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 }
274