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