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