/*
Copyright (C) 2008 Richard Gomes
This source code is release under the BSD License.
This file is part of JQuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://jquantlib.org/
JQuantLib is free software: you can redistribute it and/or modify it
under the terms of the JQuantLib license. You should have received a
copy of the license along with this program; if not, please email
. The license is also available online at
.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
JQuantLib is based on QuantLib. http://quantlib.org/
When applicable, the original copyright notice follows this notice.
*/
package org.jquantlib.math;
/**
* In mathematics, the error function (also called the Gauss error function)
* is a non-elementary function which occurs in probability, statistics and
* partial differential equations. It is defined as:
*
* {@latex[
* \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt
* }
*
* @see Error Function on Wikipedia
*
* @author Richard Gomes
*/
public class ErrorFunction implements Ops.DoubleOp {
// x
// 2 |
// erf(x) = --------- | exp(-t*t)dt
// sqrt(pi) \|
// 0
//
// erfc(x) = 1-erf(x)
// Note that
// erf(-x) = -erf(x)
// erfc(-x) = 2 - erfc(x)
//
// Method:
// 1. For |x| in [0, 0.84375]
// erf(x) = x + x*R(x^2)
// erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
// = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
// where R = P/Q where P is an odd poly of degree 8 and
// Q is an odd poly of degree 10.
// -57.90
// | R - (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 fix
// 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.
//
// 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
// c = 0.84506291151 rounded to single (24 bits)
// erf(x) = sign(x) * (c + P1(s)/Q1(s))
// erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
// 1+(c+P1(s)/Q1(s)) if x < 0
// |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
//
// 3. For x in [1.25,1/0.35(~2.857143)],
// erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
// erf(x) = 1 - erfc(x)
// where
// R1(z) = degree 7 poly in z, (z=1/x^2)
// S1(z) = degree 8 poly in z
//
// 4. For x in [1/0.35,28]
// erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
// = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6 x >= 28
// erf(x) = sign(x) *(1 - tiny) (raise inexact)
// erfc(x) = tiny*tiny (raise underflow) if x > 0
// = 2 - tiny if x<0
//
// 7. Special case:
// erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
// erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
// erfc/erf(NaN) is NaN
private final static double tiny = Constants.QL_EPSILON;
private final static double one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
/* c = (float)0.84506291151 */
private final static double erx = 8.45062911510467529297e-01; /* 0x3FEB0AC1, 0x60000000 */
//
// Coefficients for approximation to erf on [0,0.84375]
//
private final static double efx = 1.28379167095512586316e-01; /* 0x3FC06EBA, 0x8214DB69 */
private final static double efx8 = 1.02703333676410069053e+00; /* 0x3FF06EBA, 0x8214DB69 */
private final static double pp0 = 1.28379167095512558561e-01; /* 0x3FC06EBA, 0x8214DB68 */
private final static double pp1 = -3.25042107247001499370e-01; /* 0xBFD4CD7D, 0x691CB913 */
private final static double pp2 = -2.84817495755985104766e-02; /* 0xBF9D2A51, 0xDBD7194F */
private final static double pp3 = -5.77027029648944159157e-03; /* 0xBF77A291, 0x236668E4 */
private final static double pp4 = -2.37630166566501626084e-05; /* 0xBEF8EAD6, 0x120016AC */
private final static double qq1 = 3.97917223959155352819e-01; /* 0x3FD97779, 0xCDDADC09 */
private final static double qq2 = 6.50222499887672944485e-02; /* 0x3FB0A54C, 0x5536CEBA */
private final static double qq3 = 5.08130628187576562776e-03; /* 0x3F74D022, 0xC4D36B0F */
private final static double qq4 = 1.32494738004321644526e-04; /* 0x3F215DC9, 0x221C1A10 */
private final static double qq5 = -3.96022827877536812320e-06; /* 0xBED09C43, 0x42A26120 */
//
// Coefficients for approximation to erf in [0.84375,1.25]
//
private final static double pa0 = -2.36211856075265944077e-03; /* 0xBF6359B8, 0xBEF77538 */
private final static double pa1 = 4.14856118683748331666e-01; /* 0x3FDA8D00, 0xAD92B34D */
private final static double pa2 = -3.72207876035701323847e-01; /* 0xBFD7D240, 0xFBB8C3F1 */
private final static double pa3 = 3.18346619901161753674e-01; /* 0x3FD45FCA, 0x805120E4 */
private final static double pa4 = -1.10894694282396677476e-01; /* 0xBFBC6398, 0x3D3E28EC */
private final static double pa5 = 3.54783043256182359371e-02; /* 0x3FA22A36, 0x599795EB */
private final static double pa6 = -2.16637559486879084300e-03; /* 0xBF61BF38, 0x0A96073F */
private final static double qa1 = 1.06420880400844228286e-01; /* 0x3FBB3E66, 0x18EEE323 */
private final static double qa2 = 5.40397917702171048937e-01; /* 0x3FE14AF0, 0x92EB6F33 */
private final static double qa3 = 7.18286544141962662868e-02; /* 0x3FB2635C, 0xD99FE9A7 */
private final static double qa4 = 1.26171219808761642112e-01; /* 0x3FC02660, 0xE763351F */
private final static double qa5 = 1.36370839120290507362e-02; /* 0x3F8BEDC2, 0x6B51DD1C */
private final static double qa6 = 1.19844998467991074170e-02; /* 0x3F888B54, 0x5735151D */
//
// Coefficients for approximation to erfc in [1.25,1/0.35]
//
private final static double ra0 = -9.86494403484714822705e-03; /* 0xBF843412, 0x600D6435 */
private final static double ra1 = -6.93858572707181764372e-01; /* 0xBFE63416, 0xE4BA7360 */
private final static double ra2 = -1.05586262253232909814e+01; /* 0xC0251E04, 0x41B0E726 */
private final static double ra3 = -6.23753324503260060396e+01; /* 0xC04F300A, 0xE4CBA38D */
private final static double ra4 = -1.62396669462573470355e+02; /* 0xC0644CB1, 0x84282266 */
private final static double ra5 = -1.84605092906711035994e+02; /* 0xC067135C, 0xEBCCABB2 */
private final static double ra6 = -8.12874355063065934246e+01; /* 0xC0545265, 0x57E4D2F2 */
private final static double ra7 = -9.81432934416914548592e+00; /* 0xC023A0EF, 0xC69AC25C */
private final static double sa1 = 1.96512716674392571292e+01; /* 0x4033A6B9, 0xBD707687 */
private final static double sa2 = 1.37657754143519042600e+02; /* 0x4061350C, 0x526AE721 */
private final static double sa3 = 4.34565877475229228821e+02; /* 0x407B290D, 0xD58A1A71 */
private final static double sa4 = 6.45387271733267880336e+02; /* 0x40842B19, 0x21EC2868 */
private final static double sa5 = 4.29008140027567833386e+02; /* 0x407AD021, 0x57700314 */
private final static double sa6 = 1.08635005541779435134e+02; /* 0x405B28A3, 0xEE48AE2C */
private final static double sa7 = 6.57024977031928170135e+00; /* 0x401A47EF, 0x8E484A93 */
private final static double sa8 = -6.04244152148580987438e-02; /* 0xBFAEEFF2, 0xEE749A62 */
//
// Coefficients for approximation to erfc in [1/.35,28]
//
private final static double rb0 = -9.86494292470009928597e-03; /* 0xBF843412, 0x39E86F4A */
private final static double rb1 = -7.99283237680523006574e-01; /* 0xBFE993BA, 0x70C285DE */
private final static double rb2 = -1.77579549177547519889e+01; /* 0xC031C209, 0x555F995A */
private final static double rb3 = -1.60636384855821916062e+02; /* 0xC064145D, 0x43C5ED98 */
private final static double rb4 = -6.37566443368389627722e+02; /* 0xC083EC88, 0x1375F228 */
private final static double rb5 = -1.02509513161107724954e+03; /* 0xC0900461, 0x6A2E5992 */
private final static double rb6 = -4.83519191608651397019e+02; /* 0xC07E384E, 0x9BDC383F */
private final static double sb1 = 3.03380607434824582924e+01; /* 0x403E568B, 0x261D5190 */
private final static double sb2 = 3.25792512996573918826e+02; /* 0x40745CAE, 0x221B9F0A */
private final static double sb3 = 1.53672958608443695994e+03; /* 0x409802EB, 0x189D5118 */
private final static double sb4 = 3.19985821950859553908e+03; /* 0x40A8FFB7, 0x688C246A */
private final static double sb5 = 2.55305040643316442583e+03; /* 0x40A3F219, 0xCEDF3BE6 */
private final static double sb6 = 4.74528541206955367215e+02; /* 0x407DA874, 0xE79FE763 */
private final static double sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
public double op(final double x) {
double R,S,P,Q,s,y,z,r, ax;
/* not portable!
// The finite() function returns a non-zero value if value is
// neither infinite nor a "not-a-number" (NaN) value,
// and 0 otherwise.
if (!_finite(x)) {
// The isnan() function returns a non-zero value if value is
// "not-a-number" (NaN), and 0 otherwise.
if (_isnan(x))
return x;
else
return ( x > 0 ? 1 : -1);
}
*/
ax = Math.abs(x);
if(ax < 0.84375) { /* |x|<0.84375 */
if(ax < 3.7252902984e-09) { /* |x|<2**-28 */
if (ax < Constants.DBL_MIN*16)
return 0.125*(8.0*x+efx8*x); /*avoid underflow */
return x + efx*x;
}
z = x*x;
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
return x + x*y;
}
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) return erx + P/Q; else return -erx - P/Q;
}
if (ax >= 6) { /* inf>|x|>=6 */
if(x>=0) return one-tiny; else return tiny-one;
}
/* Starts to lose accuracy when ax~5 */
s = one/(ax*ax);
if(ax < 2.85714285714285) { /* |x| < 1/0.35 */
R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
} else { /* |x| >= 1/0.35 */
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
}
r = Math.exp( -ax*ax-0.5625 +R/S);
if(x>=0) return one-r/ax; else return r/ax-one;
}
}
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