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//===========================================================================
2
//
3
//      s_erf.c
4
//
5
//      Part of the standard mathematical function library
6
//
7
//===========================================================================
8
//####ECOSGPLCOPYRIGHTBEGIN####
9
// -------------------------------------------
10
// This file is part of eCos, the Embedded Configurable Operating System.
11
// Copyright (C) 1998, 1999, 2000, 2001, 2002 Red Hat, Inc.
12
//
13
// eCos is free software; you can redistribute it and/or modify it under
14
// the terms of the GNU General Public License as published by the Free
15
// Software Foundation; either version 2 or (at your option) any later version.
16
//
17
// eCos is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or
19
// FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
20
// for more details.
21
//
22
// You should have received a copy of the GNU General Public License along
23
// with eCos; if not, write to the Free Software Foundation, Inc.,
24
// 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.
25
//
26
// As a special exception, if other files instantiate templates or use macros
27
// or inline functions from this file, or you compile this file and link it
28
// with other works to produce a work based on this file, this file does not
29
// by itself cause the resulting work to be covered by the GNU General Public
30
// License. However the source code for this file must still be made available
31
// in accordance with section (3) of the GNU General Public License.
32
//
33
// This exception does not invalidate any other reasons why a work based on
34
// this file might be covered by the GNU General Public License.
35
//
36
// Alternative licenses for eCos may be arranged by contacting Red Hat, Inc.
37
// at http://sources.redhat.com/ecos/ecos-license/
38
// -------------------------------------------
39
//####ECOSGPLCOPYRIGHTEND####
40
//===========================================================================
41
//#####DESCRIPTIONBEGIN####
42
//
43
// Author(s):   jlarmour
44
// Contributors:  jlarmour
45
// Date:        1998-02-13
46
// Purpose:     
47
// Description: 
48
// Usage:       
49
//
50
//####DESCRIPTIONEND####
51
//
52
//===========================================================================
53
 
54
// CONFIGURATION
55
 
56
#include <pkgconf/libm.h>   // Configuration header
57
 
58
// Include the Math library?
59
#ifdef CYGPKG_LIBM     
60
 
61
// Derived from code with the following copyright
62
 
63
 
64
/* @(#)s_erf.c 1.3 95/01/18 */
65
/*
66
 * ====================================================
67
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
68
 *
69
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
70
 * Permission to use, copy, modify, and distribute this
71
 * software is freely granted, provided that this notice
72
 * is preserved.
73
 * ====================================================
74
 */
75
 
76
/* double erf(double x)
77
 * double erfc(double x)
78
 *                           x
79
 *                    2      |\
80
 *     erf(x)  =  ---------  | exp(-t*t)dt
81
 *                 sqrt(pi) \|
82
 *                           0
83
 *
84
 *     erfc(x) =  1-erf(x)
85
 *  Note that
86
 *              erf(-x) = -erf(x)
87
 *              erfc(-x) = 2 - erfc(x)
88
 *
89
 * Method:
90
 *      1. For |x| in [0, 0.84375]
91
 *          erf(x)  = x + x*R(x^2)
92
 *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
93
 *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
94
 *         where R = P/Q where P is an odd poly of degree 8 and
95
 *         Q is an odd poly of degree 10.
96
 *                                               -57.90
97
 *                      | R - (erf(x)-x)/x | <= 2
98
 *
99
 *
100
 *         Remark. The formula is derived by noting
101
 *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
102
 *         and that
103
 *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
104
 *         is close to one. The interval is chosen because the fix
105
 *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
106
 *         near 0.6174), and by some experiment, 0.84375 is chosen to
107
 *         guarantee the error is less than one ulp for erf.
108
 *
109
 *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
110
 *         c = 0.84506291151 rounded to single (24 bits)
111
 *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
112
 *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
113
 *                        1+(c+P1(s)/Q1(s))    if x < 0
114
 *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
115
 *         Remark: here we use the taylor series expansion at x=1.
116
 *              erf(1+s) = erf(1) + s*Poly(s)
117
 *                       = 0.845.. + P1(s)/Q1(s)
118
 *         That is, we use rational approximation to approximate
119
 *                      erf(1+s) - (c = (single)0.84506291151)
120
 *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
121
 *         where
122
 *              P1(s) = degree 6 poly in s
123
 *              Q1(s) = degree 6 poly in s
124
 *
125
 *      3. For x in [1.25,1/0.35(~2.857143)],
126
 *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
127
 *              erf(x)  = 1 - erfc(x)
128
 *         where
129
 *              R1(z) = degree 7 poly in z, (z=1/x^2)
130
 *              S1(z) = degree 8 poly in z
131
 *
132
 *      4. For x in [1/0.35,28]
133
 *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
134
 *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
135
 *                      = 2.0 - tiny            (if x <= -6)
136
 *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
137
 *              erf(x)  = sign(x)*(1.0 - tiny)
138
 *         where
139
 *              R2(z) = degree 6 poly in z, (z=1/x^2)
140
 *              S2(z) = degree 7 poly in z
141
 *
142
 *      Note1:
143
 *         To compute exp(-x*x-0.5625+R/S), let s be a single
144
 *         precision number and s := x; then
145
 *              -x*x = -s*s + (s-x)*(s+x)
146
 *              exp(-x*x-0.5626+R/S) =
147
 *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
148
 *      Note2:
149
 *         Here 4 and 5 make use of the asymptotic series
150
 *                        exp(-x*x)
151
 *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
152
 *                        x*sqrt(pi)
153
 *         We use rational approximation to approximate
154
 *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
155
 *         Here is the error bound for R1/S1 and R2/S2
156
 *              |R1/S1 - f(x)|  < 2**(-62.57)
157
 *              |R2/S2 - f(x)|  < 2**(-61.52)
158
 *
159
 *      5. For inf > x >= 28
160
 *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
161
 *              erfc(x) = tiny*tiny (raise underflow) if x > 0
162
 *                      = 2 - tiny if x<0
163
 *
164
 *      7. Special case:
165
 *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
166
 *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
167
 *              erfc/erf(NaN) is NaN
168
 */
169
 
170
 
171
#include "mathincl/fdlibm.h"
172
 
173
static const double
174
tiny        = 1e-300,
175
half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
176
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
177
two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
178
        /* c = (float)0.84506291151 */
179
erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
180
/*
181
 * Coefficients for approximation to  erf on [0,0.84375]
182
 */
183
efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
184
efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
185
pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
186
pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
187
pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
188
pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
189
pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
190
qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
191
qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
192
qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
193
qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
194
qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
195
/*
196
 * Coefficients for approximation to  erf  in [0.84375,1.25]
197
 */
198
pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
199
pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
200
pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
201
pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
202
pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
203
pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
204
pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
205
qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
206
qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
207
qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
208
qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
209
qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
210
qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
211
/*
212
 * Coefficients for approximation to  erfc in [1.25,1/0.35]
213
 */
214
ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
215
ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
216
ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
217
ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
218
ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
219
ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
220
ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
221
ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
222
sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
223
sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
224
sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
225
sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
226
sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
227
sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
228
sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
229
sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
230
/*
231
 * Coefficients for approximation to  erfc in [1/.35,28]
232
 */
233
rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
234
rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
235
rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
236
rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
237
rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
238
rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
239
rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
240
sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
241
sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
242
sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
243
sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
244
sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
245
sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
246
sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
247
 
248
        double erf(double x)
249
{
250
        int hx,ix,i;
251
        double R,S,P,Q,s,y,z,r;
252
        hx = CYG_LIBM_HI(x);
253
        ix = hx&0x7fffffff;
254
        if(ix>=0x7ff00000) {            /* erf(nan)=nan */
255
            i = ((unsigned)hx>>31)<<1;
256
            return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
257
        }
258
 
259
        if(ix < 0x3feb0000) {           /* |x|<0.84375 */
260
            if(ix < 0x3e300000) {       /* |x|<2**-28 */
261
                if (ix < 0x00800000)
262
                    return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
263
                return x + efx*x;
264
            }
265
            z = x*x;
266
            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
267
            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
268
            y = r/s;
269
            return x + x*y;
270
        }
271
        if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
272
            s = fabs(x)-one;
273
            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
274
            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
275
            if(hx>=0) return erx + P/Q; else return -erx - P/Q;
276
        }
277
        if (ix >= 0x40180000) {         /* inf>|x|>=6 */
278
            if(hx>=0) return one-tiny; else return tiny-one;
279
        }
280
        x = fabs(x);
281
        s = one/(x*x);
282
        if(ix< 0x4006DB6E) {    /* |x| < 1/0.35 */
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| >= 1/0.35 */
288
            R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
289
                                rb5+s*rb6)))));
290
            S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
291
                                sb5+s*(sb6+s*sb7))))));
292
        }
293
        z  = x;
294
        CYG_LIBM_LO(z) = 0;
295
        r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
296
        if(hx>=0) return one-r/x; else return  r/x-one;
297
}
298
 
299
        double erfc(double x)
300
{
301
        int hx,ix;
302
        double R,S,P,Q,s,y,z,r;
303
        hx = CYG_LIBM_HI(x);
304
        ix = hx&0x7fffffff;
305
        if(ix>=0x7ff00000) {                    /* erfc(nan)=nan */
306
                                                /* erfc(+-inf)=0,2 */
307
            return (double)(((unsigned)hx>>31)<<1)+one/x;
308
        }
309
 
310
        if(ix < 0x3feb0000) {           /* |x|<0.84375 */
311
            if(ix < 0x3c700000)         /* |x|<2**-56 */
312
                return one-x;
313
            z = x*x;
314
            r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
315
            s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
316
            y = r/s;
317
            if(hx < 0x3fd00000) {       /* x<1/4 */
318
                return one-(x+x*y);
319
            } else {
320
                r = x*y;
321
                r += (x-half);
322
                return half - r ;
323
            }
324
        }
325
        if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
326
            s = fabs(x)-one;
327
            P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
328
            Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
329
            if(hx>=0) {
330
                z  = one-erx; return z - P/Q;
331
            } else {
332
                z = erx+P/Q; return one+z;
333
            }
334
        }
335
        if (ix < 0x403c0000) {          /* |x|<28 */
336
            x = fabs(x);
337
            s = one/(x*x);
338
            if(ix< 0x4006DB6D) {        /* |x| < 1/.35 ~ 2.857143*/
339
                R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
340
                                ra5+s*(ra6+s*ra7))))));
341
                S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
342
                                sa5+s*(sa6+s*(sa7+s*sa8)))))));
343
            } else {                    /* |x| >= 1/.35 ~ 2.857143 */
344
                if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
345
                R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
346
                                rb5+s*rb6)))));
347
                S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
348
                                sb5+s*(sb6+s*sb7))))));
349
            }
350
            z  = x;
351
            CYG_LIBM_LO(z)  = 0;
352
            r  =  __ieee754_exp(-z*z-0.5625)*
353
                        __ieee754_exp((z-x)*(z+x)+R/S);
354
            if(hx>0) return r/x; else return two-r/x;
355
        } else {
356
            if(hx>0) return tiny*tiny; else return two-tiny;
357
        }
358
}
359
 
360
#endif // ifdef CYGPKG_LIBM     
361
 
362
// EOF s_erf.c

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