ceph/src/boost/boost/math/special_functions/detail/bessel_k0.hpp

   1 //  Copyright (c) 2006 Xiaogang Zhang
   2 //  Copyright (c) 2017 John Maddock
   3 //  Use, modification and distribution are subject to the
   4 //  Boost Software License, Version 1.0. (See accompanying file
   5 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   6
   7 #ifndef BOOST_MATH_BESSEL_K0_HPP
   8 #define BOOST_MATH_BESSEL_K0_HPP
   9
  10 #ifdef _MSC_VER
  11 #pragma once
  12 #pragma warning(push)
  13 #pragma warning(disable:4702) // Unreachable code (release mode only warning)
  14 #endif
  15
  16 #include <boost/math/tools/rational.hpp>
  17 #include <boost/math/tools/big_constant.hpp>
  18 #include <boost/math/policies/error_handling.hpp>
  19 #include <boost/assert.hpp>
  20
  21 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
  22 //
  23 // This is the only way we can avoid
  24 // warning: non-standard suffix on floating constant [-Wpedantic]
  25 // when building with -Wall -pedantic.  Neither __extension__
  26 // nor #pragma dianostic ignored work :(
  27 //
  28 #pragma GCC system_header
  29 #endif
  30
  31 // Modified Bessel function of the second kind of order zero
  32 // minimax rational approximations on intervals, see
  33 // Russon and Blair, Chalk River Report AECL-3461, 1969,
  34 // as revised by Pavel Holoborodko in "Rational Approximations
  35 // for the Modified Bessel Function of the Second Kind - K0(x)
  36 // for Computations with Double Precision", see
  37 // http://www.advanpix.com/2015/11/25/rational-approximations-for-the-modified-bessel-function-of-the-second-kind-k0-for-computations-with-double-precision/
  38 //
  39 // The actual coefficients used are our own derivation (by JM)
  40 // since we extend to both greater and lesser precision than the
  41 // references above.  We can also improve performance WRT to
  42 // Holoborodko without loss of precision.
  43
  44 namespace boost { namespace math { namespace detail{
  45
  46 template <typename T>
  47 T bessel_k0(const T& x);
  48
  49 template <class T, class tag>
  50 struct bessel_k0_initializer
  51 {
  52    struct init
  53    {
  54       init()
  55       {
  56          do_init(tag());
  57       }
  58       static void do_init(const mpl::int_<113>&)
  59       {
  60          bessel_k0(T(0.5));
  61          bessel_k0(T(1.5));
  62       }
  63       static void do_init(const mpl::int_<64>&)
  64       {
  65          bessel_k0(T(0.5));
  66          bessel_k0(T(1.5));
  67       }
  68       template <class U>
  69       static void do_init(const U&){}
  70       void force_instantiate()const{}
  71    };
  72    static const init initializer;
  73    static void force_instantiate()
  74    {
  75       initializer.force_instantiate();
  76    }
  77 };
  78
  79 template <class T, class tag>
  80 const typename bessel_k0_initializer<T, tag>::init bessel_k0_initializer<T, tag>::initializer;
  81
  82
  83 template <typename T, int N>
  84 T bessel_k0_imp(const T& x, const mpl::int_<N>&)
  85 {
  86    BOOST_ASSERT(0);
  87    return 0;
  88 }
  89
  90 template <typename T>
  91 T bessel_k0_imp(const T& x, const mpl::int_<24>&)
  92 {
  93    BOOST_MATH_STD_USING
  94    if(x <= 1)
  95    {
  96       // Maximum Deviation Found : 2.358e-09
  97       // Expected Error Term : -2.358e-09
  98       // Maximum Relative Change in Control Points : 9.552e-02
  99       // Max Error found at float precision = Poly : 4.448220e-08
 100       static const T Y = 1.137250900268554688f;
 101       static const T P[] =
 102       {
 103          -1.372508979104259711e-01f,
 104          2.622545986273687617e-01f,
 105          5.047103728247919836e-03f
 106       };
 107       static const T Q[] =
 108       {
 109          1.000000000000000000e+00f,
 110          -8.928694018000029415e-02f,
 111          2.985980684180969241e-03f
 112       };
 113       T a = x * x / 4;
 114       a = (tools::evaluate_rational(P, Q, a) + Y) * a + 1;
 115
 116       // Maximum Deviation Found:                     1.346e-09
 117       // Expected Error Term : -1.343e-09
 118       // Maximum Relative Change in Control Points : 2.405e-02
 119       // Max Error found at float precision = Poly : 1.354814e-07
 120       static const T P2[] = {
 121          1.159315158e-01f,
 122          2.789828686e-01f,
 123          2.524902861e-02f,
 124          8.457241514e-04f,
 125          1.530051997e-05f
 126       };
 127       return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a;
 128    }
 129    else
 130    {
 131       // Maximum Deviation Found:                     1.587e-08
 132       // Expected Error Term : 1.531e-08
 133       // Maximum Relative Change in Control Points : 9.064e-02
 134       // Max Error found at float precision = Poly : 5.065020e-08
 135
 136       static const T P[] =
 137       {
 138          2.533141220e-01f,
 139          5.221502603e-01f,
 140          6.380180669e-02f,
 141          -5.934976547e-02f
 142       };
 143       static const T Q[] =
 144       {
 145          1.000000000e+00f,
 146          2.679722431e+00f,
 147          1.561635813e+00f,
 148          1.573660661e-01f
 149       };
 150       if(x < tools::log_max_value<T>())
 151          return ((tools::evaluate_rational(P, Q, T(1 / x)) + 1) * exp(-x) / sqrt(x));
 152       else
 153       {
 154          T ex = exp(-x / 2);
 155          return ((tools::evaluate_rational(P, Q, T(1 / x)) + 1) * ex / sqrt(x)) * ex;
 156       }
 157    }
 158 }
 159
 160 template <typename T>
 161 T bessel_k0_imp(const T& x, const mpl::int_<53>&)
 162 {
 163    BOOST_MATH_STD_USING
 164    if(x <= 1)
 165    {
 166       // Maximum Deviation Found:                     6.077e-17
 167       // Expected Error Term : -6.077e-17
 168       // Maximum Relative Change in Control Points : 7.797e-02
 169       // Max Error found at double precision = Poly : 1.003156e-16
 170       static const T Y = 1.137250900268554688;
 171       static const T P[] =
 172       {
 173          -1.372509002685546267e-01,
 174          2.574916117833312855e-01,
 175          1.395474602146869316e-02,
 176          5.445476986653926759e-04,
 177          7.125159422136622118e-06
 178       };
 179       static const T Q[] =
 180       {
 181          1.000000000000000000e+00,
 182          -5.458333438017788530e-02,
 183          1.291052816975251298e-03,
 184          -1.367653946978586591e-05
 185       };
 186
 187       T a = x * x / 4;
 188       a = (tools::evaluate_polynomial(P, a) / tools::evaluate_polynomial(Q, a) + Y) * a + 1;
 189
 190       // Maximum Deviation Found:                     3.429e-18
 191       // Expected Error Term : 3.392e-18
 192       // Maximum Relative Change in Control Points : 2.041e-02
 193       // Max Error found at double precision = Poly : 2.513112e-16
 194       static const T P2[] =
 195       {
 196          1.159315156584124484e-01,
 197          2.789828789146031732e-01,
 198          2.524892993216121934e-02,
 199          8.460350907213637784e-04,
 200          1.491471924309617534e-05,
 201          1.627106892422088488e-07,
 202          1.208266102392756055e-09,
 203          6.611686391749704310e-12
 204       };
 205
 206       return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a;
 207    }
 208    else
 209    {
 210       // Maximum Deviation Found:                     4.316e-17
 211       // Expected Error Term : 9.570e-18
 212       // Maximum Relative Change in Control Points : 2.757e-01
 213       // Max Error found at double precision = Poly : 1.001560e-16
 214
 215       static const T Y = 1;
 216       static const T P[] =
 217       {
 218          2.533141373155002416e-01,
 219          3.628342133984595192e+00,
 220          1.868441889406606057e+01,
 221          4.306243981063412784e+01,
 222          4.424116209627428189e+01,
 223          1.562095339356220468e+01,
 224          -1.810138978229410898e+00,
 225          -1.414237994269995877e+00,
 226          -9.369168119754924625e-02
 227       };
 228       static const T Q[] =
 229       {
 230          1.000000000000000000e+00,
 231          1.494194694879908328e+01,
 232          8.265296455388554217e+01,
 233          2.162779506621866970e+02,
 234          2.845145155184222157e+02,
 235          1.851714491916334995e+02,
 236          5.486540717439723515e+01,
 237          6.118075837628957015e+00,
 238          1.586261269326235053e-01
 239       };
 240       if(x < tools::log_max_value<T>())
 241          return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 242       else
 243       {
 244          T ex = exp(-x / 2);
 245          return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
 246       }
 247    }
 248 }
 249
 250 template <typename T>
 251 T bessel_k0_imp(const T& x, const mpl::int_<64>&)
 252 {
 253    BOOST_MATH_STD_USING
 254       if(x <= 1)
 255       {
 256          // Maximum Deviation Found:                     2.180e-22
 257          // Expected Error Term : 2.180e-22
 258          // Maximum Relative Change in Control Points : 2.943e-01
 259          // Max Error found at float80 precision = Poly : 3.923207e-20
 260          static const T Y = 1.137250900268554687500e+00;
 261          static const T P[] =
 262          {
 263             BOOST_MATH_BIG_CONSTANT(T, 64, -1.372509002685546875002e-01),
 264             BOOST_MATH_BIG_CONSTANT(T, 64, 2.566481981037407600436e-01),
 265             BOOST_MATH_BIG_CONSTANT(T, 64, 1.551881122448948854873e-02),
 266             BOOST_MATH_BIG_CONSTANT(T, 64, 6.646112454323276529650e-04),
 267             BOOST_MATH_BIG_CONSTANT(T, 64, 1.213747930378196492543e-05),
 268             BOOST_MATH_BIG_CONSTANT(T, 64, 9.423709328020389560844e-08)
 269          };
 270          static const T Q[] =
 271          {
 272             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
 273             BOOST_MATH_BIG_CONSTANT(T, 64, -4.843828412587773008342e-02),
 274             BOOST_MATH_BIG_CONSTANT(T, 64, 1.088484822515098936140e-03),
 275             BOOST_MATH_BIG_CONSTANT(T, 64, -1.374724008530702784829e-05),
 276             BOOST_MATH_BIG_CONSTANT(T, 64, 8.452665455952581680339e-08)
 277          };
 278
 279
 280          T a = x * x / 4;
 281          a = (tools::evaluate_polynomial(P, a) / tools::evaluate_polynomial(Q, a) + Y) * a + 1;
 282
 283          // Maximum Deviation Found:                     2.440e-21
 284          // Expected Error Term : -2.434e-21
 285          // Maximum Relative Change in Control Points : 2.459e-02
 286          // Max Error found at float80 precision = Poly : 1.482487e-19
 287          static const T P2[] =
 288          {
 289             BOOST_MATH_BIG_CONSTANT(T, 64, 1.159315156584124488110e-01),
 290             BOOST_MATH_BIG_CONSTANT(T, 64, 2.764832791416047889734e-01),
 291             BOOST_MATH_BIG_CONSTANT(T, 64, 1.926062887220923354112e-02),
 292             BOOST_MATH_BIG_CONSTANT(T, 64, 3.660777862036966089410e-04),
 293             BOOST_MATH_BIG_CONSTANT(T, 64, 2.094942446930673386849e-06)
 294          };
 295          static const T Q2[] =
 296          {
 297             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
 298             BOOST_MATH_BIG_CONSTANT(T, 64, -2.156100313881251616320e-02),
 299             BOOST_MATH_BIG_CONSTANT(T, 64, 2.315993873344905957033e-04),
 300             BOOST_MATH_BIG_CONSTANT(T, 64, -1.529444499350703363451e-06),
 301             BOOST_MATH_BIG_CONSTANT(T, 64, 5.524988589917857531177e-09)
 302          };
 303          return tools::evaluate_rational(P2, Q2, T(x * x)) - log(x) * a;
 304       }
 305       else
 306       {
 307          // Maximum Deviation Found:                     4.291e-20
 308          // Expected Error Term : 2.236e-21
 309          // Maximum Relative Change in Control Points : 3.021e-01
 310          //Max Error found at float80 precision = Poly : 8.727378e-20
 311          static const T Y = 1;
 312          static const T P[] =
 313          {
 314             BOOST_MATH_BIG_CONSTANT(T, 64, 2.533141373155002512056e-01),
 315             BOOST_MATH_BIG_CONSTANT(T, 64, 5.417942070721928652715e+00),
 316             BOOST_MATH_BIG_CONSTANT(T, 64, 4.477464607463971754433e+01),
 317             BOOST_MATH_BIG_CONSTANT(T, 64, 1.838745728725943889876e+02),
 318             BOOST_MATH_BIG_CONSTANT(T, 64, 4.009736314927811202517e+02),
 319             BOOST_MATH_BIG_CONSTANT(T, 64, 4.557411293123609803452e+02),
 320             BOOST_MATH_BIG_CONSTANT(T, 64, 2.360222564015361268955e+02),
 321             BOOST_MATH_BIG_CONSTANT(T, 64, 2.385435333168505701022e+01),
 322             BOOST_MATH_BIG_CONSTANT(T, 64, -1.750195760942181592050e+01),
 323             BOOST_MATH_BIG_CONSTANT(T, 64, -4.059789241612946683713e+00),
 324             BOOST_MATH_BIG_CONSTANT(T, 64, -1.612783121537333908889e-01)
 325          };
 326          static const T Q[] =
 327          {
 328             BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
 329             BOOST_MATH_BIG_CONSTANT(T, 64, 2.200669254769325861404e+01),
 330             BOOST_MATH_BIG_CONSTANT(T, 64, 1.900177593527144126549e+02),
 331             BOOST_MATH_BIG_CONSTANT(T, 64, 8.361003989965786932682e+02),
 332             BOOST_MATH_BIG_CONSTANT(T, 64, 2.041319870804843395893e+03),
 333             BOOST_MATH_BIG_CONSTANT(T, 64, 2.828491555113790345068e+03),
 334             BOOST_MATH_BIG_CONSTANT(T, 64, 2.190342229261529076624e+03),
 335             BOOST_MATH_BIG_CONSTANT(T, 64, 9.003330795963812219852e+02),
 336             BOOST_MATH_BIG_CONSTANT(T, 64, 1.773371397243777891569e+02),
 337             BOOST_MATH_BIG_CONSTANT(T, 64, 1.368634935531158398439e+01),
 338             BOOST_MATH_BIG_CONSTANT(T, 64, 2.543310879400359967327e-01)
 339          };
 340          if(x < tools::log_max_value<T>())
 341             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 342          else
 343          {
 344             T ex = exp(-x / 2);
 345             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
 346          }
 347       }
 348 }
 349
 350 template <typename T>
 351 T bessel_k0_imp(const T& x, const mpl::int_<113>&)
 352 {
 353    BOOST_MATH_STD_USING
 354       if(x <= 1)
 355       {
 356          // Maximum Deviation Found:                     5.682e-37
 357          // Expected Error Term : 5.682e-37
 358          // Maximum Relative Change in Control Points : 6.094e-04
 359          // Max Error found at float128 precision = Poly : 5.338213e-35
 360          static const T Y = 1.137250900268554687500000000000000000e+00f;
 361          static const T P[] =
 362          {
 363             BOOST_MATH_BIG_CONSTANT(T, 113, -1.372509002685546875000000000000000006e-01),
 364             BOOST_MATH_BIG_CONSTANT(T, 113, 2.556212905071072782462974351698081303e-01),
 365             BOOST_MATH_BIG_CONSTANT(T, 113, 1.742459135264203478530904179889103929e-02),
 366             BOOST_MATH_BIG_CONSTANT(T, 113, 8.077860530453688571555479526961318918e-04),
 367             BOOST_MATH_BIG_CONSTANT(T, 113, 1.868173911669241091399374307788635148e-05),
 368             BOOST_MATH_BIG_CONSTANT(T, 113, 2.496405768838992243478709145123306602e-07),
 369             BOOST_MATH_BIG_CONSTANT(T, 113, 1.752489221949580551692915881999762125e-09),
 370             BOOST_MATH_BIG_CONSTANT(T, 113, 5.243010555737173524710512824955368526e-12)
 371          };
 372          static const T Q[] =
 373          {
 374             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
 375             BOOST_MATH_BIG_CONSTANT(T, 113, -4.095631064064621099785696980653193721e-02),
 376             BOOST_MATH_BIG_CONSTANT(T, 113, 8.313880983725212151967078809725835532e-04),
 377             BOOST_MATH_BIG_CONSTANT(T, 113, -1.095229912293480063501285562382835142e-05),
 378             BOOST_MATH_BIG_CONSTANT(T, 113, 1.022828799511943141130509410251996277e-07),
 379             BOOST_MATH_BIG_CONSTANT(T, 113, -6.860874007419812445494782795829046836e-10),
 380             BOOST_MATH_BIG_CONSTANT(T, 113, 3.107297802344970725756092082686799037e-12),
 381             BOOST_MATH_BIG_CONSTANT(T, 113, -7.460529579244623559164763757787600944e-15)
 382          };
 383          T a = x * x / 4;
 384          a = (tools::evaluate_rational(P, Q, a) + Y) * a + 1;
 385
 386          // Maximum Deviation Found:                     5.173e-38
 387          // Expected Error Term : 5.105e-38
 388          // Maximum Relative Change in Control Points : 9.734e-03
 389          // Max Error found at float128 precision = Poly : 1.688806e-34
 390          static const T P2[] =
 391          {
 392             BOOST_MATH_BIG_CONSTANT(T, 113, 1.159315156584124488107200313757741370e-01),
 393             BOOST_MATH_BIG_CONSTANT(T, 113, 2.789828789146031122026800078439435369e-01),
 394             BOOST_MATH_BIG_CONSTANT(T, 113, 2.524892993216269451266750049024628432e-02),
 395             BOOST_MATH_BIG_CONSTANT(T, 113, 8.460350907082229957222453839935101823e-04),
 396             BOOST_MATH_BIG_CONSTANT(T, 113, 1.491471929926042875260452849503857976e-05),
 397             BOOST_MATH_BIG_CONSTANT(T, 113, 1.627105610481598430816014719558896866e-07),
 398             BOOST_MATH_BIG_CONSTANT(T, 113, 1.208426165007797264194914898538250281e-09),
 399             BOOST_MATH_BIG_CONSTANT(T, 113, 6.508697838747354949164182457073784117e-12),
 400             BOOST_MATH_BIG_CONSTANT(T, 113, 2.659784680639805301101014383907273109e-14),
 401             BOOST_MATH_BIG_CONSTANT(T, 113, 8.531090131964391104248859415958109654e-17),
 402             BOOST_MATH_BIG_CONSTANT(T, 113, 2.205195117066478034260323124669936314e-19),
 403             BOOST_MATH_BIG_CONSTANT(T, 113, 4.692219280289030165761119775783115426e-22),
 404             BOOST_MATH_BIG_CONSTANT(T, 113, 8.362350161092532344171965861545860747e-25),
 405             BOOST_MATH_BIG_CONSTANT(T, 113, 1.277990623924628999539014980773738258e-27)
 406          };
 407
 408          return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a;
 409       }
 410       else
 411       {
 412          // Maximum Deviation Found:                     1.462e-34
 413          // Expected Error Term : 4.917e-40
 414          // Maximum Relative Change in Control Points : 3.385e-01
 415          // Max Error found at float128 precision = Poly : 1.567573e-34
 416          static const T Y = 1;
 417          static const T P[] =
 418          {
 419             BOOST_MATH_BIG_CONSTANT(T, 113, 2.533141373155002512078826424055226265e-01),
 420             BOOST_MATH_BIG_CONSTANT(T, 113, 2.001949740768235770078339977110749204e+01),
 421             BOOST_MATH_BIG_CONSTANT(T, 113, 6.991516715983883248363351472378349986e+02),
 422             BOOST_MATH_BIG_CONSTANT(T, 113, 1.429587951594593159075690819360687720e+04),
 423             BOOST_MATH_BIG_CONSTANT(T, 113, 1.911933815201948768044660065771258450e+05),
 424             BOOST_MATH_BIG_CONSTANT(T, 113, 1.769943016204926614862175317962439875e+06),
 425             BOOST_MATH_BIG_CONSTANT(T, 113, 1.170866154649560750500954150401105606e+07),
 426             BOOST_MATH_BIG_CONSTANT(T, 113, 5.634687099724383996792011977705727661e+07),
 427             BOOST_MATH_BIG_CONSTANT(T, 113, 1.989524036456492581597607246664394014e+08),
 428             BOOST_MATH_BIG_CONSTANT(T, 113, 5.160394785715328062088529400178080360e+08),
 429             BOOST_MATH_BIG_CONSTANT(T, 113, 9.778173054417826368076483100902201433e+08),
 430             BOOST_MATH_BIG_CONSTANT(T, 113, 1.335667778588806892764139643950439733e+09),
 431             BOOST_MATH_BIG_CONSTANT(T, 113, 1.283635100080306980206494425043706838e+09),
 432             BOOST_MATH_BIG_CONSTANT(T, 113, 8.300616188213640626577036321085025855e+08),
 433             BOOST_MATH_BIG_CONSTANT(T, 113, 3.277591957076162984986406540894621482e+08),
 434             BOOST_MATH_BIG_CONSTANT(T, 113, 5.564360536834214058158565361486115932e+07),
 435             BOOST_MATH_BIG_CONSTANT(T, 113, -1.043505161612403359098596828115690596e+07),
 436             BOOST_MATH_BIG_CONSTANT(T, 113, -7.217035248223503605127967970903027314e+06),
 437             BOOST_MATH_BIG_CONSTANT(T, 113, -1.422938158797326748375799596769964430e+06),
 438             BOOST_MATH_BIG_CONSTANT(T, 113, -1.229125746200586805278634786674745210e+05),
 439             BOOST_MATH_BIG_CONSTANT(T, 113, -4.201632288615609937883545928660649813e+03),
 440             BOOST_MATH_BIG_CONSTANT(T, 113, -3.690820607338480548346746717311811406e+01)
 441          };
 442          static const T Q[] =
 443          {
 444             BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
 445             BOOST_MATH_BIG_CONSTANT(T, 113, 7.964877874035741452203497983642653107e+01),
 446             BOOST_MATH_BIG_CONSTANT(T, 113, 2.808929943826193766839360018583294769e+03),
 447             BOOST_MATH_BIG_CONSTANT(T, 113, 5.814524004679994110944366890912384139e+04),
 448             BOOST_MATH_BIG_CONSTANT(T, 113, 7.897794522506725610540209610337355118e+05),
 449             BOOST_MATH_BIG_CONSTANT(T, 113, 7.456339470955813675629523617440433672e+06),
 450             BOOST_MATH_BIG_CONSTANT(T, 113, 5.057818717813969772198911392875127212e+07),
 451             BOOST_MATH_BIG_CONSTANT(T, 113, 2.513821619536852436424913886081133209e+08),
 452             BOOST_MATH_BIG_CONSTANT(T, 113, 9.255938846873380596038513316919990776e+08),
 453             BOOST_MATH_BIG_CONSTANT(T, 113, 2.537077551699028079347581816919572141e+09),
 454             BOOST_MATH_BIG_CONSTANT(T, 113, 5.176769339768120752974843214652367321e+09),
 455             BOOST_MATH_BIG_CONSTANT(T, 113, 7.828722317390455845253191337207432060e+09),
 456             BOOST_MATH_BIG_CONSTANT(T, 113, 8.698864296569996402006511705803675890e+09),
 457             BOOST_MATH_BIG_CONSTANT(T, 113, 7.007803261356636409943826918468544629e+09),
 458             BOOST_MATH_BIG_CONSTANT(T, 113, 4.016564631288740308993071395104715469e+09),
 459             BOOST_MATH_BIG_CONSTANT(T, 113, 1.595893010619754750655947035567624730e+09),
 460             BOOST_MATH_BIG_CONSTANT(T, 113, 4.241241839120481076862742189989406856e+08),
 461             BOOST_MATH_BIG_CONSTANT(T, 113, 7.168778094393076220871007550235840858e+07),
 462             BOOST_MATH_BIG_CONSTANT(T, 113, 7.156200301360388147635052029404211109e+06),
 463             BOOST_MATH_BIG_CONSTANT(T, 113, 3.752130382550379886741949463587008794e+05),
 464             BOOST_MATH_BIG_CONSTANT(T, 113, 8.370574966987293592457152146806662562e+03),
 465             BOOST_MATH_BIG_CONSTANT(T, 113, 4.871254714311063594080644835895740323e+01)
 466          };
 467          if(x < tools::log_max_value<T>())
 468             return  ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
 469          else
 470          {
 471             T ex = exp(-x / 2);
 472             return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
 473          }
 474       }
 475 }
 476
 477 template <typename T>
 478 T bessel_k0_imp(const T& x, const mpl::int_<0>&)
 479 {
 480    if(boost::math::tools::digits<T>() <= 24)
 481       return bessel_k0_imp(x, mpl::int_<24>());
 482    else if(boost::math::tools::digits<T>() <= 53)
 483       return bessel_k0_imp(x, mpl::int_<53>());
 484    else if(boost::math::tools::digits<T>() <= 64)
 485       return bessel_k0_imp(x, mpl::int_<64>());
 486    else if(boost::math::tools::digits<T>() <= 113)
 487       return bessel_k0_imp(x, mpl::int_<113>());
 488    BOOST_ASSERT(0);
 489    return 0;
 490 }
 491
 492 template <typename T>
 493 inline T bessel_k0(const T& x)
 494 {
 495    typedef mpl::int_<
 496       ((std::numeric_limits<T>::digits == 0) || (std::numeric_limits<T>::radix != 2)) ?
 497       0 :
 498       std::numeric_limits<T>::digits <= 24 ?
 499       24 :
 500       std::numeric_limits<T>::digits <= 53 ?
 501       53 :
 502       std::numeric_limits<T>::digits <= 64 ?
 503       64 :
 504       std::numeric_limits<T>::digits <= 113 ?
 505       113 : -1
 506    > tag_type;
 507
 508    bessel_k0_initializer<T, tag_type>::force_instantiate();
 509    return bessel_k0_imp(x, tag_type());
 510 }
 511
 512 }}} // namespaces
 513
 514 #ifdef _MSC_VER
 515 #pragma warning(pop)
 516 #endif
 517
 518 #endif // BOOST_MATH_BESSEL_K0_HPP
 519