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authormatz <matz@b2dd03c8-39d4-4d8f-98ff-823fe69b080e>2005-07-01 03:25:46 +0000
committermatz <matz@b2dd03c8-39d4-4d8f-98ff-823fe69b080e>2005-07-01 03:25:46 +0000
commit70db83a8c3e56a821fec46195216bf9cd78d9b6a (patch)
tree75d83759150203309452cf1d9dafe014bcbe8941 /missing/erf.c
parent9d4392ee6cf09fd143d6295d15b1f4de2e2b2930 (diff)
* missing/crypt.c: replaced with 4.4BSD version.
* missing/erf.c: ditto. * missing/vsnprintf.c: removed the third provision from the old BSD license. [ruby-core:05177] git-svn-id: svn+ssh://ci.ruby-lang.org/ruby/branches/ruby_1_8@8687 b2dd03c8-39d4-4d8f-98ff-823fe69b080e
Diffstat (limited to 'missing/erf.c')
-rw-r--r--missing/erf.c457
1 files changed, 379 insertions, 78 deletions
diff --git a/missing/erf.c b/missing/erf.c
index e30a90051e5..541972667a3 100644
--- a/missing/erf.c
+++ b/missing/erf.c
@@ -1,91 +1,392 @@
-/* erf.c
-reference - Haruhiko Okumura: C-gengo niyoru saishin algorithm jiten
- (New Algorithm handbook in C language) (Gijyutsu hyouron
- sha, Tokyo, 1991) p.227 [in Japanese] */
-#include <stdio.h>
-#include <math.h>
+/*-
+ * 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. 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.
+ */
-#ifdef _WIN32
-# include <float.h>
-# if !defined __MINGW32__ || defined __NO_ISOCEXT
-# ifndef isnan
-# define isnan(x) _isnan(x)
-# endif
-# ifndef isinf
-# define isinf(x) (!_finite(x) && !_isnan(x))
-# endif
-# ifndef finite
-# define finite(x) _finite(x)
-# endif
-# endif
-#endif
-
-static double q_gamma(double, double, double);
+#ifndef lint
+static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93";
+#endif /* not lint */
-/* Incomplete gamma function
- 1 / Gamma(a) * Int_0^x exp(-t) t^(a-1) dt */
-static double p_gamma(a, x, loggamma_a)
- double a, x, loggamma_a;
-{
- int k;
- double result, term, previous;
+/* 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.
+ */
- if (x >= 1 + a) return 1 - q_gamma(a, x, loggamma_a);
- if (x == 0) return 0;
- result = term = exp(a * log(x) - x - loggamma_a) / a;
- for (k = 1; k < 1000; k++) {
- term *= x / (a + k);
- previous = result; result += term;
- if (result == previous) return result;
- }
- fprintf(stderr, "erf.c:%d:p_gamma() could not converge.", __LINE__);
- return result;
-}
+/* 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
+ */
-/* Incomplete gamma function
- 1 / Gamma(a) * Int_x^inf exp(-t) t^(a-1) dt */
-static double q_gamma(a, x, loggamma_a)
- double a, x, loggamma_a;
-{
- int k;
- double result, w, temp, previous;
- double la = 1, lb = 1 + x - a; /* Laguerre polynomial */
+#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
- if (x < 1 + a) return 1 - p_gamma(a, x, loggamma_a);
- w = exp(a * log(x) - x - loggamma_a);
- result = w / lb;
- for (k = 2; k < 1000; k++) {
- temp = ((k - 1 - a) * (lb - la) + (k + x) * lb) / k;
- la = lb; lb = temp;
- w *= (k - 1 - a) / k;
- temp = w / (la * lb);
- previous = result; result += temp;
- if (result == previous) return result;
- }
- fprintf(stderr, "erf.c:%d:q_gamma() could not converge.", __LINE__);
- return result;
-}
+#ifdef _IEEE_LIBM
+/*
+ * redefining "___function" to "function" in _IEEE_LIBM mode
+ */
+#include "ieee_libm.h"
+#endif
-#define LOG_PI_OVER_2 0.572364942924700087071713675675 /* log_e(PI)/2 */
+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 x;
{
- if (!finite(x)) {
- if (isnan(x)) return x; /* erf(NaN) = NaN */
- return (x>0 ? 1.0 : -1.0); /* erf(+-inf) = +-1.0 */
- }
- if (x >= 0) return p_gamma(0.5, x * x, LOG_PI_OVER_2);
- else return - p_gamma(0.5, x * x, LOG_PI_OVER_2);
+ 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 erfc(x)
+ double x;
{
- if (!finite(x)) {
- if (isnan(x)) return x; /* erfc(NaN) = NaN */
- return (x>0 ? 0.0 : 2.0); /* erfc(+-inf) = 0.0, 2.0 */
- }
- if (x >= 0) return q_gamma(0.5, x * x, LOG_PI_OVER_2);
- else return 1 + p_gamma(0.5, x * x, LOG_PI_OVER_2);
+ 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;
}