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s_erf.c

/* @(#)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 = pa[6];
          Q4 = 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 = pa[6];
          Q4 = 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

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