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/* @(#)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.
* ====================================================
*/
#ifndef lint
static char rcsid
[] = "$\Id: s_erf.c,v 1.2 1995/05/30 05:49:31 rgrimes 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 */
pp0
= 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
pp1
= -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
pp2
= -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
pp3
= -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
pp4
= -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
qq1
= 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
qq2
= 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
qq3
= 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
qq4
= 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
qq5
= -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
/*
* Coefficients for approximation to erf in [0.84375,1.25]
*/
pa0
= -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
pa1
= 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
pa2
= -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
pa3
= 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
pa4
= -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
pa5
= 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
pa6
= -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
qa1
= 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
qa2
= 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
qa3
= 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
qa4
= 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
qa5
= 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
qa6
= 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
/*
* Coefficients for approximation to erfc in [1.25,1/0.35]
*/
ra0
= -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
ra1
= -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
ra2
= -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
ra3
= -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
ra4
= -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
ra5
= -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
ra6
= -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
ra7
= -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
sa1
= 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
sa2
= 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
sa3
= 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
sa4
= 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
sa5
= 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
sa6
= 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
sa7
= 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
sa8
= -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
/*
* Coefficients for approximation to erfc in [1/.35,28]
*/
rb0
= -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
rb1
= -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
rb2
= -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
rb3
= -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
rb4
= -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
rb5
= -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
rb6
= -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
sb1
= 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
sb2
= 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
sb3
= 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
sb4
= 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
sb5
= 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
sb6
= 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
sb7
= -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 */
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
;
r
= pp0
+z
*(pp1
+z
*(pp2
+z
*(pp3
+z
*pp4
)));
s
= one
+z
*(qq1
+z
*(qq2
+z
*(qq3
+z
*(qq4
+z
*qq5
))));
y
= r
/s
;
return x
+ x
*y
;
}
if(ix
< 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
s
= fabs(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
)))));
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 */
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 { /* |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
))))));
}
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
;
}
#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 */
if(ix
< 0x3c700000) /* |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
))));
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 */
s
= fabs(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
)))));
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*/
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 { /* |x| >= 1/.35 ~ 2.857143 */
if(hx
<0&&ix
>=0x40180000) 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
))))));
}
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
;
}
}