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author | bde <bde@FreeBSD.org> | 2002-03-21 00:42:48 +0000 |
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committer | bde <bde@FreeBSD.org> | 2002-03-21 00:42:48 +0000 |
commit | be2cf84b24905468b296d0b27d7c0459acd4dec8 (patch) | |
tree | 57bd79659e7f6862429527509385c95663b5c33b /lib/libm/common_source/erf.c | |
parent | e1ec4d77dcf9547ad4c02a371fcae1d9f176c338 (diff) | |
download | FreeBSD-src-be2cf84b24905468b296d0b27d7c0459acd4dec8.zip FreeBSD-src-be2cf84b24905468b296d0b27d7c0459acd4dec8.tar.gz |
Removed all files in libm except README-FREEBSD and files needed to
implement tgamma().
Diffstat (limited to 'lib/libm/common_source/erf.c')
-rw-r--r-- | lib/libm/common_source/erf.c | 399 |
1 files changed, 0 insertions, 399 deletions
diff --git a/lib/libm/common_source/erf.c b/lib/libm/common_source/erf.c deleted file mode 100644 index 5b7b725..0000000 --- a/lib/libm/common_source/erf.c +++ /dev/null @@ -1,399 +0,0 @@ -/*- - * 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; -} |