1/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
2
3// NOTE: The following copyright notice
4// applies only to the (modified) code of erff.
5//
6
7// erff
8// ====
9//
10// Based on code from the gnu C library, originally written by Sun.
11// Modified to remove reliance on features of gcc and 64-bit width
12// of doubles. No doubt this results in some slight deterioration
13// of efficiency, but this is not really noticeable in testing.
14//
15
16//
17// ====================================================
18// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
19//
20// Developed at SunPro, a Sun Microsystems, Inc. business.
21// Permission to use, copy, modify, and distribute this
22// software is freely granted, provided that this notice
23// is preserved.
24// ====================================================
25
26
27#include <ql/math/errorfunction.hpp>
28#include <cfloat>
29
30namespace QuantLib {
31
32 // x
33 // 2 |
34 // erf(x) = --------- | exp(-t*t)dt
35 // sqrt(pi) \|
36 // 0
37 //
38 // erfc(x) = 1-erf(x)
39 // Note that
40 // erf(-x) = -erf(x)
41 // erfc(-x) = 2 - erfc(x)
42 //
43 // Method:
44 // 1. For |x| in [0, 0.84375]
45 // erf(x) = x + x*R(x^2)
46 // erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
47 // = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
48 // where R = P/Q where P is an odd poly of degree 8 and
49 // Q is an odd poly of degree 10.
50 // -57.90
51 // | R - (erf(x)-x)/x | <= 2
52 //
53 //
54 // Remark. The formula is derived by noting
55 // erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
56 // and that
57 // 2/sqrt(pi) = 1.128379167095512573896158903121545171688
58 // is close to one. The interval is chosen because the fix
59 // point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
60 // near 0.6174), and by some experiment, 0.84375 is chosen to
61 // guarantee the error is less than one ulp for erf.
62 //
63 // 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
64 // c = 0.84506291151 rounded to single (24 bits)
65 // erf(x) = sign(x) * (c + P1(s)/Q1(s))
66 // erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
67 // 1+(c+P1(s)/Q1(s)) if x < 0
68 // |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
69 // Remark: here we use the taylor series expansion at x=1.
70 // erf(1+s) = erf(1) + s*Poly(s)
71 // = 0.845.. + P1(s)/Q1(s)
72 // That is, we use rational approximation to approximate
73 // erf(1+s) - (c = (single)0.84506291151)
74 // Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
75 // where
76 // P1(s) = degree 6 poly in s
77 // Q1(s) = degree 6 poly in s
78 //
79 // 3. For x in [1.25,1/0.35(~2.857143)],
80 // erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
81 // erf(x) = 1 - erfc(x)
82 // where
83 // R1(z) = degree 7 poly in z, (z=1/x^2)
84 // S1(z) = degree 8 poly in z
85 //
86 // 4. For x in [1/0.35,28]
87 // erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
88 // = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
89 // = 2.0 - tiny (if x <= -6)
90 // erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
91 // erf(x) = sign(x)*(1.0 - tiny)
92 // where
93 // R2(z) = degree 6 poly in z, (z=1/x^2)
94 // S2(z) = degree 7 poly in z
95 //
96 // Note1:
97 // To compute exp(-x*x-0.5625+R/S), let s be a single
98 // precision number and s := x; then
99 // -x*x = -s*s + (s-x)*(s+x)
100 // exp(-x*x-0.5626+R/S) =
101 // exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
102 // Note2:
103 // Here 4 and 5 make use of the asymptotic series
104 // exp(-x*x)
105 // erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
106 // x*sqrt(pi)
107 // We use rational approximation to approximate
108 // g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
109 // Here is the error bound for R1/S1 and R2/S2
110 // |R1/S1 - f(x)| < 2**(-62.57)
111 // |R2/S2 - f(x)| < 2**(-61.52)
112 //
113 // 5. For inf > x >= 28
114 // erf(x) = sign(x) *(1 - tiny) (raise inexact)
115 // erfc(x) = tiny*tiny (raise underflow) if x > 0
116 // = 2 - tiny if x<0
117 //
118 // 7. Special case:
119 // erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
120 // erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
121 // erfc/erf(NaN) is NaN
122
123 const Real
124 ErrorFunction::tiny = QL_EPSILON,
125 ErrorFunction::one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
126 /* c = (float)0.84506291151 */
127 ErrorFunction::erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
128 //
129 // Coefficients for approximation to erf on [0,0.84375]
130 //
131 ErrorFunction::efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
132 ErrorFunction::efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
133 ErrorFunction::pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
134 ErrorFunction::pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
135 ErrorFunction::pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
136 ErrorFunction::pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
137 ErrorFunction::pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
138 ErrorFunction::qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
139 ErrorFunction::qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
140 ErrorFunction::qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
141 ErrorFunction::qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
142 ErrorFunction::qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
143 //
144 // Coefficients for approximation to erf in [0.84375,1.25]
145 //
146 ErrorFunction::pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
147 ErrorFunction::pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
148 ErrorFunction::pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
149 ErrorFunction::pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
150 ErrorFunction::pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
151 ErrorFunction::pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
152 ErrorFunction::pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
153 ErrorFunction::qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
154 ErrorFunction::qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
155 ErrorFunction::qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
156 ErrorFunction::qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
157 ErrorFunction::qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
158 ErrorFunction::qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
159 //
160 // Coefficients for approximation to erfc in [1.25,1/0.35]
161 //
162 ErrorFunction::ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
163 ErrorFunction::ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
164 ErrorFunction::ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
165 ErrorFunction::ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
166 ErrorFunction::ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
167 ErrorFunction::ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
168 ErrorFunction::ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
169 ErrorFunction::ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
170 ErrorFunction::sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
171 ErrorFunction::sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
172 ErrorFunction::sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
173 ErrorFunction::sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
174 ErrorFunction::sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
175 ErrorFunction::sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
176 ErrorFunction::sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
177 ErrorFunction::sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
178 //
179 // Coefficients for approximation to erfc in [1/.35,28]
180 //
181 ErrorFunction::rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
182 ErrorFunction::rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
183 ErrorFunction::rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
184 ErrorFunction::rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
185 ErrorFunction::rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
186 ErrorFunction::rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
187 ErrorFunction::rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
188 ErrorFunction::sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
189 ErrorFunction::sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
190 ErrorFunction::sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
191 ErrorFunction::sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
192 ErrorFunction::sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
193 ErrorFunction::sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
194 ErrorFunction::sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
195
196 Real ErrorFunction::operator()(Real x) const {
197
198 Real R,S,P,Q,s,y,z,r, ax;
199
200 if (!std::isfinite(x: x)) {
201 if (std::isnan(x: x))
202 return x;
203 else
204 return ( x > 0 ? 1 : -1);
205 }
206
207 ax = std::fabs(x: x);
208
209 if(ax < 0.84375) { /* |x|<0.84375 */
210 if(ax < 3.7252902984e-09) { /* |x|<2**-28 */
211 if (ax < DBL_MIN*16)
212 return 0.125*(8.0*x+efx8*x); /*avoid underflow */
213 return x + efx*x;
214 }
215 z = x*x;
216 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
217 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
218 y = r/s;
219 return x + x*y;
220 }
221 if(ax <1.25) { /* 0.84375 <= |x| < 1.25 */
222 s = ax-one;
223 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
224 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
225 if(x>=0) return erx + P/Q; else return -erx - P/Q;
226 }
227 if (ax >= 6) { /* inf>|x|>=6 */
228 if(x>=0) return one-tiny; else return tiny-one;
229 }
230
231 /* Starts to lose accuracy when ax~5 */
232 s = one/(ax*ax);
233
234 if(ax < 2.85714285714285) { /* |x| < 1/0.35 */
235 R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))));
236 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))));
237 } else { /* |x| >= 1/0.35 */
238 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))));
239 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))));
240 }
241 r = std::exp( x: -ax*ax-0.5625 +R/S);
242 if(x>=0) return one-r/ax; else return r/ax-one;
243
244 }
245
246}
247

source code of quantlib/ql/math/errorfunction.cpp

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