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tgamma.c
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‎tgamma.c

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/*
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* Implementation of the gamma function based on CPython's implementation.
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* This is a standalone version without dependencies on CPython.
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*/
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#include <math.h>
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#include <errno.h>
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#include <float.h>
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#include <assert.h>
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#include <stdbool.h>
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/* Constants and definitions needed for gamma calculation */
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static const double pi = 3.141592653589793238462643383279502884197;
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static const double logpi = 1.144729885849400174143427351353058711647;
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/* Check if a value is finite */
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#ifndef Py_IS_FINITE
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#define Py_IS_FINITE(x) isfinite(x)
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#endif
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/* Check if a value is infinity */
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#ifndef Py_IS_INFINITY
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#define Py_IS_INFINITY(x) isinf(x)
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#endif
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/* Check if a value is NaN */
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#ifndef Py_IS_NAN
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#define Py_IS_NAN(x) isnan(x)
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#endif
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/* Define HUGE_VAL if not available */
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#ifndef Py_HUGE_VAL
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#define Py_HUGE_VAL HUGE_VAL
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#endif
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/* Define mathematical constants required for Lanczos approximation */
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#define LANCZOS_N 13
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static const double lanczos_g = 6.024680040776729583740234375;
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static const double lanczos_g_minus_half = 5.524680040776729583740234375;
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static const double lanczos_num_coeffs[LANCZOS_N] = {
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23531376880.410759688572007674451636754734846804940,
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42919803642.649098768957899047001988850926355848959,
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35711959237.355668049440185451547166705960488635843,
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17921034426.037209699919755754458931112671403265390,
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6039542586.3520280050642916443072979210699388420708,
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1439720407.3117216736632230727949123939715485786772,
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248874557.86205415651146038641322942321632125127801,
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31426415.585400194380614231628318205362874684987640,
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2876370.6289353724412254090516208496135991145378768,
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186056.26539522349504029498971604569928220784236328,
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8071.6720023658162106380029022722506138218516325024,
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210.82427775157934587250973392071336271166969580291,
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2.5066282746310002701649081771338373386264310793408
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};
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/* denominator is x*(x+1)*...*(x+LANCZOS_N-2) */
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static const double lanczos_den_coeffs[LANCZOS_N] = {
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0.0, 39916800.0, 120543840.0, 150917976.0, 105258076.0, 45995730.0,
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13339535.0, 2637558.0, 357423.0, 32670.0, 1925.0, 66.0, 1.0};
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/* gamma values for small positive integers, 1 though NGAMMA_INTEGRAL */
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#define NGAMMA_INTEGRAL 23
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static const double gamma_integral[NGAMMA_INTEGRAL] = {
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1.0, 1.0, 2.0, 6.0, 24.0, 120.0, 720.0, 5040.0, 40320.0, 362880.0,
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3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0,
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1307674368000.0, 20922789888000.0, 355687428096000.0,
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6402373705728000.0, 121645100408832000.0, 2432902008176640000.0,
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51090942171709440000.0, 1124000727777607680000.0,
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};
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/* Helper function for sin(pi*x) used in gamma calculation */
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static double m_sinpi(double x)
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{
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double y, r;
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int n;
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/* this function should only ever be called for finite arguments */
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assert(Py_IS_FINITE(x));
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y = fmod(fabs(x), 2.0);
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n = (int)round(2.0*y);
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assert(0 <= n && n <= 4);
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switch (n) {
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case 0:
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r = sin(pi*y);
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break;
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case 1:
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r = cos(pi*(y-0.5));
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break;
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case 2:
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/* N.B. -sin(pi*(y-1.0)) is *not* equivalent: it would give
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-0.0 instead of 0.0 when y == 1.0. */
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r = sin(pi*(1.0-y));
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break;
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case 3:
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r = -cos(pi*(y-1.5));
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break;
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case 4:
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r = sin(pi*(y-2.0));
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break;
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default:
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/* Should never happen due to the assert above */
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r = 0.0;
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}
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return copysign(1.0, x)*r;
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}
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/* Implementation of Lanczos sum for gamma function */
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static double lanczos_sum(double x)
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{
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double num = 0.0, den = 0.0;
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int i;
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assert(x > 0.0);
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/* evaluate the rational function lanczos_sum(x). For large
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x, the obvious algorithm risks overflow, so we instead
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rescale the denominator and numerator of the rational
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function by x**(1-LANCZOS_N) and treat this as a
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rational function in 1/x. This also reduces the error for
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larger x values. The choice of cutoff point (5.0 below) is
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somewhat arbitrary; in tests, smaller cutoff values than
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this resulted in lower accuracy. */
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if (x < 5.0) {
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for (i = LANCZOS_N; --i >= 0; ) {
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num = num * x + lanczos_num_coeffs[i];
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den = den * x + lanczos_den_coeffs[i];
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}
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}
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else {
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for (i = 0; i < LANCZOS_N; i++) {
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num = num / x + lanczos_num_coeffs[i];
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den = den / x + lanczos_den_coeffs[i];
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}
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}
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return num/den;
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}
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/**
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* Computes the gamma function value for x.
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*
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* The implementation is based on the Lanczos approximation with parameters
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* N=13 and g=6.024680040776729583740234375, which provides excellent accuracy
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* across the domain of the function.
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*
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* @param x The input value
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* @return The gamma function value
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*
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* Special cases:
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* - If x is NaN, returns NaN
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* - If x is +Inf, returns +Inf
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* - If x is -Inf, sets errno to EDOM and returns NaN
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* - If x is 0, sets errno to EDOM and returns +/-Inf (with the sign of x)
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* - If x is a negative integer, sets errno to EDOM and returns NaN
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*/
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double tgamma(double x) {
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double absx, r, y, z, sqrtpow;
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/* special cases */
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if (!Py_IS_FINITE(x)) {
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if (Py_IS_NAN(x) || x > 0.0)
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return x; /* tgamma(nan) = nan, tgamma(inf) = inf */
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else {
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errno = EDOM;
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return NAN; /* tgamma(-inf) = nan, invalid */
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}
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}
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if (x == 0.0) {
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errno = EDOM;
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/* tgamma(+-0.0) = +-inf, divide-by-zero */
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return copysign(INFINITY, x);
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}
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/* integer arguments */
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if (x == floor(x)) {
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if (x < 0.0) {
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errno = EDOM; /* tgamma(n) = nan, invalid for */
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return NAN; /* negative integers n */
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}
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if (x <= NGAMMA_INTEGRAL)
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return gamma_integral[(int)x - 1];
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}
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absx = fabs(x);
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/* tiny arguments: tgamma(x) ~ 1/x for x near 0 */
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if (absx < 1e-20) {
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r = 1.0/x;
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if (Py_IS_INFINITY(r))
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errno = ERANGE;
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return r;
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}
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/* large arguments: assuming IEEE 754 doubles, tgamma(x) overflows for
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x > 200, and underflows to +-0.0 for x < -200, not a negative
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integer. */
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if (absx > 200.0) {
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if (x < 0.0) {
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return 0.0/m_sinpi(x);
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}
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else {
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errno = ERANGE;
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return Py_HUGE_VAL;
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}
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}
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y = absx + lanczos_g_minus_half;
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/* compute error in sum */
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if (absx > lanczos_g_minus_half) {
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/* note: the correction can be foiled by an optimizing
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compiler that (incorrectly) thinks that an expression like
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a + b - a - b can be optimized to 0.0. This shouldn't
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happen in a standards-conforming compiler. */
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double q = y - absx;
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z = q - lanczos_g_minus_half;
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}
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else {
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double q = y - lanczos_g_minus_half;
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z = q - absx;
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}
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z = z * lanczos_g / y;
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if (x < 0.0) {
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r = -pi / m_sinpi(absx) / absx * exp(y) / lanczos_sum(absx);
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r -= z * r;
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if (absx < 140.0) {
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r /= pow(y, absx - 0.5);
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}
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else {
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sqrtpow = pow(y, absx / 2.0 - 0.25);
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r /= sqrtpow;
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r /= sqrtpow;
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}
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}
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else {
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r = lanczos_sum(absx) / exp(y);
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r += z * r;
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if (absx < 140.0) {
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r *= pow(y, absx - 0.5);
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}
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else {
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sqrtpow = pow(y, absx / 2.0 - 0.25);
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r *= sqrtpow;
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r *= sqrtpow;
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}
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}
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if (Py_IS_INFINITY(r))
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errno = ERANGE;
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return r;
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}
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/**
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* Computes the natural logarithm of the absolute value of the gamma function.
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*
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* @param x The input value
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* @return The log of the absolute gamma function value
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*
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* Special cases:
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* - If x is NaN, returns NaN
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* - If x is +/-Inf, returns +Inf
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* - If x is a non-positive integer, sets errno to EDOM and returns +Inf
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* - If x is 1 or 2, returns 0.0
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*/
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double lgamma(double x) {
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double r;
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double absx;
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/* special cases */
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if (!Py_IS_FINITE(x)) {
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if (Py_IS_NAN(x))
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return x; /* lgamma(nan) = nan */
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else
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return HUGE_VAL; /* lgamma(+-inf) = +inf */
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}
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/* integer arguments */
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if (x == floor(x) && x <= 2.0) {
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if (x <= 0.0) {
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errno = EDOM; /* lgamma(n) = inf, divide-by-zero for */
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return HUGE_VAL; /* integers n <= 0 */
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}
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else {
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return 0.0; /* lgamma(1) = lgamma(2) = 0.0 */
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}
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}
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absx = fabs(x);
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/* tiny arguments: lgamma(x) ~ -log(fabs(x)) for small x */
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if (absx < 1e-20)
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return -log(absx);
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/* Lanczos' formula. We could save a fraction of a ulp in accuracy by
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having a second set of numerator coefficients for lanczos_sum that
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absorbed the exp(-lanczos_g) term, and throwing out the lanczos_g
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subtraction below; it's probably not worth it. */
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r = log(lanczos_sum(absx)) - lanczos_g;
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r += (absx - 0.5) * (log(absx + lanczos_g - 0.5) - 1);
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if (x < 0.0)
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/* Use reflection formula to get value for negative x. */
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r = logpi - log(fabs(m_sinpi(absx))) - log(absx) - r;
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if (Py_IS_INFINITY(r))
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errno = ERANGE;
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return r;
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}
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#include <stdio.h>
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#include <stdint.h>
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union Result {
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double d;
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uint64_t u;
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};
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int main() {
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union Result result;
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result.d = tgamma(-3.8510064710745118);
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printf("The result of tgamma(-3.8510064710745118) is: %f\n", result.d);
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// Print the binary representation of a double
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printf("Bit representation of result: %llu\n", result.u);
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return 0;
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}

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