ceph/src/boost/boost/math/special_functions/detail/bessel_i0.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_I0_HPP
   8 #define BOOST_MATH_BESSEL_I0_HPP
   9
  10 #ifdef _MSC_VER
  11 #pragma once
  12 #endif
  13
  14 #include <boost/math/tools/rational.hpp>
  15 #include <boost/math/tools/big_constant.hpp>
  16 #include <boost/assert.hpp>
  17
  18 #if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
  19 //
  20 // This is the only way we can avoid
  21 // warning: non-standard suffix on floating constant [-Wpedantic]
  22 // when building with -Wall -pedantic.  Neither __extension__
  23 // nor #pragma dianostic ignored work :(
  24 //
  25 #pragma GCC system_header
  26 #endif
  27
  28 // Modified Bessel function of the first kind of order zero
  29 // we use the approximating forms derived in:
  30 // "Rational Approximations for the Modified Bessel Function of the First Kind - I0(x) for Computations with Double Precision"
  31 // by Pavel Holoborodko,
  32 // see http://www.advanpix.com/2015/11/11/rational-approximations-for-the-modified-bessel-function-of-the-first-kind-i0-computations-double-precision
  33 // The actual coefficients used are our own, and extend Pavel's work to precision's other than double.
  34
  35 namespace boost { namespace math { namespace detail{
  36
  37 template <typename T>
  38 T bessel_i0(const T& x);
  39
  40 template <class T, class tag>
  41 struct bessel_i0_initializer
  42 {
  43    struct init
  44    {
  45       init()
  46       {
  47          do_init(tag());
  48       }
  49       static void do_init(const mpl::int_<64>&)
  50       {
  51          bessel_i0(T(1));
  52          bessel_i0(T(8));
  53          bessel_i0(T(12));
  54          bessel_i0(T(40));
  55          bessel_i0(T(101));
  56       }
  57       static void do_init(const mpl::int_<113>&)
  58       {
  59          bessel_i0(T(1));
  60          bessel_i0(T(10));
  61          bessel_i0(T(20));
  62          bessel_i0(T(40));
  63          bessel_i0(T(101));
  64       }
  65       template <class U>
  66       static void do_init(const U&) {}
  67       void force_instantiate()const {}
  68    };
  69    static const init initializer;
  70    static void force_instantiate()
  71    {
  72       initializer.force_instantiate();
  73    }
  74 };
  75
  76 template <class T, class tag>
  77 const typename bessel_i0_initializer<T, tag>::init bessel_i0_initializer<T, tag>::initializer;
  78
  79 template <typename T, int N>
  80 T bessel_i0_imp(const T&, const mpl::int_<N>&)
  81 {
  82    BOOST_ASSERT(0);
  83    return 0;
  84 }
  85
  86 template <typename T>
  87 T bessel_i0_imp(const T& x, const mpl::int_<24>&)
  88 {
  89    BOOST_MATH_STD_USING
  90    if(x < 7.75)
  91    {
  92       // Max error in interpolated form: 3.929e-08
  93       // Max Error found at float precision = Poly: 1.991226e-07
  94       static const float P[] = {
  95          1.00000003928615375e+00f,
  96          2.49999576572179639e-01f,
  97          2.77785268558399407e-02f,
  98          1.73560257755821695e-03f,
  99          6.96166518788906424e-05f,
 100          1.89645733877137904e-06f,
 101          4.29455004657565361e-08f,
 102          3.90565476357034480e-10f,
 103          1.48095934745267240e-11f
 104       };
 105       T a = x * x / 4;
 106       return a * boost::math::tools::evaluate_polynomial(P, a) + 1;
 107    }
 108    else if(x < 50)
 109    {
 110       // Max error in interpolated form: 5.195e-08
 111       // Max Error found at float precision = Poly: 8.502534e-08
 112       static const float P[] = {
 113          3.98942651588301770e-01f,
 114          4.98327234176892844e-02f,
 115          2.91866904423115499e-02f,
 116          1.35614940793742178e-02f,
 117          1.31409251787866793e-01f
 118       };
 119       return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 120    }
 121    else
 122    {
 123       // Max error in interpolated form: 1.782e-09
 124       // Max Error found at float precision = Poly: 6.473568e-08
 125       static const float P[] = {
 126          3.98942391532752700e-01f,
 127          4.98455950638200020e-02f,
 128          2.94835666900682535e-02f
 129       };
 130       T ex = exp(x / 2);
 131       T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 132       result *= ex;
 133       return result;
 134    }
 135 }
 136
 137 template <typename T>
 138 T bessel_i0_imp(const T& x, const mpl::int_<53>&)
 139 {
 140    BOOST_MATH_STD_USING
 141    if(x < 7.75)
 142    {
 143       // Bessel I0 over[10 ^ -16, 7.75]
 144       // Max error in interpolated form : 3.042e-18
 145       // Max Error found at double precision = Poly : 5.106609e-16 Cheb : 5.239199e-16
 146       static const double P[] = {
 147          1.00000000000000000e+00,
 148          2.49999999999999909e-01,
 149          2.77777777777782257e-02,
 150          1.73611111111023792e-03,
 151          6.94444444453352521e-05,
 152          1.92901234513219920e-06,
 153          3.93675991102510739e-08,
 154          6.15118672704439289e-10,
 155          7.59407002058973446e-12,
 156          7.59389793369836367e-14,
 157          6.27767773636292611e-16,
 158          4.34709704153272287e-18,
 159          2.63417742690109154e-20,
 160          1.13943037744822825e-22,
 161          9.07926920085624812e-25
 162       };
 163       T a = x * x / 4;
 164       return a * boost::math::tools::evaluate_polynomial(P, a) + 1;
 165    }
 166    else if(x < 500)
 167    {
 168       // Max error in interpolated form : 1.685e-16
 169       // Max Error found at double precision = Poly : 2.575063e-16 Cheb : 2.247615e+00
 170       static const double P[] = {
 171          3.98942280401425088e-01,
 172          4.98677850604961985e-02,
 173          2.80506233928312623e-02,
 174          2.92211225166047873e-02,
 175          4.44207299493659561e-02,
 176          1.30970574605856719e-01,
 177          -3.35052280231727022e+00,
 178          2.33025711583514727e+02,
 179          -1.13366350697172355e+04,
 180          4.24057674317867331e+05,
 181          -1.23157028595698731e+07,
 182          2.80231938155267516e+08,
 183          -5.01883999713777929e+09,
 184          7.08029243015109113e+10,
 185          -7.84261082124811106e+11,
 186          6.76825737854096565e+12,
 187          -4.49034849696138065e+13,
 188          2.24155239966958995e+14,
 189          -8.13426467865659318e+14,
 190          2.02391097391687777e+15,
 191          -3.08675715295370878e+15,
 192          2.17587543863819074e+15
 193       };
 194       return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 195    }
 196    else
 197    {
 198       // Max error in interpolated form : 2.437e-18
 199       // Max Error found at double precision = Poly : 1.216719e-16
 200       static const double P[] = {
 201          3.98942280401432905e-01,
 202          4.98677850491434560e-02,
 203          2.80506308916506102e-02,
 204          2.92179096853915176e-02,
 205          4.53371208762579442e-02
 206       };
 207       T ex = exp(x / 2);
 208       T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 209       result *= ex;
 210       return result;
 211    }
 212 }
 213
 214 template <typename T>
 215 T bessel_i0_imp(const T& x, const mpl::int_<64>&)
 216 {
 217    BOOST_MATH_STD_USING
 218    if(x < 7.75)
 219    {
 220       // Bessel I0 over[10 ^ -16, 7.75]
 221       // Max error in interpolated form : 3.899e-20
 222       // Max Error found at float80 precision = Poly : 1.770840e-19
 223       static const T P[] = {
 224          BOOST_MATH_BIG_CONSTANT(T, 64, 9.99999999999999999961011629e-01),
 225          BOOST_MATH_BIG_CONSTANT(T, 64, 2.50000000000000001321873912e-01),
 226          BOOST_MATH_BIG_CONSTANT(T, 64, 2.77777777777777703400424216e-02),
 227          BOOST_MATH_BIG_CONSTANT(T, 64, 1.73611111111112764793802701e-03),
 228          BOOST_MATH_BIG_CONSTANT(T, 64, 6.94444444444251461247253525e-05),
 229          BOOST_MATH_BIG_CONSTANT(T, 64, 1.92901234569262206386118739e-06),
 230          BOOST_MATH_BIG_CONSTANT(T, 64, 3.93675988851131457141005209e-08),
 231          BOOST_MATH_BIG_CONSTANT(T, 64, 6.15118734688297476454205352e-10),
 232          BOOST_MATH_BIG_CONSTANT(T, 64, 7.59405797058091016449222685e-12),
 233          BOOST_MATH_BIG_CONSTANT(T, 64, 7.59406599631719800679835140e-14),
 234          BOOST_MATH_BIG_CONSTANT(T, 64, 6.27598961062070013516660425e-16),
 235          BOOST_MATH_BIG_CONSTANT(T, 64, 4.35920318970387940278362992e-18),
 236          BOOST_MATH_BIG_CONSTANT(T, 64, 2.57372492687715452949437981e-20),
 237          BOOST_MATH_BIG_CONSTANT(T, 64, 1.33908663475949906992942204e-22),
 238          BOOST_MATH_BIG_CONSTANT(T, 64, 5.15976668870980234582896010e-25),
 239          BOOST_MATH_BIG_CONSTANT(T, 64, 3.46240478946376069211156548e-27)
 240       };
 241       T a = x * x / 4;
 242       return a * boost::math::tools::evaluate_polynomial(P, a) + 1;
 243    }
 244    else if(x < 10)
 245    {
 246       // Maximum Deviation Found:                     6.906e-21
 247       // Expected Error Term : -6.903e-21
 248       // Maximum Relative Change in Control Points : 1.631e-04
 249       // Max Error found at float80 precision = Poly : 7.811948e-21
 250       static const T Y = 4.051098823547363281250e-01f;
 251       static const T P[] = {
 252          BOOST_MATH_BIG_CONSTANT(T, 64, -6.158081780620616479492e-03),
 253          BOOST_MATH_BIG_CONSTANT(T, 64, 4.883635969834048766148e-02),
 254          BOOST_MATH_BIG_CONSTANT(T, 64, 7.892782002476195771920e-02),
 255          BOOST_MATH_BIG_CONSTANT(T, 64, -1.478784996478070170327e+00),
 256          BOOST_MATH_BIG_CONSTANT(T, 64, 2.988611837308006851257e+01),
 257          BOOST_MATH_BIG_CONSTANT(T, 64, -4.140133766747436806179e+02),
 258          BOOST_MATH_BIG_CONSTANT(T, 64, 4.117316447921276453271e+03),
 259          BOOST_MATH_BIG_CONSTANT(T, 64, -2.942353667455141676001e+04),
 260          BOOST_MATH_BIG_CONSTANT(T, 64, 1.493482682461387081534e+05),
 261          BOOST_MATH_BIG_CONSTANT(T, 64, -5.228100538921466124653e+05),
 262          BOOST_MATH_BIG_CONSTANT(T, 64, 1.195279248600467989454e+06),
 263          BOOST_MATH_BIG_CONSTANT(T, 64, -1.601530760654337045917e+06),
 264          BOOST_MATH_BIG_CONSTANT(T, 64, 9.504921137873298402679e+05)
 265       };
 266       return exp(x) * (boost::math::tools::evaluate_polynomial(P, T(1 / x)) + Y) / sqrt(x);
 267    }
 268    else if(x < 15)
 269    {
 270       // Maximum Deviation Found:                     4.083e-21
 271       // Expected Error Term : -4.025e-21
 272       // Maximum Relative Change in Control Points : 1.304e-03
 273       // Max Error found at float80 precision = Poly : 2.303527e-20
 274       static const T Y = 4.033188819885253906250e-01f;
 275       static const T P[] = {
 276          BOOST_MATH_BIG_CONSTANT(T, 64, -4.376373876116109401062e-03),
 277          BOOST_MATH_BIG_CONSTANT(T, 64, 4.982899138682911273321e-02),
 278          BOOST_MATH_BIG_CONSTANT(T, 64, 3.109477529533515397644e-02),
 279          BOOST_MATH_BIG_CONSTANT(T, 64, -1.163760580110576407673e-01),
 280          BOOST_MATH_BIG_CONSTANT(T, 64, 4.776501832837367371883e+00),
 281          BOOST_MATH_BIG_CONSTANT(T, 64, -1.101478069227776656318e+02),
 282          BOOST_MATH_BIG_CONSTANT(T, 64, 1.892071912448960299773e+03),
 283          BOOST_MATH_BIG_CONSTANT(T, 64, -2.417739279982328117483e+04),
 284          BOOST_MATH_BIG_CONSTANT(T, 64, 2.296963447724067390552e+05),
 285          BOOST_MATH_BIG_CONSTANT(T, 64, -1.598589306710589358747e+06),
 286          BOOST_MATH_BIG_CONSTANT(T, 64, 7.903662411851774878322e+06),
 287          BOOST_MATH_BIG_CONSTANT(T, 64, -2.622677059040339516093e+07),
 288          BOOST_MATH_BIG_CONSTANT(T, 64, 5.227776578828667629347e+07),
 289          BOOST_MATH_BIG_CONSTANT(T, 64, -4.727797957441040896878e+07)
 290       };
 291       return exp(x) * (boost::math::tools::evaluate_polynomial(P, T(1 / x)) + Y) / sqrt(x);
 292    }
 293    else if(x < 50)
 294    {
 295       // Max error in interpolated form: 1.035e-21
 296       // Max Error found at float80 precision = Poly: 1.885872e-21
 297       static const T Y = 4.011702537536621093750e-01f;
 298       static const T P[] = {
 299          BOOST_MATH_BIG_CONSTANT(T, 64, -2.227973351806078464328e-03),
 300          BOOST_MATH_BIG_CONSTANT(T, 64, 4.986778486088017419036e-02),
 301          BOOST_MATH_BIG_CONSTANT(T, 64, 2.805066823812285310011e-02),
 302          BOOST_MATH_BIG_CONSTANT(T, 64, 2.921443721160964964623e-02),
 303          BOOST_MATH_BIG_CONSTANT(T, 64, 4.517504941996594744052e-02),
 304          BOOST_MATH_BIG_CONSTANT(T, 64, 6.316922639868793684401e-02),
 305          BOOST_MATH_BIG_CONSTANT(T, 64, 1.535891099168810015433e+00),
 306          BOOST_MATH_BIG_CONSTANT(T, 64, -4.706078229522448308087e+01),
 307          BOOST_MATH_BIG_CONSTANT(T, 64, 1.351015763079160914632e+03),
 308          BOOST_MATH_BIG_CONSTANT(T, 64, -2.948809013999277355098e+04),
 309          BOOST_MATH_BIG_CONSTANT(T, 64, 4.967598958582595361757e+05),
 310          BOOST_MATH_BIG_CONSTANT(T, 64, -6.346924657995383019558e+06),
 311          BOOST_MATH_BIG_CONSTANT(T, 64, 5.998794574259956613472e+07),
 312          BOOST_MATH_BIG_CONSTANT(T, 64, -4.016371355801690142095e+08),
 313          BOOST_MATH_BIG_CONSTANT(T, 64, 1.768791455631826490838e+09),
 314          BOOST_MATH_BIG_CONSTANT(T, 64, -4.441995678177349895640e+09),
 315          BOOST_MATH_BIG_CONSTANT(T, 64, 4.482292669974971387738e+09)
 316       };
 317       return exp(x) * (boost::math::tools::evaluate_polynomial(P, T(1 / x)) + Y) / sqrt(x);
 318    }
 319    else
 320    {
 321       // Bessel I0 over[50, INF]
 322       // Max error in interpolated form : 5.587e-20
 323       // Max Error found at float80 precision = Poly : 8.776852e-20
 324       static const T P[] = {
 325          BOOST_MATH_BIG_CONSTANT(T, 64, 3.98942280401432677955074061e-01),
 326          BOOST_MATH_BIG_CONSTANT(T, 64, 4.98677850501789875615574058e-02),
 327          BOOST_MATH_BIG_CONSTANT(T, 64, 2.80506290908675604202206833e-02),
 328          BOOST_MATH_BIG_CONSTANT(T, 64, 2.92194052159035901631494784e-02),
 329          BOOST_MATH_BIG_CONSTANT(T, 64, 4.47422430732256364094681137e-02),
 330          BOOST_MATH_BIG_CONSTANT(T, 64, 9.05971614435738691235525172e-02),
 331          BOOST_MATH_BIG_CONSTANT(T, 64, 2.29180522595459823234266708e-01),
 332          BOOST_MATH_BIG_CONSTANT(T, 64, 6.15122547776140254569073131e-01),
 333          BOOST_MATH_BIG_CONSTANT(T, 64, 7.48491812136365376477357324e+00),
 334          BOOST_MATH_BIG_CONSTANT(T, 64, -2.45569740166506688169730713e+02),
 335          BOOST_MATH_BIG_CONSTANT(T, 64, 9.66857566379480730407063170e+03),
 336          BOOST_MATH_BIG_CONSTANT(T, 64, -2.71924083955641197750323901e+05),
 337          BOOST_MATH_BIG_CONSTANT(T, 64, 5.74276685704579268845870586e+06),
 338          BOOST_MATH_BIG_CONSTANT(T, 64, -8.89753803265734681907148778e+07),
 339          BOOST_MATH_BIG_CONSTANT(T, 64, 9.82590905134996782086242180e+08),
 340          BOOST_MATH_BIG_CONSTANT(T, 64, -7.30623197145529889358596301e+09),
 341          BOOST_MATH_BIG_CONSTANT(T, 64, 3.27310000726207055200805893e+10),
 342          BOOST_MATH_BIG_CONSTANT(T, 64, -6.64365417189215599168817064e+10)
 343       };
 344       T ex = exp(x / 2);
 345       T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 346       result *= ex;
 347       return result;
 348    }
 349 }
 350
 351 template <typename T>
 352 T bessel_i0_imp(const T& x, const mpl::int_<113>&)
 353 {
 354    BOOST_MATH_STD_USING
 355    if(x < 7.75)
 356    {
 357       // Bessel I0 over[10 ^ -34, 7.75]
 358       // Max error in interpolated form : 1.274e-34
 359       // Max Error found at float128 precision = Poly : 3.096091e-34
 360       static const T P[] = {
 361          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0000000000000000000000000000000001273856e+00),
 362          BOOST_MATH_BIG_CONSTANT(T, 113, 2.4999999999999999999999999999999107477496e-01),
 363          BOOST_MATH_BIG_CONSTANT(T, 113, 2.7777777777777777777777777777881795230918e-02),
 364          BOOST_MATH_BIG_CONSTANT(T, 113, 1.7361111111111111111111111106290091648808e-03),
 365          BOOST_MATH_BIG_CONSTANT(T, 113, 6.9444444444444444444444445629960334523101e-05),
 366          BOOST_MATH_BIG_CONSTANT(T, 113, 1.9290123456790123456790105563456483249753e-06),
 367          BOOST_MATH_BIG_CONSTANT(T, 113, 3.9367598891408415217940836339080514004844e-08),
 368          BOOST_MATH_BIG_CONSTANT(T, 113, 6.1511873267825648777900014857992724731476e-10),
 369          BOOST_MATH_BIG_CONSTANT(T, 113, 7.5940584281266233066162999610732449709209e-12),
 370          BOOST_MATH_BIG_CONSTANT(T, 113, 7.5940584281266232783124723601470051895304e-14),
 371          BOOST_MATH_BIG_CONSTANT(T, 113, 6.2760813455591936763439337059117957836078e-16),
 372          BOOST_MATH_BIG_CONSTANT(T, 113, 4.3583898233049738471136482147779094353096e-18),
 373          BOOST_MATH_BIG_CONSTANT(T, 113, 2.5789288895299965395422423848480340736308e-20),
 374          BOOST_MATH_BIG_CONSTANT(T, 113, 1.3157800456718804437960453545507623434606e-22),
 375          BOOST_MATH_BIG_CONSTANT(T, 113, 5.8479113149412360748032684260932041506493e-25),
 376          BOOST_MATH_BIG_CONSTANT(T, 113, 2.2843403488398038539283241944594140493394e-27),
 377          BOOST_MATH_BIG_CONSTANT(T, 113, 7.9042925594356556196790242908697582021825e-30),
 378          BOOST_MATH_BIG_CONSTANT(T, 113, 2.4395919891312152120710245152115597111101e-32),
 379          BOOST_MATH_BIG_CONSTANT(T, 113, 6.7580986145276689333214547502373003196707e-35),
 380          BOOST_MATH_BIG_CONSTANT(T, 113, 1.6886514018062348877723837017198859723889e-37),
 381          BOOST_MATH_BIG_CONSTANT(T, 113, 3.8540558465757554512570197585002702777999e-40),
 382          BOOST_MATH_BIG_CONSTANT(T, 113, 7.4684706070226893763741850944911705726436e-43),
 383          BOOST_MATH_BIG_CONSTANT(T, 113, 2.0210715309399646335858150349406935414314e-45)
 384       };
 385       T a = x * x / 4;
 386       return a * boost::math::tools::evaluate_polynomial(P, a) + 1;
 387    }
 388    else if(x < 15)
 389    {
 390       // Bessel I0 over[7.75, 15]
 391       // Max error in interpolated form : 7.534e-35
 392       // Max Error found at float128 precision = Poly : 6.123912e-34
 393       static const T P[] = {
 394          BOOST_MATH_BIG_CONSTANT(T, 113, 9.9999999999999999992388573069504617493518e-01),
 395          BOOST_MATH_BIG_CONSTANT(T, 113, 2.5000000000000000007304739268173096975340e-01),
 396          BOOST_MATH_BIG_CONSTANT(T, 113, 2.7777777777777777744261405400543564492074e-02),
 397          BOOST_MATH_BIG_CONSTANT(T, 113, 1.7361111111111111209006987259719750726867e-03),
 398          BOOST_MATH_BIG_CONSTANT(T, 113, 6.9444444444444442399703186871329381908321e-05),
 399          BOOST_MATH_BIG_CONSTANT(T, 113, 1.9290123456790126709286741580242189785431e-06),
 400          BOOST_MATH_BIG_CONSTANT(T, 113, 3.9367598891408374246503061422528266924389e-08),
 401          BOOST_MATH_BIG_CONSTANT(T, 113, 6.1511873267826068395343047827801353170966e-10),
 402          BOOST_MATH_BIG_CONSTANT(T, 113, 7.5940584281262673459688011737168286944521e-12),
 403          BOOST_MATH_BIG_CONSTANT(T, 113, 7.5940584281291583769928563167645746144508e-14),
 404          BOOST_MATH_BIG_CONSTANT(T, 113, 6.2760813455438840231126529638737436950274e-16),
 405          BOOST_MATH_BIG_CONSTANT(T, 113, 4.3583898233839583885132809584770578894948e-18),
 406          BOOST_MATH_BIG_CONSTANT(T, 113, 2.5789288891798658971960571838369339742994e-20),
 407          BOOST_MATH_BIG_CONSTANT(T, 113, 1.3157800470129311623308216856009970266088e-22),
 408          BOOST_MATH_BIG_CONSTANT(T, 113, 5.8479112701534604520063520412207286692581e-25),
 409          BOOST_MATH_BIG_CONSTANT(T, 113, 2.2843404822552330714586265081801727491890e-27),
 410          BOOST_MATH_BIG_CONSTANT(T, 113, 7.9042888166225242675881424439818162458179e-30),
 411          BOOST_MATH_BIG_CONSTANT(T, 113, 2.4396027771820721384198604723320045236973e-32),
 412          BOOST_MATH_BIG_CONSTANT(T, 113, 6.7577659910606076328136207973456511895030e-35),
 413          BOOST_MATH_BIG_CONSTANT(T, 113, 1.6896548123724136624716224328803899914646e-37),
 414          BOOST_MATH_BIG_CONSTANT(T, 113, 3.8285850162160539150210466453921758781984e-40),
 415          BOOST_MATH_BIG_CONSTANT(T, 113, 7.9419071894227736216423562425429524883562e-43),
 416          BOOST_MATH_BIG_CONSTANT(T, 113, 1.4720374049498608905571855665134539425038e-45),
 417          BOOST_MATH_BIG_CONSTANT(T, 113, 2.7763533278527958112907118930154738930378e-48),
 418          BOOST_MATH_BIG_CONSTANT(T, 113, 3.1213839473168678646697528580511702663617e-51),
 419          BOOST_MATH_BIG_CONSTANT(T, 113, 1.0648035313124146852372607519737686740964e-53),
 420          -BOOST_MATH_BIG_CONSTANT(T, 113, 5.1255595184052024349371058585102280860878e-57),
 421          BOOST_MATH_BIG_CONSTANT(T, 113, 3.4652470895944157957727948355523715335882e-59)
 422       };
 423       T a = x * x / 4;
 424       return a * boost::math::tools::evaluate_polynomial(P, a) + 1;
 425    }
 426    else if(x < 30)
 427    {
 428       // Max error in interpolated form : 1.808e-34
 429       // Max Error found at float128 precision = Poly : 2.399403e-34
 430       static const T P[] = {
 431          BOOST_MATH_BIG_CONSTANT(T, 113, 3.9894228040870793650581242239624530714032e-01),
 432          BOOST_MATH_BIG_CONSTANT(T, 113, 4.9867780576714783790784348982178607842250e-02),
 433          BOOST_MATH_BIG_CONSTANT(T, 113, 2.8051948347934462928487999569249907599510e-02),
 434          BOOST_MATH_BIG_CONSTANT(T, 113, 2.8971143420388958551176254291160976367263e-02),
 435          BOOST_MATH_BIG_CONSTANT(T, 113, 7.8197359701715582763961322341827341098897e-02),
 436          BOOST_MATH_BIG_CONSTANT(T, 113, -3.3430484862908317377522273217643346601271e+00),
 437          BOOST_MATH_BIG_CONSTANT(T, 113, 2.7884507603213662610604413960838990199224e+02),
 438          BOOST_MATH_BIG_CONSTANT(T, 113, -1.8304926482356755790062999202373909300514e+04),
 439          BOOST_MATH_BIG_CONSTANT(T, 113, 9.8867173178574875515293357145875120137676e+05),
 440          BOOST_MATH_BIG_CONSTANT(T, 113, -4.4261178812193528551544261731796888257644e+07),
 441          BOOST_MATH_BIG_CONSTANT(T, 113, 1.6453010340778116475788083817762403540097e+09),
 442          BOOST_MATH_BIG_CONSTANT(T, 113, -5.0432401330113978669454035365747869477960e+10),
 443          BOOST_MATH_BIG_CONSTANT(T, 113, 1.2462165331309799059332310595587606836357e+12),
 444          BOOST_MATH_BIG_CONSTANT(T, 113, -2.3299800389951335932792950236410844978273e+13),
 445          BOOST_MATH_BIG_CONSTANT(T, 113, 2.5748218240248714177527965706790413406639e+14),
 446          BOOST_MATH_BIG_CONSTANT(T, 113, 1.8330014378766930869945511450377736037385e+15),
 447          BOOST_MATH_BIG_CONSTANT(T, 113, -1.8494610073827453236940544799030787866218e+17),
 448          BOOST_MATH_BIG_CONSTANT(T, 113, 5.7244661371420647691301043350229977856476e+18),
 449          BOOST_MATH_BIG_CONSTANT(T, 113, -1.2386378807889388140099109087465781254321e+20),
 450          BOOST_MATH_BIG_CONSTANT(T, 113, 2.1104000573102013529518477353943384110982e+21),
 451          BOOST_MATH_BIG_CONSTANT(T, 113, -2.9426541092239879262282594572224300191016e+22),
 452          BOOST_MATH_BIG_CONSTANT(T, 113, 3.4061439136301913488512592402635688101020e+23),
 453          BOOST_MATH_BIG_CONSTANT(T, 113, -3.2836554760521986358980180942859101564671e+24),
 454          BOOST_MATH_BIG_CONSTANT(T, 113, 2.6270285589905206294944214795661236766988e+25),
 455          BOOST_MATH_BIG_CONSTANT(T, 113, -1.7278631455211972017740134341610659484259e+26),
 456          BOOST_MATH_BIG_CONSTANT(T, 113, 9.1971734473772196124736986948034978906801e+26),
 457          BOOST_MATH_BIG_CONSTANT(T, 113, -3.8669270707172568763908838463689093500098e+27),
 458          BOOST_MATH_BIG_CONSTANT(T, 113, 1.2368879358870281916900125550129211146626e+28),
 459          BOOST_MATH_BIG_CONSTANT(T, 113, -2.8296235063297831758204519071113999839858e+28),
 460          BOOST_MATH_BIG_CONSTANT(T, 113, 4.1253861666023020670144616019148954773662e+28),
 461          BOOST_MATH_BIG_CONSTANT(T, 113, -2.8809536950051955163648980306847791014734e+28) };
 462       return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 463    }
 464    else if(x < 100)
 465    {
 466       // Bessel I0 over[30, 100]
 467       // Max error in interpolated form : 1.487e-34
 468       // Max Error found at float128 precision = Poly : 1.929924e-34
 469       static const T P[] = {
 470          BOOST_MATH_BIG_CONSTANT(T, 113, 3.9894228040143267793996798658172135362278e-01),
 471          BOOST_MATH_BIG_CONSTANT(T, 113, 4.9867785050179084714910130342157246539820e-02),
 472          BOOST_MATH_BIG_CONSTANT(T, 113, 2.8050629090725751585266360464766768437048e-02),
 473          BOOST_MATH_BIG_CONSTANT(T, 113, 2.9219405302833158254515212437025679637597e-02),
 474          BOOST_MATH_BIG_CONSTANT(T, 113, 4.4742214371598631578107310396249912330627e-02),
 475          BOOST_MATH_BIG_CONSTANT(T, 113, 9.0602983776478659136184969363625092585520e-02),
 476          BOOST_MATH_BIG_CONSTANT(T, 113, 2.2839507231977478205885469900971893734770e-01),
 477          BOOST_MATH_BIG_CONSTANT(T, 113, 6.8925739165733823730525449511456529001868e-01),
 478          BOOST_MATH_BIG_CONSTANT(T, 113, 2.4238082222874015159424842335385854632223e+00),
 479          BOOST_MATH_BIG_CONSTANT(T, 113, 9.6759648427182491050716309699208988458050e+00),
 480          BOOST_MATH_BIG_CONSTANT(T, 113, 4.7292246491169360014875196108746167872215e+01),
 481          BOOST_MATH_BIG_CONSTANT(T, 113, 3.1001411442786230340015781205680362993575e+01),
 482          BOOST_MATH_BIG_CONSTANT(T, 113, 9.8277628835804873490331739499978938078848e+03),
 483          BOOST_MATH_BIG_CONSTANT(T, 113, -3.1208326312801432038715638596517882759639e+05),
 484          BOOST_MATH_BIG_CONSTANT(T, 113, 9.4813611580683862051838126076298945680803e+06),
 485          BOOST_MATH_BIG_CONSTANT(T, 113, -2.1278197693321821164135890132925119054391e+08),
 486          BOOST_MATH_BIG_CONSTANT(T, 113, 3.3190303792682886967459489059860595063574e+09),
 487          BOOST_MATH_BIG_CONSTANT(T, 113, -2.1580767338646580750893606158043485767644e+10),
 488          BOOST_MATH_BIG_CONSTANT(T, 113, -5.0256008808415702780816006134784995506549e+11),
 489          BOOST_MATH_BIG_CONSTANT(T, 113, 1.9044186472918017896554580836514681614475e+13),
 490          BOOST_MATH_BIG_CONSTANT(T, 113, -3.2521078890073151875661384381880225635135e+14),
 491          BOOST_MATH_BIG_CONSTANT(T, 113, 3.3620352486836976842181057590770636605454e+15),
 492          BOOST_MATH_BIG_CONSTANT(T, 113, -2.0375525734060401555856465179734887312420e+16),
 493          BOOST_MATH_BIG_CONSTANT(T, 113, 5.6392664899881014534361728644608549445131e+16)
 494       };
 495       return exp(x) * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 496    }
 497    else
 498    {
 499       // Bessel I0 over[100, INF]
 500       // Max error in interpolated form : 5.459e-35
 501       // Max Error found at float128 precision = Poly : 1.472240e-34
 502       static const T P[] = {
 503          BOOST_MATH_BIG_CONSTANT(T, 113, 3.9894228040143267793994605993438166526772e-01),
 504          BOOST_MATH_BIG_CONSTANT(T, 113, 4.9867785050179084742493257495245185241487e-02),
 505          BOOST_MATH_BIG_CONSTANT(T, 113, 2.8050629090725735167652437695397756897920e-02),
 506          BOOST_MATH_BIG_CONSTANT(T, 113, 2.9219405302839307466358297347675795965363e-02),
 507          BOOST_MATH_BIG_CONSTANT(T, 113, 4.4742214369972689474366968442268908028204e-02),
 508          BOOST_MATH_BIG_CONSTANT(T, 113, 9.0602984099194778006610058410222616383078e-02),
 509          BOOST_MATH_BIG_CONSTANT(T, 113, 2.2839502241666629677015839125593079416327e-01),
 510          BOOST_MATH_BIG_CONSTANT(T, 113, 6.8926354981801627920292655818232972385750e-01),
 511          BOOST_MATH_BIG_CONSTANT(T, 113, 2.4231921590621824187100989532173995000655e+00),
 512          BOOST_MATH_BIG_CONSTANT(T, 113, 9.7264260959693775207585700654645245723497e+00),
 513          BOOST_MATH_BIG_CONSTANT(T, 113, 4.3890136225398811195878046856373030127018e+01),
 514          BOOST_MATH_BIG_CONSTANT(T, 113, 2.1999720924619285464910452647408431234369e+02),
 515          BOOST_MATH_BIG_CONSTANT(T, 113, 1.2076909538525038580501368530598517194748e+03),
 516          BOOST_MATH_BIG_CONSTANT(T, 113, 7.5684635141332367730007149159063086133399e+03),
 517          BOOST_MATH_BIG_CONSTANT(T, 113, 3.5178192543258299267923025833141286569141e+04),
 518          BOOST_MATH_BIG_CONSTANT(T, 113, 6.2966297919851965784482163987240461837728e+05) };
 519       T ex = exp(x / 2);
 520       T result = ex * boost::math::tools::evaluate_polynomial(P, T(1 / x)) / sqrt(x);
 521       result *= ex;
 522       return result;
 523    }
 524 }
 525
 526 template <typename T>
 527 T bessel_i0_imp(const T& x, const mpl::int_<0>&)
 528 {
 529    if(boost::math::tools::digits<T>() <= 24)
 530       return bessel_i0_imp(x, mpl::int_<24>());
 531    else if(boost::math::tools::digits<T>() <= 53)
 532       return bessel_i0_imp(x, mpl::int_<53>());
 533    else if(boost::math::tools::digits<T>() <= 64)
 534       return bessel_i0_imp(x, mpl::int_<64>());
 535    else if(boost::math::tools::digits<T>() <= 113)
 536       return bessel_i0_imp(x, mpl::int_<113>());
 537    BOOST_ASSERT(0);
 538    return 0;
 539 }
 540
 541 template <typename T>
 542 inline T bessel_i0(const T& x)
 543 {
 544    typedef mpl::int_<
 545       ((std::numeric_limits<T>::digits == 0) || (std::numeric_limits<T>::radix != 2)) ?
 546       0 :
 547       std::numeric_limits<T>::digits <= 24 ?
 548       24 :
 549       std::numeric_limits<T>::digits <= 53 ?
 550       53 :
 551       std::numeric_limits<T>::digits <= 64 ?
 552       64 :
 553       std::numeric_limits<T>::digits <= 113 ?
 554       113 : -1
 555    > tag_type;
 556
 557    bessel_i0_initializer<T, tag_type>::force_instantiate();
 558    return bessel_i0_imp(x, tag_type());
 559 }
 560
 561 }}} // namespaces
 562
 563 #endif // BOOST_MATH_BESSEL_I0_HPP
 564