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-/*-
- * 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.
- */
-
-#include <sys/cdefs.h>
-__FBSDID("$FreeBSD$");
-
-#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;
-}
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