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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
// NOTE: The following copyright notice
// applies only to the (modified) code of erff.
//
// erff
// ====
//
// Based on code from the gnu C library, originally written by Sun.
// Modified to remove reliance on features of gcc and 64-bit width
// of doubles. No doubt this results in some slight deterioration
// of efficiency, but this is not really noticeable in testing.
//
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
#include <ql/math/errorfunction.hpp>
#include <cfloat>
namespace QuantLib {
// 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<0
// = 2.0 - tiny (if x <= -6)
// erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
// erf(x) = sign(x)*(1.0 - tiny)
// where
// R2(z) = degree 6 poly in z, (z=1/x^2)
// S2(z) = degree 7 poly in z
//
// Note1:
// To compute exp(-x*x-0.5625+R/S), let s be a single
// precision number and s := x; then
// -x*x = -s*s + (s-x)*(s+x)
// exp(-x*x-0.5626+R/S) =
// exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
// Note2:
// Here 4 and 5 make use of the asymptotic series
// exp(-x*x)
// erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
// x*sqrt(pi)
// We use rational approximation to approximate
// g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
// Here is the error bound for R1/S1 and R2/S2
// |R1/S1 - f(x)| < 2**(-62.57)
// |R2/S2 - f(x)| < 2**(-61.52)
//
// 5. For inf > 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
const Real
ErrorFunction::tiny = QL_EPSILON,
ErrorFunction::one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
/* c = (float)0.84506291151 */
ErrorFunction::erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
//
// Coefficients for approximation to erf on [0,0.84375]
//
ErrorFunction::efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
ErrorFunction::efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
ErrorFunction::pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
ErrorFunction::pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
ErrorFunction::pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
ErrorFunction::pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
ErrorFunction::pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
ErrorFunction::qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
ErrorFunction::qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
ErrorFunction::qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
ErrorFunction::qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
ErrorFunction::qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
//
// Coefficients for approximation to erf in [0.84375,1.25]
//
ErrorFunction::pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
ErrorFunction::pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
ErrorFunction::pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
ErrorFunction::pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
ErrorFunction::pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
ErrorFunction::pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
ErrorFunction::pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
ErrorFunction::qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
ErrorFunction::qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
ErrorFunction::qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
ErrorFunction::qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
ErrorFunction::qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
ErrorFunction::qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
//
// Coefficients for approximation to erfc in [1.25,1/0.35]
//
ErrorFunction::ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
ErrorFunction::ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
ErrorFunction::ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
ErrorFunction::ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
ErrorFunction::ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
ErrorFunction::ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
ErrorFunction::ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
ErrorFunction::ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
ErrorFunction::sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
ErrorFunction::sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
ErrorFunction::sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
ErrorFunction::sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
ErrorFunction::sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
ErrorFunction::sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
ErrorFunction::sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
ErrorFunction::sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
//
// Coefficients for approximation to erfc in [1/.35,28]
//
ErrorFunction::rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
ErrorFunction::rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
ErrorFunction::rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
ErrorFunction::rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
ErrorFunction::rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
ErrorFunction::rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
ErrorFunction::rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
ErrorFunction::sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
ErrorFunction::sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
ErrorFunction::sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
ErrorFunction::sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
ErrorFunction::sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
ErrorFunction::sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
ErrorFunction::sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
Real ErrorFunction::operator()(Real x) const {
Real R,S,P,Q,s,y,z,r, ax;
if (!std::isfinite(x)) {
if (std::isnan(x))
return x;
else
return ( x > 0 ? 1 : -1);
}
ax = std::fabs(x);
if(ax < 0.84375) { /* |x|<0.84375 */
if(ax < 3.7252902984e-09) { /* |x|<2**-28 */
if (ax < 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 = std::exp( -ax*ax-0.5625 +R/S);
if(x>=0) return one-r/ax; else return r/ax-one;
}
}
|
