venv/lib/python3.11/site-packages/scipy/special/xsf/cephes/gamma.h

/* Translated into C++ by SciPy developers in 2024.
 * Original header with Copyright information appears below.
 */

/*
 *     Gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, Gamma();
 *
 * y = Gamma( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns Gamma function of the argument.  The result is
 * correctly signed.
 *
 * Arguments |x| <= 34 are reduced by recurrence and the function
 * approximated by a rational function of degree 6/7 in the
 * interval (2,3).  Large arguments are handled by Stirling's
 * formula. Large negative arguments are made positive using
 * a reflection formula.
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE    -170,-33      20000       2.3e-15     3.3e-16
 *    IEEE     -33,  33     20000       9.4e-16     2.2e-16
 *    IEEE      33, 171.6   20000       2.3e-15     3.2e-16
 *
 * Error for arguments outside the test range will be larger
 * owing to error amplification by the exponential function.
 *
 */

/*                                                     lgam()
 *
 *     Natural logarithm of Gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, lgam();
 *
 * y = lgam( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the base e (2.718...) logarithm of the absolute
 * value of the Gamma function of the argument.
 *
 * For arguments greater than 13, the logarithm of the Gamma
 * function is approximated by the logarithmic version of
 * Stirling's formula using a polynomial approximation of
 * degree 4. Arguments between -33 and +33 are reduced by
 * recurrence to the interval [2,3] of a rational approximation.
 * The cosecant reflection formula is employed for arguments
 * less than -33.
 *
 * Arguments greater than MAXLGM return INFINITY and an error
 * message.  MAXLGM = 2.556348e305 for IEEE arithmetic.
 *
 *
 *
 * ACCURACY:
 *
 *
 * arithmetic      domain        # trials     peak         rms
 *    IEEE    0, 3                 28000     5.4e-16     1.1e-16
 *    IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
 * The error criterion was relative when the function magnitude
 * was greater than one but absolute when it was less than one.
 *
 * The following test used the relative error criterion, though
 * at certain points the relative error could be much higher than
 * indicated.
 *    IEEE    -200, -4             10000     4.8e-16     1.3e-16
 *
 */

/*
 * Cephes Math Library Release 2.2:  July, 1992
 * Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
 * Direct inquiries to 30 Frost Street, Cambridge, MA 02140
 */
#pragma once

#include "../config.h"
#include "../error.h"
#include "const.h"
#include "polevl.h"
#include "trig.h"

namespace xsf {
namespace cephes {
    namespace detail {
        constexpr double gamma_P[] = {1.60119522476751861407E-4, 1.19135147006586384913E-3, 1.04213797561761569935E-2,
                                      4.76367800457137231464E-2, 2.07448227648435975150E-1, 4.94214826801497100753E-1,
                                      9.99999999999999996796E-1};

        constexpr double gamma_Q[] = {-2.31581873324120129819E-5, 5.39605580493303397842E-4, -4.45641913851797240494E-3,
                                      1.18139785222060435552E-2,  3.58236398605498653373E-2, -2.34591795718243348568E-1,
                                      7.14304917030273074085E-2,  1.00000000000000000320E0};

        /* Stirling's formula for the Gamma function */
        constexpr double gamma_STIR[5] = {
            7.87311395793093628397E-4, -2.29549961613378126380E-4, -2.68132617805781232825E-3,
            3.47222221605458667310E-3, 8.33333333333482257126E-2,
        };

        constexpr double MAXSTIR = 143.01608;

        /* Gamma function computed by Stirling's formula.
         * The polynomial STIR is valid for 33 <= x <= 172.
         */
        XSF_HOST_DEVICE inline double stirf(double x) {
            double y, w, v;

            if (x >= MAXGAM) {
                return (std::numeric_limits<double>::infinity());
            }
            w = 1.0 / x;
            w = 1.0 + w * xsf::cephes::polevl(w, gamma_STIR, 4);
            y = std::exp(x);
            if (x > MAXSTIR) { /* Avoid overflow in pow() */
                v = std::pow(x, 0.5 * x - 0.25);
                y = v * (v / y);
            } else {
                y = std::pow(x, x - 0.5) / y;
            }
            y = SQRTPI * y * w;
            return (y);
        }
    } // namespace detail

    XSF_HOST_DEVICE inline double Gamma(double x) {
        double p, q, z;
        int i;
        int sgngam = 1;

        if (!std::isfinite(x)) {
	    if (x > 0) {
		// gamma(+inf) = +inf
		return x;
	    }
	    // gamma(NaN) and gamma(-inf) both should equal NaN.
            return std::numeric_limits<double>::quiet_NaN();
        }

	if (x == 0) {
	    /* For pole at zero, value depends on sign of zero.
	     * +inf when approaching from right, -inf when approaching
	     * from left. */
	    return std::copysign(std::numeric_limits<double>::infinity(), x);
	}

        q = std::abs(x);

        if (q > 33.0) {
            if (x < 0.0) {
                p = std::floor(q);
                if (p == q) {
		    // x is a negative integer. This is a pole.
                    set_error("Gamma", SF_ERROR_SINGULAR, NULL);
                    return (std::numeric_limits<double>::quiet_NaN());
                }
                i = p;
                if ((i & 1) == 0) {
                    sgngam = -1;
                }
                z = q - p;
                if (z > 0.5) {
                    p += 1.0;
                    z = q - p;
                }
                z = q * sinpi(z);
                if (z == 0.0) {
                    return (sgngam * std::numeric_limits<double>::infinity());
                }
                z = std::abs(z);
                z = M_PI / (z * detail::stirf(q));
            } else {
                z = detail::stirf(x);
            }
            return (sgngam * z);
        }

        z = 1.0;
        while (x >= 3.0) {
            x -= 1.0;
            z *= x;
        }

        while (x < 0.0) {
            if (x > -1.E-9) {
                goto small;
            }
            z /= x;
            x += 1.0;
        }

        while (x < 2.0) {
            if (x < 1.e-9) {
                goto small;
            }
            z /= x;
            x += 1.0;
        }

        if (x == 2.0) {
            return (z);
        }

        x -= 2.0;
        p = polevl(x, detail::gamma_P, 6);
        q = polevl(x, detail::gamma_Q, 7);
        return (z * p / q);

    small:
        if (x == 0.0) {
	    /* For this to have happened, x must have started as a negative integer. */
	    set_error("Gamma", SF_ERROR_SINGULAR, NULL);
	    return (std::numeric_limits<double>::quiet_NaN());
        } else
            return (z / ((1.0 + 0.5772156649015329 * x) * x));
    }

    namespace detail {
        /* A[]: Stirling's formula expansion of log Gamma
         * B[], C[]: log Gamma function between 2 and 3
         */
        constexpr double gamma_A[] = {8.11614167470508450300E-4, -5.95061904284301438324E-4, 7.93650340457716943945E-4,
                                      -2.77777777730099687205E-3, 8.33333333333331927722E-2};

        constexpr double gamma_B[] = {-1.37825152569120859100E3, -3.88016315134637840924E4, -3.31612992738871184744E5,
                                      -1.16237097492762307383E6, -1.72173700820839662146E6, -8.53555664245765465627E5};

        constexpr double gamma_C[] = {
            /* 1.00000000000000000000E0, */
            -3.51815701436523470549E2, -1.70642106651881159223E4, -2.20528590553854454839E5,
            -1.13933444367982507207E6, -2.53252307177582951285E6, -2.01889141433532773231E6};

        /* log( sqrt( 2*pi ) ) */
        constexpr double LS2PI = 0.91893853320467274178;

        constexpr double MAXLGM = 2.556348e305;

        /* Disable optimizations for this function on 32 bit systems when compiling with GCC.
         * We've found that enabling optimizations can result in degraded precision
         * for this asymptotic approximation in that case. */
#if defined(__GNUC__) && defined(__i386__)
#pragma GCC push_options
#pragma GCC optimize("00")
#endif
        XSF_HOST_DEVICE inline double lgam_large_x(double x) {
            double q = (x - 0.5) * std::log(x) - x + LS2PI;
            if (x > 1.0e8) {
                return (q);
            }
            double p = 1.0 / (x * x);
            p = ((7.9365079365079365079365e-4 * p - 2.7777777777777777777778e-3) * p + 0.0833333333333333333333) / x;
            return q + p;
        }
#if defined(__GNUC__) && defined(__i386__)
#pragma GCC pop_options
#endif

        XSF_HOST_DEVICE inline double lgam_sgn(double x, int *sign) {
            double p, q, u, w, z;
            int i;

            *sign = 1;

            if (!std::isfinite(x)) {
                return x;
            }

            if (x < -34.0) {
                q = -x;
                w = lgam_sgn(q, sign);
                p = std::floor(q);
                if (p == q) {
                lgsing:
                    set_error("lgam", SF_ERROR_SINGULAR, NULL);
                    return (std::numeric_limits<double>::infinity());
                }
                i = p;
                if ((i & 1) == 0) {
                    *sign = -1;
                } else {
                    *sign = 1;
                }
                z = q - p;
                if (z > 0.5) {
                    p += 1.0;
                    z = p - q;
                }
                z = q * sinpi(z);
                if (z == 0.0) {
                    goto lgsing;
                }
                /*     z = log(M_PI) - log( z ) - w; */
                z = LOGPI - std::log(z) - w;
                return (z);
            }

            if (x < 13.0) {
                z = 1.0;
                p = 0.0;
                u = x;
                while (u >= 3.0) {
                    p -= 1.0;
                    u = x + p;
                    z *= u;
                }
                while (u < 2.0) {
                    if (u == 0.0) {
                        goto lgsing;
                    }
                    z /= u;
                    p += 1.0;
                    u = x + p;
                }
                if (z < 0.0) {
                    *sign = -1;
                    z = -z;
                } else {
                    *sign = 1;
                }
                if (u == 2.0) {
                    return (std::log(z));
                }
                p -= 2.0;
                x = x + p;
                p = x * polevl(x, gamma_B, 5) / p1evl(x, gamma_C, 6);
                return (std::log(z) + p);
            }

            if (x > MAXLGM) {
                return (*sign * std::numeric_limits<double>::infinity());
            }

            if (x >= 1000.0) {
                return lgam_large_x(x);
            }

            q = (x - 0.5) * std::log(x) - x + LS2PI;
            p = 1.0 / (x * x);
            return q + polevl(p, gamma_A, 4) / x;
        }
    } // namespace detail

    /* Logarithm of Gamma function */
    XSF_HOST_DEVICE inline double lgam(double x) {
        int sign;
        return detail::lgam_sgn(x, &sign);
    }

    /* Sign of the Gamma function */
    XSF_HOST_DEVICE inline double gammasgn(double x) {
        double fx;

        if (std::isnan(x)) {
            return x;
        }
        if (x > 0) {
            return 1.0;
	}
	if (x == 0) {
	    return std::copysign(1.0, x);
	}
	if (std::isinf(x)) {
	    // x > 0 case handled, so x must be negative infinity.
	    return std::numeric_limits<double>::quiet_NaN();
	}
	fx = std::floor(x);
	if (x - fx == 0.0) {
	    return std::numeric_limits<double>::quiet_NaN();
	}
	// sign of gamma for x in (-n, -n+1) for positive integer n is (-1)^n.
	if (static_cast<int>(fx) % 2) {
	    return -1.0;
	}
	return 1.0;
    }

} // namespace cephes
} // namespace xsf