s_erf.c

Go to the documentation of this file.

/* @(#)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.
 * ====================================================
 */

/* 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"

static const double
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 */

double
erf(double x) 
{
    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  =  exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
    if(hx>=0) return one-r/x; else return  r/x-one;
}

double
erfc(double x) 
{
    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  =  exp(-z*z-0.5625) * 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;
    }
}