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1 siliconforks 2 /* -*- 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     }

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