354 lines
12 KiB
C
354 lines
12 KiB
C
/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_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.
|
|
* ====================================================
|
|
*/
|
|
/*
|
|
* 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 "libc/math/libm.h"
|
|
|
|
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
|
|
long double erfl(long double x)
|
|
{
|
|
return erf(x);
|
|
}
|
|
long double erfcl(long double x)
|
|
{
|
|
return erfc(x);
|
|
}
|
|
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
|
|
static const long double
|
|
erx = 0.845062911510467529296875L,
|
|
|
|
/*
|
|
* Coefficients for approximation to erf on [0,0.84375]
|
|
*/
|
|
/* 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 */
|
|
};
|
|
|
|
static long double erfc1(long double x)
|
|
{
|
|
long double s,P,Q;
|
|
|
|
s = fabsl(x) - 1;
|
|
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))))));
|
|
return 1 - erx - P / Q;
|
|
}
|
|
|
|
static long double erfc2(uint32_t ix, long double x)
|
|
{
|
|
union ldshape u;
|
|
long double s,z,R,S;
|
|
|
|
if (ix < 0x3fffa000) /* 0.84375 <= |x| < 1.25 */
|
|
return erfc1(x);
|
|
|
|
x = fabsl(x);
|
|
s = 1 / (x * x);
|
|
if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.857 ~ 1/.35 */
|
|
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) { /* 2.857 <= |x| < 6.6666259765625 */
|
|
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 { /* 6.666 <= |x| < 107 (erfc only) */
|
|
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))));
|
|
}
|
|
u.f = x;
|
|
u.i.m &= -1ULL << 40;
|
|
z = u.f;
|
|
return expl(-z*z - 0.5625) * expl((z - x) * (z + x) + R / S) / x;
|
|
}
|
|
|
|
long double erfl(long double x)
|
|
{
|
|
long double r, s, z, y;
|
|
union ldshape u = {x};
|
|
uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
|
|
int sign = u.i.se >> 15;
|
|
|
|
if (ix >= 0x7fff0000)
|
|
/* erf(nan)=nan, erf(+-inf)=+-1 */
|
|
return 1 - 2*sign + 1/x;
|
|
if (ix < 0x3ffed800) { /* |x| < 0.84375 */
|
|
if (ix < 0x3fde8000) { /* |x| < 2**-33 */
|
|
return 0.125 * (8 * x + efx8 * x); /* avoid underflow */
|
|
}
|
|
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 < 0x4001d555) /* |x| < 6.6666259765625 */
|
|
y = 1 - erfc2(ix,x);
|
|
else
|
|
y = 1 - 0x1p-16382L;
|
|
return sign ? -y : y;
|
|
}
|
|
|
|
long double erfcl(long double x)
|
|
{
|
|
long double r, s, z, y;
|
|
union ldshape u = {x};
|
|
uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48;
|
|
int sign = u.i.se >> 15;
|
|
|
|
if (ix >= 0x7fff0000)
|
|
/* erfc(nan) = nan, erfc(+-inf) = 0,2 */
|
|
return 2*sign + 1/x;
|
|
if (ix < 0x3ffed800) { /* |x| < 0.84375 */
|
|
if (ix < 0x3fbe0000) /* |x| < 2**-65 */
|
|
return 1.0 - 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 1.0 - (x + x * y);
|
|
return 0.5 - (x - 0.5 + x * y);
|
|
}
|
|
if (ix < 0x4005d600) /* |x| < 107 */
|
|
return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
|
|
y = 0x1p-16382L;
|
|
return sign ? 2 - y : y*y;
|
|
}
|
|
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
|
|
// TODO: broken implementation to make things compile
|
|
long double erfl(long double x)
|
|
{
|
|
return erf(x);
|
|
}
|
|
long double erfcl(long double x)
|
|
{
|
|
return erfc(x);
|
|
}
|
|
#endif
|