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OpenLibm/ld80/s_erfl.c
Viral B. Shah 9ecf223fc1 Get the ld80 routines from OpenBSD to build on mac and linux. 
Bump version number and SO major version, since we have
introduced new long double APIs.
2014-12-04 23:56:11 +05:30

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/*
* ====================================================
* 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.
* ====================================================
*/
/*
* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
*
* Permission to use, copy, modify, and distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
*/
/* 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 <openlibm.h>
#include "math_private.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,
/* 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 */
};
long double
erfl(long double x)
{
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 =
expl (-z * z - 0.5625) * expl ((z - x) * (z + x) + R / S);
if ((se & 0x8000) == 0)
return one - r / x;
else
return r / x - one;
}
long double
erfcl(long double x)
{
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 = expl (-z * z - 0.5625) *
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;
}
}
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