1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */
|
---|
2 | /*
|
---|
3 | * ====================================================
|
---|
4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
---|
5 | *
|
---|
6 | * Developed at SunPro, a Sun Microsystems, Inc. business.
|
---|
7 | * Permission to use, copy, modify, and distribute this
|
---|
8 | * software is freely granted, provided that this notice
|
---|
9 | * is preserved.
|
---|
10 | * ====================================================
|
---|
11 | */
|
---|
12 | /* double erf(double x)
|
---|
13 | * double erfc(double x)
|
---|
14 | * x
|
---|
15 | * 2 |\
|
---|
16 | * erf(x) = --------- | exp(-t*t)dt
|
---|
17 | * sqrt(pi) \|
|
---|
18 | * 0
|
---|
19 | *
|
---|
20 | * erfc(x) = 1-erf(x)
|
---|
21 | * Note that
|
---|
22 | * erf(-x) = -erf(x)
|
---|
23 | * erfc(-x) = 2 - erfc(x)
|
---|
24 | *
|
---|
25 | * Method:
|
---|
26 | * 1. For |x| in [0, 0.84375]
|
---|
27 | * erf(x) = x + x*R(x^2)
|
---|
28 | * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
|
---|
29 | * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
|
---|
30 | * where R = P/Q where P is an odd poly of degree 8 and
|
---|
31 | * Q is an odd poly of degree 10.
|
---|
32 | * -57.90
|
---|
33 | * | R - (erf(x)-x)/x | <= 2
|
---|
34 | *
|
---|
35 | *
|
---|
36 | * Remark. The formula is derived by noting
|
---|
37 | * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
|
---|
38 | * and that
|
---|
39 | * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
|
---|
40 | * is close to one. The interval is chosen because the fix
|
---|
41 | * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
|
---|
42 | * near 0.6174), and by some experiment, 0.84375 is chosen to
|
---|
43 | * guarantee the error is less than one ulp for erf.
|
---|
44 | *
|
---|
45 | * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
|
---|
46 | * c = 0.84506291151 rounded to single (24 bits)
|
---|
47 | * erf(x) = sign(x) * (c + P1(s)/Q1(s))
|
---|
48 | * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
|
---|
49 | * 1+(c+P1(s)/Q1(s)) if x < 0
|
---|
50 | * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
|
---|
51 | * Remark: here we use the taylor series expansion at x=1.
|
---|
52 | * erf(1+s) = erf(1) + s*Poly(s)
|
---|
53 | * = 0.845.. + P1(s)/Q1(s)
|
---|
54 | * That is, we use rational approximation to approximate
|
---|
55 | * erf(1+s) - (c = (single)0.84506291151)
|
---|
56 | * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
|
---|
57 | * where
|
---|
58 | * P1(s) = degree 6 poly in s
|
---|
59 | * Q1(s) = degree 6 poly in s
|
---|
60 | *
|
---|
61 | * 3. For x in [1.25,1/0.35(~2.857143)],
|
---|
62 | * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
|
---|
63 | * erf(x) = 1 - erfc(x)
|
---|
64 | * where
|
---|
65 | * R1(z) = degree 7 poly in z, (z=1/x^2)
|
---|
66 | * S1(z) = degree 8 poly in z
|
---|
67 | *
|
---|
68 | * 4. For x in [1/0.35,28]
|
---|
69 | * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
|
---|
70 | * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
|
---|
71 | * = 2.0 - tiny (if x <= -6)
|
---|
72 | * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
|
---|
73 | * erf(x) = sign(x)*(1.0 - tiny)
|
---|
74 | * where
|
---|
75 | * R2(z) = degree 6 poly in z, (z=1/x^2)
|
---|
76 | * S2(z) = degree 7 poly in z
|
---|
77 | *
|
---|
78 | * Note1:
|
---|
79 | * To compute exp(-x*x-0.5625+R/S), let s be a single
|
---|
80 | * precision number and s := x; then
|
---|
81 | * -x*x = -s*s + (s-x)*(s+x)
|
---|
82 | * exp(-x*x-0.5626+R/S) =
|
---|
83 | * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
|
---|
84 | * Note2:
|
---|
85 | * Here 4 and 5 make use of the asymptotic series
|
---|
86 | * exp(-x*x)
|
---|
87 | * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
|
---|
88 | * x*sqrt(pi)
|
---|
89 | * We use rational approximation to approximate
|
---|
90 | * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
|
---|
91 | * Here is the error bound for R1/S1 and R2/S2
|
---|
92 | * |R1/S1 - f(x)| < 2**(-62.57)
|
---|
93 | * |R2/S2 - f(x)| < 2**(-61.52)
|
---|
94 | *
|
---|
95 | * 5. For inf > x >= 28
|
---|
96 | * erf(x) = sign(x) *(1 - tiny) (raise inexact)
|
---|
97 | * erfc(x) = tiny*tiny (raise underflow) if x > 0
|
---|
98 | * = 2 - tiny if x<0
|
---|
99 | *
|
---|
100 | * 7. Special case:
|
---|
101 | * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
|
---|
102 | * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
|
---|
103 | * erfc/erf(NaN) is NaN
|
---|
104 | */
|
---|
105 |
|
---|
106 | #include "libm.h"
|
---|
107 |
|
---|
108 | static const double
|
---|
109 | erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
|
---|
110 | /*
|
---|
111 | * Coefficients for approximation to erf on [0,0.84375]
|
---|
112 | */
|
---|
113 | efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
|
---|
114 | pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
|
---|
115 | pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
|
---|
116 | pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
|
---|
117 | pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
|
---|
118 | pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
|
---|
119 | qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
|
---|
120 | qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
|
---|
121 | qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
|
---|
122 | qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
|
---|
123 | qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
|
---|
124 | /*
|
---|
125 | * Coefficients for approximation to erf in [0.84375,1.25]
|
---|
126 | */
|
---|
127 | pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
|
---|
128 | pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
|
---|
129 | pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
|
---|
130 | pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
|
---|
131 | pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
|
---|
132 | pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
|
---|
133 | pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
|
---|
134 | qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
|
---|
135 | qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
|
---|
136 | qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
|
---|
137 | qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
|
---|
138 | qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
|
---|
139 | qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
|
---|
140 | /*
|
---|
141 | * Coefficients for approximation to erfc in [1.25,1/0.35]
|
---|
142 | */
|
---|
143 | ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
|
---|
144 | ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
|
---|
145 | ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
|
---|
146 | ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
|
---|
147 | ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
|
---|
148 | ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
|
---|
149 | ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
|
---|
150 | ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
|
---|
151 | sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
|
---|
152 | sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
|
---|
153 | sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
|
---|
154 | sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
|
---|
155 | sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
|
---|
156 | sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
|
---|
157 | sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
|
---|
158 | sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
|
---|
159 | /*
|
---|
160 | * Coefficients for approximation to erfc in [1/.35,28]
|
---|
161 | */
|
---|
162 | rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
|
---|
163 | rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
|
---|
164 | rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
|
---|
165 | rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
|
---|
166 | rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
|
---|
167 | rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
|
---|
168 | rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
|
---|
169 | sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
|
---|
170 | sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
|
---|
171 | sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
|
---|
172 | sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
|
---|
173 | sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
|
---|
174 | sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
|
---|
175 | sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
|
---|
176 |
|
---|
177 | static double erfc1(double x)
|
---|
178 | {
|
---|
179 | double_t s,P,Q;
|
---|
180 |
|
---|
181 | s = fabs(x) - 1;
|
---|
182 | P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
|
---|
183 | Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
|
---|
184 | return 1 - erx - P/Q;
|
---|
185 | }
|
---|
186 |
|
---|
187 | static double erfc2(uint32_t ix, double x)
|
---|
188 | {
|
---|
189 | double_t s,R,S;
|
---|
190 | double z;
|
---|
191 |
|
---|
192 | if (ix < 0x3ff40000) /* |x| < 1.25 */
|
---|
193 | return erfc1(x);
|
---|
194 |
|
---|
195 | x = fabs(x);
|
---|
196 | s = 1/(x*x);
|
---|
197 | if (ix < 0x4006db6d) { /* |x| < 1/.35 ~ 2.85714 */
|
---|
198 | R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
|
---|
199 | ra5+s*(ra6+s*ra7))))));
|
---|
200 | S = 1.0+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
|
---|
201 | sa5+s*(sa6+s*(sa7+s*sa8)))))));
|
---|
202 | } else { /* |x| > 1/.35 */
|
---|
203 | R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
|
---|
204 | rb5+s*rb6)))));
|
---|
205 | S = 1.0+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
|
---|
206 | sb5+s*(sb6+s*sb7))))));
|
---|
207 | }
|
---|
208 | z = x;
|
---|
209 | SET_LOW_WORD(z,0);
|
---|
210 | return exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)/x;
|
---|
211 | }
|
---|
212 |
|
---|
213 | double erf(double x)
|
---|
214 | {
|
---|
215 | double r,s,z,y;
|
---|
216 | uint32_t ix;
|
---|
217 | int sign;
|
---|
218 |
|
---|
219 | GET_HIGH_WORD(ix, x);
|
---|
220 | sign = ix>>31;
|
---|
221 | ix &= 0x7fffffff;
|
---|
222 | if (ix >= 0x7ff00000) {
|
---|
223 | /* erf(nan)=nan, erf(+-inf)=+-1 */
|
---|
224 | return 1-2*sign + 1/x;
|
---|
225 | }
|
---|
226 | if (ix < 0x3feb0000) { /* |x| < 0.84375 */
|
---|
227 | if (ix < 0x3e300000) { /* |x| < 2**-28 */
|
---|
228 | /* avoid underflow */
|
---|
229 | return 0.125*(8*x + efx8*x);
|
---|
230 | }
|
---|
231 | z = x*x;
|
---|
232 | r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
---|
233 | s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
---|
234 | y = r/s;
|
---|
235 | return x + x*y;
|
---|
236 | }
|
---|
237 | if (ix < 0x40180000) /* 0.84375 <= |x| < 6 */
|
---|
238 | y = 1 - erfc2(ix,x);
|
---|
239 | else
|
---|
240 | y = 1 - 0x1p-1022;
|
---|
241 | return sign ? -y : y;
|
---|
242 | }
|
---|
243 |
|
---|
244 | double erfc(double x)
|
---|
245 | {
|
---|
246 | double r,s,z,y;
|
---|
247 | uint32_t ix;
|
---|
248 | int sign;
|
---|
249 |
|
---|
250 | GET_HIGH_WORD(ix, x);
|
---|
251 | sign = ix>>31;
|
---|
252 | ix &= 0x7fffffff;
|
---|
253 | if (ix >= 0x7ff00000) {
|
---|
254 | /* erfc(nan)=nan, erfc(+-inf)=0,2 */
|
---|
255 | return 2*sign + 1/x;
|
---|
256 | }
|
---|
257 | if (ix < 0x3feb0000) { /* |x| < 0.84375 */
|
---|
258 | if (ix < 0x3c700000) /* |x| < 2**-56 */
|
---|
259 | return 1.0 - x;
|
---|
260 | z = x*x;
|
---|
261 | r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
|
---|
262 | s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
|
---|
263 | y = r/s;
|
---|
264 | if (sign || ix < 0x3fd00000) { /* x < 1/4 */
|
---|
265 | return 1.0 - (x+x*y);
|
---|
266 | }
|
---|
267 | return 0.5 - (x - 0.5 + x*y);
|
---|
268 | }
|
---|
269 | if (ix < 0x403c0000) { /* 0.84375 <= |x| < 28 */
|
---|
270 | return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
|
---|
271 | }
|
---|
272 | return sign ? 2 - 0x1p-1022 : 0x1p-1022*0x1p-1022;
|
---|
273 | }
|
---|