4.4BSD/usr/src/lib/libm/common_source/erf.c
/*-
* Copyright (c) 1992, 1993
* The Regents of the University of California. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
* 3. All advertising materials mentioning features or use of this software
* must display the following acknowledgement:
* This product includes software developed by the University of
* California, Berkeley and its contributors.
* 4. Neither the name of the University nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#ifndef lint
static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93";
#endif /* not lint */
/* Modified Nov 30, 1992 P. McILROY:
* Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp)
* Replaced even+odd with direct calculation for x < .84375,
* to avoid destructive cancellation.
*
* Performance of erfc(x):
* In 300000 trials in the range [.83, .84375] the
* maximum observed error was 3.6ulp.
*
* In [.84735,1.25] the maximum observed error was <2.5ulp in
* 100000 runs in the range [1.2, 1.25].
*
* In [1.25,26] (Not including subnormal results)
* the error is < 1.7ulp.
*/
/* double erf(double x)
* double erfc(double x)
* x
* 2 |\
* erf(x) = --------- | exp(-t*t)dt
* sqrt(pi) \|
* 0
*
* erfc(x) = 1-erf(x)
*
* Method:
* 1. Reduce x to |x| by erf(-x) = -erf(x)
* 2. For x in [0, 0.84375]
* erf(x) = x + x*P(x^2)
* erfc(x) = 1 - erf(x) if x<=0.25
* = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375]
* where
* 2 2 4 20
* P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x )
* is an approximation to (erf(x)-x)/x with precision
*
* -56.45
* | P - (erf(x)-x)/x | <= 2
*
*
* 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 fixed
* 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.
*
* 3. For x in [0.84375,1.25], let s = x - 1, and
* c = 0.84506291151 rounded to single (24 bits)
* erf(x) = c + P1(s)/Q1(s)
* erfc(x) = (1-c) - P1(s)/Q1(s)
* |P1/Q1 - (erf(x)-c)| <= 2**-59.06
* 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)
* That is, we use rational approximation to approximate
* erf(1+s) - (c = (single)0.84506291151)
* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
* where
* P1(s) = degree 6 poly in s
* Q1(s) = degree 6 poly in s
*
* 4. For x in [1.25, 2]; [2, 4]
* erf(x) = 1.0 - tiny
* erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z))
*
* Where z = 1/(x*x), R is degree 9, and S is degree 3;
*
* 5. For x in [4,28]
* erf(x) = 1.0 - tiny
* erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z))
*
* Where P is degree 14 polynomial in 1/(x*x).
*
* Notes:
* Here 4 and 5 make use of the asymptotic series
* exp(-x*x)
* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) );
* x*sqrt(pi)
*
* where for z = 1/(x*x)
* P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...))))
*
* Thus we use rational approximation to approximate
* erfc*x*exp(x*x) ~ 1/sqrt(pi);
*
* The error bound for the target function, G(z) for
* the interval
* [4, 28]:
* |eps + 1/(z)P(z) - G(z)| < 2**(-56.61)
* for [2, 4]:
* |R(z)/S(z) - G(z)| < 2**(-58.24)
* for [1.25, 2]:
* |R(z)/S(z) - G(z)| < 2**(-58.12)
*
* 6. For inf > x >= 28
* erf(x) = 1 - tiny (raise inexact)
* erfc(x) = tiny*tiny (raise underflow)
*
* 7. Special cases:
* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
* erfc/erf(NaN) is NaN
*/
#if defined(vax) || defined(tahoe)
#define _IEEE 0
#define TRUNC(x) (double) (float) (x)
#else
#define _IEEE 1
#define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000
#define infnan(x) 0.0
#endif
#ifdef _IEEE_LIBM
/*
* redefining "___function" to "function" in _IEEE_LIBM mode
*/
#include "ieee_libm.h"
#endif
static double
tiny = 1e-300,
half = 0.5,
one = 1.0,
two = 2.0,
c = 8.45062911510467529297e-01, /* (float)0.84506291151 */
/*
* Coefficients for approximation to erf in [0,0.84375]
*/
p0t8 = 1.02703333676410051049867154944018394163280,
p0 = 1.283791670955125638123339436800229927041e-0001,
p1 = -3.761263890318340796574473028946097022260e-0001,
p2 = 1.128379167093567004871858633779992337238e-0001,
p3 = -2.686617064084433642889526516177508374437e-0002,
p4 = 5.223977576966219409445780927846432273191e-0003,
p5 = -8.548323822001639515038738961618255438422e-0004,
p6 = 1.205520092530505090384383082516403772317e-0004,
p7 = -1.492214100762529635365672665955239554276e-0005,
p8 = 1.640186161764254363152286358441771740838e-0006,
p9 = -1.571599331700515057841960987689515895479e-0007,
p10= 1.073087585213621540635426191486561494058e-0008;
/*
* Coefficients for approximation to erf in [0.84375,1.25]
*/
static double
pa0 = -2.362118560752659485957248365514511540287e-0003,
pa1 = 4.148561186837483359654781492060070469522e-0001,
pa2 = -3.722078760357013107593507594535478633044e-0001,
pa3 = 3.183466199011617316853636418691420262160e-0001,
pa4 = -1.108946942823966771253985510891237782544e-0001,
pa5 = 3.547830432561823343969797140537411825179e-0002,
pa6 = -2.166375594868790886906539848893221184820e-0003,
qa1 = 1.064208804008442270765369280952419863524e-0001,
qa2 = 5.403979177021710663441167681878575087235e-0001,
qa3 = 7.182865441419627066207655332170665812023e-0002,
qa4 = 1.261712198087616469108438860983447773726e-0001,
qa5 = 1.363708391202905087876983523620537833157e-0002,
qa6 = 1.198449984679910764099772682882189711364e-0002;
/*
* log(sqrt(pi)) for large x expansions.
* The tail (lsqrtPI_lo) is included in the rational
* approximations.
*/
static double
lsqrtPI_hi = .5723649429247000819387380943226;
/*
* lsqrtPI_lo = .000000000000000005132975581353913;
*
* Coefficients for approximation to erfc in [2, 4]
*/
static double
rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */
rb1 = 2.15592846101742183841910806188e-008,
rb2 = 6.24998557732436510470108714799e-001,
rb3 = 8.24849222231141787631258921465e+000,
rb4 = 2.63974967372233173534823436057e+001,
rb5 = 9.86383092541570505318304640241e+000,
rb6 = -7.28024154841991322228977878694e+000,
rb7 = 5.96303287280680116566600190708e+000,
rb8 = -4.40070358507372993983608466806e+000,
rb9 = 2.39923700182518073731330332521e+000,
rb10 = -6.89257464785841156285073338950e-001,
sb1 = 1.56641558965626774835300238919e+001,
sb2 = 7.20522741000949622502957936376e+001,
sb3 = 9.60121069770492994166488642804e+001;
/*
* Coefficients for approximation to erfc in [1.25, 2]
*/
static double
rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */
rc1 = 1.28735722546372485255126993930e-005,
rc2 = 6.24664954087883916855616917019e-001,
rc3 = 4.69798884785807402408863708843e+000,
rc4 = 7.61618295853929705430118701770e+000,
rc5 = 9.15640208659364240872946538730e-001,
rc6 = -3.59753040425048631334448145935e-001,
rc7 = 1.42862267989304403403849619281e-001,
rc8 = -4.74392758811439801958087514322e-002,
rc9 = 1.09964787987580810135757047874e-002,
rc10 = -1.28856240494889325194638463046e-003,
sc1 = 9.97395106984001955652274773456e+000,
sc2 = 2.80952153365721279953959310660e+001,
sc3 = 2.19826478142545234106819407316e+001;
/*
* Coefficients for approximation to erfc in [4,28]
*/
static double
rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */
rd1 = -4.99999999999640086151350330820e-001,
rd2 = 6.24999999772906433825880867516e-001,
rd3 = -1.54166659428052432723177389562e+000,
rd4 = 5.51561147405411844601985649206e+000,
rd5 = -2.55046307982949826964613748714e+001,
rd6 = 1.43631424382843846387913799845e+002,
rd7 = -9.45789244999420134263345971704e+002,
rd8 = 6.94834146607051206956384703517e+003,
rd9 = -5.27176414235983393155038356781e+004,
rd10 = 3.68530281128672766499221324921e+005,
rd11 = -2.06466642800404317677021026611e+006,
rd12 = 7.78293889471135381609201431274e+006,
rd13 = -1.42821001129434127360582351685e+007;
double erf(x)
double x;
{
double R,S,P,Q,ax,s,y,z,r,fabs(),exp();
if(!finite(x)) { /* erf(nan)=nan */
if (isnan(x))
return(x);
return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */
}
if ((ax = x) < 0)
ax = - ax;
if (ax < .84375) {
if (ax < 3.7e-09) {
if (ax < 1.0e-308)
return 0.125*(8.0*x+p0t8*x); /*avoid underflow */
return x + p0*x;
}
y = x*x;
r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
return x + x*(p0+r);
}
if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */
s = fabs(x)-one;
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
if (x>=0)
return (c + P/Q);
else
return (-c - P/Q);
}
if (ax >= 6.0) { /* inf>|x|>=6 */
if (x >= 0.0)
return (one-tiny);
else
return (tiny-one);
}
/* 1.25 <= |x| < 6 */
z = -ax*ax;
s = -one/z;
if (ax < 2.0) {
R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
S = one+s*(sc1+s*(sc2+s*sc3));
} else {
R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
S = one+s*(sb1+s*(sb2+s*sb3));
}
y = (R/S -.5*s) - lsqrtPI_hi;
z += y;
z = exp(z)/ax;
if (x >= 0)
return (one-z);
else
return (z-one);
}
double erfc(x)
double x;
{
double R,S,P,Q,s,ax,y,z,r,fabs(),__exp__D();
if (!finite(x)) {
if (isnan(x)) /* erfc(NaN) = NaN */
return(x);
else if (x > 0) /* erfc(+-inf)=0,2 */
return 0.0;
else
return 2.0;
}
if ((ax = x) < 0)
ax = -ax;
if (ax < .84375) { /* |x|<0.84375 */
if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */
return one-x;
y = x*x;
r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
if (ax < .0625) { /* |x|<2**-4 */
return (one-(x+x*(p0+r)));
} else {
r = x*(p0+r);
r += (x-half);
return (half - r);
}
}
if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */
s = ax-one;
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
if (x>=0) {
z = one-c; return z - P/Q;
} else {
z = c+P/Q; return one+z;
}
}
if (ax >= 28) /* Out of range */
if (x>0)
return (tiny*tiny);
else
return (two-tiny);
z = ax;
TRUNC(z);
y = z - ax; y *= (ax+z);
z *= -z; /* Here z + y = -x^2 */
s = one/(-z-y); /* 1/(x*x) */
if (ax >= 4) { /* 6 <= ax */
R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+
s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10
+s*(rd11+s*(rd12+s*rd13))))))))))));
y += rd0;
} else if (ax >= 2) {
R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
S = one+s*(sb1+s*(sb2+s*sb3));
y += R/S;
R = -.5*s;
} else {
R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
S = one+s*(sc1+s*(sc2+s*sc3));
y += R/S;
R = -.5*s;
}
/* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */
s = ((R + y) - lsqrtPI_hi) + z;
y = (((z-s) - lsqrtPI_hi) + R) + y;
r = __exp__D(s, y)/x;
if (x>0)
return r;
else
return two-r;
}