/[jscoverage]/trunk/js/src/fdlibm/s_erf.c
ViewVC logotype

Contents of /trunk/js/src/fdlibm/s_erf.c

Parent Directory Parent Directory | Revision Log Revision Log


Revision 2 - (show annotations)
Wed Aug 1 13:51:53 2007 UTC (12 years, 5 months ago) by siliconforks
File MIME type: text/plain
File size: 13133 byte(s)
Initial import.

1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
2 *
3 * ***** BEGIN LICENSE BLOCK *****
4 * Version: MPL 1.1/GPL 2.0/LGPL 2.1
5 *
6 * The contents of this file are subject to the Mozilla Public License Version
7 * 1.1 (the "License"); you may not use this file except in compliance with
8 * the License. You may obtain a copy of the License at
9 * http://www.mozilla.org/MPL/
10 *
11 * Software distributed under the License is distributed on an "AS IS" basis,
12 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
13 * for the specific language governing rights and limitations under the
14 * License.
15 *
16 * The Original Code is Mozilla Communicator client code, released
17 * March 31, 1998.
18 *
19 * The Initial Developer of the Original Code is
20 * Sun Microsystems, Inc.
21 * Portions created by the Initial Developer are Copyright (C) 1998
22 * the Initial Developer. All Rights Reserved.
23 *
24 * Contributor(s):
25 *
26 * Alternatively, the contents of this file may be used under the terms of
27 * either of the GNU General Public License Version 2 or later (the "GPL"),
28 * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
29 * in which case the provisions of the GPL or the LGPL are applicable instead
30 * of those above. If you wish to allow use of your version of this file only
31 * under the terms of either the GPL or the LGPL, and not to allow others to
32 * use your version of this file under the terms of the MPL, indicate your
33 * decision by deleting the provisions above and replace them with the notice
34 * and other provisions required by the GPL or the LGPL. If you do not delete
35 * the provisions above, a recipient may use your version of this file under
36 * the terms of any one of the MPL, the GPL or the LGPL.
37 *
38 * ***** END LICENSE BLOCK ***** */
39
40 /* @(#)s_erf.c 1.3 95/01/18 */
41 /*
42 * ====================================================
43 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
44 *
45 * Developed at SunSoft, a Sun Microsystems, Inc. business.
46 * Permission to use, copy, modify, and distribute this
47 * software is freely granted, provided that this notice
48 * is preserved.
49 * ====================================================
50 */
51
52 /* double erf(double x)
53 * double erfc(double x)
54 * x
55 * 2 |\
56 * erf(x) = --------- | exp(-t*t)dt
57 * sqrt(pi) \|
58 * 0
59 *
60 * erfc(x) = 1-erf(x)
61 * Note that
62 * erf(-x) = -erf(x)
63 * erfc(-x) = 2 - erfc(x)
64 *
65 * Method:
66 * 1. For |x| in [0, 0.84375]
67 * erf(x) = x + x*R(x^2)
68 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
69 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
70 * where R = P/Q where P is an odd poly of degree 8 and
71 * Q is an odd poly of degree 10.
72 * -57.90
73 * | R - (erf(x)-x)/x | <= 2
74 *
75 *
76 * Remark. The formula is derived by noting
77 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
78 * and that
79 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
80 * is close to one. The interval is chosen because the fix
81 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
82 * near 0.6174), and by some experiment, 0.84375 is chosen to
83 * guarantee the error is less than one ulp for erf.
84 *
85 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
86 * c = 0.84506291151 rounded to single (24 bits)
87 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
88 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
89 * 1+(c+P1(s)/Q1(s)) if x < 0
90 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
91 * Remark: here we use the taylor series expansion at x=1.
92 * erf(1+s) = erf(1) + s*Poly(s)
93 * = 0.845.. + P1(s)/Q1(s)
94 * That is, we use rational approximation to approximate
95 * erf(1+s) - (c = (single)0.84506291151)
96 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
97 * where
98 * P1(s) = degree 6 poly in s
99 * Q1(s) = degree 6 poly in s
100 *
101 * 3. For x in [1.25,1/0.35(~2.857143)],
102 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
103 * erf(x) = 1 - erfc(x)
104 * where
105 * R1(z) = degree 7 poly in z, (z=1/x^2)
106 * S1(z) = degree 8 poly in z
107 *
108 * 4. For x in [1/0.35,28]
109 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
110 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
111 * = 2.0 - tiny (if x <= -6)
112 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
113 * erf(x) = sign(x)*(1.0 - tiny)
114 * where
115 * R2(z) = degree 6 poly in z, (z=1/x^2)
116 * S2(z) = degree 7 poly in z
117 *
118 * Note1:
119 * To compute exp(-x*x-0.5625+R/S), let s be a single
120 * precision number and s := x; then
121 * -x*x = -s*s + (s-x)*(s+x)
122 * exp(-x*x-0.5626+R/S) =
123 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
124 * Note2:
125 * Here 4 and 5 make use of the asymptotic series
126 * exp(-x*x)
127 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
128 * x*sqrt(pi)
129 * We use rational approximation to approximate
130 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
131 * Here is the error bound for R1/S1 and R2/S2
132 * |R1/S1 - f(x)| < 2**(-62.57)
133 * |R2/S2 - f(x)| < 2**(-61.52)
134 *
135 * 5. For inf > x >= 28
136 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
137 * erfc(x) = tiny*tiny (raise underflow) if x > 0
138 * = 2 - tiny if x<0
139 *
140 * 7. Special case:
141 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
142 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
143 * erfc/erf(NaN) is NaN
144 */
145
146
147 #include "fdlibm.h"
148
149 #ifdef __STDC__
150 static const double
151 #else
152 static double
153 #endif
154 tiny = 1e-300,
155 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
156 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
157 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
158 /* c = (float)0.84506291151 */
159 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
160 /*
161 * Coefficients for approximation to erf on [0,0.84375]
162 */
163 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
164 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
165 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
166 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
167 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
168 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
169 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
170 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
171 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
172 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
173 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
174 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
175 /*
176 * Coefficients for approximation to erf in [0.84375,1.25]
177 */
178 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
179 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
180 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
181 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
182 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
183 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
184 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
185 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
186 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
187 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
188 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
189 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
190 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
191 /*
192 * Coefficients for approximation to erfc in [1.25,1/0.35]
193 */
194 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
195 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
196 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
197 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
198 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
199 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
200 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
201 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
202 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
203 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
204 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
205 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
206 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
207 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
208 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
209 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
210 /*
211 * Coefficients for approximation to erfc in [1/.35,28]
212 */
213 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
214 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
215 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
216 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
217 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
218 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
219 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
220 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
221 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
222 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
223 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
224 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
225 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
226 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
227
228 #ifdef __STDC__
229 double fd_erf(double x)
230 #else
231 double fd_erf(x)
232 double x;
233 #endif
234 {
235 fd_twoints u;
236 int hx,ix,i;
237 double R,S,P,Q,s,y,z,r;
238 u.d = x;
239 hx = __HI(u);
240 ix = hx&0x7fffffff;
241 if(ix>=0x7ff00000) { /* erf(nan)=nan */
242 i = ((unsigned)hx>>31)<<1;
243 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
244 }
245
246 if(ix < 0x3feb0000) { /* |x|<0.84375 */
247 if(ix < 0x3e300000) { /* |x|<2**-28 */
248 if (ix < 0x00800000)
249 return 0.125*(8.0*x+efx8*x); /*avoid underflow */
250 return x + efx*x;
251 }
252 z = x*x;
253 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
254 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
255 y = r/s;
256 return x + x*y;
257 }
258 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
259 s = fd_fabs(x)-one;
260 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
261 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
262 if(hx>=0) return erx + P/Q; else return -erx - P/Q;
263 }
264 if (ix >= 0x40180000) { /* inf>|x|>=6 */
265 if(hx>=0) return one-tiny; else return tiny-one;
266 }
267 x = fd_fabs(x);
268 s = one/(x*x);
269 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */
270 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
271 ra5+s*(ra6+s*ra7))))));
272 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
273 sa5+s*(sa6+s*(sa7+s*sa8)))))));
274 } else { /* |x| >= 1/0.35 */
275 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
276 rb5+s*rb6)))));
277 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
278 sb5+s*(sb6+s*sb7))))));
279 }
280 z = x;
281 u.d = z;
282 __LO(u) = 0;
283 z = u.d;
284 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
285 if(hx>=0) return one-r/x; else return r/x-one;
286 }
287
288 #ifdef __STDC__
289 double erfc(double x)
290 #else
291 double erfc(x)
292 double x;
293 #endif
294 {
295 fd_twoints u;
296 int hx,ix;
297 double R,S,P,Q,s,y,z,r;
298 u.d = x;
299 hx = __HI(u);
300 ix = hx&0x7fffffff;
301 if(ix>=0x7ff00000) { /* erfc(nan)=nan */
302 /* erfc(+-inf)=0,2 */
303 return (double)(((unsigned)hx>>31)<<1)+one/x;
304 }
305
306 if(ix < 0x3feb0000) { /* |x|<0.84375 */
307 if(ix < 0x3c700000) /* |x|<2**-56 */
308 return one-x;
309 z = x*x;
310 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
311 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
312 y = r/s;
313 if(hx < 0x3fd00000) { /* x<1/4 */
314 return one-(x+x*y);
315 } else {
316 r = x*y;
317 r += (x-half);
318 return half - r ;
319 }
320 }
321 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
322 s = fd_fabs(x)-one;
323 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
324 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
325 if(hx>=0) {
326 z = one-erx; return z - P/Q;
327 } else {
328 z = erx+P/Q; return one+z;
329 }
330 }
331 if (ix < 0x403c0000) { /* |x|<28 */
332 x = fd_fabs(x);
333 s = one/(x*x);
334 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/
335 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
336 ra5+s*(ra6+s*ra7))))));
337 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
338 sa5+s*(sa6+s*(sa7+s*sa8)))))));
339 } else { /* |x| >= 1/.35 ~ 2.857143 */
340 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
341 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
342 rb5+s*rb6)))));
343 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
344 sb5+s*(sb6+s*sb7))))));
345 }
346 z = x;
347 u.d = z;
348 __LO(u) = 0;
349 z = u.d;
350 r = __ieee754_exp(-z*z-0.5625)*
351 __ieee754_exp((z-x)*(z+x)+R/S);
352 if(hx>0) return r/x; else return two-r/x;
353 } else {
354 if(hx>0) return tiny*tiny; else return two-tiny;
355 }
356 }

  ViewVC Help
Powered by ViewVC 1.1.24