1141296Sdas/* @(#)e_lgamma_r.c 1.3 95/01/18 */ 22116Sjkh/* 32116Sjkh * ==================================================== 42116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 52116Sjkh * 6141296Sdas * Developed at SunSoft, a Sun Microsystems, Inc. business. 72116Sjkh * Permission to use, copy, modify, and distribute this 8272138Skargl * software is freely granted, provided that this notice 92116Sjkh * is preserved. 102116Sjkh * ==================================================== 112116Sjkh */ 122116Sjkh 13176451Sdas#include <sys/cdefs.h> 14176451Sdas__FBSDID("$FreeBSD$"); 152116Sjkh 162116Sjkh/* __ieee754_lgamma_r(x, signgamp) 17272457Skargl * Reentrant version of the logarithm of the Gamma function 18272457Skargl * with user provide pointer for the sign of Gamma(x). 192116Sjkh * 202116Sjkh * Method: 212116Sjkh * 1. Argument Reduction for 0 < x <= 8 22272457Skargl * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 232116Sjkh * reduce x to a number in [1.5,2.5] by 242116Sjkh * lgamma(1+s) = log(s) + lgamma(s) 252116Sjkh * for example, 262116Sjkh * lgamma(7.3) = log(6.3) + lgamma(6.3) 272116Sjkh * = log(6.3*5.3) + lgamma(5.3) 282116Sjkh * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 292116Sjkh * 2. Polynomial approximation of lgamma around its 302116Sjkh * minimun ymin=1.461632144968362245 to maintain monotonicity. 312116Sjkh * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 322116Sjkh * Let z = x-ymin; 332116Sjkh * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 342116Sjkh * where 352116Sjkh * poly(z) is a 14 degree polynomial. 362116Sjkh * 2. Rational approximation in the primary interval [2,3] 372116Sjkh * We use the following approximation: 382116Sjkh * s = x-2.0; 392116Sjkh * lgamma(x) = 0.5*s + s*P(s)/Q(s) 402116Sjkh * with accuracy 412116Sjkh * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 422116Sjkh * Our algorithms are based on the following observation 432116Sjkh * 442116Sjkh * zeta(2)-1 2 zeta(3)-1 3 452116Sjkh * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 462116Sjkh * 2 3 472116Sjkh * 482116Sjkh * where Euler = 0.5771... is the Euler constant, which is very 492116Sjkh * close to 0.5. 502116Sjkh * 512116Sjkh * 3. For x>=8, we have 522116Sjkh * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 532116Sjkh * (better formula: 542116Sjkh * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 552116Sjkh * Let z = 1/x, then we approximation 562116Sjkh * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 572116Sjkh * by 582116Sjkh * 3 5 11 592116Sjkh * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 60272457Skargl * where 612116Sjkh * |w - f(z)| < 2**-58.74 62272457Skargl * 632116Sjkh * 4. For negative x, since (G is gamma function) 642116Sjkh * -x*G(-x)*G(x) = pi/sin(pi*x), 652116Sjkh * we have 662116Sjkh * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 672116Sjkh * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 68272457Skargl * Hence, for x<0, signgam = sign(sin(pi*x)) and 692116Sjkh * lgamma(x) = log(|Gamma(x)|) 702116Sjkh * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 71272457Skargl * Note: one should avoid compute pi*(-x) directly in the 722116Sjkh * computation of sin(pi*(-x)). 73272457Skargl * 742116Sjkh * 5. Special Cases 752116Sjkh * lgamma(2+s) ~ s*(1-Euler) for tiny s 76169220Sbde * lgamma(1) = lgamma(2) = 0 77169220Sbde * lgamma(x) ~ -log(|x|) for tiny x 78169220Sbde * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero 79169220Sbde * lgamma(inf) = inf 80169220Sbde * lgamma(-inf) = inf (bug for bug compatible with C99!?) 812116Sjkh */ 822116Sjkh 83271651Skargl#include <float.h> 84271651Skargl 852116Sjkh#include "math.h" 862116Sjkh#include "math_private.h" 872116Sjkh 88270947Skarglstatic const volatile double vzero = 0; 89270947Skargl 908870Srgrimesstatic const double 91270947Skarglzero= 0.00000000000000000000e+00, 922116Sjkhhalf= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 932116Sjkhone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 942116Sjkhpi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ 952116Sjkha0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ 962116Sjkha1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ 972116Sjkha2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ 982116Sjkha3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ 992116Sjkha4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ 1002116Sjkha5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ 1012116Sjkha6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ 1022116Sjkha7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ 1032116Sjkha8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ 1042116Sjkha9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ 1052116Sjkha10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ 1062116Sjkha11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ 1072116Sjkhtc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ 1082116Sjkhtf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ 1092116Sjkh/* tt = -(tail of tf) */ 1102116Sjkhtt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ 1112116Sjkht0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ 1122116Sjkht1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ 1132116Sjkht2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ 1142116Sjkht3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ 1152116Sjkht4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ 1162116Sjkht5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ 1172116Sjkht6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ 1182116Sjkht7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ 1192116Sjkht8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ 1202116Sjkht9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ 1212116Sjkht10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ 1222116Sjkht11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ 1232116Sjkht12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ 1242116Sjkht13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ 1252116Sjkht14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ 1262116Sjkhu0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 1272116Sjkhu1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ 1282116Sjkhu2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ 1292116Sjkhu3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ 1302116Sjkhu4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ 1312116Sjkhu5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ 1322116Sjkhv1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ 1332116Sjkhv2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ 1342116Sjkhv3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ 1352116Sjkhv4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ 1362116Sjkhv5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ 1372116Sjkhs0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 1382116Sjkhs1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ 1392116Sjkhs2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ 1402116Sjkhs3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ 1412116Sjkhs4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ 1422116Sjkhs5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ 1432116Sjkhs6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ 1442116Sjkhr1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ 1452116Sjkhr2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ 1462116Sjkhr3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ 1472116Sjkhr4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ 1482116Sjkhr5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ 1492116Sjkhr6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ 1502116Sjkhw0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ 1512116Sjkhw1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ 1522116Sjkhw2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ 1532116Sjkhw3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ 1542116Sjkhw4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ 1552116Sjkhw5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ 1562116Sjkhw6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ 1572116Sjkh 158270893Skargl/* 159270893Skargl * Compute sin(pi*x) without actually doing the pi*x multiplication. 160270893Skargl * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is 161270893Skargl * the precision of x. 162270893Skargl */ 163270893Skarglstatic double 164270893Skarglsin_pi(double x) 1652116Sjkh{ 166270893Skargl volatile double vz; 1672116Sjkh double y,z; 168270893Skargl int n; 1692116Sjkh 170270893Skargl y = -x; 1712116Sjkh 172270893Skargl vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */ 173270932Skargl z = vz-0x1p52; /* rint(y) for the above range */ 174270893Skargl if (z == y) 175270932Skargl return zero; 1762116Sjkh 177270893Skargl vz = y+0x1p50; 178270893Skargl GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */ 179270893Skargl z = vz-0x1p50; /* y rounded to a multiple of 0.25 */ 180270893Skargl if (z > y) { 181270893Skargl z -= 0.25; /* adjust to round down */ 182270893Skargl n--; 183270893Skargl } 184270893Skargl n &= 7; /* octant of y mod 2 */ 185270893Skargl y = y - z + n * 0.25; /* y mod 2 */ 186270893Skargl 1872116Sjkh switch (n) { 1882116Sjkh case 0: y = __kernel_sin(pi*y,zero,0); break; 189272457Skargl case 1: 1902116Sjkh case 2: y = __kernel_cos(pi*(0.5-y),zero); break; 191272457Skargl case 3: 1922116Sjkh case 4: y = __kernel_sin(pi*(one-y),zero,0); break; 1932116Sjkh case 5: 1942116Sjkh case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; 1952116Sjkh default: y = __kernel_sin(pi*(y-2.0),zero,0); break; 1962116Sjkh } 1972116Sjkh return -y; 1982116Sjkh} 1992116Sjkh 2002116Sjkh 20197413Salfreddouble 20297413Salfred__ieee754_lgamma_r(double x, int *signgamp) 2032116Sjkh{ 204272845Skargl double nadj,p,p1,p2,p3,q,r,t,w,y,z; 205169220Sbde int32_t hx; 206270947Skargl int i,ix,lx; 2072116Sjkh 2082116Sjkh EXTRACT_WORDS(hx,lx,x); 2092116Sjkh 210272845Skargl /* purge +-Inf and NaNs */ 2112116Sjkh *signgamp = 1; 2122116Sjkh ix = hx&0x7fffffff; 2132116Sjkh if(ix>=0x7ff00000) return x*x; 214272845Skargl 215272845Skargl /* purge +-0 and tiny arguments */ 216272845Skargl *signgamp = 1-2*((uint32_t)hx>>31); 217272845Skargl if(ix<0x3c700000) { /* |x|<2**-56, return -log(|x|) */ 218272845Skargl if((ix|lx)==0) 219272845Skargl return one/vzero; 220272845Skargl return -__ieee754_log(fabs(x)); 221271719Skargl } 222272845Skargl 223272845Skargl /* purge negative integers and start evaluation for other x < 0 */ 2242116Sjkh if(hx<0) { 225272845Skargl *signgamp = 1; 2262116Sjkh if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ 227270947Skargl return one/vzero; 2282116Sjkh t = sin_pi(x); 229270947Skargl if(t==zero) return one/vzero; /* -integer */ 2302116Sjkh nadj = __ieee754_log(pi/fabs(t*x)); 2312116Sjkh if(t<zero) *signgamp = -1; 2322116Sjkh x = -x; 2332116Sjkh } 2342116Sjkh 235272845Skargl /* purge 1 and 2 */ 2362116Sjkh if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; 2372116Sjkh /* for x < 2.0 */ 2382116Sjkh else if(ix<0x40000000) { 2392116Sjkh if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ 2402116Sjkh r = -__ieee754_log(x); 2412116Sjkh if(ix>=0x3FE76944) {y = one-x; i= 0;} 2422116Sjkh else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} 2432116Sjkh else {y = x; i=2;} 2442116Sjkh } else { 2452116Sjkh r = zero; 2462116Sjkh if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ 2472116Sjkh else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ 2482116Sjkh else {y=x-one;i=2;} 2492116Sjkh } 2502116Sjkh switch(i) { 2512116Sjkh case 0: 2522116Sjkh z = y*y; 2532116Sjkh p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); 2542116Sjkh p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); 2552116Sjkh p = y*p1+p2; 256272845Skargl r += p-y/2; break; 2572116Sjkh case 1: 2582116Sjkh z = y*y; 2592116Sjkh w = z*y; 2602116Sjkh p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ 2612116Sjkh p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); 2622116Sjkh p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); 2632116Sjkh p = z*p1-(tt-w*(p2+y*p3)); 264272845Skargl r += tf + p; break; 265272457Skargl case 2: 2662116Sjkh p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); 2672116Sjkh p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); 268272845Skargl r += p1/p2-y/2; 2692116Sjkh } 2702116Sjkh } 271272845Skargl /* x < 8.0 */ 272272845Skargl else if(ix<0x40200000) { 273272845Skargl i = x; 274272845Skargl y = x-i; 2752116Sjkh p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); 2762116Sjkh q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); 277272845Skargl r = y/2+p/q; 2782116Sjkh z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ 2792116Sjkh switch(i) { 280271651Skargl case 7: z *= (y+6); /* FALLTHRU */ 281271651Skargl case 6: z *= (y+5); /* FALLTHRU */ 282271651Skargl case 5: z *= (y+4); /* FALLTHRU */ 283271651Skargl case 4: z *= (y+3); /* FALLTHRU */ 284271651Skargl case 3: z *= (y+2); /* FALLTHRU */ 2852116Sjkh r += __ieee754_log(z); break; 2862116Sjkh } 287272845Skargl /* 8.0 <= x < 2**56 */ 288272845Skargl } else if (ix < 0x43700000) { 2892116Sjkh t = __ieee754_log(x); 2902116Sjkh z = one/x; 2912116Sjkh y = z*z; 2922116Sjkh w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); 2932116Sjkh r = (x-half)*(t-one)+w; 294272457Skargl } else 295272845Skargl /* 2**56 <= x <= inf */ 2962116Sjkh r = x*(__ieee754_log(x)-one); 2972116Sjkh if(hx<0) r = nadj - r; 2982116Sjkh return r; 2992116Sjkh} 300271651Skargl 301271651Skargl#if (LDBL_MANT_DIG == 53) 302271651Skargl__weak_reference(lgamma_r, lgammal_r); 303271651Skargl#endif 304