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math: erf and erfc cleanup

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common part of erf and erfc was put in a separate function which
saved some space and the new code is using unsigned arithmetics

erfcf had a bug: for some inputs in [7.95,8] the result had
more than 60ulp error: in expf(-z*z - 0.5625f) the argument
must be exact but not enough lowbits of z were zeroed,
-SET_FLOAT_WORD(z, ix&0xfffff000);
+SET_FLOAT_WORD(z, ix&0xffffe000);
fixed the issue
This commit is contained in:
Szabolcs Nagy
2013-01-07 23:54:13 +01:00
parent d84923d89e
commit 121e3a38a1
3 changed files with 208 additions and 298 deletions
Download Patch File Download Diff File Expand all files Collapse all files

154
src/math/erff.c
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@ -16,13 +16,10 @@
#include "libm.h"
static const float
tiny = 1e-30,
/* c = (subfloat)0.84506291151 */
erx = 8.4506291151e-01, /* 0x3f58560b */
/*
* Coefficients for approximation to erf on [0,0.84375]
*/
efx = 1.2837916613e-01, /* 0x3e0375d4 */
efx8 = 1.0270333290e+00, /* 0x3f8375d4 */
pp0 = 1.2837916613e-01, /* 0x3e0375d4 */
pp1 = -3.2504209876e-01, /* 0xbea66beb */
@ -87,47 +84,26 @@ sb5 = 2.5530502930e+03, /* 0x451f90ce */
sb6 = 4.7452853394e+02, /* 0x43ed43a7 */
sb7 = -2.2440952301e+01; /* 0xc1b38712 */
float erff(float x)
static float erfc1(float x)
{
int32_t hx,ix,i;
float R,S,P,Q,s,y,z,r;
float s,P,Q;
s = fabsf(x) - 1;
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
return 1 - erx - P/Q;
}
static float erfc2(uint32_t ix, float x)
{
float s,z,R,S;
if (ix < 0x3fa00000) /* |x| < 1.25 */
return erfc1(x);
GET_FLOAT_WORD(hx, x);
ix = hx & 0x7fffffff;
if (ix >= 0x7f800000) {
/* erf(nan)=nan, erf(+-inf)=+-1 */
i = ((uint32_t)hx>>31)<<1;
return (float)(1-i)+1.0f/x;
}
if (ix < 0x3f580000) { /* |x| < 0.84375 */
if (ix < 0x31800000) { /* |x| < 2**-28 */
if (ix < 0x04000000)
/*avoid underflow */
return 0.125f*(8.0f*x + efx8*x);
return x + efx*x;
}
z = x*x;
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = 1.0f+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
return x + x*y;
}
if (ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-1.0f;
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
Q = 1.0f+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
if (hx >= 0)
return erx + P/Q;
return -erx - P/Q;
}
if (ix >= 0x40c00000) { /* inf > |x| >= 6 */
if (hx >= 0)
return 1.0f - tiny;
return tiny - 1.0f;
}
x = fabsf(x);
s = 1.0f/(x*x);
if (ix < 0x4036DB6E) { /* |x| < 1/0.35 */
s = 1/(x*x);
if (ix < 0x4036db6d) { /* |x| < 1/0.35 */
R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
ra5+s*(ra6+s*ra7))))));
S = 1.0f+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
@ -139,23 +115,53 @@ float erff(float x)
sb5+s*(sb6+s*sb7))))));
}
GET_FLOAT_WORD(ix, x);
SET_FLOAT_WORD(z, ix&0xfffff000);
r = expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S);
if (hx >= 0)
return 1.0f - r/x;
return r/x - 1.0f;
SET_FLOAT_WORD(z, ix&0xffffe000);
return expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S)/x;
}
float erff(float x)
{
float r,s,z,y;
uint32_t ix;
int sign;
GET_FLOAT_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x7f800000) {
/* erf(nan)=nan, erf(+-inf)=+-1 */
return 1-2*sign + 1/x;
}
if (ix < 0x3f580000) { /* |x| < 0.84375 */
if (ix < 0x31800000) { /* |x| < 2**-28 */
/*avoid underflow */
return 0.125f*(8*x + efx8*x);
}
z = x*x;
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = 1+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
return x + x*y;
}
if (ix < 0x40c00000) /* |x| < 6 */
y = 1 - erfc2(ix,x);
else
y = 1 - 0x1p-120f;
return sign ? -y : y;
}
float erfcf(float x)
{
int32_t hx,ix;
float R,S,P,Q,s,y,z,r;
float r,s,z,y;
uint32_t ix;
int sign;
GET_FLOAT_WORD(hx, x);
ix = hx & 0x7fffffff;
GET_FLOAT_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x7f800000) {
/* erfc(nan)=nan, erfc(+-inf)=0,2 */
return (float)(((uint32_t)hx>>31)<<1) + 1.0f/x;
return 2*sign + 1/x;
}
if (ix < 0x3f580000) { /* |x| < 0.84375 */
@ -165,50 +171,12 @@ float erfcf(float x)
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = 1.0f+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
if (hx < 0x3e800000) { /* x<1/4 */
if (sign || ix < 0x3e800000) /* x < 1/4 */
return 1.0f - (x+x*y);
} else {
r = x*y;
r += (x-0.5f);
return 0.5f - r ;
}
}
if (ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
s = fabsf(x)-1.0f;
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
Q = 1.0f+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
if(hx >= 0) {
z = 1.0f - erx;
return z - P/Q;
} else {
z = erx + P/Q;
return 1.0f + z;
}
return 0.5f - (x - 0.5f + x*y);
}
if (ix < 0x41e00000) { /* |x| < 28 */
x = fabsf(x);
s = 1.0f/(x*x);
if (ix < 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/
R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
ra5+s*(ra6+s*ra7))))));
S = 1.0f+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
sa5+s*(sa6+s*(sa7+s*sa8)))))));
} else { /* |x| >= 1/.35 ~ 2.857143 */
if (hx < 0 && ix >= 0x40c00000) /* x < -6 */
return 2.0f-tiny;
R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
rb5+s*rb6)))));
S = 1.0f+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
sb5+s*(sb6+s*sb7))))));
}
GET_FLOAT_WORD(ix, x);
SET_FLOAT_WORD(z, ix&0xfffff000);
r = expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S);
if (hx > 0)
return r/x;
return 2.0f - r/x;
return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
}
if (hx > 0)
return tiny*tiny;
return 2.0f - tiny;
return sign ? 2 - 0x1p-120f : 0x1p-120f*0x1p-120f;
}