c2307b7d21
* Update the domain and range of comments for the polynomial
approximations, including using the the correct variable names
(e.g., pp(x) instead of p(x)).
* Use hex values of the form 0x3e0375d4 instead of 0x1.06eba8p-3,
which was obtained from printf("%.6a").
* In the domain [0.84375, 1.25], qa(x) can be reduced from a 4th
order polynomial to 3rd order.
* In the domain [1.25,1/0.35], sa(x) can be reduced from a 4th
order polynomial to 3rd order.
* In the domain [1/0.35, 11], the 4th order polynomials rb(x) and
sb(x) can be reduced to 2nd and 3rd order, respectively.
182 lines
5.1 KiB
C
182 lines
5.1 KiB
C
/* s_erff.c -- float version of s_erf.c.
|
|
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
|
|
*/
|
|
|
|
/*
|
|
* ====================================================
|
|
* 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.
|
|
* ====================================================
|
|
*/
|
|
|
|
#include <sys/cdefs.h>
|
|
__FBSDID("$FreeBSD$");
|
|
|
|
#include "math.h"
|
|
#include "math_private.h"
|
|
|
|
/* XXX Prevent compilers from erroneously constant folding: */
|
|
static const volatile float tiny = 1e-30;
|
|
|
|
static const float
|
|
half= 0.5,
|
|
one = 1,
|
|
two = 2,
|
|
erx = 8.42697144e-01, /* 0x3f57bb00 */
|
|
/*
|
|
* In the domain [0, 2**-14], only the first term in the power series
|
|
* expansion of erf(x) is used. The magnitude of the first neglected
|
|
* terms is less than 2**-42.
|
|
*/
|
|
efx = 1.28379166e-01, /* 0x3e0375d4 */
|
|
efx8= 1.02703333e+00, /* 0x3f8375d4 */
|
|
/*
|
|
* Domain [0, 0.84375], range ~[-5.4419e-10, 5.5179e-10]:
|
|
* |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-31
|
|
*/
|
|
pp0 = 1.28379166e-01, /* 0x3e0375d4 */
|
|
pp1 = -3.36030394e-01, /* 0xbeac0c2d */
|
|
pp2 = -1.86261395e-03, /* 0xbaf422f4 */
|
|
qq1 = 3.12324315e-01, /* 0x3e9fe8f9 */
|
|
qq2 = 2.16070414e-02, /* 0x3cb10140 */
|
|
qq3 = -1.98859372e-03, /* 0xbb025311 */
|
|
/*
|
|
* Domain [0.84375, 1.25], range ~[-1.023e-9, 1.023e-9]:
|
|
* |(erf(x) - erx) - pa(x)/qa(x)| < 2**-31
|
|
*/
|
|
pa0 = 3.65041046e-06, /* 0x3674f993 */
|
|
pa1 = 4.15109307e-01, /* 0x3ed48935 */
|
|
pa2 = -2.09395722e-01, /* 0xbe566bd5 */
|
|
pa3 = 8.67677554e-02, /* 0x3db1b34b */
|
|
qa1 = 4.95560974e-01, /* 0x3efdba2b */
|
|
qa2 = 3.71248513e-01, /* 0x3ebe1449 */
|
|
qa3 = 3.92478965e-02, /* 0x3d20c267 */
|
|
/*
|
|
* Domain [1.25,1/0.35], range ~[-4.821e-9, 4.927e-9]:
|
|
* |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-28
|
|
*/
|
|
ra0 = -9.88156721e-03, /* 0xbc21e64c */
|
|
ra1 = -5.43658376e-01, /* 0xbf0b2d32 */
|
|
ra2 = -1.66828310e+00, /* 0xbfd58a4d */
|
|
ra3 = -6.91554189e-01, /* 0xbf3109b2 */
|
|
sa1 = 4.48581553e+00, /* 0x408f8bcd */
|
|
sa2 = 4.10799170e+00, /* 0x408374ab */
|
|
sa3 = 5.53855181e-01, /* 0x3f0dc974 */
|
|
/*
|
|
* Domain [2.85715, 11], range ~[-1.484e-9, 1.505e-9]:
|
|
* |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-30
|
|
*/
|
|
rb0 = -9.86496918e-03, /* 0xbc21a0ae */
|
|
rb1 = -5.48049808e-01, /* 0xbf0c4cfe */
|
|
rb2 = -1.84115684e+00, /* 0xbfebab07 */
|
|
sb1 = 4.87132740e+00, /* 0x409be1ea */
|
|
sb2 = 3.04982710e+00, /* 0x4043305e */
|
|
sb3 = -7.61900663e-01; /* 0xbf430bec */
|
|
|
|
float
|
|
erff(float x)
|
|
{
|
|
int32_t hx,ix,i;
|
|
float R,S,P,Q,s,y,z,r;
|
|
GET_FLOAT_WORD(hx,x);
|
|
ix = hx&0x7fffffff;
|
|
if(ix>=0x7f800000) { /* erff(nan)=nan */
|
|
i = ((u_int32_t)hx>>31)<<1;
|
|
return (float)(1-i)+one/x; /* erff(+-inf)=+-1 */
|
|
}
|
|
|
|
if(ix < 0x3f580000) { /* |x|<0.84375 */
|
|
if(ix < 0x38800000) { /* |x|<2**-14 */
|
|
if (ix < 0x04000000) /* |x|<0x1p-119 */
|
|
return (8*x+efx8*x)/8; /* avoid spurious underflow */
|
|
return x + efx*x;
|
|
}
|
|
z = x*x;
|
|
r = pp0+z*(pp1+z*pp2);
|
|
s = one+z*(qq1+z*(qq2+z*qq3));
|
|
y = r/s;
|
|
return x + x*y;
|
|
}
|
|
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
|
|
s = fabsf(x)-one;
|
|
P = pa0+s*(pa1+s*(pa2+s*pa3));
|
|
Q = one+s*(qa1+s*(qa2+s*qa3));
|
|
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
|
|
}
|
|
if (ix >= 0x40800000) { /* inf>|x|>=4 */
|
|
if(hx>=0) return one-tiny; else return tiny-one;
|
|
}
|
|
x = fabsf(x);
|
|
s = one/(x*x);
|
|
if(ix< 0x4036db8c) { /* |x| < 2.85715 ~ 1/0.35 */
|
|
R=ra0+s*(ra1+s*(ra2+s*ra3));
|
|
S=one+s*(sa1+s*(sa2+s*sa3));
|
|
} else { /* |x| >= 2.85715 ~ 1/0.35 */
|
|
R=rb0+s*(rb1+s*rb2);
|
|
S=one+s*(sb1+s*(sb2+s*sb3));
|
|
}
|
|
SET_FLOAT_WORD(z,hx&0xffffe000);
|
|
r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
|
|
if(hx>=0) return one-r/x; else return r/x-one;
|
|
}
|
|
|
|
float
|
|
erfcf(float x)
|
|
{
|
|
int32_t hx,ix;
|
|
float R,S,P,Q,s,y,z,r;
|
|
GET_FLOAT_WORD(hx,x);
|
|
ix = hx&0x7fffffff;
|
|
if(ix>=0x7f800000) { /* erfcf(nan)=nan */
|
|
/* erfcf(+-inf)=0,2 */
|
|
return (float)(((u_int32_t)hx>>31)<<1)+one/x;
|
|
}
|
|
|
|
if(ix < 0x3f580000) { /* |x|<0.84375 */
|
|
if(ix < 0x33800000) /* |x|<2**-24 */
|
|
return one-x;
|
|
z = x*x;
|
|
r = pp0+z*(pp1+z*pp2);
|
|
s = one+z*(qq1+z*(qq2+z*qq3));
|
|
y = r/s;
|
|
if(hx < 0x3e800000) { /* x<1/4 */
|
|
return one-(x+x*y);
|
|
} else {
|
|
r = x*y;
|
|
r += (x-half);
|
|
return half - r ;
|
|
}
|
|
}
|
|
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
|
|
s = fabsf(x)-one;
|
|
P = pa0+s*(pa1+s*(pa2+s*pa3));
|
|
Q = one+s*(qa1+s*(qa2+s*qa3));
|
|
if(hx>=0) {
|
|
z = one-erx; return z - P/Q;
|
|
} else {
|
|
z = erx+P/Q; return one+z;
|
|
}
|
|
}
|
|
if (ix < 0x41300000) { /* |x|<11 */
|
|
x = fabsf(x);
|
|
s = one/(x*x);
|
|
if(ix< 0x4036db8c) { /* |x| < 2.85715 ~ 1/.35 */
|
|
R=ra0+s*(ra1+s*(ra2+s*ra3));
|
|
S=one+s*(sa1+s*(sa2+s*sa3));
|
|
} else { /* |x| >= 2.85715 ~ 1/.35 */
|
|
if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
|
|
R=rb0+s*(rb1+s*rb2);
|
|
S=one+s*(sb1+s*(sb2+s*sb3));
|
|
}
|
|
SET_FLOAT_WORD(z,hx&0xffffe000);
|
|
r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
|
|
if(hx>0) return r/x; else return two-r/x;
|
|
} else {
|
|
if(hx>0) return tiny*tiny; else return two-tiny;
|
|
}
|
|
}
|