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erf.c

#include "FEATURE/uwin"

#if !_UWIN || _lib_erf

void _STUB_erf(){}

#else

/*-
 * Copyright (c) 1992, 1993
 *    The Regents of the University of California.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 */

#ifndef lint
static char sccsid[] = "@(#)erf.c   8.1 (Berkeley) 6/4/93";
#endif /* not lint */

/* Modified Nov 30, 1992 P. McILROY:
 *    Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp)
 * Replaced even+odd with direct calculation for x < .84375,
 * to avoid destructive cancellation.
 *
 * Performance of erfc(x):
 * In 300000 trials in the range [.83, .84375] the
 * maximum observed error was 3.6ulp.
 *
 * In [.84735,1.25] the maximum observed error was <2.5ulp in
 * 100000 runs in the range [1.2, 1.25].
 *
 * In [1.25,26] (Not including subnormal results)
 * the error is < 1.7ulp.
 */

/* double erf(double x)
 * double erfc(double x)
 *                     x
 *                2      |\
 *     erf(x)  =  ---------  | exp(-t*t)dt
 *             sqrt(pi) \|
 *                     0
 *
 *     erfc(x) =  1-erf(x)
 *
 * Method:
 *      1. Reduce x to |x| by erf(-x) = -erf(x)
 *    2. For x in [0, 0.84375]
 *        erf(x)  = x + x*P(x^2)
 *          erfc(x) = 1 - erf(x)           if x<=0.25
 *                  = 0.5 + ((0.5-x)-x*P)  if x in [0.25,0.84375]
 *       where
 *                2            2      4           20  
 *              P =  P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x  )
 *       is an approximation to (erf(x)-x)/x with precision
 *
 *                                   -56.45
 *                | P - (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 fixed
 *       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.
 *
 *      3. For x in [0.84375,1.25], let s = x - 1, and
 *         c = 0.84506291151 rounded to single (24 bits)
 *          erf(x)  = c  + P1(s)/Q1(s)
 *          erfc(x) = (1-c)  - P1(s)/Q1(s)
 *          |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
 *
 *    4. For x in [1.25, 2]; [2, 4]
 *          erf(x)  = 1.0 - tiny
 *          erfc(x)     = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z))
 *
 *    Where z = 1/(x*x), R is degree 9, and S is degree 3;
 *    
 *      5. For x in [4,28]
 *          erf(x)  = 1.0 - tiny
 *          erfc(x)     = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z))
 *
 *    Where P is degree 14 polynomial in 1/(x*x).
 *
 *      Notes:
 *       Here 4 and 5 make use of the asymptotic series
 *                  exp(-x*x)
 *          erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) );
 *                  x*sqrt(pi)
 *
 *          where for z = 1/(x*x)
 *          P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...))))
 *
 *       Thus we use rational approximation to approximate
 *              erfc*x*exp(x*x) ~ 1/sqrt(pi);
 *
 *          The error bound for the target function, G(z) for
 *          the interval
 *          [4, 28]:
 *          |eps + 1/(z)P(z) - G(z)| < 2**(-56.61)
 *          for [2, 4]:
 *          |R(z)/S(z) - G(z)|       < 2**(-58.24)
 *          for [1.25, 2]:
 *          |R(z)/S(z) - G(z)|       < 2**(-58.12)
 *
 *      6. For inf > x >= 28
 *          erf(x)  = 1 - tiny  (raise inexact)
 *          erfc(x) = tiny*tiny (raise underflow)
 *
 *      7. Special cases:
 *          erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
 *          erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 
 *          erfc/erf(NaN) is NaN
 */

#if defined(vax) || defined(tahoe)
#define _IEEE     0
#define TRUNC(x) (double) (float) (x)
#else
#define _IEEE     1
#define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000
#define infnan(x) 0.0
#endif

#ifdef _IEEE_LIBM
/*
 * redefining "___function" to "function" in _IEEE_LIBM mode
 */
#include "ieee_libm.h"
#endif
#include "mathimpl.h"

static double
tiny      = 1e-300,
half      = 0.5,
one       = 1.0,
two       = 2.0,
c         = 8.45062911510467529297e-01, /* (float)0.84506291151 */
/*
 * Coefficients for approximation to erf in [0,0.84375]
 */
p0t8 = 1.02703333676410051049867154944018394163280,
p0 =   1.283791670955125638123339436800229927041e-0001,
p1 =  -3.761263890318340796574473028946097022260e-0001,
p2 =   1.128379167093567004871858633779992337238e-0001,
p3 =  -2.686617064084433642889526516177508374437e-0002,
p4 =   5.223977576966219409445780927846432273191e-0003,
p5 =  -8.548323822001639515038738961618255438422e-0004,
p6 =   1.205520092530505090384383082516403772317e-0004,
p7 =  -1.492214100762529635365672665955239554276e-0005,
p8 =   1.640186161764254363152286358441771740838e-0006,
p9 =  -1.571599331700515057841960987689515895479e-0007,
p10=   1.073087585213621540635426191486561494058e-0008;
/*
 * Coefficients for approximation to erf in [0.84375,1.25] 
 */
static double
pa0 =  -2.362118560752659485957248365514511540287e-0003,
pa1 =   4.148561186837483359654781492060070469522e-0001,
pa2 =  -3.722078760357013107593507594535478633044e-0001,
pa3 =   3.183466199011617316853636418691420262160e-0001,
pa4 =  -1.108946942823966771253985510891237782544e-0001,
pa5 =   3.547830432561823343969797140537411825179e-0002,
pa6 =  -2.166375594868790886906539848893221184820e-0003,
qa1 =   1.064208804008442270765369280952419863524e-0001,
qa2 =   5.403979177021710663441167681878575087235e-0001,
qa3 =   7.182865441419627066207655332170665812023e-0002,
qa4 =   1.261712198087616469108438860983447773726e-0001,
qa5 =   1.363708391202905087876983523620537833157e-0002,
qa6 =   1.198449984679910764099772682882189711364e-0002;
/*
 * log(sqrt(pi)) for large x expansions.
 * The tail (lsqrtPI_lo) is included in the rational
 * approximations.
*/
static double
   lsqrtPI_hi = .5723649429247000819387380943226;
/*
 * lsqrtPI_lo = .000000000000000005132975581353913;
 *
 * Coefficients for approximation to erfc in [2, 4]
*/
static double
rb0  =      -1.5306508387410807582e-010,  /* includes lsqrtPI_lo */
rb1  =       2.15592846101742183841910806188e-008,
rb2  =       6.24998557732436510470108714799e-001,
rb3  =       8.24849222231141787631258921465e+000,
rb4  =       2.63974967372233173534823436057e+001,
rb5  =       9.86383092541570505318304640241e+000,
rb6  =      -7.28024154841991322228977878694e+000,
rb7  =       5.96303287280680116566600190708e+000,
rb8  =      -4.40070358507372993983608466806e+000,
rb9  =       2.39923700182518073731330332521e+000,
rb10 =      -6.89257464785841156285073338950e-001,
sb1  =       1.56641558965626774835300238919e+001,
sb2  =       7.20522741000949622502957936376e+001,
sb3  =       9.60121069770492994166488642804e+001;
/*
 * Coefficients for approximation to erfc in [1.25, 2]
*/
static double
rc0  =      -2.47925334685189288817e-007, /* includes lsqrtPI_lo */
rc1  =       1.28735722546372485255126993930e-005,
rc2  =       6.24664954087883916855616917019e-001,
rc3  =       4.69798884785807402408863708843e+000,
rc4  =       7.61618295853929705430118701770e+000,
rc5  =       9.15640208659364240872946538730e-001,
rc6  =      -3.59753040425048631334448145935e-001,
rc7  =       1.42862267989304403403849619281e-001,
rc8  =      -4.74392758811439801958087514322e-002,
rc9  =       1.09964787987580810135757047874e-002,
rc10 =      -1.28856240494889325194638463046e-003,
sc1  =       9.97395106984001955652274773456e+000,
sc2  =       2.80952153365721279953959310660e+001,
sc3  =       2.19826478142545234106819407316e+001;
/*
 * Coefficients for approximation to  erfc in [4,28]
 */
static double
rd0  =      -2.1491361969012978677e-016,  /* includes lsqrtPI_lo */
rd1  =      -4.99999999999640086151350330820e-001,
rd2  =       6.24999999772906433825880867516e-001,
rd3  =      -1.54166659428052432723177389562e+000,
rd4  =       5.51561147405411844601985649206e+000,
rd5  =      -2.55046307982949826964613748714e+001,
rd6  =       1.43631424382843846387913799845e+002,
rd7  =      -9.45789244999420134263345971704e+002,
rd8  =       6.94834146607051206956384703517e+003,
rd9  =      -5.27176414235983393155038356781e+004,
rd10 =       3.68530281128672766499221324921e+005,
rd11 =      -2.06466642800404317677021026611e+006,
rd12 =       7.78293889471135381609201431274e+006,
rd13 =      -1.42821001129434127360582351685e+007;

extern double erf(x)
      double x;
{
      double R,S,P,Q,ax,s,y,z,r,fabs(),exp();
      if(!finite(x)) {        /* erf(nan)=nan */
          if (isnan(x))
            return(x);
          return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */
      }
      if ((ax = x) < 0)
            ax = - ax;
      if (ax < .84375) {
          if (ax < 3.7e-09) {
            if (ax < 1.0e-308)
                return 0.125*(8.0*x+p0t8*x);  /*avoid underflow */
            return x + p0*x;
          }
          y = x*x;
          r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
                  y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
          return x + x*(p0+r);
      }
      if (ax < 1.25) {        /* 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 (x>=0)
            return (c + P/Q);
          else
            return (-c - P/Q);
      }
      if (ax >= 6.0) {        /* inf>|x|>=6 */
          if (x >= 0.0)
            return (one-tiny);
          else
            return (tiny-one);
      }
    /* 1.25 <= |x| < 6 */
      z = -ax*ax;
      s = -one/z;
      if (ax < 2.0) {
            R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
                  s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
            S = one+s*(sc1+s*(sc2+s*sc3));
      } else {
            R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
                  s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
            S = one+s*(sb1+s*(sb2+s*sb3));
      }
      y = (R/S -.5*s) - lsqrtPI_hi;
      z += y;
      z = exp(z)/ax;
      if (x >= 0)
            return (one-z);
      else
            return (z-one);
}

extern double erfc(x) 
      double x;
{
      double R,S,P,Q,s,ax,y,z,r,fabs(),__exp__D();
      if (!finite(x)) {
            if (isnan(x))           /* erfc(NaN) = NaN */
                  return(x);
            else if (x > 0)         /* erfc(+-inf)=0,2 */
                  return 0.0;
            else
                  return 2.0;
      }
      if ((ax = x) < 0)
            ax = -ax;
      if (ax < .84375) {                  /* |x|<0.84375 */
          if (ax < 1.38777878078144568e-17)     /* |x|<2**-56 */
            return one-x;
          y = x*x;
          r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
                  y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
          if (ax < .0625) {   /* |x|<2**-4 */
            return (one-(x+x*(p0+r)));
          } else {
            r = x*(p0+r);
            r += (x-half);
              return (half - r);
          }
      }
      if (ax < 1.25) {        /* 0.84375 <= |x| < 1.25 */
          s = ax-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 (x>=0) {
              z  = one-c; return z - P/Q; 
          } else {
            z = c+P/Q; return one+z;
          }
      }
      if (ax >= 28)     /* Out of range */
            if (x>0)
                  return (tiny*tiny);
            else
                  return (two-tiny);
      z = ax;
      TRUNC(z);
      y = z - ax; y *= (ax+z);
      z *= -z;                /* Here z + y = -x^2 */
            s = one/(-z-y);         /* 1/(x*x) */
      if (ax >= 4) {                /* 6 <= ax */
            R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+
                  s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10
                  +s*(rd11+s*(rd12+s*rd13))))))))))));
            y += rd0;
      } else if (ax >= 2) {
            R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
                  s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
            S = one+s*(sb1+s*(sb2+s*sb3));
            y += R/S;
            R = -.5*s;
      } else {
            R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
                  s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
            S = one+s*(sc1+s*(sc2+s*sc3));
            y += R/S;
            R = -.5*s;
      }
      /* return exp(-x^2 - lsqrtPI_hi + R + y)/x;     */
      s = ((R + y) - lsqrtPI_hi) + z;
      y = (((z-s) - lsqrtPI_hi) + R) + y;
      r = __exp__D(s, y)/x;
      if (x>0)
            return r;
      else
            return two-r;
}

#endif

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