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s_erf.c (11154B)
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1 /* @(#)s_erf.c 5.1 93/09/24 */
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
13 #ifndef lint
14 static char rcsid[] = "$FreeBSD: src/lib/msun/src/s_erf.c,v 1.7 2002/05/28 18:15:04 alfred Exp $";
15 #endif
16
17 /* double erf(double x)
18 * double erfc(double x)
19 * x
20 * 2 |\
21 * erf(x) = --------- | exp(-t*t)dt
22 * sqrt(pi) \|
23 * 0
24 *
25 * erfc(x) = 1-erf(x)
26 * Note that
27 * erf(-x) = -erf(x)
28 * erfc(-x) = 2 - erfc(x)
29 *
30 * Method:
31 * 1. For |x| in [0, 0.84375]
32 * erf(x) = x + x*R(x^2)
33 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
34 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
35 * where R = P/Q where P is an odd poly of degree 8 and
36 * Q is an odd poly of degree 10.
37 * -57.90
38 * | R - (erf(x)-x)/x | <= 2
39 *
40 *
41 * Remark. The formula is derived by noting
42 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
43 * and that
44 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
45 * is close to one. The interval is chosen because the fix
46 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
47 * near 0.6174), and by some experiment, 0.84375 is chosen to
48 * guarantee the error is less than one ulp for erf.
49 *
50 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
51 * c = 0.84506291151 rounded to single (24 bits)
52 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
53 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
54 * 1+(c+P1(s)/Q1(s)) if x < 0
55 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
56 * Remark: here we use the taylor series expansion at x=1.
57 * erf(1+s) = erf(1) + s*Poly(s)
58 * = 0.845.. + P1(s)/Q1(s)
59 * That is, we use rational approximation to approximate
60 * erf(1+s) - (c = (single)0.84506291151)
61 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
62 * where
63 * P1(s) = degree 6 poly in s
64 * Q1(s) = degree 6 poly in s
65 *
66 * 3. For x in [1.25,1/0.35(~2.857143)],
67 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
68 * erf(x) = 1 - erfc(x)
69 * where
70 * R1(z) = degree 7 poly in z, (z=1/x^2)
71 * S1(z) = degree 8 poly in z
72 *
73 * 4. For x in [1/0.35,28]
74 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
75 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
76 * = 2.0 - tiny (if x <= -6)
77 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
78 * erf(x) = sign(x)*(1.0 - tiny)
79 * where
80 * R2(z) = degree 6 poly in z, (z=1/x^2)
81 * S2(z) = degree 7 poly in z
82 *
83 * Note1:
84 * To compute exp(-x*x-0.5625+R/S), let s be a single
85 * precision number and s := x; then
86 * -x*x = -s*s + (s-x)*(s+x)
87 * exp(-x*x-0.5626+R/S) =
88 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
89 * Note2:
90 * Here 4 and 5 make use of the asymptotic series
91 * exp(-x*x)
92 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
93 * x*sqrt(pi)
94 * We use rational approximation to approximate
95 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
96 * Here is the error bound for R1/S1 and R2/S2
97 * |R1/S1 - f(x)| < 2**(-62.57)
98 * |R2/S2 - f(x)| < 2**(-61.52)
99 *
100 * 5. For inf > x >= 28
101 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
102 * erfc(x) = tiny*tiny (raise underflow) if x > 0
103 * = 2 - tiny if x<0
104 *
105 * 7. Special case:
106 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
107 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
108 * erfc/erf(NaN) is NaN
109 */
110
111
112 #include "math.h"
113 #include "math_private.h"
114
115 static const double
116 tiny = 1e-300,
117 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
118 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
119 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
120 /* c = (float)0.84506291151 */
121 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
122 /*
123 * Coefficients for approximation to erf on [0,0.84375]
124 */
125 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
126 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
127 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
128 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
129 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
130 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
131 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
132 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
133 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
134 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
135 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
136 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
137 /*
138 * Coefficients for approximation to erf in [0.84375,1.25]
139 */
140 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
141 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
142 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
143 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
144 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
145 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
146 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
147 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
148 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
149 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
150 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
151 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
152 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
153 /*
154 * Coefficients for approximation to erfc in [1.25,1/0.35]
155 */
156 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
157 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
158 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
159 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
160 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
161 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
162 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
163 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
164 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
165 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
166 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
167 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
168 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
169 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
170 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
171 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
172 /*
173 * Coefficients for approximation to erfc in [1/.35,28]
174 */
175 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
176 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
177 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
178 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
179 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
180 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
181 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
182 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
183 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
184 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
185 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
186 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
187 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
188 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
189
190 double
191 erf(double x)
192 {
193 int32_t hx,ix,i;
194 double R,S,P,Q,s,y,z,r;
195 GET_HIGH_WORD(hx,x);
196 ix = hx&0x7fffffff;
197 if(ix>=0x7ff00000) { /* erf(nan)=nan */
198 i = ((u_int32_t)hx>>31)<<1;
199 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
200 }
201
202 if(ix < 0x3feb0000) { /* |x|<0.84375 */
203 if(ix < 0x3e300000) { /* |x|<2**-28 */
204 if (ix < 0x00800000)
205 return 0.125*(8.0*x+efx8*x); /*avoid underflow */
206 return x + efx*x;
207 }
208 z = x*x;
209 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
210 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
211 y = r/s;
212 return x + x*y;
213 }
214 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
215 s = fabs(x)-one;
216 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
217 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
218 if(hx>=0) return erx + P/Q; else return -erx - P/Q;
219 }
220 if (ix >= 0x40180000) { /* inf>|x|>=6 */
221 if(hx>=0) return one-tiny; else return tiny-one;
222 }
223 x = fabs(x);
224 s = one/(x*x);
225 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
226 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
227 ra5+s*(ra6+s*ra7))))));
228 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
229 sa5+s*(sa6+s*(sa7+s*sa8)))))));
230 } else { /* |x| >= 1/0.35 */
231 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
232 rb5+s*rb6)))));
233 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
234 sb5+s*(sb6+s*sb7))))));
235 }
236 z = x;
237 SET_LOW_WORD(z,0);
238 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
239 if(hx>=0) return one-r/x; else return r/x-one;
240 }
241
242 double
243 erfc(double x)
244 {
245 int32_t hx,ix;
246 double R,S,P,Q,s,y,z,r;
247 GET_HIGH_WORD(hx,x);
248 ix = hx&0x7fffffff;
249 if(ix>=0x7ff00000) { /* erfc(nan)=nan */
250 /* erfc(+-inf)=0,2 */
251 return (double)(((u_int32_t)hx>>31)<<1)+one/x;
252 }
253
254 if(ix < 0x3feb0000) { /* |x|<0.84375 */
255 if(ix < 0x3c700000) /* |x|<2**-56 */
256 return one-x;
257 z = x*x;
258 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
259 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
260 y = r/s;
261 if(hx < 0x3fd00000) { /* x<1/4 */
262 return one-(x+x*y);
263 } else {
264 r = x*y;
265 r += (x-half);
266 return half - r ;
267 }
268 }
269 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
270 s = fabs(x)-one;
271 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
272 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
273 if(hx>=0) {
274 z = one-erx; return z - P/Q;
275 } else {
276 z = erx+P/Q; return one+z;
277 }
278 }
279 if (ix < 0x403c0000) { /* |x|<28 */
280 x = fabs(x);
281 s = one/(x*x);
282 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
283 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
284 ra5+s*(ra6+s*ra7))))));
285 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
286 sa5+s*(sa6+s*(sa7+s*sa8)))))));
287 } else { /* |x| >= 1/.35 ~ 2.857143 */
288 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
289 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
290 rb5+s*rb6)))));
291 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
292 sb5+s*(sb6+s*sb7))))));
293 }
294 z = x;
295 SET_LOW_WORD(z,0);
296 r = __ieee754_exp(-z*z-0.5625)*
297 __ieee754_exp((z-x)*(z+x)+R/S);
298 if(hx>0) return r/x; else return two-r/x;
299 } else {
300 if(hx>0) return tiny*tiny; else return two-tiny;
301 }
302 }