/*- * Copyright (c) 1992 The Regents of the University of California. * All rights reserved. * * %sccs.include.redist.c% */ #ifndef lint static char sccsid[] = "@(#)lgamma.c 5.8 (Berkeley) 12/14/92"; #endif /* not lint */ #include #include #include "mathimpl.h" /* Log gamma function. * Error: x > 0 error < 1.3ulp. * x > 4, error < 1ulp. * x > 9, error < .6ulp. * x < 0, all bets are off. * Method: * x > 6: * Use the asymptotic expansion (Stirling's Formula) * 0 < x < 6: * Use gamma(x+1) = x*gamma(x) * Use rational approximation in * the range 1.2, 2.5 * x < 0: * Use the reflection formula, * G(1-x)G(x) = PI/sin(PI*x) * Special values: * non-positive integer returns +Inf. * NaN returns NaN */ #if defined(vax) || defined(tahoe) /* double and float have same size exponent field */ #define TRUNC(x) (double) (float) (x) #define _IEEE 0 #else #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000 #define _IEEE 1 #define infnan(x) (zero/zero) #endif extern double log1p(double); static double small_lgam(double); static double large_lgam(double); static double neg_lgam(double); static double zero = 0.0, one = 1.0; int signgam; #define lns2pi .418938533204672741780329736405 #define UNDERFL (1e-1020 * 1e-1020) #define LEFT (1.0 - (x0 + .25)) #define RIGHT (x0 - .218) /* /* Constants for approximation in [1.244,1.712] */ #define x0 0.461632144968362356785 #define x0_lo -.000000000000000015522348162858676890521 #define a0_hi -0.12148629128932952880859 #define a0_lo .0000000007534799204229502 #define r0 -2.771227512955130520e-002 #define r1 -2.980729795228150847e-001 #define r2 -3.257411333183093394e-001 #define r3 -1.126814387531706041e-001 #define r4 -1.129130057170225562e-002 #define r5 -2.259650588213369095e-005 #define s0 1.714457160001714442e+000 #define s1 2.786469504618194648e+000 #define s2 1.564546365519179805e+000 #define s3 3.485846389981109850e-001 #define s4 2.467759345363656348e-002 /* * Constants for approximation in [1.71, 2.5] */ #define a1_hi 4.227843350984671344505727574870e-01 #define a1_lo 4.670126436531227189e-18 #define p0 3.224670334241133695662995251041e-01 #define p1 3.569659696950364669021382724168e-01 #define p2 1.342918716072560025853732668111e-01 #define p3 1.950702176409779831089963408886e-02 #define p4 8.546740251667538090796227834289e-04 #define q0 1.000000000000000444089209850062e+00 #define q1 1.315850076960161985084596381057e+00 #define q2 6.274644311862156431658377186977e-01 #define q3 1.304706631926259297049597307705e-01 #define q4 1.102815279606722369265536798366e-02 #define q5 2.512690594856678929537585620579e-04 #define q6 -1.003597548112371003358107325598e-06 /* * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf]. */ #define pb0 .0833333333333333148296162562474 #define pb1 -.00277777777774548123579378966497 #define pb2 .000793650778754435631476282786423 #define pb3 -.000595235082566672847950717262222 #define pb4 .000841428560346653702135821806252 #define pb5 -.00189773526463879200348872089421 #define pb6 .00569394463439411649408050664078 #define pb7 -.0144705562421428915453880392761 double lgamma(double x) { double r; signgam = 1; if (!finite(x)) if (_IEEE) return (x+x); else return (infnan(EDOM)); if (x > 6 + RIGHT) { r = large_lgam(x); return (r); } else if (x > 1e-16) return (small_lgam(x)); else if (x > -1e-16) { if (x < 0) signgam = -1, x = -x; return (-log(x)); } else return (neg_lgam(x)); } static double large_lgam(double x) { double z, p, x1; int i; struct Double t, u, v; u = log__D(x); u.a -= 1.0; if (x > 1e15) { v.a = x - 0.5; TRUNC(v.a); v.b = (x - v.a) - 0.5; t.a = u.a*v.a; t.b = x*u.b + v.b*u.a; if (_IEEE == 0 && !finite(t.a)) return(infnan(ERANGE)); return(t.a + t.b); } x1 = 1./x; z = x1*x1; p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7)))))); /* error in approximation = 2.8e-19 */ p = p*x1; /* error < 2.3e-18 absolute */ /* 0 < p < 1/64 (at x = 5.5) */ x = x - 0.5; TRUNC(v.a); /* truncate v.a to 26 bits. */ v.b = x - v.a; t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */ t.b = v.b*u.a + x*u.b; t.b += p; t.b += lns2pi; /* return t + lns2pi + p */ return (t.a + t.b); } static double small_lgam(double x) { int x_int; double y, z, t, r = 0, p, q, hi, lo; struct Double rr; x_int = (x + .5); y = x - x_int; if (x_int <= 2 && y > RIGHT) { t = y - x0; y--; x_int++; goto CONTINUE; } else if (y < -LEFT) { t = y +(1.0-x0); CONTINUE: z = t - x0_lo; p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5)))); q = s0+z*(s1+z*(s2+z*(s3+z*s4))); r = t*(z*(p/q) - x0_lo); t = .5*t*t; z = 1.0; switch (x_int) { case 6: z = (y + 5); case 5: z *= (y + 4); case 4: z *= (y + 3); case 3: z *= (y + 2); rr = log__D(z); rr.b += a0_lo; rr.a += a0_hi; return(((r+rr.b)+t+rr.a)); case 2: return(((r+a0_lo)+t)+a0_hi); case 0: r -= log1p(x); default: rr = log__D(x); rr.a -= a0_hi; rr.b -= a0_lo; return(((r - rr.b) + t) - rr.a); } } else { p = p0+y*(p1+y*(p2+y*(p3+y*p4))); q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6))))); p = p*(y/q); t = (double)(float) y; z = y-t; hi = (double)(float) (p+a1_hi); lo = a1_hi - hi; lo += p; lo += a1_lo; r = lo*y + z*hi; /* q + r = y*(a0+p/q) */ q = hi*t; z = 1.0; switch (x_int) { case 6: z = (y + 5); case 5: z *= (y + 4); case 4: z *= (y + 3); case 3: z *= (y + 2); rr = log__D(z); r += rr.b; r += q; return(rr.a + r); case 2: return (q+ r); case 0: rr = log__D(x); r -= rr.b; r -= log1p(x); r += q; r-= rr.a; return(r); default: rr = log__D(x); r -= rr.b; q -= rr.a; return (r+q); } } } #define lpi_hi 1.1447298858494001638 #define lpi_lo .0000000000000000102659511627078262 /* Error: within 3.5 ulp for x < 171. For large x, see lgamma. */ static double neg_lgam(double x) { double y, z, one = 1.0, zero = 0.0; z = floor(x + .5); if (z == x) { /* convention: G(-(integer)) -> +Inf */ if (_IEEE) return (one/zero); else return (infnan(ERANGE)); } y = .5*ceil(x); if (y == ceil(y)) signgam = -1; x = -x; z = fabs(x + z); /* 0 < z <= .5 */ if (z < .25) z = sin(M_PI*z); else z = cos(M_PI*(0.5-z)); z = -log(z*x/M_PI); if (x > 6. + RIGHT) y -= large_lgam(x); else y = -small_lgam (x); return (y + z); }