* ====================================================
* 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.
* ====================================================
*/
* 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 "libm.h"
static const double
erx = 8.45062911510467529297e-01,
* Coefficients for approximation to erf on [0,0.84375]
*/
efx8 = 1.02703333676410069053e+00,
pp0 = 1.28379167095512558561e-01,
pp1 = -3.25042107247001499370e-01,
pp2 = -2.84817495755985104766e-02,
pp3 = -5.77027029648944159157e-03,
pp4 = -2.37630166566501626084e-05,
qq1 = 3.97917223959155352819e-01,
qq2 = 6.50222499887672944485e-02,
qq3 = 5.08130628187576562776e-03,
qq4 = 1.32494738004321644526e-04,
qq5 = -3.96022827877536812320e-06,
* Coefficients for approximation to erf in [0.84375,1.25]
*/
pa0 = -2.36211856075265944077e-03,
pa1 = 4.14856118683748331666e-01,
pa2 = -3.72207876035701323847e-01,
pa3 = 3.18346619901161753674e-01,
pa4 = -1.10894694282396677476e-01,
pa5 = 3.54783043256182359371e-02,
pa6 = -2.16637559486879084300e-03,
qa1 = 1.06420880400844228286e-01,
qa2 = 5.40397917702171048937e-01,
qa3 = 7.18286544141962662868e-02,
qa4 = 1.26171219808761642112e-01,
qa5 = 1.36370839120290507362e-02,
qa6 = 1.19844998467991074170e-02,
* Coefficients for approximation to erfc in [1.25,1/0.35]
*/
ra0 = -9.86494403484714822705e-03,
ra1 = -6.93858572707181764372e-01,
ra2 = -1.05586262253232909814e+01,
ra3 = -6.23753324503260060396e+01,
ra4 = -1.62396669462573470355e+02,
ra5 = -1.84605092906711035994e+02,
ra6 = -8.12874355063065934246e+01,
ra7 = -9.81432934416914548592e+00,
sa1 = 1.96512716674392571292e+01,
sa2 = 1.37657754143519042600e+02,
sa3 = 4.34565877475229228821e+02,
sa4 = 6.45387271733267880336e+02,
sa5 = 4.29008140027567833386e+02,
sa6 = 1.08635005541779435134e+02,
sa7 = 6.57024977031928170135e+00,
sa8 = -6.04244152148580987438e-02,
* Coefficients for approximation to erfc in [1/.35,28]
*/
rb0 = -9.86494292470009928597e-03,
rb1 = -7.99283237680523006574e-01,
rb2 = -1.77579549177547519889e+01,
rb3 = -1.60636384855821916062e+02,
rb4 = -6.37566443368389627722e+02,
rb5 = -1.02509513161107724954e+03,
rb6 = -4.83519191608651397019e+02,
sb1 = 3.03380607434824582924e+01,
sb2 = 3.25792512996573918826e+02,
sb3 = 1.53672958608443695994e+03,
sb4 = 3.19985821950859553908e+03,
sb5 = 2.55305040643316442583e+03,
sb6 = 4.74528541206955367215e+02,
sb7 = -2.24409524465858183362e+01;
static double erfc1(double x)
{
double_t s,P,Q;
s = fabs(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 double erfc2(uint32_t ix, double x)
{
double_t s,R,S;
double z;
if (ix < 0x3ff40000)
return erfc1(x);
x = fabs(x);
s = 1/(x*x);
if (ix < 0x4006db6d) {
R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
ra5+s*(ra6+s*ra7))))));
S = 1.0+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
sa5+s*(sa6+s*(sa7+s*sa8)))))));
} else {
R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
rb5+s*rb6)))));
S = 1.0+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
sb5+s*(sb6+s*sb7))))));
}
z = x;
SET_LOW_WORD(z,0);
return exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)/x;
}
double erf(double x)
{
double r,s,z,y;
uint32_t ix;
int sign;
GET_HIGH_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x7ff00000) {
return 1-2*sign + 1/x;
}
if (ix < 0x3feb0000) {
if (ix < 0x3e300000) {
return 0.125*(8*x + efx8*x);
}
z = x*x;
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
return x + x*y;
}
if (ix < 0x40180000)
y = 1 - erfc2(ix,x);
else
y = 1 - 0x1p-1022;
return sign ? -y : y;
}
double erfc(double x)
{
double r,s,z,y;
uint32_t ix;
int sign;
GET_HIGH_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x7ff00000) {
return 2*sign + 1/x;
}
if (ix < 0x3feb0000) {
if (ix < 0x3c700000)
return 1.0 - x;
z = x*x;
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
if (sign || ix < 0x3fd00000) {
return 1.0 - (x+x*y);
}
return 0.5 - (x - 0.5 + x*y);
}
if (ix < 0x403c0000) {
return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
}
return sign ? 2 - 0x1p-1022 : 0x1p-1022*0x1p-1022;
}
