399 lines
12 KiB
C++
399 lines
12 KiB
C++
/* Translated into C++ by SciPy developers in 2024.
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* Original header with Copyright information appears below.
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*/
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/*
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* Gamma function
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*
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*
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*
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* SYNOPSIS:
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*
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* double x, y, Gamma();
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*
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* y = Gamma( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns Gamma function of the argument. The result is
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* correctly signed.
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*
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* Arguments |x| <= 34 are reduced by recurrence and the function
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* approximated by a rational function of degree 6/7 in the
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* interval (2,3). Large arguments are handled by Stirling's
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* formula. Large negative arguments are made positive using
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* a reflection formula.
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* IEEE -170,-33 20000 2.3e-15 3.3e-16
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* IEEE -33, 33 20000 9.4e-16 2.2e-16
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* IEEE 33, 171.6 20000 2.3e-15 3.2e-16
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*
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* Error for arguments outside the test range will be larger
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* owing to error amplification by the exponential function.
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*
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*/
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/* lgam()
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*
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* Natural logarithm of Gamma function
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*
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*
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*
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* SYNOPSIS:
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*
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* double x, y, lgam();
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*
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* y = lgam( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns the base e (2.718...) logarithm of the absolute
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* value of the Gamma function of the argument.
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*
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* For arguments greater than 13, the logarithm of the Gamma
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* function is approximated by the logarithmic version of
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* Stirling's formula using a polynomial approximation of
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* degree 4. Arguments between -33 and +33 are reduced by
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* recurrence to the interval [2,3] of a rational approximation.
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* The cosecant reflection formula is employed for arguments
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* less than -33.
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*
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* Arguments greater than MAXLGM return INFINITY and an error
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* message. MAXLGM = 2.556348e305 for IEEE arithmetic.
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*
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*
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*
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* ACCURACY:
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*
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*
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* arithmetic domain # trials peak rms
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* IEEE 0, 3 28000 5.4e-16 1.1e-16
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* IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
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* The error criterion was relative when the function magnitude
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* was greater than one but absolute when it was less than one.
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*
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* The following test used the relative error criterion, though
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* at certain points the relative error could be much higher than
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* indicated.
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* IEEE -200, -4 10000 4.8e-16 1.3e-16
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*
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*/
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/*
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* Cephes Math Library Release 2.2: July, 1992
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* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
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* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
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*/
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#pragma once
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#include "../config.h"
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#include "../error.h"
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#include "const.h"
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#include "polevl.h"
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#include "trig.h"
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namespace xsf {
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namespace cephes {
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namespace detail {
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constexpr double gamma_P[] = {1.60119522476751861407E-4, 1.19135147006586384913E-3, 1.04213797561761569935E-2,
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4.76367800457137231464E-2, 2.07448227648435975150E-1, 4.94214826801497100753E-1,
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9.99999999999999996796E-1};
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constexpr double gamma_Q[] = {-2.31581873324120129819E-5, 5.39605580493303397842E-4, -4.45641913851797240494E-3,
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1.18139785222060435552E-2, 3.58236398605498653373E-2, -2.34591795718243348568E-1,
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7.14304917030273074085E-2, 1.00000000000000000320E0};
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/* Stirling's formula for the Gamma function */
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constexpr double gamma_STIR[5] = {
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7.87311395793093628397E-4, -2.29549961613378126380E-4, -2.68132617805781232825E-3,
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3.47222221605458667310E-3, 8.33333333333482257126E-2,
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};
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constexpr double MAXSTIR = 143.01608;
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/* Gamma function computed by Stirling's formula.
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* The polynomial STIR is valid for 33 <= x <= 172.
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*/
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XSF_HOST_DEVICE inline double stirf(double x) {
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double y, w, v;
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if (x >= MAXGAM) {
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return (std::numeric_limits<double>::infinity());
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}
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w = 1.0 / x;
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w = 1.0 + w * xsf::cephes::polevl(w, gamma_STIR, 4);
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y = std::exp(x);
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if (x > MAXSTIR) { /* Avoid overflow in pow() */
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v = std::pow(x, 0.5 * x - 0.25);
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y = v * (v / y);
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} else {
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y = std::pow(x, x - 0.5) / y;
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}
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y = SQRTPI * y * w;
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return (y);
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}
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} // namespace detail
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XSF_HOST_DEVICE inline double Gamma(double x) {
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double p, q, z;
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int i;
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int sgngam = 1;
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if (!std::isfinite(x)) {
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if (x > 0) {
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// gamma(+inf) = +inf
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return x;
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}
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// gamma(NaN) and gamma(-inf) both should equal NaN.
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return std::numeric_limits<double>::quiet_NaN();
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}
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if (x == 0) {
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/* For pole at zero, value depends on sign of zero.
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* +inf when approaching from right, -inf when approaching
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* from left. */
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return std::copysign(std::numeric_limits<double>::infinity(), x);
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}
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q = std::abs(x);
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if (q > 33.0) {
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if (x < 0.0) {
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p = std::floor(q);
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if (p == q) {
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// x is a negative integer. This is a pole.
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set_error("Gamma", SF_ERROR_SINGULAR, NULL);
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return (std::numeric_limits<double>::quiet_NaN());
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}
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i = p;
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if ((i & 1) == 0) {
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sgngam = -1;
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}
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z = q - p;
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if (z > 0.5) {
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p += 1.0;
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z = q - p;
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}
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z = q * sinpi(z);
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if (z == 0.0) {
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return (sgngam * std::numeric_limits<double>::infinity());
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}
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z = std::abs(z);
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z = M_PI / (z * detail::stirf(q));
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} else {
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z = detail::stirf(x);
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}
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return (sgngam * z);
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}
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z = 1.0;
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while (x >= 3.0) {
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x -= 1.0;
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z *= x;
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}
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while (x < 0.0) {
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if (x > -1.E-9) {
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goto small;
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}
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z /= x;
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x += 1.0;
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}
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while (x < 2.0) {
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if (x < 1.e-9) {
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goto small;
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}
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z /= x;
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x += 1.0;
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}
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if (x == 2.0) {
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return (z);
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}
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x -= 2.0;
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p = polevl(x, detail::gamma_P, 6);
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q = polevl(x, detail::gamma_Q, 7);
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return (z * p / q);
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small:
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if (x == 0.0) {
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/* For this to have happened, x must have started as a negative integer. */
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set_error("Gamma", SF_ERROR_SINGULAR, NULL);
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return (std::numeric_limits<double>::quiet_NaN());
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} else
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return (z / ((1.0 + 0.5772156649015329 * x) * x));
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}
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namespace detail {
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/* A[]: Stirling's formula expansion of log Gamma
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* B[], C[]: log Gamma function between 2 and 3
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*/
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constexpr double gamma_A[] = {8.11614167470508450300E-4, -5.95061904284301438324E-4, 7.93650340457716943945E-4,
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-2.77777777730099687205E-3, 8.33333333333331927722E-2};
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constexpr double gamma_B[] = {-1.37825152569120859100E3, -3.88016315134637840924E4, -3.31612992738871184744E5,
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-1.16237097492762307383E6, -1.72173700820839662146E6, -8.53555664245765465627E5};
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constexpr double gamma_C[] = {
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/* 1.00000000000000000000E0, */
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-3.51815701436523470549E2, -1.70642106651881159223E4, -2.20528590553854454839E5,
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-1.13933444367982507207E6, -2.53252307177582951285E6, -2.01889141433532773231E6};
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/* log( sqrt( 2*pi ) ) */
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constexpr double LS2PI = 0.91893853320467274178;
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constexpr double MAXLGM = 2.556348e305;
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/* Disable optimizations for this function on 32 bit systems when compiling with GCC.
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* We've found that enabling optimizations can result in degraded precision
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* for this asymptotic approximation in that case. */
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#if defined(__GNUC__) && defined(__i386__)
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#pragma GCC push_options
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#pragma GCC optimize("00")
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#endif
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XSF_HOST_DEVICE inline double lgam_large_x(double x) {
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double q = (x - 0.5) * std::log(x) - x + LS2PI;
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if (x > 1.0e8) {
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return (q);
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}
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double p = 1.0 / (x * x);
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p = ((7.9365079365079365079365e-4 * p - 2.7777777777777777777778e-3) * p + 0.0833333333333333333333) / x;
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return q + p;
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}
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#if defined(__GNUC__) && defined(__i386__)
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#pragma GCC pop_options
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#endif
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XSF_HOST_DEVICE inline double lgam_sgn(double x, int *sign) {
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double p, q, u, w, z;
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int i;
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*sign = 1;
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if (!std::isfinite(x)) {
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return x;
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}
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if (x < -34.0) {
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q = -x;
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w = lgam_sgn(q, sign);
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p = std::floor(q);
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if (p == q) {
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lgsing:
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set_error("lgam", SF_ERROR_SINGULAR, NULL);
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return (std::numeric_limits<double>::infinity());
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}
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i = p;
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if ((i & 1) == 0) {
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*sign = -1;
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} else {
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*sign = 1;
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}
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z = q - p;
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if (z > 0.5) {
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p += 1.0;
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z = p - q;
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}
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z = q * sinpi(z);
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if (z == 0.0) {
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goto lgsing;
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}
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/* z = log(M_PI) - log( z ) - w; */
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z = LOGPI - std::log(z) - w;
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return (z);
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}
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if (x < 13.0) {
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z = 1.0;
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p = 0.0;
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u = x;
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while (u >= 3.0) {
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p -= 1.0;
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u = x + p;
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z *= u;
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}
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while (u < 2.0) {
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if (u == 0.0) {
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goto lgsing;
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}
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z /= u;
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p += 1.0;
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u = x + p;
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}
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if (z < 0.0) {
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*sign = -1;
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z = -z;
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} else {
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*sign = 1;
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}
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if (u == 2.0) {
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return (std::log(z));
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}
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p -= 2.0;
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x = x + p;
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p = x * polevl(x, gamma_B, 5) / p1evl(x, gamma_C, 6);
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return (std::log(z) + p);
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}
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if (x > MAXLGM) {
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return (*sign * std::numeric_limits<double>::infinity());
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}
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if (x >= 1000.0) {
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return lgam_large_x(x);
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}
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q = (x - 0.5) * std::log(x) - x + LS2PI;
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p = 1.0 / (x * x);
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return q + polevl(p, gamma_A, 4) / x;
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}
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} // namespace detail
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/* Logarithm of Gamma function */
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XSF_HOST_DEVICE inline double lgam(double x) {
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int sign;
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return detail::lgam_sgn(x, &sign);
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}
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/* Sign of the Gamma function */
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XSF_HOST_DEVICE inline double gammasgn(double x) {
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double fx;
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if (std::isnan(x)) {
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return x;
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}
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if (x > 0) {
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return 1.0;
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}
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if (x == 0) {
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return std::copysign(1.0, x);
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}
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if (std::isinf(x)) {
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// x > 0 case handled, so x must be negative infinity.
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return std::numeric_limits<double>::quiet_NaN();
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}
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fx = std::floor(x);
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if (x - fx == 0.0) {
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return std::numeric_limits<double>::quiet_NaN();
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}
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// sign of gamma for x in (-n, -n+1) for positive integer n is (-1)^n.
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if (static_cast<int>(fx) % 2) {
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return -1.0;
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}
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return 1.0;
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}
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} // namespace cephes
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} // namespace xsf
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