/*
* ====================================================
* 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, see
<http://www.gnu.org/licenses/>. */
/* 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 <errno.h>
#include <float.h>
#include <math.h>
#include <math_private.h>
#include <math-underflow.h>
#include <libm-alias-ldouble.h>
static const long double
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,
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 */
};
long double
__erfl (long double x)
{
long double R, S, P, Q, s, y, z, r;
int32_t ix, i;
uint32_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)
{
/* Avoid spurious underflow. */
long double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
math_check_force_underflow (ret);
return ret;
}
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;
}
libm_alias_ldouble (__erf, erf)
long double
__erfcl (long double x)
{
int32_t hx, ix;
long double R, S, P, Q, s, y, z, r;
uint32_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)
{
long double ret = r / x;
if (ret == 0)
__set_errno (ERANGE);
return ret;
}
else
return two - r / x;
}
else
{
if ((se & 0x8000) == 0)
{
__set_errno (ERANGE);
return tiny * tiny;
}
else
return two - tiny;
}
}
libm_alias_ldouble (__erfc, erfc)