e_lgamma_r.c revision 270947
1141296Sdas 2141296Sdas/* @(#)e_lgamma_r.c 1.3 95/01/18 */ 32116Sjkh/* 42116Sjkh * ==================================================== 52116Sjkh * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 62116Sjkh * 7141296Sdas * Developed at SunSoft, a Sun Microsystems, Inc. business. 82116Sjkh * Permission to use, copy, modify, and distribute this 9141296Sdas * software is freely granted, provided that this notice 102116Sjkh * is preserved. 112116Sjkh * ==================================================== 12141296Sdas * 132116Sjkh */ 142116Sjkh 15176451Sdas#include <sys/cdefs.h> 16176451Sdas__FBSDID("$FreeBSD: head/lib/msun/src/e_lgamma_r.c 270947 2014-09-01 18:57:13Z kargl $"); 172116Sjkh 182116Sjkh/* __ieee754_lgamma_r(x, signgamp) 19141296Sdas * Reentrant version of the logarithm of the Gamma function 20141296Sdas * with user provide pointer for the sign of Gamma(x). 212116Sjkh * 222116Sjkh * Method: 232116Sjkh * 1. Argument Reduction for 0 < x <= 8 24141296Sdas * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 252116Sjkh * reduce x to a number in [1.5,2.5] by 262116Sjkh * lgamma(1+s) = log(s) + lgamma(s) 272116Sjkh * for example, 282116Sjkh * lgamma(7.3) = log(6.3) + lgamma(6.3) 292116Sjkh * = log(6.3*5.3) + lgamma(5.3) 302116Sjkh * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 312116Sjkh * 2. Polynomial approximation of lgamma around its 322116Sjkh * minimun ymin=1.461632144968362245 to maintain monotonicity. 332116Sjkh * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 342116Sjkh * Let z = x-ymin; 352116Sjkh * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 362116Sjkh * where 372116Sjkh * poly(z) is a 14 degree polynomial. 382116Sjkh * 2. Rational approximation in the primary interval [2,3] 392116Sjkh * We use the following approximation: 402116Sjkh * s = x-2.0; 412116Sjkh * lgamma(x) = 0.5*s + s*P(s)/Q(s) 422116Sjkh * with accuracy 432116Sjkh * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 442116Sjkh * Our algorithms are based on the following observation 452116Sjkh * 462116Sjkh * zeta(2)-1 2 zeta(3)-1 3 472116Sjkh * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 482116Sjkh * 2 3 492116Sjkh * 502116Sjkh * where Euler = 0.5771... is the Euler constant, which is very 512116Sjkh * close to 0.5. 522116Sjkh * 532116Sjkh * 3. For x>=8, we have 542116Sjkh * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 552116Sjkh * (better formula: 562116Sjkh * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 572116Sjkh * Let z = 1/x, then we approximation 582116Sjkh * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 592116Sjkh * by 602116Sjkh * 3 5 11 612116Sjkh * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 62141296Sdas * where 632116Sjkh * |w - f(z)| < 2**-58.74 64141296Sdas * 652116Sjkh * 4. For negative x, since (G is gamma function) 662116Sjkh * -x*G(-x)*G(x) = pi/sin(pi*x), 672116Sjkh * we have 682116Sjkh * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 692116Sjkh * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 70141296Sdas * Hence, for x<0, signgam = sign(sin(pi*x)) and 712116Sjkh * lgamma(x) = log(|Gamma(x)|) 722116Sjkh * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 73141296Sdas * Note: one should avoid compute pi*(-x) directly in the 742116Sjkh * computation of sin(pi*(-x)). 75141296Sdas * 762116Sjkh * 5. Special Cases 772116Sjkh * lgamma(2+s) ~ s*(1-Euler) for tiny s 78169220Sbde * lgamma(1) = lgamma(2) = 0 79169220Sbde * lgamma(x) ~ -log(|x|) for tiny x 80169220Sbde * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero 81169220Sbde * lgamma(inf) = inf 82169220Sbde * lgamma(-inf) = inf (bug for bug compatible with C99!?) 83141296Sdas * 842116Sjkh */ 852116Sjkh 862116Sjkh#include "math.h" 872116Sjkh#include "math_private.h" 882116Sjkh 89270947Skarglstatic const volatile double vzero = 0; 90270947Skargl 918870Srgrimesstatic const double 92270947Skarglzero= 0.00000000000000000000e+00, 932116Sjkhtwo52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ 942116Sjkhhalf= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 952116Sjkhone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 962116Sjkhpi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ 972116Sjkha0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ 982116Sjkha1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ 992116Sjkha2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ 1002116Sjkha3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ 1012116Sjkha4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ 1022116Sjkha5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ 1032116Sjkha6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ 1042116Sjkha7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ 1052116Sjkha8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ 1062116Sjkha9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ 1072116Sjkha10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ 1082116Sjkha11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ 1092116Sjkhtc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ 1102116Sjkhtf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ 1112116Sjkh/* tt = -(tail of tf) */ 1122116Sjkhtt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ 1132116Sjkht0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ 1142116Sjkht1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ 1152116Sjkht2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ 1162116Sjkht3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ 1172116Sjkht4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ 1182116Sjkht5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ 1192116Sjkht6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ 1202116Sjkht7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ 1212116Sjkht8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ 1222116Sjkht9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ 1232116Sjkht10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ 1242116Sjkht11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ 1252116Sjkht12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ 1262116Sjkht13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ 1272116Sjkht14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ 1282116Sjkhu0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 1292116Sjkhu1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ 1302116Sjkhu2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ 1312116Sjkhu3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ 1322116Sjkhu4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ 1332116Sjkhu5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ 1342116Sjkhv1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ 1352116Sjkhv2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ 1362116Sjkhv3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ 1372116Sjkhv4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ 1382116Sjkhv5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ 1392116Sjkhs0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ 1402116Sjkhs1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ 1412116Sjkhs2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ 1422116Sjkhs3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ 1432116Sjkhs4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ 1442116Sjkhs5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ 1452116Sjkhs6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ 1462116Sjkhr1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ 1472116Sjkhr2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ 1482116Sjkhr3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ 1492116Sjkhr4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ 1502116Sjkhr5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ 1512116Sjkhr6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ 1522116Sjkhw0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ 1532116Sjkhw1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ 1542116Sjkhw2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ 1552116Sjkhw3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ 1562116Sjkhw4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ 1572116Sjkhw5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ 1582116Sjkhw6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ 1592116Sjkh 160270893Skargl/* 161270893Skargl * Compute sin(pi*x) without actually doing the pi*x multiplication. 162270893Skargl * sin_pi(x) is only called for x < 0 and |x| < 2**(p-1) where p is 163270893Skargl * the precision of x. 164270893Skargl */ 165270893Skarglstatic double 166270893Skarglsin_pi(double x) 1672116Sjkh{ 168270893Skargl volatile double vz; 1692116Sjkh double y,z; 170270893Skargl int n; 1712116Sjkh 172270893Skargl y = -x; 1732116Sjkh 174270893Skargl vz = y+0x1p52; /* depend on 0 <= y < 0x1p52 */ 175270932Skargl z = vz-0x1p52; /* rint(y) for the above range */ 176270893Skargl if (z == y) 177270932Skargl return zero; 1782116Sjkh 179270893Skargl vz = y+0x1p50; 180270893Skargl GET_LOW_WORD(n,vz); /* bits for rounded y (units 0.25) */ 181270893Skargl z = vz-0x1p50; /* y rounded to a multiple of 0.25 */ 182270893Skargl if (z > y) { 183270893Skargl z -= 0.25; /* adjust to round down */ 184270893Skargl n--; 185270893Skargl } 186270893Skargl n &= 7; /* octant of y mod 2 */ 187270893Skargl y = y - z + n * 0.25; /* y mod 2 */ 188270893Skargl 1892116Sjkh switch (n) { 1902116Sjkh case 0: y = __kernel_sin(pi*y,zero,0); break; 191141296Sdas case 1: 1922116Sjkh case 2: y = __kernel_cos(pi*(0.5-y),zero); break; 193141296Sdas case 3: 1942116Sjkh case 4: y = __kernel_sin(pi*(one-y),zero,0); break; 1952116Sjkh case 5: 1962116Sjkh case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; 1972116Sjkh default: y = __kernel_sin(pi*(y-2.0),zero,0); break; 1982116Sjkh } 1992116Sjkh return -y; 2002116Sjkh} 2012116Sjkh 2022116Sjkh 20397413Salfreddouble 20497413Salfred__ieee754_lgamma_r(double x, int *signgamp) 2052116Sjkh{ 2062116Sjkh double t,y,z,nadj,p,p1,p2,p3,q,r,w; 207169220Sbde int32_t hx; 208270947Skargl int i,ix,lx; 2092116Sjkh 2102116Sjkh EXTRACT_WORDS(hx,lx,x); 2112116Sjkh 212169220Sbde /* purge off +-inf, NaN, +-0, tiny and negative arguments */ 2132116Sjkh *signgamp = 1; 2142116Sjkh ix = hx&0x7fffffff; 2152116Sjkh if(ix>=0x7ff00000) return x*x; 216270947Skargl if((ix|lx)==0) return one/vzero; 2172116Sjkh if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ 2182116Sjkh if(hx<0) { 2192116Sjkh *signgamp = -1; 2202116Sjkh return -__ieee754_log(-x); 2212116Sjkh } else return -__ieee754_log(x); 2222116Sjkh } 2232116Sjkh if(hx<0) { 2242116Sjkh if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ 225270947Skargl return one/vzero; 2262116Sjkh t = sin_pi(x); 227270947Skargl if(t==zero) return one/vzero; /* -integer */ 2282116Sjkh nadj = __ieee754_log(pi/fabs(t*x)); 2292116Sjkh if(t<zero) *signgamp = -1; 2302116Sjkh x = -x; 2312116Sjkh } 2322116Sjkh 2332116Sjkh /* purge off 1 and 2 */ 2342116Sjkh if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; 2352116Sjkh /* for x < 2.0 */ 2362116Sjkh else if(ix<0x40000000) { 2372116Sjkh if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ 2382116Sjkh r = -__ieee754_log(x); 2392116Sjkh if(ix>=0x3FE76944) {y = one-x; i= 0;} 2402116Sjkh else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} 2412116Sjkh else {y = x; i=2;} 2422116Sjkh } else { 2432116Sjkh r = zero; 2442116Sjkh if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ 2452116Sjkh else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ 2462116Sjkh else {y=x-one;i=2;} 2472116Sjkh } 2482116Sjkh switch(i) { 2492116Sjkh case 0: 2502116Sjkh z = y*y; 2512116Sjkh p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); 2522116Sjkh p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); 2532116Sjkh p = y*p1+p2; 2542116Sjkh r += (p-0.5*y); break; 2552116Sjkh case 1: 2562116Sjkh z = y*y; 2572116Sjkh w = z*y; 2582116Sjkh p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ 2592116Sjkh p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); 2602116Sjkh p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); 2612116Sjkh p = z*p1-(tt-w*(p2+y*p3)); 2622116Sjkh r += (tf + p); break; 263141296Sdas case 2: 2642116Sjkh p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); 2652116Sjkh p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); 2662116Sjkh r += (-0.5*y + p1/p2); 2672116Sjkh } 2682116Sjkh } 2692116Sjkh else if(ix<0x40200000) { /* x < 8.0 */ 2702116Sjkh i = (int)x; 2712116Sjkh y = x-(double)i; 2722116Sjkh p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); 2732116Sjkh q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); 2742116Sjkh r = half*y+p/q; 2752116Sjkh z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ 2762116Sjkh switch(i) { 2772116Sjkh case 7: z *= (y+6.0); /* FALLTHRU */ 2782116Sjkh case 6: z *= (y+5.0); /* FALLTHRU */ 2792116Sjkh case 5: z *= (y+4.0); /* FALLTHRU */ 2802116Sjkh case 4: z *= (y+3.0); /* FALLTHRU */ 2812116Sjkh case 3: z *= (y+2.0); /* FALLTHRU */ 2822116Sjkh r += __ieee754_log(z); break; 2832116Sjkh } 2842116Sjkh /* 8.0 <= x < 2**58 */ 2852116Sjkh } else if (ix < 0x43900000) { 2862116Sjkh t = __ieee754_log(x); 2872116Sjkh z = one/x; 2882116Sjkh y = z*z; 2892116Sjkh w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); 2902116Sjkh r = (x-half)*(t-one)+w; 291141296Sdas } else 2922116Sjkh /* 2**58 <= x <= inf */ 2932116Sjkh r = x*(__ieee754_log(x)-one); 2942116Sjkh if(hx<0) r = nadj - r; 2952116Sjkh return r; 2962116Sjkh} 297