| /* @(#)er_lgamma.c 5.1 93/09/24 */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunPro, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| |
| /* __ieee754_lgamma_r(x, signgamp) |
| * Reentrant version of the logarithm of the Gamma function |
| * with user provide pointer for the sign of Gamma(x). |
| * |
| * Method: |
| * 1. Argument Reduction for 0 < x <= 8 |
| * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may |
| * reduce x to a number in [1.5,2.5] by |
| * lgamma(1+s) = log(s) + lgamma(s) |
| * for example, |
| * lgamma(7.3) = log(6.3) + lgamma(6.3) |
| * = log(6.3*5.3) + lgamma(5.3) |
| * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) |
| * 2. Polynomial approximation of lgamma around its |
| * minimun ymin=1.461632144968362245 to maintain monotonicity. |
| * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use |
| * Let z = x-ymin; |
| * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) |
| * where |
| * poly(z) is a 14 degree polynomial. |
| * 2. Rational approximation in the primary interval [2,3] |
| * We use the following approximation: |
| * s = x-2.0; |
| * lgamma(x) = 0.5*s + s*P(s)/Q(s) |
| * with accuracy |
| * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 |
| * Our algorithms are based on the following observation |
| * |
| * zeta(2)-1 2 zeta(3)-1 3 |
| * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... |
| * 2 3 |
| * |
| * where Euler = 0.5771... is the Euler constant, which is very |
| * close to 0.5. |
| * |
| * 3. For x>=8, we have |
| * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... |
| * (better formula: |
| * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) |
| * Let z = 1/x, then we approximation |
| * f(z) = lgamma(x) - (x-0.5)(log(x)-1) |
| * by |
| * 3 5 11 |
| * w = w0 + w1*z + w2*z + w3*z + ... + w6*z |
| * where |
| * |w - f(z)| < 2**-58.74 |
| * |
| * 4. For negative x, since (G is gamma function) |
| * -x*G(-x)*G(x) = pi/sin(pi*x), |
| * we have |
| * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) |
| * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 |
| * Hence, for x<0, signgam = sign(sin(pi*x)) and |
| * lgamma(x) = log(|Gamma(x)|) |
| * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); |
| * Note: one should avoid compute pi*(-x) directly in the |
| * computation of sin(pi*(-x)). |
| * |
| * 5. Special Cases |
| * lgamma(2+s) ~ s*(1-Euler) for tiny s |
| * lgamma(1)=lgamma(2)=0 |
| * lgamma(x) ~ -log(x) for tiny x |
| * lgamma(0) = lgamma(inf) = inf |
| * lgamma(-integer) = +-inf |
| * |
| */ |
| |
| #include <math.h> |
| #include <math_private.h> |
| #include <libc-diag.h> |
| |
| static const double |
| two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ |
| half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
| one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
| pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ |
| a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ |
| a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ |
| a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ |
| a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ |
| a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ |
| a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ |
| a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ |
| a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ |
| a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ |
| a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ |
| a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ |
| a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ |
| tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ |
| tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ |
| /* tt = -(tail of tf) */ |
| tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ |
| t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ |
| t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ |
| t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ |
| t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ |
| t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ |
| t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ |
| t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ |
| t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ |
| t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ |
| t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ |
| t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ |
| t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ |
| t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ |
| t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ |
| t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ |
| u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
| u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ |
| u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ |
| u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ |
| u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ |
| u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ |
| v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ |
| v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ |
| v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ |
| v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ |
| v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ |
| s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
| s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ |
| s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ |
| s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ |
| s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ |
| s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ |
| s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ |
| r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ |
| r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ |
| r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ |
| r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ |
| r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ |
| r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ |
| w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ |
| w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ |
| w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ |
| w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ |
| w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ |
| w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ |
| w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ |
| |
| static const double zero= 0.00000000000000000000e+00; |
| |
| static double |
| sin_pi(double x) |
| { |
| double y,z; |
| int n,ix; |
| |
| GET_HIGH_WORD(ix,x); |
| ix &= 0x7fffffff; |
| |
| if(ix<0x3fd00000) return __sin(pi*x); |
| y = -x; /* x is assume negative */ |
| |
| /* |
| * argument reduction, make sure inexact flag not raised if input |
| * is an integer |
| */ |
| z = __floor(y); |
| if(z!=y) { /* inexact anyway */ |
| y *= 0.5; |
| y = 2.0*(y - __floor(y)); /* y = |x| mod 2.0 */ |
| n = (int) (y*4.0); |
| } else { |
| if(ix>=0x43400000) { |
| y = zero; n = 0; /* y must be even */ |
| } else { |
| if(ix<0x43300000) z = y+two52; /* exact */ |
| GET_LOW_WORD(n,z); |
| n &= 1; |
| y = n; |
| n<<= 2; |
| } |
| } |
| switch (n) { |
| case 0: y = __sin(pi*y); break; |
| case 1: |
| case 2: y = __cos(pi*(0.5-y)); break; |
| case 3: |
| case 4: y = __sin(pi*(one-y)); break; |
| case 5: |
| case 6: y = -__cos(pi*(y-1.5)); break; |
| default: y = __sin(pi*(y-2.0)); break; |
| } |
| return -y; |
| } |
| |
| |
| double |
| __ieee754_lgamma_r(double x, int *signgamp) |
| { |
| double t,y,z,nadj,p,p1,p2,p3,q,r,w; |
| int i,hx,lx,ix; |
| |
| EXTRACT_WORDS(hx,lx,x); |
| |
| /* purge off +-inf, NaN, +-0, and negative arguments */ |
| *signgamp = 1; |
| ix = hx&0x7fffffff; |
| if(__builtin_expect(ix>=0x7ff00000, 0)) return x*x; |
| if(__builtin_expect((ix|lx)==0, 0)) |
| { |
| if (hx < 0) |
| *signgamp = -1; |
| return one/fabs(x); |
| } |
| if(__builtin_expect(ix<0x3b900000, 0)) { |
| /* |x|<2**-70, return -log(|x|) */ |
| if(hx<0) { |
| *signgamp = -1; |
| return -__ieee754_log(-x); |
| } else return -__ieee754_log(x); |
| } |
| if(hx<0) { |
| if(__builtin_expect(ix>=0x43300000, 0)) |
| /* |x|>=2**52, must be -integer */ |
| return fabs (x)/zero; |
| if (x < -2.0 && x > -28.0) |
| return __lgamma_neg (x, signgamp); |
| t = sin_pi(x); |
| if(t==zero) return one/fabsf(t); /* -integer */ |
| nadj = __ieee754_log(pi/fabs(t*x)); |
| if(t<zero) *signgamp = -1; |
| x = -x; |
| } |
| |
| /* purge off 1 and 2 */ |
| if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; |
| /* for x < 2.0 */ |
| else if(ix<0x40000000) { |
| if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ |
| r = -__ieee754_log(x); |
| if(ix>=0x3FE76944) {y = one-x; i= 0;} |
| else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} |
| else {y = x; i=2;} |
| } else { |
| r = zero; |
| if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ |
| else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ |
| else {y=x-one;i=2;} |
| } |
| switch(i) { |
| case 0: |
| z = y*y; |
| p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); |
| p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); |
| p = y*p1+p2; |
| r += (p-0.5*y); break; |
| case 1: |
| z = y*y; |
| w = z*y; |
| p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ |
| p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); |
| p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); |
| p = z*p1-(tt-w*(p2+y*p3)); |
| r += (tf + p); break; |
| case 2: |
| p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); |
| p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); |
| r += (-0.5*y + p1/p2); |
| } |
| } |
| else if(ix<0x40200000) { /* x < 8.0 */ |
| i = (int)x; |
| t = zero; |
| y = x-(double)i; |
| p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); |
| q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); |
| r = half*y+p/q; |
| z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ |
| switch(i) { |
| case 7: z *= (y+6.0); /* FALLTHRU */ |
| case 6: z *= (y+5.0); /* FALLTHRU */ |
| case 5: z *= (y+4.0); /* FALLTHRU */ |
| case 4: z *= (y+3.0); /* FALLTHRU */ |
| case 3: z *= (y+2.0); /* FALLTHRU */ |
| r += __ieee754_log(z); break; |
| } |
| /* 8.0 <= x < 2**58 */ |
| } else if (ix < 0x43900000) { |
| t = __ieee754_log(x); |
| z = one/x; |
| y = z*z; |
| w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); |
| r = (x-half)*(t-one)+w; |
| } else |
| /* 2**58 <= x <= inf */ |
| r = math_narrow_eval (x*(__ieee754_log(x)-one)); |
| /* NADJ is set for negative arguments but not otherwise, |
| resulting in warnings that it may be used uninitialized |
| although in the cases where it is used it has always been |
| set. */ |
| DIAG_PUSH_NEEDS_COMMENT; |
| DIAG_IGNORE_NEEDS_COMMENT (4.9, "-Wmaybe-uninitialized"); |
| if(hx<0) r = nadj - r; |
| DIAG_POP_NEEDS_COMMENT; |
| return r; |
| } |
| strong_alias (__ieee754_lgamma_r, __lgamma_r_finite) |
