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