399 lines
12 KiB
C
399 lines
12 KiB
C
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/* $NetBSD: n_erf.c,v 1.7 2005/05/03 04:18:32 matt Exp $ */
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/*-
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* Copyright (c) 1992, 1993
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* The Regents of the University of California. All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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* 3. Neither the name of the University nor the names of its contributors
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* may be used to endorse or promote products derived from this software
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* without specific prior written permission.
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*
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* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
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* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
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* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
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* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
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* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
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* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
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* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
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* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
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* SUCH DAMAGE.
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*/
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#ifndef lint
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#if 0
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static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93";
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#endif
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#endif /* not lint */
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#include "mathimpl.h"
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/* Modified Nov 30, 1992 P. McILROY:
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* Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp)
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* Replaced even+odd with direct calculation for x < .84375,
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* to avoid destructive cancellation.
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*
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* Performance of erfc(x):
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* In 300000 trials in the range [.83, .84375] the
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* maximum observed error was 3.6ulp.
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*
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* In [.84735,1.25] the maximum observed error was <2.5ulp in
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* 100000 runs in the range [1.2, 1.25].
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*
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* In [1.25,26] (Not including subnormal results)
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* the error is < 1.7ulp.
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*/
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/* double erf(double x)
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* double erfc(double x)
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* x
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* 2 |\
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* erf(x) = --------- | exp(-t*t)dt
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* sqrt(pi) \|
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* 0
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*
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* erfc(x) = 1-erf(x)
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*
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* Method:
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* 1. Reduce x to |x| by erf(-x) = -erf(x)
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* 2. For x in [0, 0.84375]
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* erf(x) = x + x*P(x^2)
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* erfc(x) = 1 - erf(x) if x<=0.25
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* = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375]
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* where
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* 2 2 4 20
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* P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x )
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* is an approximation to (erf(x)-x)/x with precision
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*
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* -56.45
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* | P - (erf(x)-x)/x | <= 2
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*
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*
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* Remark. The formula is derived by noting
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* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
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* and that
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* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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* is close to one. The interval is chosen because the fixed
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* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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* near 0.6174), and by some experiment, 0.84375 is chosen to
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* guarantee the error is less than one ulp for erf.
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*
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* 3. For x in [0.84375,1.25], let s = x - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(x) = c + P1(s)/Q1(s)
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* erfc(x) = (1-c) - P1(s)/Q1(s)
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* |P1/Q1 - (erf(x)-c)| <= 2**-59.06
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* Remark: here we use the taylor series expansion at x=1.
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* erf(1+s) = erf(1) + s*Poly(s)
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* = 0.845.. + P1(s)/Q1(s)
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* That is, we use rational approximation to approximate
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* erf(1+s) - (c = (single)0.84506291151)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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* where
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* P1(s) = degree 6 poly in s
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* Q1(s) = degree 6 poly in s
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*
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* 4. For x in [1.25, 2]; [2, 4]
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* erf(x) = 1.0 - tiny
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* erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z))
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*
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* Where z = 1/(x*x), R is degree 9, and S is degree 3;
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*
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* 5. For x in [4,28]
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* erf(x) = 1.0 - tiny
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* erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z))
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*
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* Where P is degree 14 polynomial in 1/(x*x).
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*
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* Notes:
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* Here 4 and 5 make use of the asymptotic series
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* exp(-x*x)
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* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) );
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* x*sqrt(pi)
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*
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* where for z = 1/(x*x)
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* P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...))))
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*
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* Thus we use rational approximation to approximate
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* erfc*x*exp(x*x) ~ 1/sqrt(pi);
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*
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* The error bound for the target function, G(z) for
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* the interval
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* [4, 28]:
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* |eps + 1/(z)P(z) - G(z)| < 2**(-56.61)
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* for [2, 4]:
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* |R(z)/S(z) - G(z)| < 2**(-58.24)
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* for [1.25, 2]:
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* |R(z)/S(z) - G(z)| < 2**(-58.12)
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*
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* 6. For inf > x >= 28
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* erf(x) = 1 - tiny (raise inexact)
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* erfc(x) = tiny*tiny (raise underflow)
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*
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* 7. Special cases:
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* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
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* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
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* erfc/erf(NaN) is NaN
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*/
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#if defined(__vax__) || defined(tahoe)
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#define _IEEE 0
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#define TRUNC(x) (x) = (float)(x)
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#else
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#define _IEEE 1
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#define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000
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#define infnan(x) 0.0
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#endif
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#ifdef _IEEE_LIBM
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/*
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* redefining "___function" to "function" in _IEEE_LIBM mode
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*/
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#include "ieee_libm.h"
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#endif
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static const double
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tiny = 1e-300,
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half = 0.5,
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one = 1.0,
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two = 2.0,
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c = 8.45062911510467529297e-01, /* (float)0.84506291151 */
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/*
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* Coefficients for approximation to erf in [0,0.84375]
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*/
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p0t8 = 1.02703333676410051049867154944018394163280,
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p0 = 1.283791670955125638123339436800229927041e-0001,
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p1 = -3.761263890318340796574473028946097022260e-0001,
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p2 = 1.128379167093567004871858633779992337238e-0001,
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p3 = -2.686617064084433642889526516177508374437e-0002,
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p4 = 5.223977576966219409445780927846432273191e-0003,
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p5 = -8.548323822001639515038738961618255438422e-0004,
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p6 = 1.205520092530505090384383082516403772317e-0004,
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p7 = -1.492214100762529635365672665955239554276e-0005,
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p8 = 1.640186161764254363152286358441771740838e-0006,
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p9 = -1.571599331700515057841960987689515895479e-0007,
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p10= 1.073087585213621540635426191486561494058e-0008;
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/*
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* Coefficients for approximation to erf in [0.84375,1.25]
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*/
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static const double
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pa0 = -2.362118560752659485957248365514511540287e-0003,
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pa1 = 4.148561186837483359654781492060070469522e-0001,
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pa2 = -3.722078760357013107593507594535478633044e-0001,
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pa3 = 3.183466199011617316853636418691420262160e-0001,
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pa4 = -1.108946942823966771253985510891237782544e-0001,
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pa5 = 3.547830432561823343969797140537411825179e-0002,
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pa6 = -2.166375594868790886906539848893221184820e-0003,
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qa1 = 1.064208804008442270765369280952419863524e-0001,
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qa2 = 5.403979177021710663441167681878575087235e-0001,
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qa3 = 7.182865441419627066207655332170665812023e-0002,
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qa4 = 1.261712198087616469108438860983447773726e-0001,
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qa5 = 1.363708391202905087876983523620537833157e-0002,
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qa6 = 1.198449984679910764099772682882189711364e-0002;
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/*
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* log(sqrt(pi)) for large x expansions.
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* The tail (lsqrtPI_lo) is included in the rational
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* approximations.
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*/
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static const double
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lsqrtPI_hi = .5723649429247000819387380943226;
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/*
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* lsqrtPI_lo = .000000000000000005132975581353913;
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*
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* Coefficients for approximation to erfc in [2, 4]
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*/
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static const double
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rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */
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rb1 = 2.15592846101742183841910806188e-008,
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rb2 = 6.24998557732436510470108714799e-001,
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rb3 = 8.24849222231141787631258921465e+000,
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rb4 = 2.63974967372233173534823436057e+001,
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rb5 = 9.86383092541570505318304640241e+000,
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rb6 = -7.28024154841991322228977878694e+000,
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rb7 = 5.96303287280680116566600190708e+000,
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rb8 = -4.40070358507372993983608466806e+000,
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rb9 = 2.39923700182518073731330332521e+000,
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rb10 = -6.89257464785841156285073338950e-001,
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sb1 = 1.56641558965626774835300238919e+001,
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sb2 = 7.20522741000949622502957936376e+001,
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sb3 = 9.60121069770492994166488642804e+001;
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/*
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* Coefficients for approximation to erfc in [1.25, 2]
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*/
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static const double
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rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */
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rc1 = 1.28735722546372485255126993930e-005,
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rc2 = 6.24664954087883916855616917019e-001,
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rc3 = 4.69798884785807402408863708843e+000,
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rc4 = 7.61618295853929705430118701770e+000,
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rc5 = 9.15640208659364240872946538730e-001,
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rc6 = -3.59753040425048631334448145935e-001,
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rc7 = 1.42862267989304403403849619281e-001,
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rc8 = -4.74392758811439801958087514322e-002,
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rc9 = 1.09964787987580810135757047874e-002,
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rc10 = -1.28856240494889325194638463046e-003,
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sc1 = 9.97395106984001955652274773456e+000,
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sc2 = 2.80952153365721279953959310660e+001,
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sc3 = 2.19826478142545234106819407316e+001;
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/*
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* Coefficients for approximation to erfc in [4,28]
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*/
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static const double
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rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */
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rd1 = -4.99999999999640086151350330820e-001,
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rd2 = 6.24999999772906433825880867516e-001,
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rd3 = -1.54166659428052432723177389562e+000,
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rd4 = 5.51561147405411844601985649206e+000,
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rd5 = -2.55046307982949826964613748714e+001,
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rd6 = 1.43631424382843846387913799845e+002,
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rd7 = -9.45789244999420134263345971704e+002,
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rd8 = 6.94834146607051206956384703517e+003,
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rd9 = -5.27176414235983393155038356781e+004,
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rd10 = 3.68530281128672766499221324921e+005,
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rd11 = -2.06466642800404317677021026611e+006,
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rd12 = 7.78293889471135381609201431274e+006,
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rd13 = -1.42821001129434127360582351685e+007;
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double
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erf(double x)
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{
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double R,S,P,Q,ax,s,y,z,r;
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if(!finite(x)) { /* erf(nan)=nan */
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if (isnan(x))
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return(x);
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return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */
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}
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if ((ax = x) < 0)
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ax = - ax;
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if (ax < .84375) {
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if (ax < 3.7e-09) {
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if (ax < 1.0e-308)
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return 0.125*(8.0*x+p0t8*x); /*avoid underflow */
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return x + p0*x;
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}
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y = x*x;
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r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
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y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
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return x + x*(p0+r);
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}
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if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */
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s = fabs(x)-one;
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P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
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Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
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if (x>=0)
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return (c + P/Q);
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else
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return (-c - P/Q);
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}
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if (ax >= 6.0) { /* inf>|x|>=6 */
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if (x >= 0.0)
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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;
|
||
|
|
}
|