1 #include "FEATURE/uwin"
2
3 #if !_UWIN || _lib_erf
4
_STUB_erf()5 void _STUB_erf(){}
6
7 #else
8
9 /*-
10 * Copyright (c) 1992, 1993
11 * The Regents of the University of California. All rights reserved.
12 *
13 * Redistribution and use in source and binary forms, with or without
14 * modification, are permitted provided that the following conditions
15 * are met:
16 * 1. Redistributions of source code must retain the above copyright
17 * notice, this list of conditions and the following disclaimer.
18 * 2. Redistributions in binary form must reproduce the above copyright
19 * notice, this list of conditions and the following disclaimer in the
20 * documentation and/or other materials provided with the distribution.
21 * 3. Neither the name of the University nor the names of its contributors
22 * may be used to endorse or promote products derived from this software
23 * without specific prior written permission.
24 *
25 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
26 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
27 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
28 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
29 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
30 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
31 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
32 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
33 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
34 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
35 * SUCH DAMAGE.
36 */
37
38 #ifndef lint
39 static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93";
40 #endif /* not lint */
41
42 /* Modified Nov 30, 1992 P. McILROY:
43 * Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp)
44 * Replaced even+odd with direct calculation for x < .84375,
45 * to avoid destructive cancellation.
46 *
47 * Performance of erfc(x):
48 * In 300000 trials in the range [.83, .84375] the
49 * maximum observed error was 3.6ulp.
50 *
51 * In [.84735,1.25] the maximum observed error was <2.5ulp in
52 * 100000 runs in the range [1.2, 1.25].
53 *
54 * In [1.25,26] (Not including subnormal results)
55 * the error is < 1.7ulp.
56 */
57
58 /* double erf(double x)
59 * double erfc(double x)
60 * x
61 * 2 |\
62 * erf(x) = --------- | exp(-t*t)dt
63 * sqrt(pi) \|
64 * 0
65 *
66 * erfc(x) = 1-erf(x)
67 *
68 * Method:
69 * 1. Reduce x to |x| by erf(-x) = -erf(x)
70 * 2. For x in [0, 0.84375]
71 * erf(x) = x + x*P(x^2)
72 * erfc(x) = 1 - erf(x) if x<=0.25
73 * = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375]
74 * where
75 * 2 2 4 20
76 * P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x )
77 * is an approximation to (erf(x)-x)/x with precision
78 *
79 * -56.45
80 * | P - (erf(x)-x)/x | <= 2
81 *
82 *
83 * Remark. The formula is derived by noting
84 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
85 * and that
86 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
87 * is close to one. The interval is chosen because the fixed
88 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
89 * near 0.6174), and by some experiment, 0.84375 is chosen to
90 * guarantee the error is less than one ulp for erf.
91 *
92 * 3. For x in [0.84375,1.25], let s = x - 1, and
93 * c = 0.84506291151 rounded to single (24 bits)
94 * erf(x) = c + P1(s)/Q1(s)
95 * erfc(x) = (1-c) - P1(s)/Q1(s)
96 * |P1/Q1 - (erf(x)-c)| <= 2**-59.06
97 * Remark: here we use the taylor series expansion at x=1.
98 * erf(1+s) = erf(1) + s*Poly(s)
99 * = 0.845.. + P1(s)/Q1(s)
100 * That is, we use rational approximation to approximate
101 * erf(1+s) - (c = (single)0.84506291151)
102 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
103 * where
104 * P1(s) = degree 6 poly in s
105 * Q1(s) = degree 6 poly in s
106 *
107 * 4. For x in [1.25, 2]; [2, 4]
108 * erf(x) = 1.0 - tiny
109 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z))
110 *
111 * Where z = 1/(x*x), R is degree 9, and S is degree 3;
112 *
113 * 5. For x in [4,28]
114 * erf(x) = 1.0 - tiny
115 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z))
116 *
117 * Where P is degree 14 polynomial in 1/(x*x).
118 *
119 * Notes:
120 * Here 4 and 5 make use of the asymptotic series
121 * exp(-x*x)
122 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) );
123 * x*sqrt(pi)
124 *
125 * where for z = 1/(x*x)
126 * P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...))))
127 *
128 * Thus we use rational approximation to approximate
129 * erfc*x*exp(x*x) ~ 1/sqrt(pi);
130 *
131 * The error bound for the target function, G(z) for
132 * the interval
133 * [4, 28]:
134 * |eps + 1/(z)P(z) - G(z)| < 2**(-56.61)
135 * for [2, 4]:
136 * |R(z)/S(z) - G(z)| < 2**(-58.24)
137 * for [1.25, 2]:
138 * |R(z)/S(z) - G(z)| < 2**(-58.12)
139 *
140 * 6. For inf > x >= 28
141 * erf(x) = 1 - tiny (raise inexact)
142 * erfc(x) = tiny*tiny (raise underflow)
143 *
144 * 7. Special cases:
145 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
146 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
147 * erfc/erf(NaN) is NaN
148 */
149
150 #if defined(vax) || defined(tahoe)
151 #define _IEEE 0
152 #define TRUNC(x) (double) (float) (x)
153 #else
154 #define _IEEE 1
155 #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000
156 #define infnan(x) 0.0
157 #endif
158
159 #ifdef _IEEE_LIBM
160 /*
161 * redefining "___function" to "function" in _IEEE_LIBM mode
162 */
163 #include "ieee_libm.h"
164 #endif
165 #include "mathimpl.h"
166
167 static double
168 tiny = 1e-300,
169 half = 0.5,
170 one = 1.0,
171 two = 2.0,
172 c = 8.45062911510467529297e-01, /* (float)0.84506291151 */
173 /*
174 * Coefficients for approximation to erf in [0,0.84375]
175 */
176 p0t8 = 1.02703333676410051049867154944018394163280,
177 p0 = 1.283791670955125638123339436800229927041e-0001,
178 p1 = -3.761263890318340796574473028946097022260e-0001,
179 p2 = 1.128379167093567004871858633779992337238e-0001,
180 p3 = -2.686617064084433642889526516177508374437e-0002,
181 p4 = 5.223977576966219409445780927846432273191e-0003,
182 p5 = -8.548323822001639515038738961618255438422e-0004,
183 p6 = 1.205520092530505090384383082516403772317e-0004,
184 p7 = -1.492214100762529635365672665955239554276e-0005,
185 p8 = 1.640186161764254363152286358441771740838e-0006,
186 p9 = -1.571599331700515057841960987689515895479e-0007,
187 p10= 1.073087585213621540635426191486561494058e-0008;
188 /*
189 * Coefficients for approximation to erf in [0.84375,1.25]
190 */
191 static double
192 pa0 = -2.362118560752659485957248365514511540287e-0003,
193 pa1 = 4.148561186837483359654781492060070469522e-0001,
194 pa2 = -3.722078760357013107593507594535478633044e-0001,
195 pa3 = 3.183466199011617316853636418691420262160e-0001,
196 pa4 = -1.108946942823966771253985510891237782544e-0001,
197 pa5 = 3.547830432561823343969797140537411825179e-0002,
198 pa6 = -2.166375594868790886906539848893221184820e-0003,
199 qa1 = 1.064208804008442270765369280952419863524e-0001,
200 qa2 = 5.403979177021710663441167681878575087235e-0001,
201 qa3 = 7.182865441419627066207655332170665812023e-0002,
202 qa4 = 1.261712198087616469108438860983447773726e-0001,
203 qa5 = 1.363708391202905087876983523620537833157e-0002,
204 qa6 = 1.198449984679910764099772682882189711364e-0002;
205 /*
206 * log(sqrt(pi)) for large x expansions.
207 * The tail (lsqrtPI_lo) is included in the rational
208 * approximations.
209 */
210 static double
211 lsqrtPI_hi = .5723649429247000819387380943226;
212 /*
213 * lsqrtPI_lo = .000000000000000005132975581353913;
214 *
215 * Coefficients for approximation to erfc in [2, 4]
216 */
217 static double
218 rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */
219 rb1 = 2.15592846101742183841910806188e-008,
220 rb2 = 6.24998557732436510470108714799e-001,
221 rb3 = 8.24849222231141787631258921465e+000,
222 rb4 = 2.63974967372233173534823436057e+001,
223 rb5 = 9.86383092541570505318304640241e+000,
224 rb6 = -7.28024154841991322228977878694e+000,
225 rb7 = 5.96303287280680116566600190708e+000,
226 rb8 = -4.40070358507372993983608466806e+000,
227 rb9 = 2.39923700182518073731330332521e+000,
228 rb10 = -6.89257464785841156285073338950e-001,
229 sb1 = 1.56641558965626774835300238919e+001,
230 sb2 = 7.20522741000949622502957936376e+001,
231 sb3 = 9.60121069770492994166488642804e+001;
232 /*
233 * Coefficients for approximation to erfc in [1.25, 2]
234 */
235 static double
236 rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */
237 rc1 = 1.28735722546372485255126993930e-005,
238 rc2 = 6.24664954087883916855616917019e-001,
239 rc3 = 4.69798884785807402408863708843e+000,
240 rc4 = 7.61618295853929705430118701770e+000,
241 rc5 = 9.15640208659364240872946538730e-001,
242 rc6 = -3.59753040425048631334448145935e-001,
243 rc7 = 1.42862267989304403403849619281e-001,
244 rc8 = -4.74392758811439801958087514322e-002,
245 rc9 = 1.09964787987580810135757047874e-002,
246 rc10 = -1.28856240494889325194638463046e-003,
247 sc1 = 9.97395106984001955652274773456e+000,
248 sc2 = 2.80952153365721279953959310660e+001,
249 sc3 = 2.19826478142545234106819407316e+001;
250 /*
251 * Coefficients for approximation to erfc in [4,28]
252 */
253 static double
254 rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */
255 rd1 = -4.99999999999640086151350330820e-001,
256 rd2 = 6.24999999772906433825880867516e-001,
257 rd3 = -1.54166659428052432723177389562e+000,
258 rd4 = 5.51561147405411844601985649206e+000,
259 rd5 = -2.55046307982949826964613748714e+001,
260 rd6 = 1.43631424382843846387913799845e+002,
261 rd7 = -9.45789244999420134263345971704e+002,
262 rd8 = 6.94834146607051206956384703517e+003,
263 rd9 = -5.27176414235983393155038356781e+004,
264 rd10 = 3.68530281128672766499221324921e+005,
265 rd11 = -2.06466642800404317677021026611e+006,
266 rd12 = 7.78293889471135381609201431274e+006,
267 rd13 = -1.42821001129434127360582351685e+007;
268
269 extern double erf(x)
270 double x;
271 {
272 double R,S,P,Q,ax,s,y,z,r,fabs(),exp();
273 if(!finite(x)) { /* erf(nan)=nan */
274 if (isnan(x))
275 return(x);
276 return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */
277 }
278 if ((ax = x) < 0)
279 ax = - ax;
280 if (ax < .84375) {
281 if (ax < 3.7e-09) {
282 if (ax < 1.0e-308)
283 return 0.125*(8.0*x+p0t8*x); /*avoid underflow */
284 return x + p0*x;
285 }
286 y = x*x;
287 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
288 y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
289 return x + x*(p0+r);
290 }
291 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */
292 s = fabs(x)-one;
293 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
294 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
295 if (x>=0)
296 return (c + P/Q);
297 else
298 return (-c - P/Q);
299 }
300 if (ax >= 6.0) { /* inf>|x|>=6 */
301 if (x >= 0.0)
302 return (one-tiny);
303 else
304 return (tiny-one);
305 }
306 /* 1.25 <= |x| < 6 */
307 z = -ax*ax;
308 s = -one/z;
309 if (ax < 2.0) {
310 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
311 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
312 S = one+s*(sc1+s*(sc2+s*sc3));
313 } else {
314 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
315 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
316 S = one+s*(sb1+s*(sb2+s*sb3));
317 }
318 y = (R/S -.5*s) - lsqrtPI_hi;
319 z += y;
320 z = exp(z)/ax;
321 if (x >= 0)
322 return (one-z);
323 else
324 return (z-one);
325 }
326
327 extern double erfc(x)
328 double x;
329 {
330 double R,S,P,Q,s,ax,y,z,r,fabs(),__exp__D();
331 if (!finite(x)) {
332 if (isnan(x)) /* erfc(NaN) = NaN */
333 return(x);
334 else if (x > 0) /* erfc(+-inf)=0,2 */
335 return 0.0;
336 else
337 return 2.0;
338 }
339 if ((ax = x) < 0)
340 ax = -ax;
341 if (ax < .84375) { /* |x|<0.84375 */
342 if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */
343 return one-x;
344 y = x*x;
345 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
346 y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
347 if (ax < .0625) { /* |x|<2**-4 */
348 return (one-(x+x*(p0+r)));
349 } else {
350 r = x*(p0+r);
351 r += (x-half);
352 return (half - r);
353 }
354 }
355 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */
356 s = ax-one;
357 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
358 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
359 if (x>=0) {
360 z = one-c; return z - P/Q;
361 } else {
362 z = c+P/Q; return one+z;
363 }
364 }
365 if (ax >= 28) /* Out of range */
366 if (x>0)
367 return (tiny*tiny);
368 else
369 return (two-tiny);
370 z = ax;
371 TRUNC(z);
372 y = z - ax; y *= (ax+z);
373 z *= -z; /* Here z + y = -x^2 */
374 s = one/(-z-y); /* 1/(x*x) */
375 if (ax >= 4) { /* 6 <= ax */
376 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+
377 s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10
378 +s*(rd11+s*(rd12+s*rd13))))))))))));
379 y += rd0;
380 } else if (ax >= 2) {
381 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
382 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
383 S = one+s*(sb1+s*(sb2+s*sb3));
384 y += R/S;
385 R = -.5*s;
386 } else {
387 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
388 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
389 S = one+s*(sc1+s*(sc2+s*sc3));
390 y += R/S;
391 R = -.5*s;
392 }
393 /* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */
394 s = ((R + y) - lsqrtPI_hi) + z;
395 y = (((z-s) - lsqrtPI_hi) + R) + y;
396 r = __exp__D(s, y)/x;
397 if (x>0)
398 return r;
399 else
400 return two-r;
401 }
402
403 #endif
404