// Copyright (c) 2006 Xiaogang Zhang
+// Copyright (c) 2017 John Maddock
// Use, modification and distribution are subject to the
// Boost Software License, Version 1.0. (See accompanying file
// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
// Modified Bessel function of the second kind of order zero
// minimax rational approximations on intervals, see
-// Russon and Blair, Chalk River Report AECL-3461, 1969
+// Russon and Blair, Chalk River Report AECL-3461, 1969,
+// as revised by Pavel Holoborodko in "Rational Approximations
+// for the Modified Bessel Function of the Second Kind - K0(x)
+// for Computations with Double Precision", see
+// http://www.advanpix.com/2015/11/25/rational-approximations-for-the-modified-bessel-function-of-the-second-kind-k0-for-computations-with-double-precision/
+//
+// The actual coefficients used are our own derivation (by JM)
+// since we extend to both greater and lesser precision than the
+// references above. We can also improve performance WRT to
+// Holoborodko without loss of precision.
namespace boost { namespace math { namespace detail{
-template <typename T, typename Policy>
-T bessel_k0(T x, const Policy&);
+template <typename T>
+T bessel_k0(const T& x);
-template <class T, class Policy>
+template <class T, class tag>
struct bessel_k0_initializer
{
struct init
{
init()
{
- do_init();
+ do_init(tag());
}
- static void do_init()
+ static void do_init(const mpl::int_<113>&)
{
- bessel_k0(T(1), Policy());
+ bessel_k0(T(0.5));
+ bessel_k0(T(1.5));
}
+ static void do_init(const mpl::int_<64>&)
+ {
+ bessel_k0(T(0.5));
+ bessel_k0(T(1.5));
+ }
+ template <class U>
+ static void do_init(const U&){}
void force_instantiate()const{}
};
static const init initializer;
}
};
-template <class T, class Policy>
-const typename bessel_k0_initializer<T, Policy>::init bessel_k0_initializer<T, Policy>::initializer;
+template <class T, class tag>
+const typename bessel_k0_initializer<T, tag>::init bessel_k0_initializer<T, tag>::initializer;
+
+
+template <typename T, int N>
+T bessel_k0_imp(const T& x, const mpl::int_<N>&)
+{
+ BOOST_ASSERT(0);
+ return 0;
+}
+
+template <typename T>
+T bessel_k0_imp(const T& x, const mpl::int_<24>&)
+{
+ BOOST_MATH_STD_USING
+ if(x <= 1)
+ {
+ // Maximum Deviation Found : 2.358e-09
+ // Expected Error Term : -2.358e-09
+ // Maximum Relative Change in Control Points : 9.552e-02
+ // Max Error found at float precision = Poly : 4.448220e-08
+ static const T Y = 1.137250900268554688f;
+ static const T P[] =
+ {
+ -1.372508979104259711e-01f,
+ 2.622545986273687617e-01f,
+ 5.047103728247919836e-03f
+ };
+ static const T Q[] =
+ {
+ 1.000000000000000000e+00f,
+ -8.928694018000029415e-02f,
+ 2.985980684180969241e-03f
+ };
+ T a = x * x / 4;
+ a = (tools::evaluate_rational(P, Q, a) + Y) * a + 1;
+
+ // Maximum Deviation Found: 1.346e-09
+ // Expected Error Term : -1.343e-09
+ // Maximum Relative Change in Control Points : 2.405e-02
+ // Max Error found at float precision = Poly : 1.354814e-07
+ static const T P2[] = {
+ 1.159315158e-01f,
+ 2.789828686e-01f,
+ 2.524902861e-02f,
+ 8.457241514e-04f,
+ 1.530051997e-05f
+ };
+ return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a;
+ }
+ else
+ {
+ // Maximum Deviation Found: 1.587e-08
+ // Expected Error Term : 1.531e-08
+ // Maximum Relative Change in Control Points : 9.064e-02
+ // Max Error found at float precision = Poly : 5.065020e-08
+
+ static const T P[] =
+ {
+ 2.533141220e-01,
+ 5.221502603e-01,
+ 6.380180669e-02,
+ -5.934976547e-02
+ };
+ static const T Q[] =
+ {
+ 1.000000000e+00,
+ 2.679722431e+00,
+ 1.561635813e+00,
+ 1.573660661e-01
+ };
+ if(x < tools::log_max_value<T>())
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + 1) * exp(-x) / sqrt(x));
+ else
+ {
+ T ex = exp(-x / 2);
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + 1) * ex / sqrt(x)) * ex;
+ }
+ }
+}
+
+template <typename T>
+T bessel_k0_imp(const T& x, const mpl::int_<53>&)
+{
+ BOOST_MATH_STD_USING
+ if(x <= 1)
+ {
+ // Maximum Deviation Found: 6.077e-17
+ // Expected Error Term : -6.077e-17
+ // Maximum Relative Change in Control Points : 7.797e-02
+ // Max Error found at double precision = Poly : 1.003156e-16
+ static const T Y = 1.137250900268554688;
+ static const T P[] =
+ {
+ -1.372509002685546267e-01,
+ 2.574916117833312855e-01,
+ 1.395474602146869316e-02,
+ 5.445476986653926759e-04,
+ 7.125159422136622118e-06
+ };
+ static const T Q[] =
+ {
+ 1.000000000000000000e+00,
+ -5.458333438017788530e-02,
+ 1.291052816975251298e-03,
+ -1.367653946978586591e-05
+ };
+
+ T a = x * x / 4;
+ a = (tools::evaluate_polynomial(P, a) / tools::evaluate_polynomial(Q, a) + Y) * a + 1;
+
+ // Maximum Deviation Found: 3.429e-18
+ // Expected Error Term : 3.392e-18
+ // Maximum Relative Change in Control Points : 2.041e-02
+ // Max Error found at double precision = Poly : 2.513112e-16
+ static const T P2[] =
+ {
+ 1.159315156584124484e-01,
+ 2.789828789146031732e-01,
+ 2.524892993216121934e-02,
+ 8.460350907213637784e-04,
+ 1.491471924309617534e-05,
+ 1.627106892422088488e-07,
+ 1.208266102392756055e-09,
+ 6.611686391749704310e-12
+ };
+
+ return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a;
+ }
+ else
+ {
+ // Maximum Deviation Found: 4.316e-17
+ // Expected Error Term : 9.570e-18
+ // Maximum Relative Change in Control Points : 2.757e-01
+ // Max Error found at double precision = Poly : 1.001560e-16
+
+ static const T Y = 1;
+ static const T P[] =
+ {
+ 2.533141373155002416e-01,
+ 3.628342133984595192e+00,
+ 1.868441889406606057e+01,
+ 4.306243981063412784e+01,
+ 4.424116209627428189e+01,
+ 1.562095339356220468e+01,
+ -1.810138978229410898e+00,
+ -1.414237994269995877e+00,
+ -9.369168119754924625e-02
+ };
+ static const T Q[] =
+ {
+ 1.000000000000000000e+00,
+ 1.494194694879908328e+01,
+ 8.265296455388554217e+01,
+ 2.162779506621866970e+02,
+ 2.845145155184222157e+02,
+ 1.851714491916334995e+02,
+ 5.486540717439723515e+01,
+ 6.118075837628957015e+00,
+ 1.586261269326235053e-01
+ };
+ if(x < tools::log_max_value<T>())
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
+ else
+ {
+ T ex = exp(-x / 2);
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
+ }
+ }
+}
-template <typename T, typename Policy>
-T bessel_k0(T x, const Policy& pol)
+template <typename T>
+T bessel_k0_imp(const T& x, const mpl::int_<64>&)
{
- BOOST_MATH_INSTRUMENT_CODE(x);
-
- bessel_k0_initializer<T, Policy>::force_instantiate();
-
- static const T P1[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.4708152720399552679e+03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.9169059852270512312e+03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.6850901201934832188e+02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1999463724910714109e+01)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.3166052564989571850e-01)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8599221412826100000e-04))
- };
- static const T Q1[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.1312714303849120380e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.4994418972832303646e+02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
- };
- static const T P2[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6128136304458193998e+06)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -3.7333769444840079748e+05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.7984434409411765813e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.9501657892958843865e+02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6414452837299064100e+00))
- };
- static const T Q2[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -1.6128136304458193998e+06)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.9865713163054025489e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.5064972445877992730e+02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
- };
- static const T P3[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1600249425076035558e+02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.3444738764199315021e+03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8321525870183537725e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 7.1557062783764037541e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5097646353289914539e+05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.7398867902565686251e+05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0577068948034021957e+05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.1075408980684392399e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.6832589957340267940e+03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.1394980557384778174e+02))
- };
- static const T Q3[] = {
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.2556599177304839811e+01)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.8821890840982713696e+03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.4847228371802360957e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5.8824616785857027752e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.2689839587977598727e+05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.5144644673520157801e+05)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.7418829762268075784e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3.1474655750295278825e+04)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.4329628889746408858e+03)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.0013443064949242491e+02)),
- static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.0))
- };
- T value, factor, r, r1, r2;
-
- BOOST_MATH_STD_USING
- using namespace boost::math::tools;
-
- static const char* function = "boost::math::bessel_k0<%1%>(%1%,%1%)";
-
- if (x < 0)
- {
- return policies::raise_domain_error<T>(function,
- "Got x = %1%, but argument x must be non-negative, complex number result not supported", x, pol);
- }
- if (x == 0)
- {
- return policies::raise_overflow_error<T>(function, 0, pol);
- }
- if (x <= 1) // x in (0, 1]
- {
- T y = x * x;
- r1 = evaluate_polynomial(P1, y) / evaluate_polynomial(Q1, y);
- r2 = evaluate_polynomial(P2, y) / evaluate_polynomial(Q2, y);
- factor = log(x);
- value = r1 - factor * r2;
- }
- else // x in (1, \infty)
- {
- T y = 1 / x;
- r = evaluate_polynomial(P3, y) / evaluate_polynomial(Q3, y);
- factor = exp(-x) / sqrt(x);
- value = factor * r;
- BOOST_MATH_INSTRUMENT_CODE("y = " << y);
- BOOST_MATH_INSTRUMENT_CODE("r = " << r);
- BOOST_MATH_INSTRUMENT_CODE("factor = " << factor);
- BOOST_MATH_INSTRUMENT_CODE("value = " << value);
- }
-
- return value;
+ BOOST_MATH_STD_USING
+ if(x <= 1)
+ {
+ // Maximum Deviation Found: 2.180e-22
+ // Expected Error Term : 2.180e-22
+ // Maximum Relative Change in Control Points : 2.943e-01
+ // Max Error found at float80 precision = Poly : 3.923207e-20
+ static const T Y = 1.137250900268554687500e+00;
+ static const T P[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.372509002685546875002e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.566481981037407600436e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.551881122448948854873e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 6.646112454323276529650e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.213747930378196492543e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 9.423709328020389560844e-08)
+ };
+ static const T Q[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.843828412587773008342e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.088484822515098936140e-03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.374724008530702784829e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 8.452665455952581680339e-08)
+ };
+
+
+ T a = x * x / 4;
+ a = (tools::evaluate_polynomial(P, a) / tools::evaluate_polynomial(Q, a) + Y) * a + 1;
+
+ // Maximum Deviation Found: 2.440e-21
+ // Expected Error Term : -2.434e-21
+ // Maximum Relative Change in Control Points : 2.459e-02
+ // Max Error found at float80 precision = Poly : 1.482487e-19
+ static const T P2[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.159315156584124488110e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.764832791416047889734e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.926062887220923354112e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 3.660777862036966089410e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.094942446930673386849e-06)
+ };
+ static const T Q2[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -2.156100313881251616320e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.315993873344905957033e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.529444499350703363451e-06),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 5.524988589917857531177e-09)
+ };
+ return tools::evaluate_rational(P2, Q2, T(x * x)) - log(x) * a;
+ }
+ else
+ {
+ // Maximum Deviation Found: 4.291e-20
+ // Expected Error Term : 2.236e-21
+ // Maximum Relative Change in Control Points : 3.021e-01
+ //Max Error found at float80 precision = Poly : 8.727378e-20
+ static const T Y = 1;
+ static const T P[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.533141373155002512056e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 5.417942070721928652715e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 4.477464607463971754433e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.838745728725943889876e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 4.009736314927811202517e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 4.557411293123609803452e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.360222564015361268955e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.385435333168505701022e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.750195760942181592050e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -4.059789241612946683713e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 64, -1.612783121537333908889e-01)
+ };
+ static const T Q[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.000000000000000000000e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.200669254769325861404e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.900177593527144126549e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 8.361003989965786932682e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.041319870804843395893e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.828491555113790345068e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.190342229261529076624e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 9.003330795963812219852e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.773371397243777891569e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 1.368634935531158398439e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 64, 2.543310879400359967327e-01)
+ };
+ if(x < tools::log_max_value<T>())
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
+ else
+ {
+ T ex = exp(-x / 2);
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
+ }
+ }
+}
+
+template <typename T>
+T bessel_k0_imp(const T& x, const mpl::int_<113>&)
+{
+ BOOST_MATH_STD_USING
+ if(x <= 1)
+ {
+ // Maximum Deviation Found: 5.682e-37
+ // Expected Error Term : 5.682e-37
+ // Maximum Relative Change in Control Points : 6.094e-04
+ // Max Error found at float128 precision = Poly : 5.338213e-35
+ static const T Y = 1.137250900268554687500000000000000000e+00f;
+ static const T P[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.372509002685546875000000000000000006e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.556212905071072782462974351698081303e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.742459135264203478530904179889103929e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.077860530453688571555479526961318918e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.868173911669241091399374307788635148e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.496405768838992243478709145123306602e-07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.752489221949580551692915881999762125e-09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.243010555737173524710512824955368526e-12)
+ };
+ static const T Q[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.095631064064621099785696980653193721e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.313880983725212151967078809725835532e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.095229912293480063501285562382835142e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.022828799511943141130509410251996277e-07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -6.860874007419812445494782795829046836e-10),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.107297802344970725756092082686799037e-12),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -7.460529579244623559164763757787600944e-15)
+ };
+ T a = x * x / 4;
+ a = (tools::evaluate_rational(P, Q, a) + Y) * a + 1;
+
+ // Maximum Deviation Found: 5.173e-38
+ // Expected Error Term : 5.105e-38
+ // Maximum Relative Change in Control Points : 9.734e-03
+ // Max Error found at float128 precision = Poly : 1.688806e-34
+ static const T P2[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.159315156584124488107200313757741370e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.789828789146031122026800078439435369e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.524892993216269451266750049024628432e-02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.460350907082229957222453839935101823e-04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.491471929926042875260452849503857976e-05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.627105610481598430816014719558896866e-07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.208426165007797264194914898538250281e-09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.508697838747354949164182457073784117e-12),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.659784680639805301101014383907273109e-14),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.531090131964391104248859415958109654e-17),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.205195117066478034260323124669936314e-19),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.692219280289030165761119775783115426e-22),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.362350161092532344171965861545860747e-25),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.277990623924628999539014980773738258e-27)
+ };
+
+ return tools::evaluate_polynomial(P2, T(x * x)) - log(x) * a;
+ }
+ else
+ {
+ // Maximum Deviation Found: 1.462e-34
+ // Expected Error Term : 4.917e-40
+ // Maximum Relative Change in Control Points : 3.385e-01
+ // Max Error found at float128 precision = Poly : 1.567573e-34
+ static const T Y = 1;
+ static const T P[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.533141373155002512078826424055226265e-01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.001949740768235770078339977110749204e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 6.991516715983883248363351472378349986e+02),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.429587951594593159075690819360687720e+04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.911933815201948768044660065771258450e+05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.769943016204926614862175317962439875e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.170866154649560750500954150401105606e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.634687099724383996792011977705727661e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.989524036456492581597607246664394014e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.160394785715328062088529400178080360e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 9.778173054417826368076483100902201433e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.335667778588806892764139643950439733e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.283635100080306980206494425043706838e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.300616188213640626577036321085025855e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.277591957076162984986406540894621482e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.564360536834214058158565361486115932e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.043505161612403359098596828115690596e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -7.217035248223503605127967970903027314e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.422938158797326748375799596769964430e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -1.229125746200586805278634786674745210e+05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -4.201632288615609937883545928660649813e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, -3.690820607338480548346746717311811406e+01)
+ };
+ static const T Q[] =
+ {
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.000000000000000000000000000000000000e+00),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.964877874035741452203497983642653107e+01),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.808929943826193766839360018583294769e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.814524004679994110944366890912384139e+04),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.897794522506725610540209610337355118e+05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.456339470955813675629523617440433672e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.057818717813969772198911392875127212e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.513821619536852436424913886081133209e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 9.255938846873380596038513316919990776e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 2.537077551699028079347581816919572141e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 5.176769339768120752974843214652367321e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.828722317390455845253191337207432060e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.698864296569996402006511705803675890e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.007803261356636409943826918468544629e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.016564631288740308993071395104715469e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 1.595893010619754750655947035567624730e+09),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.241241839120481076862742189989406856e+08),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.168778094393076220871007550235840858e+07),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 7.156200301360388147635052029404211109e+06),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 3.752130382550379886741949463587008794e+05),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 8.370574966987293592457152146806662562e+03),
+ BOOST_MATH_BIG_CONSTANT(T, 113, 4.871254714311063594080644835895740323e+01)
+ };
+ if(x < tools::log_max_value<T>())
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * exp(-x) / sqrt(x));
+ else
+ {
+ T ex = exp(-x / 2);
+ return ((tools::evaluate_rational(P, Q, T(1 / x)) + Y) * ex / sqrt(x)) * ex;
+ }
+ }
+}
+
+template <typename T>
+T bessel_k0_imp(const T& x, const mpl::int_<0>&)
+{
+ if(boost::math::tools::digits<T>() <= 24)
+ return bessel_k0_imp(x, mpl::int_<24>());
+ else if(boost::math::tools::digits<T>() <= 53)
+ return bessel_k0_imp(x, mpl::int_<53>());
+ else if(boost::math::tools::digits<T>() <= 64)
+ return bessel_k0_imp(x, mpl::int_<64>());
+ else if(boost::math::tools::digits<T>() <= 113)
+ return bessel_k0_imp(x, mpl::int_<113>());
+ BOOST_ASSERT(0);
+ return 0;
+}
+
+template <typename T>
+inline T bessel_k0(const T& x)
+{
+ typedef mpl::int_<
+ std::numeric_limits<T>::digits == 0 ?
+ 0 :
+ std::numeric_limits<T>::digits <= 24 ?
+ 24 :
+ std::numeric_limits<T>::digits <= 53 ?
+ 53 :
+ std::numeric_limits<T>::digits <= 64 ?
+ 64 :
+ std::numeric_limits<T>::digits <= 113 ?
+ 113 : -1
+ > tag_type;
+
+ bessel_k0_initializer<T, tag_type>::force_instantiate();
+ return bessel_k0_imp(x, tag_type());
}
}}} // namespaces
projects / platform / upstream / boost.git / blobdiff