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e_lgamma_r.c (10941B)


      1 /* @(#)er_lgamma.c 5.1 93/09/24 */
      2 /*
      3  * ====================================================
      4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
      5  *
      6  * Developed at SunPro, a Sun Microsystems, Inc. business.
      7  * Permission to use, copy, modify, and distribute this
      8  * software is freely granted, provided that this notice
      9  * is preserved.
     10  * ====================================================
     11  */
     12 
     13 #ifndef lint
     14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_lgamma_r.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
     15 #endif
     16 
     17 /* __ieee754_lgamma_r(x, signgamp)
     18  * Reentrant version of the logarithm of the Gamma function
     19  * with user provide pointer for the sign of Gamma(x).
     20  *
     21  * Method:
     22  *   1. Argument Reduction for 0 < x <= 8
     23  * 	Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
     24  * 	reduce x to a number in [1.5,2.5] by
     25  * 		lgamma(1+s) = log(s) + lgamma(s)
     26  *	for example,
     27  *		lgamma(7.3) = log(6.3) + lgamma(6.3)
     28  *			    = log(6.3*5.3) + lgamma(5.3)
     29  *			    = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
     30  *   2. Polynomial approximation of lgamma around its
     31  *	minimun ymin=1.461632144968362245 to maintain monotonicity.
     32  *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
     33  *		Let z = x-ymin;
     34  *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
     35  *	where
     36  *		poly(z) is a 14 degree polynomial.
     37  *   2. Rational approximation in the primary interval [2,3]
     38  *	We use the following approximation:
     39  *		s = x-2.0;
     40  *		lgamma(x) = 0.5*s + s*P(s)/Q(s)
     41  *	with accuracy
     42  *		|P/Q - (lgamma(x)-0.5s)| < 2**-61.71
     43  *	Our algorithms are based on the following observation
     44  *
     45  *                             zeta(2)-1    2    zeta(3)-1    3
     46  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
     47  *                                 2                 3
     48  *
     49  *	where Euler = 0.5771... is the Euler constant, which is very
     50  *	close to 0.5.
     51  *
     52  *   3. For x>=8, we have
     53  *	lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
     54  *	(better formula:
     55  *	   lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
     56  *	Let z = 1/x, then we approximation
     57  *		f(z) = lgamma(x) - (x-0.5)(log(x)-1)
     58  *	by
     59  *	  			    3       5             11
     60  *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
     61  *	where
     62  *		|w - f(z)| < 2**-58.74
     63  *
     64  *   4. For negative x, since (G is gamma function)
     65  *		-x*G(-x)*G(x) = pi/sin(pi*x),
     66  * 	we have
     67  * 		G(x) = pi/(sin(pi*x)*(-x)*G(-x))
     68  *	since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
     69  *	Hence, for x<0, signgam = sign(sin(pi*x)) and
     70  *		lgamma(x) = log(|Gamma(x)|)
     71  *			  = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
     72  *	Note: one should avoid compute pi*(-x) directly in the
     73  *	      computation of sin(pi*(-x)).
     74  *
     75  *   5. Special Cases
     76  *		lgamma(2+s) ~ s*(1-Euler) for tiny s
     77  *		lgamma(1)=lgamma(2)=0
     78  *		lgamma(x) ~ -log(x) for tiny x
     79  *		lgamma(0) = lgamma(inf) = inf
     80  *	 	lgamma(-integer) = +-inf
     81  *
     82  */
     83 
     84 #include "math.h"
     85 #include "math_private.h"
     86 
     87 static const double
     88 two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
     89 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
     90 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
     91 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
     92 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
     93 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
     94 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
     95 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
     96 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
     97 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
     98 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
     99 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
    100 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
    101 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
    102 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
    103 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
    104 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
    105 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
    106 /* tt = -(tail of tf) */
    107 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
    108 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
    109 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
    110 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
    111 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
    112 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
    113 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
    114 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
    115 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
    116 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
    117 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
    118 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
    119 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
    120 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
    121 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
    122 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
    123 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
    124 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
    125 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
    126 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
    127 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
    128 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
    129 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
    130 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
    131 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
    132 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
    133 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
    134 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
    135 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
    136 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
    137 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
    138 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
    139 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
    140 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
    141 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
    142 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
    143 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
    144 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
    145 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
    146 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
    147 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
    148 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
    149 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
    150 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
    151 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
    152 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
    153 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
    154 
    155 static const double zero=  0.00000000000000000000e+00;
    156 
    157 	static double sin_pi(double x)
    158 {
    159 	double y,z;
    160 	int n,ix;
    161 
    162 	GET_HIGH_WORD(ix,x);
    163 	ix &= 0x7fffffff;
    164 
    165 	if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
    166 	y = -x;		/* x is assume negative */
    167 
    168     /*
    169      * argument reduction, make sure inexact flag not raised if input
    170      * is an integer
    171      */
    172 	z = floor(y);
    173 	if(z!=y) {				/* inexact anyway */
    174 	    y  *= 0.5;
    175 	    y   = 2.0*(y - floor(y));		/* y = |x| mod 2.0 */
    176 	    n   = (int) (y*4.0);
    177 	} else {
    178             if(ix>=0x43400000) {
    179                 y = zero; n = 0;                 /* y must be even */
    180             } else {
    181                 if(ix<0x43300000) z = y+two52;	/* exact */
    182 		GET_LOW_WORD(n,z);
    183 		n &= 1;
    184                 y  = n;
    185                 n<<= 2;
    186             }
    187         }
    188 	switch (n) {
    189 	    case 0:   y =  __kernel_sin(pi*y,zero,0); break;
    190 	    case 1:
    191 	    case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
    192 	    case 3:
    193 	    case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
    194 	    case 5:
    195 	    case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
    196 	    default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
    197 	    }
    198 	return -y;
    199 }
    200 
    201 
    202 double
    203 __ieee754_lgamma_r(double x, int *signgamp)
    204 {
    205 	double t,y,z,nadj,p,p1,p2,p3,q,r,w;
    206 	int i,hx,lx,ix;
    207 
    208 	EXTRACT_WORDS(hx,lx,x);
    209 
    210     /* purge off +-inf, NaN, +-0, and negative arguments */
    211 	*signgamp = 1;
    212 	ix = hx&0x7fffffff;
    213 	if(ix>=0x7ff00000) return x*x;
    214 	if((ix|lx)==0) return one/zero;
    215 	if(ix<0x3b900000) {	/* |x|<2**-70, return -log(|x|) */
    216 	    if(hx<0) {
    217 	        *signgamp = -1;
    218 	        return -__ieee754_log(-x);
    219 	    } else return -__ieee754_log(x);
    220 	}
    221 	if(hx<0) {
    222 	    if(ix>=0x43300000) 	/* |x|>=2**52, must be -integer */
    223 		return one/zero;
    224 	    t = sin_pi(x);
    225 	    if(t==zero) return one/zero; /* -integer */
    226 	    nadj = __ieee754_log(pi/fabs(t*x));
    227 	    if(t<zero) *signgamp = -1;
    228 	    x = -x;
    229 	}
    230 
    231     /* purge off 1 and 2 */
    232 	if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
    233     /* for x < 2.0 */
    234 	else if(ix<0x40000000) {
    235 	    if(ix<=0x3feccccc) { 	/* lgamma(x) = lgamma(x+1)-log(x) */
    236 		r = -__ieee754_log(x);
    237 		if(ix>=0x3FE76944) {y = one-x; i= 0;}
    238 		else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
    239 	  	else {y = x; i=2;}
    240 	    } else {
    241 	  	r = zero;
    242 	        if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
    243 	        else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
    244 		else {y=x-one;i=2;}
    245 	    }
    246 	    switch(i) {
    247 	      case 0:
    248 		z = y*y;
    249 		p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
    250 		p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
    251 		p  = y*p1+p2;
    252 		r  += (p-0.5*y); break;
    253 	      case 1:
    254 		z = y*y;
    255 		w = z*y;
    256 		p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));	/* parallel comp */
    257 		p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
    258 		p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
    259 		p  = z*p1-(tt-w*(p2+y*p3));
    260 		r += (tf + p); break;
    261 	      case 2:
    262 		p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
    263 		p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
    264 		r += (-0.5*y + p1/p2);
    265 	    }
    266 	}
    267 	else if(ix<0x40200000) { 			/* x < 8.0 */
    268 	    i = (int)x;
    269 	    t = zero;
    270 	    y = x-(double)i;
    271 	    p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
    272 	    q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
    273 	    r = half*y+p/q;
    274 	    z = one;	/* lgamma(1+s) = log(s) + lgamma(s) */
    275 	    switch(i) {
    276 	    case 7: z *= (y+6.0);	/* FALLTHRU */
    277 	    case 6: z *= (y+5.0);	/* FALLTHRU */
    278 	    case 5: z *= (y+4.0);	/* FALLTHRU */
    279 	    case 4: z *= (y+3.0);	/* FALLTHRU */
    280 	    case 3: z *= (y+2.0);	/* FALLTHRU */
    281 		    r += __ieee754_log(z); break;
    282 	    }
    283     /* 8.0 <= x < 2**58 */
    284 	} else if (ix < 0x43900000) {
    285 	    t = __ieee754_log(x);
    286 	    z = one/x;
    287 	    y = z*z;
    288 	    w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
    289 	    r = (x-half)*(t-one)+w;
    290 	} else
    291     /* 2**58 <= x <= inf */
    292 	    r =  x*(__ieee754_log(x)-one);
    293 	if(hx<0) r = nadj - r;
    294 	return r;
    295 }