/* $NetBSD: n_erf.c,v 1.9 2013/11/24 15:16:49 martin 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.
*/
/* 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
*/
