1.641e-17
0217          // Maximum Relative Change in Control Points : 2.786e-01
0218          // Max Error found at double precision = Poly : 1.258798e-16
0219
0220          static const T Y = 1.45034217834472656f;
0221          static const T P[] =
0222          {
0223             -1.97028041029226295e-01,
0224             -2.32408961548087617e+00,
0225             -7.98269784507699938e+00,
0226             -2.39968410774221632e+00,
0227             3.28314043780858713e+01,
0228             5.67713761158496058e+01,
0229             3.30907788466509823e+01,
0230             6.62582288933739787e+00,
0231             3.08851840645286691e-01
0232          };
0233          static const T Q[] =
0234          {
0235             1.00000000000000000e+00,
0236             1.41811409298826118e+01,
0237             7.35979466317556420e+01,
0238             1.77821793937080859e+02,
0239             2.11014501598705982e+02,
0240             1.19425262951064454e+02,
0241             2.88448064302447607e+01,
0242             2.27912927104139732e+00,
0243             2.50358186953478678e-02
0244          };
0245          if(x < tools::log_max_value<T>())
0246             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
0247          else
0248          {
0249             T ex = exp(-x / 2);
0250             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
0251          }
0252       }
0253    }
0254
0255    template <typename T>
0256    T bessel_k1_imp(const T& x, const std::integral_constant<int, 64>&)
0257    {
0258       BOOST_MATH_STD_USING
0259       if(x <= 1)
0260       {
0261          // Maximum Deviation Found:                     5.549e-23
0262          // Expected Error Term : -5.548e-23
0263          // Maximum Relative Change in Control Points : 2.002e-03
0264          // Max Error found at float80 precision = Poly : 9.352785e-22
0265          static const T Y = 8.695471286773681640625e-02f;
0266          static const T P[] =
0267          {
0268             BOOST_MATH_BIG_CONSTANT(T, 64, -3.621379534403483072861e-03),
0269             BOOST_MATH_BIG_CONSTANT(T, 64, 7.102135866103952705932e-03),
0270             BOOST_MATH_BIG_CONSTANT(T, 64, 4.167545240236717601167e-05),
0271             BOOST_MATH_BIG_CONSTANT(T, 64, 2.537484002571894870830e-06),
0272             BOOST_MATH_BIG_CONSTANT(T, 64, 6.603228256820000135990e-09)
0273          };
0274          static const T Q[] =
0275          {
0276             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
0277             BOOST_MATH_BIG_CONSTANT(T, 64, -4.354457194045068370363e-02),
0278             BOOST_MATH_BIG_CONSTANT(T, 64, 8.709137201220209072820e-04),
0279             BOOST_MATH_BIG_CONSTANT(T, 64, -9.676151796359590545143e-06),
0280             BOOST_MATH_BIG_CONSTANT(T, 64, 5.162715192766245311659e-08)
0281          };
0282
0283          T a = x * x / 4;
0284          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
0285
0286          // Maximum Deviation Found:                     1.995e-23
0287          // Expected Error Term : 1.995e-23
0288          // Maximum Relative Change in Control Points : 8.174e-04
0289          // Max Error found at float80 precision = Poly : 4.137325e-20
0290          static const T P2[] =
0291          {
0292             BOOST_MATH_BIG_CONSTANT(T, 64, -3.079657578292062244054e-01),
0293             BOOST_MATH_BIG_CONSTANT(T, 64, -7.963049154965966503231e-02),
0294             BOOST_MATH_BIG_CONSTANT(T, 64, -3.103277523735639924895e-03),
0295             BOOST_MATH_BIG_CONSTANT(T, 64, -4.023052834702215699504e-05),
0296             BOOST_MATH_BIG_CONSTANT(T, 64, -1.719459155018493821839e-07)
0297          };
0298          static const T Q2[] =
0299          {
0300             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
0301             BOOST_MATH_BIG_CONSTANT(T, 64, -1.863917670410152669768e-02),
0302             BOOST_MATH_BIG_CONSTANT(T, 64, 1.699367098849735298090e-04),
0303             BOOST_MATH_BIG_CONSTANT(T, 64, -9.309358790546076298429e-07),
0304             BOOST_MATH_BIG_CONSTANT(T, 64, 2.708893480271612711933e-09)
0305          };
0306
0307          return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a;
0308       }
0309       else
0310       {
0311          // Maximum Deviation Found:                     9.785e-20
0312          // Expected Error Term : -3.302e-21
0313          // Maximum Relative Change in Control Points : 3.432e-01
0314          // Max Error found at float80 precision = Poly : 1.083755e-19
0315          static const T Y = 1.450342178344726562500e+00f;
0316          static const T P[] =
0317          {
0318             BOOST_MATH_BIG_CONSTANT(T, 64, -1.970280410292263112917e-01),
0319             BOOST_MATH_BIG_CONSTANT(T, 64, -4.058564803062959169322e+00),
0320             BOOST_MATH_BIG_CONSTANT(T, 64, -3.036658174194917777473e+01),
0321             BOOST_MATH_BIG_CONSTANT(T, 64, -9.576825392332820142173e+01),
0322             BOOST_MATH_BIG_CONSTANT(T, 64, -6.706969489248020941949e+01),
0323             BOOST_MATH_BIG_CONSTANT(T, 64, 3.264572499406168221382e+02),
0324             BOOST_MATH_BIG_CONSTANT(T, 64, 8.584972047303151034100e+02),
0325             BOOST_MATH_BIG_CONSTANT(T, 64, 8.422082733280017909550e+02),
0326             BOOST_MATH_BIG_CONSTANT(T, 64, 3.738005441471368178383e+02),
0327             BOOST_MATH_BIG_CONSTANT(T, 64, 7.016938390144121276609e+01),
0328             BOOST_MATH_BIG_CONSTANT(T, 64, 4.319614662598089438939e+00),
0329             BOOST_MATH_BIG_CONSTANT(T, 64, 3.710715864316521856193e-02)
0330          };
0331          static const T Q[] =
0332          {
0333             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
0334             BOOST_MATH_BIG_CONSTANT(T, 64, 2.298433045824439052398e+01),
0335             BOOST_MATH_BIG_CONSTANT(T, 64, 2.082047745067709230037e+02),
0336             BOOST_MATH_BIG_CONSTANT(T, 64, 9.662367854250262046592e+02),
0337             BOOST_MATH_BIG_CONSTANT(T, 64, 2.504148628460454004686e+03),
0338             BOOST_MATH_BIG_CONSTANT(T, 64, 3.712730364911389908905e+03),
0339             BOOST_MATH_BIG_CONSTANT(T, 64, 3.108002081150068641112e+03),
0340             BOOST_MATH_BIG_CONSTANT(T, 64, 1.400149940532448553143e+03),
0341             BOOST_MATH_BIG_CONSTANT(T, 64, 3.083303048095846226299e+02),
0342             BOOST_MATH_BIG_CONSTANT(T, 64, 2.748706060530351833346e+01),
0343             BOOST_MATH_BIG_CONSTANT(T, 64, 6.321900849331506946977e-01),
0344          };
0345          if(x < tools::log_max_value<T>())
0346             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
0347          else
0348          {
0349             T ex = exp(-x / 2);
0350             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
0351          }
0352       }
0353    }
0354
0355    template <typename T>
0356    T bessel_k1_imp(const T& x, const std::integral_constant<int, 113>&)
0357    {
0358       BOOST_MATH_STD_USING
0359       if(x <= 1)
0360       {
0361          // Maximum Deviation Found:                     7.120e-35
0362          // Expected Error Term : -7.119e-35
0363          // Maximum Relative Change in Control Points : 1.207e-03
0364          // Max Error found at float128 precision = Poly : 7.143688e-35
0365          static const T Y = 8.695471286773681640625000000000000000e-02f;
0366          static const T P[] =
0367          {
0368             BOOST_MATH_BIG_CONSTANT(T, 113, -3.621379534403483072916666666666595475e-03),
0369             BOOST_MATH_BIG_CONSTANT(T, 113, 7.074117676930975433219826471336547627e-03),
0370             BOOST_MATH_BIG_CONSTANT(T, 113, 9.631337631362776369069668419033041661e-05),
0371             BOOST_MATH_BIG_CONSTANT(T, 113, 3.468935967870048731821071646104412775e-06),
0372             BOOST_MATH_BIG_CONSTANT(T, 113, 2.956705020559599861444492614737168261e-08),
0373             BOOST_MATH_BIG_CONSTANT(T, 113, 2.347140307321161346703214099534250263e-10),
0374             BOOST_MATH_BIG_CONSTANT(T, 113, 5.569608494081482873946791086435679661e-13)
0375          };
0376          static const T Q[] =
0377          {
0378             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
0379             BOOST_MATH_BIG_CONSTANT(T, 113, -3.580768910152105375615558920428350204e-02),
0380             BOOST_MATH_BIG_CONSTANT(T, 113, 6.197467671701485365363068445534557369e-04),
0381             BOOST_MATH_BIG_CONSTANT(T, 113, -6.707466533308630411966030561446666237e-06),
0382             BOOST_MATH_BIG_CONSTANT(T, 113, 4.846687802282250112624373388491123527e-08),
0383             BOOST_MATH_BIG_CONSTANT(T, 113, -2.248493131151981569517383040323900343e-10),
0384             BOOST_MATH_BIG_CONSTANT(T, 113, 5.319279786372775264555728921709381080e-13)
0385          };
0386
0387          T a = x * x / 4;
0388          a = ((tools::evaluate_rational(P, Q, a) + Y) * a * a + a / 2 + 1) * x / 2;
0389
0390          // Maximum Deviation Found:                     4.473e-37
0391          // Expected Error Term : 4.473e-37
0392          // Maximum Relative Change in Control Points : 8.550e-04
0393          // Max Error found at float128 precision = Poly : 8.167701e-35
0394          static const T P2[] =
0395          {
0396             BOOST_MATH_BIG_CONSTANT(T, 113, -3.079657578292062244053600156878870690e-01),
0397             BOOST_MATH_BIG_CONSTANT(T, 113, -8.133183745732467770755578848987414875e-02),
0398             BOOST_MATH_BIG_CONSTANT(T, 113, -3.548968792764174773125420229299431951e-03),
0399             BOOST_MATH_BIG_CONSTANT(T, 113, -5.886125468718182876076972186152445490e-05),
0400             BOOST_MATH_BIG_CONSTANT(T, 113, -4.506712111733707245745396404449639865e-07),
0401             BOOST_MATH_BIG_CONSTANT(T, 113, -1.632502325880313239698965376754406011e-09),
0402             BOOST_MATH_BIG_CONSTANT(T, 113, -2.311973065898784812266544485665624227e-12)
0403          };
0404          static const T Q2[] =
0405          {
0406             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
0407             BOOST_MATH_BIG_CONSTANT(T, 113, -1.311471216733781016657962995723287450e-02),
0408             BOOST_MATH_BIG_CONSTANT(T, 113, 8.571876054797365417068164018709472969e-05),
0409             BOOST_MATH_BIG_CONSTANT(T, 113, -3.630181215268238731442496851497901293e-07),
0410             BOOST_MATH_BIG_CONSTANT(T, 113, 1.070176111227805048604885986867484807e-09),
0411             BOOST_MATH_BIG_CONSTANT(T, 113, -2.129046580769872602793220056461084761e-12),
0412             BOOST_MATH_BIG_CONSTANT(T, 113, 2.294906469421390890762001971790074432e-15)
0413          };
0414
0415          return tools::evaluate_rational(P2, Q2, T(x * x)) * x + 1 / x + log(x) * a;
0416       }
0417       else if(x < 4)
0418       {
0419          // Max error in interpolated form: 5.307e-37
0420          // Max Error found at float128 precision = Poly: 7.087862e-35
0421          static const T Y = 1.5023040771484375f;
0422          static const T P[] =
0423          {
0424             BOOST_MATH_BIG_CONSTANT(T, 113, -2.489899398329369710528254347931380044e-01),
0425             BOOST_MATH_BIG_CONSTANT(T, 113, -6.819080211203854781858815596508456873e+00),
0426             BOOST_MATH_BIG_CONSTANT(T, 113, -7.599915699069767382647695624952723034e+01),
0427             BOOST_MATH_BIG_CONSTANT(T, 113, -4.450211910821295507926582231071300718e+02),
0428             BOOST_MATH_BIG_CONSTANT(T, 113, -1.451374687870925175794150513723956533e+03),
0429             BOOST_MATH_BIG_CONSTANT(T, 113, -2.405805746895098802803503988539098226e+03),
0430             BOOST_MATH_BIG_CONSTANT(T, 113, -5.638808326778389656403861103277220518e+02),
0431             BOOST_MATH_BIG_CONSTANT(T, 113, 5.513958744081268456191778822780865708e+03),
0432             BOOST_MATH_BIG_CONSTANT(T, 113, 1.121301640926540743072258116122834804e+04),
0433             BOOST_MATH_BIG_CONSTANT(T, 113, 1.080094900175649541266613109971296190e+04),
0434             BOOST_MATH_BIG_CONSTANT(T, 113, 5.896531083639613332407534434915552429e+03),
0435             BOOST_MATH_BIG_CONSTANT(T, 113, 1.856602122319645694042555107114028437e+03),
0436             BOOST_MATH_BIG_CONSTANT(T, 113, 3.237121918853145421414003823957537419e+02),
0437             BOOST_MATH_BIG_CONSTANT(T, 113, 2.842072954561323076230238664623893504e+01),
0438             BOOST_MATH_BIG_CONSTANT(T, 113, 1.039705646510167437971862966128055524e+00),
0439             BOOST_MATH_BIG_CONSTANT(T, 113, 1.008418100718254816100425022904039530e-02)
0440          };
0441          static const T Q[] =
0442          {
0443             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
0444             BOOST_MATH_BIG_CONSTANT(T, 113, 2.927456835239137986889227412815459529e+01),
0445             BOOST_MATH_BIG_CONSTANT(T, 113, 3.598985593265577043711382994516531273e+02),
0446             BOOST_MATH_BIG_CONSTANT(T, 113, 2.449897377085510281395819892689690579e+03),
0447             BOOST_MATH_BIG_CONSTANT(T, 113, 1.025555887684561913263090023158085327e+04),
0448             BOOST_MATH_BIG_CONSTANT(T, 113, 2.774140447181062463181892531100679195e+04),
0449             BOOST_MATH_BIG_CONSTANT(T, 113, 4.962055507843204417243602332246120418e+04),
0450             BOOST_MATH_BIG_CONSTANT(T, 113, 5.908269326976180183216954452196772931e+04),
0451             BOOST_MATH_BIG_CONSTANT(T, 113, 4.655160454422016855911700790722577942e+04),
0452             BOOST_MATH_BIG_CONSTANT(T, 113, 2.383586885019548163464418964577684608e+04),
0453             BOOST_MATH_BIG_CONSTANT(T, 113, 7.679920375586960324298491662159976419e+03),
0454             BOOST_MATH_BIG_CONSTANT(T, 113, 1.478586421028842906987799049804565008e+03),
0455             BOOST_MATH_BIG_CONSTANT(T, 113, 1.565384974896746094224942654383537090e+02),
0456             BOOST_MATH_BIG_CONSTANT(T, 113, 7.902617937084010911005732488607114511e+00),
0457             BOOST_MATH_BIG_CONSTANT(T, 113, 1.429293010387921526110949911029094926e-01),
0458             BOOST_MATH_BIG_CONSTANT(T, 113, 3.880342607911083143560111853491047663e-04)
0459          };
0460          return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
0461       }
0462       else
0463       {
0464          // Maximum Deviation Found:                     4.359e-37
0465          // Expected Error Term : -6.565e-40
0466          // Maximum Relative Change in Control Points : 1.880e-01
0467          // Max Error found at float128 precision = Poly : 2.943572e-35
0468          static const T Y = 1.308816909790039062500000000000000000f;
0469          static const T P[] =
0470          {
0471             BOOST_MATH_BIG_CONSTANT(T, 113, -5.550277247453881129211735759447737350e-02),
0472             BOOST_MATH_BIG_CONSTANT(T, 113, -3.485883080219574328217554864956175929e+00),
0473             BOOST_MATH_BIG_CONSTANT(T, 113, -8.903760658131484239300875153154881958e+01),
0474             BOOST_MATH_BIG_CONSTANT(T, 113, -1.144813672213626237418235110712293337e+03),
0475             BOOST_MATH_BIG_CONSTANT(T, 113, -6.498400501156131446691826557494158173e+03),
0476             BOOST_MATH_BIG_CONSTANT(T, 113, 1.573531831870363502604119835922166116e+04),
0477             BOOST_MATH_BIG_CONSTANT(T, 113, 5.417416550054632009958262596048841154e+05),
0478             BOOST_MATH_BIG_CONSTANT(T, 113, 4.271266450613557412825896604269130661e+06),
0479             BOOST_MATH_BIG_CONSTANT(T, 113, 1.898386013314389952534433455681107783e+07),
0480             BOOST_MATH_BIG_CONSTANT(T, 113, 5.353798784656436259250791761023512750e+07),
0481             BOOST_MATH_BIG_CONSTANT(T, 113, 9.839619195427352438957774052763490067e+07),
0482             BOOST_MATH_BIG_CONSTANT(T, 113, 1.169246368651532232388152442538005637e+08),
0483             BOOST_MATH_BIG_CONSTANT(T, 113, 8.696368884166831199967845883371116431e+07),
0484             BOOST_MATH_BIG_CONSTANT(T, 113, 3.810226630422736458064005843327500169e+07),
0485             BOOST_MATH_BIG_CONSTANT(T, 113, 8.854996610560406127438950635716757614e+06),
0486             BOOST_MATH_BIG_CONSTANT(T, 113, 8.981057433937398731355768088809437625e+05),
0487             BOOST_MATH_BIG_CONSTANT(T, 113, 2.519440069856232098711793483639792952e+04)
0488          };
0489          static const T Q[] =
0490          {
0491             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
0492             BOOST_MATH_BIG_CONSTANT(T, 113, 7.127348248283623146544565916604103560e+01),
0493             BOOST_MATH_BIG_CONSTANT(T, 113, 2.205092684176906740104488180754982065e+03),
0494             BOOST_MATH_BIG_CONSTANT(T, 113, 3.911249195069050636298346469740075758e+04),
0495             BOOST_MATH_BIG_CONSTANT(T, 113, 4.426103406579046249654548481377792614e+05),
0496             BOOST_MATH_BIG_CONSTANT(T, 113, 3.365861555422488771286500241966208541e+06),
0497             BOOST_MATH_BIG_CONSTANT(T, 113, 1.765377714160383676864913709252529840e+07),
0498             BOOST_MATH_BIG_CONSTANT(T, 113, 6.453822726931857253365138260720815246e+07),
0499             BOOST_MATH_BIG_CONSTANT(T, 113, 1.643207885048369990391975749439783892e+08),
0500             BOOST_MATH_BIG_CONSTANT(T, 113, 2.882540678243694621895816336640877878e+08),
0501             BOOST_MATH_BIG_CONSTANT(T, 113, 3.410120808992380266174106812005338148e+08),
0502             BOOST_MATH_BIG_CONSTANT(T, 113, 2.628138016559335882019310900426773027e+08),
0503             BOOST_MATH_BIG_CONSTANT(T, 113, 1.250794693811010646965360198541047961e+08),
0504             BOOST_MATH_BIG_CONSTANT(T, 113, 3.378723408195485594610593014072950078e+07),
0505             BOOST_MATH_BIG_CONSTANT(T, 113, 4.488253856312453816451380319061865560e+06),
0506             BOOST_MATH_BIG_CONSTANT(T, 113, 2.202167197882689873967723350537104582e+05),
0507             BOOST_MATH_BIG_CONSTANT(T, 113, 1.673233230356966539460728211412989843e+03)
0508          };
0509          if(x < tools::log_max_value<T>())
0510             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
0511          else
0512          {
0513             T ex = exp(-x / 2);
0514             return ((tools::evaluate_polynomial(P, T(1 / x)) / tools::evaluate_polynomial(Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
0515          }
0516       }
0517     }
0518
0519     template <typename T>
0520     T bessel_k1_imp(const T& x, const std::integral_constant<int, 0>&)
0521     {
0522        if(boost::math::tools::digits<T>() <= 24)
0523           return bessel_k1_imp(x, std::integral_constant<int, 24>());
0524        else if(boost::math::tools::digits<T>() <= 53)
0525           return bessel_k1_imp(x, std::integral_constant<int, 53>());
0526        else if(boost::math::tools::digits<T>() <= 64)
0527           return bessel_k1_imp(x, std::integral_constant<int, 64>());
0528        else if(boost::math::tools::digits<T>() <= 113)
0529           return bessel_k1_imp(x, std::integral_constant<int, 113>());
0530        BOOST_MATH_ASSERT(0);
0531        return 0;
0532     }
0533
0534     template <typename T>
0535    inline T bessel_k1(const T& x)
0536    {
0537       typedef std::integral_constant<int,
0538          ((std::numeric_limits<T>::digits == 0) || (std::numeric_limits<T>::radix != 2)) ?
0539          0 :
0540          std::numeric_limits<T>::digits <= 24 ?
0541          24 :
0542          std::numeric_limits<T>::digits <= 53 ?
0543          53 :
0544          std::numeric_limits<T>::digits <= 64 ?
0545          64 :
0546          std::numeric_limits<T>::digits <= 113 ?
0547          113 : -1
0548       > tag_type;
0549
0550       bessel_k1_initializer<T, tag_type>::force_instantiate();
0551       return bessel_k1_imp(x, tag_type());
0552    }
0553
0554 }}} // namespaces
0555
0556 #ifdef _MSC_VER
0557 #pragma warning(pop)
0558 #endif
0559
0560 #endif // BOOST_MATH_BESSEL_K1_HPP
0561