s_erfl.cpp
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CPP
458 行
/* See the import.pl script for potential modifications */
/*
* ====================================================
* 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 2.1l 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, 0.84375l]
* erf(x) = x + x*R(x^2)
* erfc(x) = 1 - erf(x) if x in [-.84375,0.25l]
* = 0.5l + ((0.5l-x)-x*R) if x in [0.25l,0.84375l]
* 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) = 1.128379167095512573896158903121545171688l
* is close to one. The interval is chosen because the fix
* point of erf(x) is near 0.6174l (i.e., erf(x)=x when x is
* near 0.6174l), and by some experiment, 0.84375l is chosen to
* guarantee the error is less than one ulp for erf.
*
* 2. For |x| in [0.84375l,1.25l], let s = |x| - 1, and
* c = 0.84506291151l 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)
* = 0.845l.. + P1(s)/Q1(s)
* Note that |P1/Q1|< 0.078l for x in [0.84375l,1.25l]
*
* 3. For x in [1.25l,1/0.35l(~2.857143l)],
* erfc(x) = (1/x)*exp(-x*x-0.5625l+R1(z)/S1(z))
* z=1/x^2
* erf(x) = 1 - erfc(x)
*
* 4. For x in [1/0.35l,107]
* erfc(x) = (1/x)*exp(-x*x-0.5625l+R2/S2) if x > 0
* = 2.0l - (1/x)*exp(-x*x-0.5625l+R2(z)/S2(z))
* if -6.666l<x<0
* = 2.0l - tiny (if x <= -6.666l)
* z=1/x^2
* erf(x) = sign(x)*(1.0l - erfc(x)) if x < 6.666l, else
* erf(x) = sign(x)*(1.0l - tiny)
* Note1:
* To compute exp(-x*x-0.5625l+R/S), let s be a single
* precision number and s := x; then
* -x*x = -s*s + (s-x)*(s+x)
* exp(-x*x-0.5626l+R/S) =
* exp(-s*s-0.5625l)*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"
namespace streflop_libm {
#ifdef __STDC__
static const Extended
#else
static Extended
#endif
tiny = 1e-4931L,
half = 0.5l,
one = 1.0l,
two = 2.0l,
/* c = (Simple)0.84506291151l */
erx = 0.845062911510467529296875l,
/*
* Coefficients for approximation to erf on [0,0.84375l]
*/
/* 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,
/* 1.000000000000000000000000000000000000000E0l */
},
/*
* Coefficients for approximation to erf in [0.84375l,1.25l]
*/
/* erf(x+1) = 0.845062911510467529296875l + pa(x)/qa(x)
-0.15625l <= x <= +.25
Peak relative error 8.5e-22l */
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,
/* 1.000000000000000000000000000000000000000E0l */
},
/*
* Coefficients for approximation to erfc in [1.25l,1/0.35l]
*/
/* erfc(1/x) = x exp (-1/x^2 - 0.5625l + ra(x^2)/sa(x^2))
1/2.85711669921875l < 1/x < 1/1.25l
Peak relative error 3.1e-21l */
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,
/* 1.000000000000000000000000000000000000000E0l */
},
/*
* Coefficients for approximation to erfc in [1/.35,107]
*/
/* erfc(1/x) = x exp (-1/x^2 - 0.5625l + rb(x^2)/sb(x^2))
1/6.6666259765625l < 1/x < 1/2.85711669921875l
Peak relative error 4.2e-22l */
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,
/* 1.000000000000000000000000000000000000000E0l */
},
/* erfc(1/x) = x exp (-1/x^2 - 0.5625l + rc(x^2)/sc(x^2))
1/107 <= 1/x <= 1/6.6666259765625l
Peak relative error 1.1e-21l */
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,
/* 1.000000000000000000000000000000000000000E0l */
};
#ifdef __STDC__
Extended
__erfl (Extended x)
#else
Extended
__erfl (x)
Extended x;
#endif
{
Extended 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 (Extended) (1 - i) + one / x; /* erf(+-inf)=+-1 */
}
ix = (ix << 16) | (i0 >> 16);
if (ix < 0x3ffed800) /* |x|<0.84375l */
{
if (ix < 0x3fde8000) /* |x|<2**-33 */
{
if (ix < 0x00080000)
return 0.125l * (8.0l * 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) /* 1.25l */
{ /* 0.84375l <= |x| < 1.25l */
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) /* 6.6666259765625l */
{ /* inf>|x|>=6.666l */
if ((se & 0x8000) == 0)
return one - tiny;
else
return tiny - one;
}
x = fabsl (x);
s = one / (x * x);
if (ix < 0x4000b6db) /* 2.85711669921875l */
{
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| >= 1/0.35l */
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 - 0.5625l) * __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 __STDC__
Extended
__erfcl (Extended x)
#else
Extended
__erfcl (x)
Extended x;
#endif
{
int32_t hx, ix;
Extended 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 (Extended) (((se & 0xffff) >> 15) << 1) + one / x;
}
ix = (ix << 16) | (i0 >> 16);
if (ix < 0x3ffed800) /* |x|<0.84375l */
{
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) /* 1.25l */
{ /* 0.84375l <= |x| < 1.25l */
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) /* 2.85711669921875l */
{ /* |x| < 1/.35 ~ 2.857143l */
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) /* 6.6666259765625l */
{ /* 6.666l > |x| >= 1/.35 ~ 2.857143l */
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| >= 6.666l */
if (se & 0x8000)
return two - tiny; /* x < -6.666l */
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 - 0.5625l) *
__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)
}
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