/* @(#)s_erf.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.
* ====================================================
*/
/* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
for performance improvement on pipelined processors.
*/
#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
#endif
/* double erf(double x)
* double erfc(double x)
* x
* 2 |\
* erf(x) = --------- | exp(-t*t)dt
* sqrt(pi) \|
* 0
*
* erfc(x) = 1-erf(x)
* Note that
* erf(-x) = -erf(x)
* erfc(-x) = 2 - erfc(x)
*
* Method:
* 1. For |x| in [0, 0.84375]
* erf(x) = x + x*R(x^2)
* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
* where R = P/Q where P is an odd poly of degree 8 and
* Q is an odd poly of degree 10.
* -57.90
* | R - (erf(x)-x)/x | <= 2
*
*
* Remark. The formula is derived by noting
* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
* and that
* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
* is close to one. The interval is chosen because the fix
* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
* near 0.6174), and by some experiment, 0.84375 is chosen to
* guarantee the error is less than one ulp for erf.
*
* 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
* c = 0.84506291151 rounded to single (24 bits)
* erf(x) = sign(x) * (c + P1(s)/Q1(s))
* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
* 1+(c+P1(s)/Q1(s)) if x < 0
* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
* Remark: here we use the taylor series expansion at x=1.
* erf(1+s) = erf(1) + s*Poly(s)
* = 0.845.. + P1(s)/Q1(s)
* That is, we use rational approximation to approximate
* erf(1+s) - (c = (single)0.84506291151)
* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
* where
* P1(s) = degree 6 poly in s
* Q1(s) = degree 6 poly in s
*
* 3. For x in [1.25,1/0.35(~2.857143)],
* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
* erf(x) = 1 - erfc(x)
* where
* R1(z) = degree 7 poly in z, (z=1/x^2)
* S1(z) = degree 8 poly in z
*
* 4. For x in [1/0.35,28]
* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
* = 2.0 - tiny (if x <= -6)
* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
* erf(x) = sign(x)*(1.0 - tiny)
* where
* R2(z) = degree 6 poly in z, (z=1/x^2)
* S2(z) = degree 7 poly in z
*
* Note1:
* To compute exp(-x*x-0.5625+R/S), let s be a single
* precision number and s := x; then
* -x*x = -s*s + (s-x)*(s+x)
* exp(-x*x-0.5626+R/S) =
* exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
* Note2:
* Here 4 and 5 make use of the asymptotic series
* exp(-x*x)
* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
* x*sqrt(pi)
* We use rational approximation to approximate
* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
* Here is the error bound for R1/S1 and R2/S2
* |R1/S1 - f(x)| < 2**(-62.57)
* |R2/S2 - f(x)| < 2**(-61.52)
*
* 5. For inf > x >= 28
* erf(x) = sign(x) *(1 - tiny) (raise inexact)
* erfc(x) = tiny*tiny (raise underflow) if x > 0
* = 2 - tiny if x<0
*
* 7. Special case:
* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
* erfc/erf(NaN) is NaN
*/
#include "math.h"
#include "math_private.h"
#ifdef __STDC__
static const double
#else
static double
#endif
tiny = 1e-300,
half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
/* c = (float)0.84506291151 */
erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
/*
* Coefficients for approximation to erf on [0,0.84375]
*/
efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
pp[] = {1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
-3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
-2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
-5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
-2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */
qq[] = {0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
-3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */
/*
* Coefficients for approximation to erf in [0.84375,1.25]
*/
pa[] = {-2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
-3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
-1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
-2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */
qa[] = {0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
1.19844998467991074170e-02}, /* 0x3F888B54, 0x5735151D */
/*
* Coefficients for approximation to erfc in [1.25,1/0.35]
*/
ra[] = {-9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
-6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
-1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
-6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
-1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
-1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
-8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
-9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */
sa[] = {0.0,1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
-6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */
/*
* Coefficients for approximation to erfc in [1/.35,28]
*/
rb[] = {-9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
-7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
-1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
-1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
-6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
-1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
-4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */
sb[] = {0.0,3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
-2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */
#ifdef __STDC__
double __erf(double x)
#else
double __erf(x)
double x;
#endif
{
int32_t hx,ix,i;
double R,S,P,Q,s,y,z,r;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff;
if(ix>=0x7ff00000) { /* erf(nan)=nan */
i = ((u_int32_t)hx>>31)<<1;
return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
}
if(ix < 0x3feb0000) { /* |x|<0.84375 */
double r1,r2,s1,s2,s3,z2,z4;
if(ix < 0x3e300000) { /* |x|<2**-28 */
if (ix < 0x00800000)
return 0.125*(8.0*x+efx8*x); /*avoid underflow */
return x + efx*x;
}
z = x*x;
#ifdef DO_NOT_USE_THIS
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
#else
r1 = pp[0]+z*pp[1]; z2=z*z;
r2 = pp[2]+z*pp[3]; z4=z2*z2;
s1 = one+z*qq[1];
s2 = qq[2]+z*qq[3];
s3 = qq[4]+z*qq[5];
r = r1 + z2*r2 + z4*pp[4];
s = s1 + z2*s2 + z4*s3;
#endif
y = r/s;
return x + x*y;
}
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
s = fabs(x)-one;
#ifdef DO_NOT_USE_THIS
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)))));
#else
P1 = pa[0]+s*pa[1]; s2=s*s;
Q1 = one+s*qa[1]; s4=s2*s2;
P2 = pa[2]+s*pa[3]; s6=s4*s2;
Q2 = qa[2]+s*qa[3];
P3 = pa[4]+s*pa[5];
Q3 = qa[4]+s*qa[5];
P4 = s6*pa[6];
Q4 = s6*qa[6];
P = P1 + s2*P2 + s4*P3 + s6*P4;
Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
#endif
if(hx>=0) return erx + P/Q; else return -erx - P/Q;
}
if (ix >= 0x40180000) { /* inf>|x|>=6 */
if(hx>=0) return one-tiny; else return tiny-one;
}
x = fabs(x);
s = one/(x*x);
if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
#ifdef DO_NOT_USE_THIS
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)))))));
#else
double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
R1 = ra[0]+s*ra[1];s2 = s*s;
S1 = one+s*sa[1]; s4 = s2*s2;
R2 = ra[2]+s*ra[3];s6 = s4*s2;
S2 = sa[2]+s*sa[3];s8 = s4*s4;
R3 = ra[4]+s*ra[5];
S3 = sa[4]+s*sa[5];
R4 = ra[6]+s*ra[7];
S4 = sa[6]+s*sa[7];
R = R1 + s2*R2 + s4*R3 + s6*R4;
S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
#endif
} else { /* |x| >= 1/0.35 */
#ifdef DO_NOT_USE_THIS
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))))));
#else
double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
R1 = rb[0]+s*rb[1];s2 = s*s;
S1 = one+s*sb[1]; s4 = s2*s2;
R2 = rb[2]+s*rb[3];s6 = s4*s2;
S2 = sb[2]+s*sb[3];
R3 = rb[4]+s*rb[5];
S3 = sb[4]+s*sb[5];
S4 = sb[6]+s*sb[7];
R = R1 + s2*R2 + s4*R3 + s6*rb[6];
S = S1 + s2*S2 + s4*S3 + s6*S4;
#endif
}
z = x;
SET_LOW_WORD(z,0);
r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
if(hx>=0) return one-r/x; else return r/x-one;
}
weak_alias (__erf, erf)
#ifdef NO_LONG_DOUBLE
strong_alias (__erf, __erfl)
weak_alias (__erf, erfl)
#endif
#ifdef __STDC__
double __erfc(double x)
#else
double __erfc(x)
double x;
#endif
{
int32_t hx,ix;
double R,S,P,Q,s,y,z,r;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff;
if(ix>=0x7ff00000) { /* erfc(nan)=nan */
/* erfc(+-inf)=0,2 */
return (double)(((u_int32_t)hx>>31)<<1)+one/x;
}
if(ix < 0x3feb0000) { /* |x|<0.84375 */
double r1,r2,s1,s2,s3,z2,z4;
if(ix < 0x3c700000) /* |x|<2**-56 */
return one-x;
z = x*x;
#ifdef DO_NOT_USE_THIS
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
#else
r1 = pp[0]+z*pp[1]; z2=z*z;
r2 = pp[2]+z*pp[3]; z4=z2*z2;
s1 = one+z*qq[1];
s2 = qq[2]+z*qq[3];
s3 = qq[4]+z*qq[5];
r = r1 + z2*r2 + z4*pp[4];
s = s1 + z2*s2 + z4*s3;
#endif
y = r/s;
if(hx < 0x3fd00000) { /* x<1/4 */
return one-(x+x*y);
} else {
r = x*y;
r += (x-half);
return half - r ;
}
}
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
s = fabs(x)-one;
#ifdef DO_NOT_USE_THIS
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)))));
#else
P1 = pa[0]+s*pa[1]; s2=s*s;
Q1 = one+s*qa[1]; s4=s2*s2;
P2 = pa[2]+s*pa[3]; s6=s4*s2;
Q2 = qa[2]+s*qa[3];
P3 = pa[4]+s*pa[5];
Q3 = qa[4]+s*qa[5];
P4 = s6*pa[6];
Q4 = s6*qa[6];
P = P1 + s2*P2 + s4*P3 + s6*P4;
Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
#endif
if(hx>=0) {
z = one-erx; return z - P/Q;
} else {
z = erx+P/Q; return one+z;
}
}
if (ix < 0x403c0000) { /* |x|<28 */
x = fabs(x);
s = one/(x*x);
if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
#ifdef DO_NOT_USE_THIS
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)))))));
#else
double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
R1 = ra[0]+s*ra[1];s2 = s*s;
S1 = one+s*sa[1]; s4 = s2*s2;
R2 = ra[2]+s*ra[3];s6 = s4*s2;
S2 = sa[2]+s*sa[3];s8 = s4*s4;
R3 = ra[4]+s*ra[5];
S3 = sa[4]+s*sa[5];
R4 = ra[6]+s*ra[7];
S4 = sa[6]+s*sa[7];
R = R1 + s2*R2 + s4*R3 + s6*R4;
S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
#endif
} else { /* |x| >= 1/.35 ~ 2.857143 */
double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
#ifdef DO_NOT_USE_THIS
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))))));
#else
R1 = rb[0]+s*rb[1];s2 = s*s;
S1 = one+s*sb[1]; s4 = s2*s2;
R2 = rb[2]+s*rb[3];s6 = s4*s2;
S2 = sb[2]+s*sb[3];
R3 = rb[4]+s*rb[5];
S3 = sb[4]+s*sb[5];
S4 = sb[6]+s*sb[7];
R = R1 + s2*R2 + s4*R3 + s6*rb[6];
S = S1 + s2*S2 + s4*S3 + s6*S4;
#endif
}
z = x;
SET_LOW_WORD(z,0);
r = __ieee754_exp(-z*z-0.5625)*
__ieee754_exp((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;
}
}
weak_alias (__erfc, erfc)
#ifdef NO_LONG_DOUBLE
strong_alias (__erfc, __erfcl)
weak_alias (__erfc, erfcl)
#endif