Blame |
Last modification |
View Log
| RSS feed
/* @(#)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.
* ====================================================
*/
#ifndef lint
static char rcsid
[] = "$\Id: e_lgamma_r.c,v 1.2 1995/05/30 05:48:27 rgrimes Exp $";
#endif
/* __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"
#ifdef __STDC__
static const double
#else
static double
#endif
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 */
#ifdef __STDC__
static const double zero
= 0.00000000000000000000e+00;
#else
static double zero
= 0.00000000000000000000e+00;
#endif
#ifdef __STDC__
static double sin_pi
(double x
)
#else
static double sin_pi
(x
)
double x
;
#endif
{
double y
,z
;
int n
,ix
;
GET_HIGH_WORD
(ix
,x
);
ix
&= 0x7fffffff;
if(ix
<0x3fd00000) return __kernel_sin
(pi
*x
,zero
,0);
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
= __kernel_sin
(pi
*y
,zero
,0); break;
case 1:
case 2: y
= __kernel_cos
(pi
*(0.5-y
),zero
); break;
case 3:
case 4: y
= __kernel_sin
(pi
*(one
-y
),zero
,0); break;
case 5:
case 6: y
= -__kernel_cos
(pi
*(y
-1.5),zero
); break;
default: y
= __kernel_sin
(pi
*(y
-2.0),zero
,0); break;
}
return -y
;
}
#ifdef __STDC__
double __ieee754_lgamma_r
(double x
, int *signgamp
)
#else
double __ieee754_lgamma_r
(x
,signgamp
)
double x
; int *signgamp
;
#endif
{
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(ix
>=0x7ff00000) return x
*x
;
if((ix
|lx
)==0) return one
/zero
;
if(ix
<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
if(hx
<0) {
*signgamp
= -1;
return -__ieee754_log
(-x
);
} else return -__ieee754_log
(x
);
}
if(hx
<0) {
if(ix
>=0x43300000) /* |x|>=2**52, must be -integer */
return one
/zero
;
t
= sin_pi
(x
);
if(t
==zero
) return one
/zero
; /* -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
= x
*(__ieee754_log
(x
)-one
);
if(hx
<0) r
= nadj
- r
;
return r
;
}