1 /*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22 /*
23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 */
25
26 /*
27 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
28 * Use is subject to license terms.
29 */
30
31 #pragma weak __erf = erf
32 #pragma weak __erfc = erfc
33
34
35 /*
36 * double erf(double x)
37 * double erfc(double x)
38 * x
39 * 2 |\
40 * erf(x) = --------- | exp(-t*t)dt
41 * sqrt(pi) \|
42 * 0
43 *
44 * erfc(x) = 1-erf(x)
45 * Note that
46 * erf(-x) = -erf(x)
47 * erfc(-x) = 2 - erfc(x)
48 *
49 * Method:
50 * 1. For |x| in [0, 0.84375]
51 * erf(x) = x + x*R(x^2)
52 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
53 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
54 * where R = P/Q where P is an odd poly of degree 8 and
55 * Q is an odd poly of degree 10.
56 * -57.90
57 * | R - (erf(x)-x)/x | <= 2
58 *
59 *
60 * Remark. The formula is derived by noting
61 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
62 * and that
63 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
64 * is close to one. The interval is chosen because the fix
65 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
66 * near 0.6174), and by some experiment, 0.84375 is chosen to
67 * guarantee the error is less than one ulp for erf.
68 *
69 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
70 * c = 0.84506291151 rounded to single (24 bits)
71 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
72 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
73 * 1+(c+P1(s)/Q1(s)) if x < 0
74 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
75 * Remark: here we use the taylor series expansion at x=1.
76 * erf(1+s) = erf(1) + s*Poly(s)
77 * = 0.845.. + P1(s)/Q1(s)
78 * That is, we use rational approximation to approximate
79 * erf(1+s) - (c = (single)0.84506291151)
80 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
81 * where
82 * P1(s) = degree 6 poly in s
83 * Q1(s) = degree 6 poly in s
84 *
85 * 3. For x in [1.25,1/0.35(~2.857143)],
86 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
87 * erf(x) = 1 - erfc(x)
88 * where
89 * R1(z) = degree 7 poly in z, (z=1/x^2)
90 * S1(z) = degree 8 poly in z
91 *
92 * 4. For x in [1/0.35,28]
93 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
94 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
95 * = 2.0 - tiny (if x <= -6)
96 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
97 * erf(x) = sign(x)*(1.0 - tiny)
98 * where
99 * R2(z) = degree 6 poly in z, (z=1/x^2)
100 * S2(z) = degree 7 poly in z
101 *
102 * Note1:
103 * To compute exp(-x*x-0.5625+R/S), let s be a single
104 * precision number and s := x; then
105 * -x*x = -s*s + (s-x)*(s+x)
106 * exp(-x*x-0.5626+R/S) =
107 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
108 * Note2:
109 * Here 4 and 5 make use of the asymptotic series
110 * exp(-x*x)
111 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
112 * x*sqrt(pi)
113 * We use rational approximation to approximate
114 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
115 * Here is the error bound for R1/S1 and R2/S2
116 * |R1/S1 - f(x)| < 2**(-62.57)
117 * |R2/S2 - f(x)| < 2**(-61.52)
118 *
119 * 5. For inf > x >= 28
120 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
121 * erfc(x) = tiny*tiny (raise underflow) if x > 0
122 * = 2 - tiny if x<0
123 *
124 * 7. Special case:
125 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
126 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
127 * erfc/erf(NaN) is NaN
128 */
129
130 #include "libm_macros.h"
131 #include <math.h>
132
133 static const double xxx[] = {
134 /* tiny */
135 1e-300,
136 /* half */ 5.00000000000000000000e-01, /* 3FE00000, 00000000 */
137 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
138 /* two */ 2.00000000000000000000e+00, /* 40000000, 00000000 */
139 /* erx */ 8.45062911510467529297e-01, /* 3FEB0AC1, 60000000 */
140
141 /*
142 * Coefficients for approximation to erf on [0,0.84375]
143 */
144 /* efx */ 1.28379167095512586316e-01, /* 3FC06EBA, 8214DB69 */
145 /* efx8 */ 1.02703333676410069053e+00, /* 3FF06EBA, 8214DB69 */
146 /* pp0 */ 1.28379167095512558561e-01, /* 3FC06EBA, 8214DB68 */
147 /* pp1 */ -3.25042107247001499370e-01, /* BFD4CD7D, 691CB913 */
148 /* pp2 */ -2.84817495755985104766e-02, /* BF9D2A51, DBD7194F */
149 /* pp3 */ -5.77027029648944159157e-03, /* BF77A291, 236668E4 */
150 /* pp4 */ -2.37630166566501626084e-05, /* BEF8EAD6, 120016AC */
151 /* qq1 */ 3.97917223959155352819e-01, /* 3FD97779, CDDADC09 */
152 /* qq2 */ 6.50222499887672944485e-02, /* 3FB0A54C, 5536CEBA */
153 /* qq3 */ 5.08130628187576562776e-03, /* 3F74D022, C4D36B0F */
154 /* qq4 */ 1.32494738004321644526e-04, /* 3F215DC9, 221C1A10 */
155 /* qq5 */ -3.96022827877536812320e-06, /* BED09C43, 42A26120 */
156
157 /*
158 * Coefficients for approximation to erf in [0.84375,1.25]
159 */
160 /* pa0 */ -2.36211856075265944077e-03, /* BF6359B8, BEF77538 */
161 /* pa1 */ 4.14856118683748331666e-01, /* 3FDA8D00, AD92B34D */
162 /* pa2 */ -3.72207876035701323847e-01, /* BFD7D240, FBB8C3F1 */
163 /* pa3 */ 3.18346619901161753674e-01, /* 3FD45FCA, 805120E4 */
164 /* pa4 */ -1.10894694282396677476e-01, /* BFBC6398, 3D3E28EC */
165 /* pa5 */ 3.54783043256182359371e-02, /* 3FA22A36, 599795EB */
166 /* pa6 */ -2.16637559486879084300e-03, /* BF61BF38, 0A96073F */
167 /* qa1 */ 1.06420880400844228286e-01, /* 3FBB3E66, 18EEE323 */
168 /* qa2 */ 5.40397917702171048937e-01, /* 3FE14AF0, 92EB6F33 */
169 /* qa3 */ 7.18286544141962662868e-02, /* 3FB2635C, D99FE9A7 */
170 /* qa4 */ 1.26171219808761642112e-01, /* 3FC02660, E763351F */
171 /* qa5 */ 1.36370839120290507362e-02, /* 3F8BEDC2, 6B51DD1C */
172 /* qa6 */ 1.19844998467991074170e-02, /* 3F888B54, 5735151D */
173
174 /*
175 * Coefficients for approximation to erfc in [1.25,1/0.35]
176 */
177 /* ra0 */ -9.86494403484714822705e-03, /* BF843412, 600D6435 */
178 /* ra1 */ -6.93858572707181764372e-01, /* BFE63416, E4BA7360 */
179 /* ra2 */ -1.05586262253232909814e+01, /* C0251E04, 41B0E726 */
180 /* ra3 */ -6.23753324503260060396e+01, /* C04F300A, E4CBA38D */
181 /* ra4 */ -1.62396669462573470355e+02, /* C0644CB1, 84282266 */
182 /* ra5 */ -1.84605092906711035994e+02, /* C067135C, EBCCABB2 */
183 /* ra6 */ -8.12874355063065934246e+01, /* C0545265, 57E4D2F2 */
184 /* ra7 */ -9.81432934416914548592e+00, /* C023A0EF, C69AC25C */
185 /* sa1 */ 1.96512716674392571292e+01, /* 4033A6B9, BD707687 */
186 /* sa2 */ 1.37657754143519042600e+02, /* 4061350C, 526AE721 */
187 /* sa3 */ 4.34565877475229228821e+02, /* 407B290D, D58A1A71 */
188 /* sa4 */ 6.45387271733267880336e+02, /* 40842B19, 21EC2868 */
189 /* sa5 */ 4.29008140027567833386e+02, /* 407AD021, 57700314 */
190 /* sa6 */ 1.08635005541779435134e+02, /* 405B28A3, EE48AE2C */
191 /* sa7 */ 6.57024977031928170135e+00, /* 401A47EF, 8E484A93 */
192 /* sa8 */ -6.04244152148580987438e-02, /* BFAEEFF2, EE749A62 */
193
194 /*
195 * Coefficients for approximation to erfc in [1/.35,28]
196 */
197 /* rb0 */ -9.86494292470009928597e-03, /* BF843412, 39E86F4A */
198 /* rb1 */ -7.99283237680523006574e-01, /* BFE993BA, 70C285DE */
199 /* rb2 */ -1.77579549177547519889e+01, /* C031C209, 555F995A */
200 /* rb3 */ -1.60636384855821916062e+02, /* C064145D, 43C5ED98 */
201 /* rb4 */ -6.37566443368389627722e+02, /* C083EC88, 1375F228 */
202 /* rb5 */ -1.02509513161107724954e+03, /* C0900461, 6A2E5992 */
203 /* rb6 */ -4.83519191608651397019e+02, /* C07E384E, 9BDC383F */
204 /* sb1 */ 3.03380607434824582924e+01, /* 403E568B, 261D5190 */
205 /* sb2 */ 3.25792512996573918826e+02, /* 40745CAE, 221B9F0A */
206 /* sb3 */ 1.53672958608443695994e+03, /* 409802EB, 189D5118 */
207 /* sb4 */ 3.19985821950859553908e+03, /* 40A8FFB7, 688C246A */
208 /* sb5 */ 2.55305040643316442583e+03, /* 40A3F219, CEDF3BE6 */
209 /* sb6 */ 4.74528541206955367215e+02, /* 407DA874, E79FE763 */
210 /* sb7 */ -2.24409524465858183362e+01 /* C03670E2, 42712D62 */
211 };
212
213 #define tiny xxx[0]
214 #define half xxx[1]
215 #define one xxx[2]
216 #define two xxx[3]
217 #define erx xxx[4]
218
219 /*
220 * Coefficients for approximation to erf on [0,0.84375]
221 */
222 #define efx xxx[5]
223 #define efx8 xxx[6]
224 #define pp0 xxx[7]
225 #define pp1 xxx[8]
226 #define pp2 xxx[9]
227 #define pp3 xxx[10]
228 #define pp4 xxx[11]
229 #define qq1 xxx[12]
230 #define qq2 xxx[13]
231 #define qq3 xxx[14]
232 #define qq4 xxx[15]
233 #define qq5 xxx[16]
234
235 /*
236 * Coefficients for approximation to erf in [0.84375,1.25]
237 */
238 #define pa0 xxx[17]
239 #define pa1 xxx[18]
240 #define pa2 xxx[19]
241 #define pa3 xxx[20]
242 #define pa4 xxx[21]
243 #define pa5 xxx[22]
244 #define pa6 xxx[23]
245 #define qa1 xxx[24]
246 #define qa2 xxx[25]
247 #define qa3 xxx[26]
248 #define qa4 xxx[27]
249 #define qa5 xxx[28]
250 #define qa6 xxx[29]
251
252 /*
253 * Coefficients for approximation to erfc in [1.25,1/0.35]
254 */
255 #define ra0 xxx[30]
256 #define ra1 xxx[31]
257 #define ra2 xxx[32]
258 #define ra3 xxx[33]
259 #define ra4 xxx[34]
260 #define ra5 xxx[35]
261 #define ra6 xxx[36]
262 #define ra7 xxx[37]
263 #define sa1 xxx[38]
264 #define sa2 xxx[39]
265 #define sa3 xxx[40]
266 #define sa4 xxx[41]
267 #define sa5 xxx[42]
268 #define sa6 xxx[43]
269 #define sa7 xxx[44]
270 #define sa8 xxx[45]
271
272 /*
273 * Coefficients for approximation to erfc in [1/.35,28]
274 */
275 #define rb0 xxx[46]
276 #define rb1 xxx[47]
277 #define rb2 xxx[48]
278 #define rb3 xxx[49]
279 #define rb4 xxx[50]
280 #define rb5 xxx[51]
281 #define rb6 xxx[52]
282 #define sb1 xxx[53]
283 #define sb2 xxx[54]
284 #define sb3 xxx[55]
285 #define sb4 xxx[56]
286 #define sb5 xxx[57]
287 #define sb6 xxx[58]
288 #define sb7 xxx[59]
289
290 double
291 erf(double x)
292 {
293 int hx, ix, i;
294 double R, S, P, Q, s, y, z, r;
295
296 hx = ((int *)&x)[HIWORD];
297 ix = hx & 0x7fffffff;
298
299 if (ix >= 0x7ff00000) { /* erf(nan)=nan */
300 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
301 if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */
302 return (x);
303 #endif
304 i = ((unsigned)hx >> 31) << 1;
305 return ((double)(1 - i) + one / x); /* erf(+-inf)=+-1 */
306 }
307
308 if (ix < 0x3feb0000) { /* |x|<0.84375 */
309 if (ix < 0x3e300000) { /* |x|<2**-28 */
310 if (ix < 0x00800000) /* avoid underflow */
311 return (0.125 * (8.0 * x + efx8 * x));
312
313 return (x + efx * x);
314 }
315
316 z = x * x;
317 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
318 s = one + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z *
319 qq5))));
320 y = r / s;
321 return (x + x * y);
322 }
323
324 if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
325 s = fabs(x) - one;
326 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 +
327 s * pa6)))));
328 Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 +
329 s * qa6)))));
330
331 if (hx >= 0)
332 return (erx + P / Q);
333 else
334 return (-erx - P / Q);
335 }
336
337 if (ix >= 0x40180000) { /* inf > |x| >= 6 */
338 if (hx >= 0)
339 return (one - tiny);
340 else
341 return (tiny - one);
342 }
343
344 x = fabs(x);
345 s = one / (x * x);
346
347 if (ix < 0x4006DB6E) { /* |x| < 1/0.35 */
348 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + s * (ra5 +
349 s * (ra6 + s * ra7))))));
350 S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + s * (sa5 +
351 s * (sa6 + s * (sa7 + s * sa8)))))));
352 } else { /* |x| >= 1/0.35 */
353 R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 +
354 s * rb6)))));
355 S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 + s * (sb5 +
356 s * (sb6 + s * sb7))))));
357 }
358
359 z = x;
360 ((int *)&z)[LOWORD] = 0;
361 r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
362
363 if (hx >= 0)
364 return (one - r / x);
365 else
366 return (r / x - one);
367 }
368
369 double
370 erfc(double x)
371 {
372 int hx, ix;
373 double R, S, P, Q, s, y, z, r;
374
375 hx = ((int *)&x)[HIWORD];
376 ix = hx & 0x7fffffff;
377
378 if (ix >= 0x7ff00000) { /* erfc(nan)=nan */
379 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
380 if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */
381 return (x);
382 #endif
383 /* erfc(+-inf)=0,2 */
384 return ((double)(((unsigned)hx >> 31) << 1) + one / x);
385 }
386
387 if (ix < 0x3feb0000) { /* |x| < 0.84375 */
388 if (ix < 0x3c700000) /* |x| < 2**-56 */
389 return (one - x);
390
391 z = x * x;
392 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
393 s = one + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z *
394 qq5))));
395 y = r / s;
396
397 if (hx < 0x3fd00000) { /* x < 1/4 */
398 return (one - (x + x * y));
399 } else {
400 r = x * y;
401 r += (x - half);
402 return (half - r);
403 }
404 }
405
406 if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
407 s = fabs(x) - one;
408 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 +
409 s * pa6)))));
410 Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 +
411 s * qa6)))));
412
413 if (hx >= 0) {
414 z = one - erx;
415 return (z - P / Q);
416 } else {
417 z = erx + P / Q;
418 return (one + z);
419 }
420 }
421
422 if (ix < 0x403c0000) { /* |x|<28 */
423 x = fabs(x);
424 s = one / (x * x);
425
426 if (ix < 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143 */
427 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 +
428 s * (ra5 + s * (ra6 + s * ra7))))));
429 S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 +
430 s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
431 } else {
432 /* |x| >= 1/.35 ~ 2.857143 */
433 if (hx < 0 && ix >= 0x40180000)
434 return (two - tiny); /* x < -6 */
435
436 R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 +
437 s * (rb5 + s * rb6)))));
438 S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 +
439 s * (sb5 + s * (sb6 + s * sb7))))));
440 }
441
442 z = x;
443 ((int *)&z)[LOWORD] = 0;
444 r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
445
446 if (hx > 0)
447 return (r / x);
448 else
449 return (two - r / x);
450 } else {
451 if (hx > 0)
452 return (tiny * tiny);
453 else
454 return (two - tiny);
455 }
456 }