new polynomials for erff, by Steve Kargl
these are both faster and more accurate see http://svnweb.freebsd.org/base/head/lib/msun/src/s_erff.c?view=log
This commit is contained in:
Jeff Bezanson
parent
89d232d114
commit
3566e32d84
171
src/s_erff.c
171
src/s_erff.c
|
|
@ -24,75 +24,60 @@ tiny = 1e-30,
|
|||
half= 5.0000000000e-01, /* 0x3F000000 */
|
||||
one = 1.0000000000e+00, /* 0x3F800000 */
|
||||
two = 2.0000000000e+00, /* 0x40000000 */
|
||||
/* c = (subfloat)0.84506291151 */
|
||||
erx = 8.4506291151e-01, /* 0x3f58560b */
|
||||
/*
|
||||
* Coefficients for approximation to erf on [0,0.84375]
|
||||
* 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 */
|
||||
pp2 = -2.8481749818e-02, /* 0xbce9528f */
|
||||
pp3 = -5.7702702470e-03, /* 0xbbbd1489 */
|
||||
pp4 = -2.3763017452e-05, /* 0xb7c756b1 */
|
||||
qq1 = 3.9791721106e-01, /* 0x3ecbbbce */
|
||||
qq2 = 6.5022252500e-02, /* 0x3d852a63 */
|
||||
qq3 = 5.0813062117e-03, /* 0x3ba68116 */
|
||||
qq4 = 1.3249473704e-04, /* 0x390aee49 */
|
||||
qq5 = -3.9602282413e-06, /* 0xb684e21a */
|
||||
/*
|
||||
* Coefficients for approximation to erf in [0.84375,1.25]
|
||||
* Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
|
||||
* |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
|
||||
*/
|
||||
pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */
|
||||
pa1 = 4.1485610604e-01, /* 0x3ed46805 */
|
||||
pa2 = -3.7220788002e-01, /* 0xbebe9208 */
|
||||
pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */
|
||||
pa4 = -1.1089469492e-01, /* 0xbde31cc2 */
|
||||
pa5 = 3.5478305072e-02, /* 0x3d1151b3 */
|
||||
pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */
|
||||
qa1 = 1.0642088205e-01, /* 0x3dd9f331 */
|
||||
qa2 = 5.4039794207e-01, /* 0x3f0a5785 */
|
||||
qa3 = 7.1828655899e-02, /* 0x3d931ae7 */
|
||||
qa4 = 1.2617121637e-01, /* 0x3e013307 */
|
||||
qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */
|
||||
qa6 = 1.1984500103e-02, /* 0x3c445aa3 */
|
||||
pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */
|
||||
pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */
|
||||
pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */
|
||||
qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */
|
||||
qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */
|
||||
qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */
|
||||
/*
|
||||
* Coefficients for approximation to erfc in [1.25,1/0.35]
|
||||
* Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
|
||||
* |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
|
||||
*/
|
||||
ra0 = -9.8649440333e-03, /* 0xbc21a093 */
|
||||
ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */
|
||||
ra2 = -1.0558626175e+01, /* 0xc128f022 */
|
||||
ra3 = -6.2375331879e+01, /* 0xc2798057 */
|
||||
ra4 = -1.6239666748e+02, /* 0xc322658c */
|
||||
ra5 = -1.8460508728e+02, /* 0xc3389ae7 */
|
||||
ra6 = -8.1287437439e+01, /* 0xc2a2932b */
|
||||
ra7 = -9.8143291473e+00, /* 0xc11d077e */
|
||||
sa1 = 1.9651271820e+01, /* 0x419d35ce */
|
||||
sa2 = 1.3765776062e+02, /* 0x4309a863 */
|
||||
sa3 = 4.3456588745e+02, /* 0x43d9486f */
|
||||
sa4 = 6.4538726807e+02, /* 0x442158c9 */
|
||||
sa5 = 4.2900814819e+02, /* 0x43d6810b */
|
||||
sa6 = 1.0863500214e+02, /* 0x42d9451f */
|
||||
sa7 = 6.5702495575e+00, /* 0x40d23f7c */
|
||||
sa8 = -6.0424413532e-02, /* 0xbd777f97 */
|
||||
erx = 8.42697144e-01F, /* 0x1.af7600p-1. erf(1) rounded to 16 bits. */
|
||||
pa0 = 3.64939137e-06F, /* 0x1.e9d022p-19 */
|
||||
pa1 = 4.15109694e-01F, /* 0x1.a91284p-2 */
|
||||
pa2 = -1.65179938e-01F, /* -0x1.5249dcp-3 */
|
||||
pa3 = 1.10914491e-01F, /* 0x1.c64e46p-4 */
|
||||
qa1 = 6.02074385e-01F, /* 0x1.344318p-1 */
|
||||
qa2 = 5.35934687e-01F, /* 0x1.126608p-1 */
|
||||
qa3 = 1.68576106e-01F, /* 0x1.593e6ep-3 */
|
||||
qa4 = 5.62181212e-02F, /* 0x1.cc89f2p-5 */
|
||||
/*
|
||||
* Coefficients for approximation to erfc in [1/.35,28]
|
||||
* Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]:
|
||||
* |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30
|
||||
*/
|
||||
rb0 = -9.8649431020e-03, /* 0xbc21a092 */
|
||||
rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */
|
||||
rb2 = -1.7757955551e+01, /* 0xc18e104b */
|
||||
rb3 = -1.6063638306e+02, /* 0xc320a2ea */
|
||||
rb4 = -6.3756646729e+02, /* 0xc41f6441 */
|
||||
rb5 = -1.0250950928e+03, /* 0xc480230b */
|
||||
rb6 = -4.8351919556e+02, /* 0xc3f1c275 */
|
||||
sb1 = 3.0338060379e+01, /* 0x41f2b459 */
|
||||
sb2 = 3.2579251099e+02, /* 0x43a2e571 */
|
||||
sb3 = 1.5367296143e+03, /* 0x44c01759 */
|
||||
sb4 = 3.1998581543e+03, /* 0x4547fdbb */
|
||||
sb5 = 2.5530502930e+03, /* 0x451f90ce */
|
||||
sb6 = 4.7452853394e+02, /* 0x43ed43a7 */
|
||||
sb7 = -2.2440952301e+01; /* 0xc1b38712 */
|
||||
ra0 = -9.87132732e-03F, /* -0x1.4376b2p-7 */
|
||||
ra1 = -5.53605914e-01F, /* -0x1.1b723cp-1 */
|
||||
ra2 = -2.17589188e+00F, /* -0x1.1683a0p+1 */
|
||||
ra3 = -1.43268085e+00F, /* -0x1.6ec42cp+0 */
|
||||
sa1 = 5.45995426e+00F, /* 0x1.5d6fe4p+2 */
|
||||
sa2 = 6.69798088e+00F, /* 0x1.acabb8p+2 */
|
||||
sa3 = 1.43113089e+00F, /* 0x1.6e5e98p+0 */
|
||||
sa4 = -5.77397496e-02F, /* -0x1.d90108p-5 */
|
||||
/*
|
||||
* Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]:
|
||||
* |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42
|
||||
*/
|
||||
rb0 = -9.86494310e-03F, /* -0x1.434124p-7 */
|
||||
rb1 = -6.25171244e-01F, /* -0x1.401672p-1 */
|
||||
rb2 = -6.16498327e+00F, /* -0x1.8a8f16p+2 */
|
||||
rb3 = -1.66696873e+01F, /* -0x1.0ab70ap+4 */
|
||||
rb4 = -9.53764343e+00F, /* -0x1.313460p+3 */
|
||||
sb1 = 1.26884899e+01F, /* 0x1.96081cp+3 */
|
||||
sb2 = 4.51839523e+01F, /* 0x1.6978bcp+5 */
|
||||
sb3 = 4.72810211e+01F, /* 0x1.7a3f88p+5 */
|
||||
sb4 = 8.93033314e+00F; /* 0x1.1dc54ap+3 */
|
||||
|
||||
|
||||
DLLEXPORT float
|
||||
erff(float x)
|
||||
|
|
@ -107,43 +92,37 @@ erff(float x)
|
|||
}
|
||||
|
||||
if(ix < 0x3f580000) { /* |x|<0.84375 */
|
||||
if(ix < 0x31800000) { /* |x|<2**-28 */
|
||||
if (ix < 0x04000000)
|
||||
/*avoid underflow */
|
||||
return (float)0.125*((float)8.0*x+efx8*x);
|
||||
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+z*(pp3+z*pp4)));
|
||||
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
||||
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+s*(pa4+s*(pa5+s*pa6)))));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
||||
P = pa0+s*(pa1+s*(pa2+s*pa3));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
|
||||
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
|
||||
}
|
||||
if (ix >= 0x40c00000) { /* inf>|x|>=6 */
|
||||
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< 0x4036DB6E) { /* |x| < 1/0.35 */
|
||||
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
||||
ra5+s*(ra6+s*ra7))))));
|
||||
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
||||
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
||||
R=ra0+s*(ra1+s*(ra2+s*ra3));
|
||||
S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
|
||||
} else { /* |x| >= 1/0.35 */
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
||||
rb5+s*rb6)))));
|
||||
S=one+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 = __ieee754_expf(-z*z-(float)0.5625)*__ieee754_expf((z-x)*(z+x)+R/S);
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
|
||||
S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
|
||||
}
|
||||
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;
|
||||
}
|
||||
|
||||
|
|
@ -160,11 +139,11 @@ erfcf(float x)
|
|||
}
|
||||
|
||||
if(ix < 0x3f580000) { /* |x|<0.84375 */
|
||||
if(ix < 0x23800000) /* |x|<2**-56 */
|
||||
if(ix < 0x33800000) /* |x|<2**-56 */
|
||||
return one-x;
|
||||
z = x*x;
|
||||
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
||||
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
||||
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);
|
||||
|
|
@ -176,33 +155,27 @@ erfcf(float x)
|
|||
}
|
||||
if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */
|
||||
s = fabsf(x)-one;
|
||||
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
||||
P = pa0+s*(pa1+s*(pa2+s*pa3));
|
||||
Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
|
||||
if(hx>=0) {
|
||||
z = one-erx; return z - P/Q;
|
||||
} else {
|
||||
z = erx+P/Q; return one+z;
|
||||
}
|
||||
}
|
||||
if (ix < 0x41e00000) { /* |x|<28 */
|
||||
if (ix < 0x41300000) { /* |x|<28 */
|
||||
x = fabsf(x);
|
||||
s = one/(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=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
||||
sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
||||
R=ra0+s*(ra1+s*(ra2+s*ra3));
|
||||
S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
|
||||
} else { /* |x| >= 1/.35 ~ 2.857143 */
|
||||
if(hx<0&&ix>=0x40c00000) return two-tiny;/* x < -6 */
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
||||
rb5+s*rb6)))));
|
||||
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
||||
sb5+s*(sb6+s*sb7))))));
|
||||
if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
|
||||
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
|
||||
S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
|
||||
}
|
||||
GET_FLOAT_WORD(ix,x);
|
||||
SET_FLOAT_WORD(z,ix&0xfffff000);
|
||||
r = __ieee754_expf(-z*z-(float)0.5625)*
|
||||
__ieee754_expf((z-x)*(z+x)+R/S);
|
||||
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;
|
||||
|
|
|
|||
Loading…
Reference in New Issue