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s_erfl.c

/*
 * ====================================================
 * 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.
 * ====================================================
 */

/* Long double expansions are
  Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
  and are incorporated herein by permission of the author.  The author 
  reserves the right to distribute this material elsewhere under different
  copying permissions.  These modifications are distributed here under 
  the following terms:

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
    License as published by the Free Software Foundation; either
    version 2.1 of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this library; if not, write to the Free Software
    Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */

/* 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]
 *       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
 *       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)
 *       Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
 *
 *      3. For x in [1.25,1/0.35(~2.857143)],
 *    erfc(x) = (1/x)*exp(-x*x-0.5625+R1(z)/S1(z))
 *              z=1/x^2
 *    erf(x)  = 1 - erfc(x)
 *
 *      4. For x in [1/0.35,107]
 *    erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
 *                = 2.0 - (1/x)*exp(-x*x-0.5625+R2(z)/S2(z))
 *                             if -6.666<x<0
 *                = 2.0 - tiny            (if x <= -6.666)
 *              z=1/x^2
 *    erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6.666, else
 *    erf(x)  = sign(x)*(1.0 - tiny)
 *      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)
 *
 *      5. For inf > x >= 107
 *    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 long double
#else
static long double
#endif
tiny = 1e-4931L,
  half = 0.5L,
  one = 1.0L,
  two = 2.0L,
      /* c = (float)0.84506291151 */
  erx = 0.845062911510467529296875L,
/*
 * Coefficients for approximation to  erf on [0,0.84375]
 */
  /* 2/sqrt(pi) - 1 */
  efx = 1.2837916709551257389615890312154517168810E-1L,
  /* 8 * (2/sqrt(pi) - 1) */
  efx8 = 1.0270333367641005911692712249723613735048E0L,

  pp[6] = {
    1.122751350964552113068262337278335028553E6L,
    -2.808533301997696164408397079650699163276E6L,
    -3.314325479115357458197119660818768924100E5L,
    -6.848684465326256109712135497895525446398E4L,
    -2.657817695110739185591505062971929859314E3L,
    -1.655310302737837556654146291646499062882E2L,
  },

  qq[6] = {
    8.745588372054466262548908189000448124232E6L,
    3.746038264792471129367533128637019611485E6L,
    7.066358783162407559861156173539693900031E5L,
    7.448928604824620999413120955705448117056E4L,
    4.511583986730994111992253980546131408924E3L,
    1.368902937933296323345610240009071254014E2L,
    /* 1.000000000000000000000000000000000000000E0 */
  },

/*
 * Coefficients for approximation to  erf  in [0.84375,1.25]
 */
/* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x)
   -0.15625 <= x <= +.25
   Peak relative error 8.5e-22  */

  pa[8] = {
    -1.076952146179812072156734957705102256059E0L,
     1.884814957770385593365179835059971587220E2L,
    -5.339153975012804282890066622962070115606E1L,
     4.435910679869176625928504532109635632618E1L,
     1.683219516032328828278557309642929135179E1L,
    -2.360236618396952560064259585299045804293E0L,
     1.852230047861891953244413872297940938041E0L,
     9.394994446747752308256773044667843200719E-2L,
  },

  qa[7] =  {
    4.559263722294508998149925774781887811255E2L,
    3.289248982200800575749795055149780689738E2L,
    2.846070965875643009598627918383314457912E2L,
    1.398715859064535039433275722017479994465E2L,
    6.060190733759793706299079050985358190726E1L,
    2.078695677795422351040502569964299664233E1L,
    4.641271134150895940966798357442234498546E0L,
    /* 1.000000000000000000000000000000000000000E0 */
  },

/*
 * Coefficients for approximation to  erfc in [1.25,1/0.35]
 */
/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2))
   1/2.85711669921875 < 1/x < 1/1.25
   Peak relative error 3.1e-21  */

    ra[] = {
      1.363566591833846324191000679620738857234E-1L,
      1.018203167219873573808450274314658434507E1L,
      1.862359362334248675526472871224778045594E2L,
      1.411622588180721285284945138667933330348E3L,
      5.088538459741511988784440103218342840478E3L,
      8.928251553922176506858267311750789273656E3L,
      7.264436000148052545243018622742770549982E3L,
      2.387492459664548651671894725748959751119E3L,
      2.220916652813908085449221282808458466556E2L,
    },

    sa[] = {
      -1.382234625202480685182526402169222331847E1L,
      -3.315638835627950255832519203687435946482E2L,
      -2.949124863912936259747237164260785326692E3L,
      -1.246622099070875940506391433635999693661E4L,
      -2.673079795851665428695842853070996219632E4L,
      -2.880269786660559337358397106518918220991E4L,
      -1.450600228493968044773354186390390823713E4L,
      -2.874539731125893533960680525192064277816E3L,
      -1.402241261419067750237395034116942296027E2L,
      /* 1.000000000000000000000000000000000000000E0 */
    },
/*
 * Coefficients for approximation to  erfc in [1/.35,107]
 */
/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2))
   1/6.6666259765625 < 1/x < 1/2.85711669921875
   Peak relative error 4.2e-22  */
    rb[] = {
      -4.869587348270494309550558460786501252369E-5L,
      -4.030199390527997378549161722412466959403E-3L,
      -9.434425866377037610206443566288917589122E-2L,
      -9.319032754357658601200655161585539404155E-1L,
      -4.273788174307459947350256581445442062291E0L,
      -8.842289940696150508373541814064198259278E0L,
      -7.069215249419887403187988144752613025255E0L,
      -1.401228723639514787920274427443330704764E0L,
    },

    sb[] = {
      4.936254964107175160157544545879293019085E-3L,
      1.583457624037795744377163924895349412015E-1L,
      1.850647991850328356622940552450636420484E0L,
      9.927611557279019463768050710008450625415E0L,
      2.531667257649436709617165336779212114570E1L,
      2.869752886406743386458304052862814690045E1L,
      1.182059497870819562441683560749192539345E1L,
      /* 1.000000000000000000000000000000000000000E0 */
    },
/* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2))
   1/107 <= 1/x <= 1/6.6666259765625
   Peak relative error 1.1e-21  */
    rc[] = {
      -8.299617545269701963973537248996670806850E-5L,
      -6.243845685115818513578933902532056244108E-3L,
      -1.141667210620380223113693474478394397230E-1L,
      -7.521343797212024245375240432734425789409E-1L,
      -1.765321928311155824664963633786967602934E0L,
      -1.029403473103215800456761180695263439188E0L,
    },

    sc[] = {
      8.413244363014929493035952542677768808601E-3L,
      2.065114333816877479753334599639158060979E-1L,
      1.639064941530797583766364412782135680148E0L,
      4.936788463787115555582319302981666347450E0L,
      5.005177727208955487404729933261347679090E0L,
      /* 1.000000000000000000000000000000000000000E0 */
    };

#ifdef __STDC__
long double
__erfl (long double x)
#else
long double
__erfl (x)
     long double x;
#endif
{
  long double R, S, P, Q, s, y, z, r;
  int32_t ix, i;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;

  if (ix >= 0x7fff)
    {                   /* erf(nan)=nan */
      i = ((se & 0xffff) >> 15) << 1;
      return (long double) (1 - i) + one / x;   /* erf(+-inf)=+-1 */
    }

  ix = (ix << 16) | (i0 >> 16);
  if (ix < 0x3ffed800) /* |x|<0.84375 */
    {
      if (ix < 0x3fde8000) /* |x|<2**-33 */
      {
        if (ix < 0x00080000)
          return 0.125 * (8.0 * x + efx8 * x);  /*avoid underflow */
        return x + efx * x;
      }
      z = x * x;
      r = pp[0] + z * (pp[1]
          + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
      s = qq[0] + z * (qq[1]
        + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
      y = r / s;
      return x + x * y;
    }
  if (ix < 0x3fffa000) /* 1.25 */
    {                   /* 0.84375 <= |x| < 1.25 */
      s = fabsl (x) - one;
      P = pa[0] + s * (pa[1] + s * (pa[2]
        + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
      Q = qa[0] + s * (qa[1] + s * (qa[2]
        + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
      if ((se & 0x8000) == 0)
      return erx + P / Q;
      else
      return -erx - P / Q;
    }
  if (ix >= 0x4001d555) /* 6.6666259765625 */
    {                   /* inf>|x|>=6.666 */
      if ((se & 0x8000) == 0)
      return one - tiny;
      else
      return tiny - one;
    }
  x = fabsl (x);
  s = one / (x * x);
  if (ix < 0x4000b6db) /* 2.85711669921875 */
    {
      R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
          s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
      S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
          s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
    }
  else
    {                   /* |x| >= 1/0.35 */
      R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
         s * (rb[5] + s * (rb[6] + s * rb[7]))))));
      S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
         s * (sb[5] + s * (sb[6] + s))))));
    }
  z = x;
  GET_LDOUBLE_WORDS (i, i0, i1, z);
  i1 = 0;
  SET_LDOUBLE_WORDS (z, i, i0, i1);
  r =
    __ieee754_expl (-z * z - 0.5625) * __ieee754_expl ((z - x) * (z + x) +
                                         R / S);
  if ((se & 0x8000) == 0)
    return one - r / x;
  else
    return r / x - one;
}

weak_alias (__erfl, erfl)
#ifdef NO_LONG_DOUBLE
strong_alias (__erf, __erfl)
weak_alias (__erf, erfl)
#endif
#ifdef __STDC__
     long double
     __erfcl (long double x)
#else
     long double
     __erfcl (x)
     long double x;
#endif
{
  int32_t hx, ix;
  long double R, S, P, Q, s, y, z, r;
  u_int32_t se, i0, i1;

  GET_LDOUBLE_WORDS (se, i0, i1, x);
  ix = se & 0x7fff;
  if (ix >= 0x7fff)
    {                   /* erfc(nan)=nan */
      /* erfc(+-inf)=0,2 */
      return (long double) (((se & 0xffff) >> 15) << 1) + one / x;
    }

  ix = (ix << 16) | (i0 >> 16);
  if (ix < 0x3ffed800) /* |x|<0.84375 */
    {
      if (ix < 0x3fbe0000) /* |x|<2**-65 */
      return one - x;
      z = x * x;
      r = pp[0] + z * (pp[1]
          + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5]))));
      s = qq[0] + z * (qq[1]
        + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z)))));
      y = r / s;
      if (ix < 0x3ffd8000) /* x<1/4 */
      {
        return one - (x + x * y);
      }
      else
      {
        r = x * y;
        r += (x - half);
        return half - r;
      }
    }
  if (ix < 0x3fffa000) /* 1.25 */
    {                   /* 0.84375 <= |x| < 1.25 */
      s = fabsl (x) - one;
      P = pa[0] + s * (pa[1] + s * (pa[2]
        + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7]))))));
      Q = qa[0] + s * (qa[1] + s * (qa[2]
        + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s))))));
      if ((se & 0x8000) == 0)
      {
        z = one - erx;
        return z - P / Q;
      }
      else
      {
        z = erx + P / Q;
        return one + z;
      }
    }
  if (ix < 0x4005d600) /* 107 */
    {                   /* |x|<107 */
      x = fabsl (x);
      s = one / (x * x);
      if (ix < 0x4000b6db) /* 2.85711669921875 */
      {                 /* |x| < 1/.35 ~ 2.857143 */
        R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] +
              s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8])))))));
        S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] +
              s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s))))))));
      }
      else if (ix < 0x4001d555) /* 6.6666259765625 */
      {                 /* 6.666 > |x| >= 1/.35 ~ 2.857143 */
        R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] +
            s * (rb[5] + s * (rb[6] + s * rb[7]))))));
        S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] +
              s * (sb[5] + s * (sb[6] + s))))));
      }
      else
      {                 /* |x| >= 6.666 */
        if (se & 0x8000)
          return two - tiny;  /* x < -6.666 */

        R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] +
                                        s * (rc[4] + s * rc[5]))));
        S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] +
                                        s * (sc[4] + s))));
      }
      z = x;
      GET_LDOUBLE_WORDS (hx, i0, i1, z);
      i1 = 0;
      i0 &= 0xffffff00;
      SET_LDOUBLE_WORDS (z, hx, i0, i1);
      r = __ieee754_expl (-z * z - 0.5625) *
      __ieee754_expl ((z - x) * (z + x) + R / S);
      if ((se & 0x8000) == 0)
      return r / x;
      else
      return two - r / x;
    }
  else
    {
      if ((se & 0x8000) == 0)
      return tiny * tiny;
      else
      return two - tiny;
    }
}

weak_alias (__erfcl, erfcl)
#ifdef NO_LONG_DOUBLE
strong_alias (__erfc, __erfcl)
weak_alias (__erfc, erfcl)
#endif

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