/* $OpenBSD: n_erf.c,v 1.6 2008/06/21 08:26:19 martynas Exp $ */ /* $NetBSD: n_erf.c,v 1.1 1995/10/10 23:36:43 ragge Exp $ */ /*- * 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. */ #ifndef lint static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93"; #endif /* not lint */ #include "math.h" #include "mathimpl.h" /* 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__) #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 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(double x) { double R, S, P, Q, ax, s, y, z, r; 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(double x) { double R, S, P, Q, s, ax, y, z, r; 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; }