diff options
| -rw-r--r-- | src/math/tgamma.c | 223 | ||||
| -rw-r--r-- | src/math/tgammaf.c | 12 |
2 files changed, 215 insertions, 20 deletions
diff --git a/src/math/tgamma.c b/src/math/tgamma.c index f3bbe370..a3f203c1 100644 --- a/src/math/tgamma.c +++ b/src/math/tgamma.c @@ -1,16 +1,221 @@ -#include <math.h> +/* +"A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964) +"Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001) +"An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004) -// FIXME: use lanczos approximation +approximation method: -double __lgamma_r(double, int *); + (x - 0.5) S(x) +Gamma(x) = (x + g - 0.5) * ---------------- + exp(x + g - 0.5) + +with + a1 a2 a3 aN +S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ] + x + 1 x + 2 x + 3 x + N + +with a0, a1, a2, a3,.. aN constants which depend on g. + +for x < 0 the following reflection formula is used: + +Gamma(x)*Gamma(-x) = -pi/(x sin(pi x)) + +most ideas and constants are from boost and python +*/ +#include "libm.h" + +static const double pi = 3.141592653589793238462643383279502884; + +/* sin(pi x) with x > 0 && isnormal(x) assumption */ +static double sinpi(double x) +{ + int n; + + /* argument reduction: x = |x| mod 2 */ + /* spurious inexact when x is odd int */ + x = x * 0.5; + x = 2 * (x - floor(x)); + + /* reduce x into [-.25,.25] */ + n = 4 * x; + n = (n+1)/2; + x -= n * 0.5; + + x *= pi; + switch (n) { + default: /* case 4 */ + case 0: + return __sin(x, 0, 0); + case 1: + return __cos(x, 0); + case 2: + /* sin(0-x) and -sin(x) have different sign at 0 */ + return __sin(0-x, 0, 0); + case 3: + return -__cos(x, 0); + } +} + +#define N 12 +//static const double g = 6.024680040776729583740234375; +static const double gmhalf = 5.524680040776729583740234375; +static const double Snum[N+1] = { + 23531376880.410759688572007674451636754734846804940, + 42919803642.649098768957899047001988850926355848959, + 35711959237.355668049440185451547166705960488635843, + 17921034426.037209699919755754458931112671403265390, + 6039542586.3520280050642916443072979210699388420708, + 1439720407.3117216736632230727949123939715485786772, + 248874557.86205415651146038641322942321632125127801, + 31426415.585400194380614231628318205362874684987640, + 2876370.6289353724412254090516208496135991145378768, + 186056.26539522349504029498971604569928220784236328, + 8071.6720023658162106380029022722506138218516325024, + 210.82427775157934587250973392071336271166969580291, + 2.5066282746310002701649081771338373386264310793408, +}; +static const double Sden[N+1] = { + 0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535, + 2637558, 357423, 32670, 1925, 66, 1, +}; +/* n! for small integer n */ +static const double fact[] = { + 1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0, + 479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0, + 355687428096000.0, 6402373705728000.0, 121645100408832000.0, + 2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0, +}; + +/* S(x) rational function for positive x */ +static double S(double x) +{ + double num = 0, den = 0; + int i; + + /* to avoid overflow handle large x differently */ + if (x < 8) + for (i = N; i >= 0; i--) { + num = num * x + Snum[i]; + den = den * x + Sden[i]; + } + else + for (i = 0; i <= N; i++) { + num = num / x + Snum[i]; + den = den / x + Sden[i]; + } + return num/den; +} double tgamma(double x) { - int sign; - double y; + double absx, y, dy, z, r; - y = exp(__lgamma_r(x, &sign)); - if (sign < 0) - y = -y; - return y; + /* special cases */ + if (!isfinite(x)) + /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */ + return x + INFINITY; + + /* integer arguments */ + /* raise inexact when non-integer */ + if (x == floor(x)) { + if (x == 0) + /* tgamma(+-0)=+-inf with divide-by-zero */ + return 1/x; + if (x < 0) + return 0/0.0; + if (x <= sizeof fact/sizeof *fact) + return fact[(int)x - 1]; + } + + absx = fabs(x); + + /* x ~ 0: tgamma(x) ~ 1/x */ + if (absx < 0x1p-54) + return 1/x; + + /* x >= 172: tgamma(x)=inf with overflow */ + /* x =< -184: tgamma(x)=+-0 with underflow */ + if (absx >= 184) { + if (x < 0) { + if (floor(x) * 0.5 == floor(x * 0.5)) + return 0; + return -0.0; + } + x *= 0x1p1023; + return x; + } + + /* handle the error of x + g - 0.5 */ + y = absx + gmhalf; + if (absx > gmhalf) { + dy = y - absx; + dy -= gmhalf; + } else { + dy = y - gmhalf; + dy -= absx; + } + + z = absx - 0.5; + r = S(absx) * exp(-y); + if (x < 0) { + /* reflection formula for negative x */ + r = -pi / (sinpi(absx) * absx * r); + dy = -dy; + z = -z; + } + r += dy * (gmhalf+0.5) * r / y; + z = pow(y, 0.5*z); + r = r * z * z; + return r; } + +#if 0 +double __lgamma_r(double x, int *sign) +{ + double r, absx, z, zz, w; + + *sign = 1; + + /* special cases */ + if (!isfinite(x)) + /* lgamma(nan)=nan, lgamma(+-inf)=inf */ + return x*x; + + /* integer arguments */ + if (x == floor(x) && x <= 2) { + /* n <= 0: lgamma(n)=inf with divbyzero */ + /* n == 1,2: lgamma(n)=0 */ + if (x <= 0) + return 1/0.0; + return 0; + } + + absx = fabs(x); + + /* lgamma(x) ~ -log(|x|) for tiny |x| */ + if (absx < 0x1p-54) { + *sign = 1 - 2*!!signbit(x); + return -log(absx); + } + + /* use tgamma for smaller |x| */ + if (absx < 128) { + x = tgamma(x); + *sign = 1 - 2*!!signbit(x); + return log(fabs(x)); + } + + /* second term (log(S)-g) could be more precise here.. */ + /* or with stirling: (|x|-0.5)*(log(|x|)-1) + poly(1/|x|) */ + r = (absx-0.5)*(log(absx+gmhalf)-1) + (log(S(absx)) - (gmhalf+0.5)); + if (x < 0) { + /* reflection formula for negative x */ + x = sinpi(absx); + *sign = 2*!!signbit(x) - 1; + r = log(pi/(fabs(x)*absx)) - r; + } + return r; +} + +weak_alias(__lgamma_r, lgamma_r); +#endif diff --git a/src/math/tgammaf.c b/src/math/tgammaf.c index 16df8076..b4ca51c9 100644 --- a/src/math/tgammaf.c +++ b/src/math/tgammaf.c @@ -1,16 +1,6 @@ #include <math.h> -// FIXME: use lanczos approximation - -float __lgammaf_r(float, int *); - float tgammaf(float x) { - int sign; - float y; - - y = exp(__lgammaf_r(x, &sign)); - if (sign < 0) - y = -y; - return y; + return tgamma(x); } |