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5261 libm should stop using synonyms.h
5298 fabs is 0-sized, confuses dis(1) and others
Reviewed by: Josef 'Jeff' Sipek <jeffpc@josefsipek.net>
Approved by: Gordon Ross <gwr@nexenta.com>
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--- old/usr/src/lib/libm/common/C/erf.c
+++ new/usr/src/lib/libm/common/C/erf.c
1 1 /*
2 2 * CDDL HEADER START
3 3 *
4 4 * The contents of this file are subject to the terms of the
5 5 * Common Development and Distribution License (the "License").
6 6 * You may not use this file except in compliance with the License.
7 7 *
8 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 9 * or http://www.opensolaris.org/os/licensing.
10 10 * See the License for the specific language governing permissions
11 11 * and limitations under the License.
12 12 *
13 13 * When distributing Covered Code, include this CDDL HEADER in each
14 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 15 * If applicable, add the following below this CDDL HEADER, with the
16 16 * fields enclosed by brackets "[]" replaced with your own identifying
17 17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 18 *
19 19 * CDDL HEADER END
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20 20 */
21 21
22 22 /*
23 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
24 24 */
25 25 /*
26 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
27 27 * Use is subject to license terms.
28 28 */
29 29
30 -#pragma weak erf = __erf
31 -#pragma weak erfc = __erfc
30 +#pragma weak __erf = erf
31 +#pragma weak __erfc = erfc
32 32
33 33 /* INDENT OFF */
34 34 /*
35 35 * double erf(double x)
36 36 * double erfc(double x)
37 37 * x
38 38 * 2 |\
39 39 * erf(x) = --------- | exp(-t*t)dt
40 40 * sqrt(pi) \|
41 41 * 0
42 42 *
43 43 * erfc(x) = 1-erf(x)
44 44 * Note that
45 45 * erf(-x) = -erf(x)
46 46 * erfc(-x) = 2 - erfc(x)
47 47 *
48 48 * Method:
49 49 * 1. For |x| in [0, 0.84375]
50 50 * erf(x) = x + x*R(x^2)
51 51 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
52 52 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
53 53 * where R = P/Q where P is an odd poly of degree 8 and
54 54 * Q is an odd poly of degree 10.
55 55 * -57.90
56 56 * | R - (erf(x)-x)/x | <= 2
57 57 *
58 58 *
59 59 * Remark. The formula is derived by noting
60 60 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
61 61 * and that
62 62 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
63 63 * is close to one. The interval is chosen because the fix
64 64 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
65 65 * near 0.6174), and by some experiment, 0.84375 is chosen to
66 66 * guarantee the error is less than one ulp for erf.
67 67 *
68 68 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
69 69 * c = 0.84506291151 rounded to single (24 bits)
70 70 * erf(x) = sign(x) * (c + P1(s)/Q1(s))
71 71 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
72 72 * 1+(c+P1(s)/Q1(s)) if x < 0
73 73 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
74 74 * Remark: here we use the taylor series expansion at x=1.
75 75 * erf(1+s) = erf(1) + s*Poly(s)
76 76 * = 0.845.. + P1(s)/Q1(s)
77 77 * That is, we use rational approximation to approximate
78 78 * erf(1+s) - (c = (single)0.84506291151)
79 79 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
80 80 * where
81 81 * P1(s) = degree 6 poly in s
82 82 * Q1(s) = degree 6 poly in s
83 83 *
84 84 * 3. For x in [1.25,1/0.35(~2.857143)],
85 85 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
86 86 * erf(x) = 1 - erfc(x)
87 87 * where
88 88 * R1(z) = degree 7 poly in z, (z=1/x^2)
89 89 * S1(z) = degree 8 poly in z
90 90 *
91 91 * 4. For x in [1/0.35,28]
92 92 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
93 93 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
94 94 * = 2.0 - tiny (if x <= -6)
95 95 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
96 96 * erf(x) = sign(x)*(1.0 - tiny)
97 97 * where
98 98 * R2(z) = degree 6 poly in z, (z=1/x^2)
99 99 * S2(z) = degree 7 poly in z
100 100 *
101 101 * Note1:
102 102 * To compute exp(-x*x-0.5625+R/S), let s be a single
103 103 * precision number and s := x; then
104 104 * -x*x = -s*s + (s-x)*(s+x)
105 105 * exp(-x*x-0.5626+R/S) =
106 106 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
107 107 * Note2:
108 108 * Here 4 and 5 make use of the asymptotic series
109 109 * exp(-x*x)
110 110 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
111 111 * x*sqrt(pi)
112 112 * We use rational approximation to approximate
113 113 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
114 114 * Here is the error bound for R1/S1 and R2/S2
115 115 * |R1/S1 - f(x)| < 2**(-62.57)
116 116 * |R2/S2 - f(x)| < 2**(-61.52)
117 117 *
118 118 * 5. For inf > x >= 28
119 119 * erf(x) = sign(x) *(1 - tiny) (raise inexact)
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120 120 * erfc(x) = tiny*tiny (raise underflow) if x > 0
121 121 * = 2 - tiny if x<0
122 122 *
123 123 * 7. Special case:
124 124 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
125 125 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
126 126 * erfc/erf(NaN) is NaN
127 127 */
128 128 /* INDENT ON */
129 129
130 -#include "libm_synonyms.h" /* __erf, __erfc, __exp */
131 130 #include "libm_macros.h"
132 131 #include <math.h>
133 132
134 133 static const double xxx[] = {
135 134 /* tiny */ 1e-300,
136 135 /* half */ 5.00000000000000000000e-01, /* 3FE00000, 00000000 */
137 136 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
138 137 /* two */ 2.00000000000000000000e+00, /* 40000000, 00000000 */
139 138 /* erx */ 8.45062911510467529297e-01, /* 3FEB0AC1, 60000000 */
140 139 /*
141 140 * Coefficients for approximation to erf on [0,0.84375]
142 141 */
143 142 /* efx */ 1.28379167095512586316e-01, /* 3FC06EBA, 8214DB69 */
144 143 /* efx8 */ 1.02703333676410069053e+00, /* 3FF06EBA, 8214DB69 */
145 144 /* pp0 */ 1.28379167095512558561e-01, /* 3FC06EBA, 8214DB68 */
146 145 /* pp1 */ -3.25042107247001499370e-01, /* BFD4CD7D, 691CB913 */
147 146 /* pp2 */ -2.84817495755985104766e-02, /* BF9D2A51, DBD7194F */
148 147 /* pp3 */ -5.77027029648944159157e-03, /* BF77A291, 236668E4 */
149 148 /* pp4 */ -2.37630166566501626084e-05, /* BEF8EAD6, 120016AC */
150 149 /* qq1 */ 3.97917223959155352819e-01, /* 3FD97779, CDDADC09 */
151 150 /* qq2 */ 6.50222499887672944485e-02, /* 3FB0A54C, 5536CEBA */
152 151 /* qq3 */ 5.08130628187576562776e-03, /* 3F74D022, C4D36B0F */
153 152 /* qq4 */ 1.32494738004321644526e-04, /* 3F215DC9, 221C1A10 */
154 153 /* qq5 */ -3.96022827877536812320e-06, /* BED09C43, 42A26120 */
155 154 /*
156 155 * Coefficients for approximation to erf in [0.84375,1.25]
157 156 */
158 157 /* pa0 */ -2.36211856075265944077e-03, /* BF6359B8, BEF77538 */
159 158 /* pa1 */ 4.14856118683748331666e-01, /* 3FDA8D00, AD92B34D */
160 159 /* pa2 */ -3.72207876035701323847e-01, /* BFD7D240, FBB8C3F1 */
161 160 /* pa3 */ 3.18346619901161753674e-01, /* 3FD45FCA, 805120E4 */
162 161 /* pa4 */ -1.10894694282396677476e-01, /* BFBC6398, 3D3E28EC */
163 162 /* pa5 */ 3.54783043256182359371e-02, /* 3FA22A36, 599795EB */
164 163 /* pa6 */ -2.16637559486879084300e-03, /* BF61BF38, 0A96073F */
165 164 /* qa1 */ 1.06420880400844228286e-01, /* 3FBB3E66, 18EEE323 */
166 165 /* qa2 */ 5.40397917702171048937e-01, /* 3FE14AF0, 92EB6F33 */
167 166 /* qa3 */ 7.18286544141962662868e-02, /* 3FB2635C, D99FE9A7 */
168 167 /* qa4 */ 1.26171219808761642112e-01, /* 3FC02660, E763351F */
169 168 /* qa5 */ 1.36370839120290507362e-02, /* 3F8BEDC2, 6B51DD1C */
170 169 /* qa6 */ 1.19844998467991074170e-02, /* 3F888B54, 5735151D */
171 170 /*
172 171 * Coefficients for approximation to erfc in [1.25,1/0.35]
173 172 */
174 173 /* ra0 */ -9.86494403484714822705e-03, /* BF843412, 600D6435 */
175 174 /* ra1 */ -6.93858572707181764372e-01, /* BFE63416, E4BA7360 */
176 175 /* ra2 */ -1.05586262253232909814e+01, /* C0251E04, 41B0E726 */
177 176 /* ra3 */ -6.23753324503260060396e+01, /* C04F300A, E4CBA38D */
178 177 /* ra4 */ -1.62396669462573470355e+02, /* C0644CB1, 84282266 */
179 178 /* ra5 */ -1.84605092906711035994e+02, /* C067135C, EBCCABB2 */
180 179 /* ra6 */ -8.12874355063065934246e+01, /* C0545265, 57E4D2F2 */
181 180 /* ra7 */ -9.81432934416914548592e+00, /* C023A0EF, C69AC25C */
182 181 /* sa1 */ 1.96512716674392571292e+01, /* 4033A6B9, BD707687 */
183 182 /* sa2 */ 1.37657754143519042600e+02, /* 4061350C, 526AE721 */
184 183 /* sa3 */ 4.34565877475229228821e+02, /* 407B290D, D58A1A71 */
185 184 /* sa4 */ 6.45387271733267880336e+02, /* 40842B19, 21EC2868 */
186 185 /* sa5 */ 4.29008140027567833386e+02, /* 407AD021, 57700314 */
187 186 /* sa6 */ 1.08635005541779435134e+02, /* 405B28A3, EE48AE2C */
188 187 /* sa7 */ 6.57024977031928170135e+00, /* 401A47EF, 8E484A93 */
189 188 /* sa8 */ -6.04244152148580987438e-02, /* BFAEEFF2, EE749A62 */
190 189 /*
191 190 * Coefficients for approximation to erfc in [1/.35,28]
192 191 */
193 192 /* rb0 */ -9.86494292470009928597e-03, /* BF843412, 39E86F4A */
194 193 /* rb1 */ -7.99283237680523006574e-01, /* BFE993BA, 70C285DE */
195 194 /* rb2 */ -1.77579549177547519889e+01, /* C031C209, 555F995A */
196 195 /* rb3 */ -1.60636384855821916062e+02, /* C064145D, 43C5ED98 */
197 196 /* rb4 */ -6.37566443368389627722e+02, /* C083EC88, 1375F228 */
198 197 /* rb5 */ -1.02509513161107724954e+03, /* C0900461, 6A2E5992 */
199 198 /* rb6 */ -4.83519191608651397019e+02, /* C07E384E, 9BDC383F */
200 199 /* sb1 */ 3.03380607434824582924e+01, /* 403E568B, 261D5190 */
201 200 /* sb2 */ 3.25792512996573918826e+02, /* 40745CAE, 221B9F0A */
202 201 /* sb3 */ 1.53672958608443695994e+03, /* 409802EB, 189D5118 */
203 202 /* sb4 */ 3.19985821950859553908e+03, /* 40A8FFB7, 688C246A */
204 203 /* sb5 */ 2.55305040643316442583e+03, /* 40A3F219, CEDF3BE6 */
205 204 /* sb6 */ 4.74528541206955367215e+02, /* 407DA874, E79FE763 */
206 205 /* sb7 */ -2.24409524465858183362e+01 /* C03670E2, 42712D62 */
207 206 };
208 207
209 208 #define tiny xxx[0]
210 209 #define half xxx[1]
211 210 #define one xxx[2]
212 211 #define two xxx[3]
213 212 #define erx xxx[4]
214 213 /*
215 214 * Coefficients for approximation to erf on [0,0.84375]
216 215 */
217 216 #define efx xxx[5]
218 217 #define efx8 xxx[6]
219 218 #define pp0 xxx[7]
220 219 #define pp1 xxx[8]
221 220 #define pp2 xxx[9]
222 221 #define pp3 xxx[10]
223 222 #define pp4 xxx[11]
224 223 #define qq1 xxx[12]
225 224 #define qq2 xxx[13]
226 225 #define qq3 xxx[14]
227 226 #define qq4 xxx[15]
228 227 #define qq5 xxx[16]
229 228 /*
230 229 * Coefficients for approximation to erf in [0.84375,1.25]
231 230 */
232 231 #define pa0 xxx[17]
233 232 #define pa1 xxx[18]
234 233 #define pa2 xxx[19]
235 234 #define pa3 xxx[20]
236 235 #define pa4 xxx[21]
237 236 #define pa5 xxx[22]
238 237 #define pa6 xxx[23]
239 238 #define qa1 xxx[24]
240 239 #define qa2 xxx[25]
241 240 #define qa3 xxx[26]
242 241 #define qa4 xxx[27]
243 242 #define qa5 xxx[28]
244 243 #define qa6 xxx[29]
245 244 /*
246 245 * Coefficients for approximation to erfc in [1.25,1/0.35]
247 246 */
248 247 #define ra0 xxx[30]
249 248 #define ra1 xxx[31]
250 249 #define ra2 xxx[32]
251 250 #define ra3 xxx[33]
252 251 #define ra4 xxx[34]
253 252 #define ra5 xxx[35]
254 253 #define ra6 xxx[36]
255 254 #define ra7 xxx[37]
256 255 #define sa1 xxx[38]
257 256 #define sa2 xxx[39]
258 257 #define sa3 xxx[40]
259 258 #define sa4 xxx[41]
260 259 #define sa5 xxx[42]
261 260 #define sa6 xxx[43]
262 261 #define sa7 xxx[44]
263 262 #define sa8 xxx[45]
264 263 /*
265 264 * Coefficients for approximation to erfc in [1/.35,28]
266 265 */
267 266 #define rb0 xxx[46]
268 267 #define rb1 xxx[47]
269 268 #define rb2 xxx[48]
270 269 #define rb3 xxx[49]
271 270 #define rb4 xxx[50]
272 271 #define rb5 xxx[51]
273 272 #define rb6 xxx[52]
274 273 #define sb1 xxx[53]
275 274 #define sb2 xxx[54]
276 275 #define sb3 xxx[55]
277 276 #define sb4 xxx[56]
278 277 #define sb5 xxx[57]
279 278 #define sb6 xxx[58]
280 279 #define sb7 xxx[59]
281 280
282 281 double
283 282 erf(double x) {
284 283 int hx, ix, i;
285 284 double R, S, P, Q, s, y, z, r;
286 285
287 286 hx = ((int *) &x)[HIWORD];
288 287 ix = hx & 0x7fffffff;
289 288 if (ix >= 0x7ff00000) { /* erf(nan)=nan */
290 289 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
291 290 if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */
292 291 return (x);
293 292 #endif
294 293 i = ((unsigned) hx >> 31) << 1;
295 294 return ((double) (1 - i) + one / x); /* erf(+-inf)=+-1 */
296 295 }
297 296
298 297 if (ix < 0x3feb0000) { /* |x|<0.84375 */
299 298 if (ix < 0x3e300000) { /* |x|<2**-28 */
300 299 if (ix < 0x00800000) /* avoid underflow */
301 300 return (0.125 * (8.0 * x + efx8 * x));
302 301 return (x + efx * x);
303 302 }
304 303 z = x * x;
305 304 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
306 305 s = one +
307 306 z *(qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
308 307 y = r / s;
309 308 return (x + x * y);
310 309 }
311 310 if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
312 311 s = fabs(x) - one;
313 312 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 +
314 313 s * (pa5 + s * pa6)))));
315 314 Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 +
316 315 s * (qa5 + s * qa6)))));
317 316 if (hx >= 0)
318 317 return (erx + P / Q);
319 318 else
320 319 return (-erx - P / Q);
321 320 }
322 321 if (ix >= 0x40180000) { /* inf > |x| >= 6 */
323 322 if (hx >= 0)
324 323 return (one - tiny);
325 324 else
326 325 return (tiny - one);
327 326 }
328 327 x = fabs(x);
329 328 s = one / (x * x);
330 329 if (ix < 0x4006DB6E) { /* |x| < 1/0.35 */
331 330 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 +
332 331 s * (ra5 + s * (ra6 + s * ra7))))));
333 332 S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 +
334 333 s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
335 334 } else { /* |x| >= 1/0.35 */
336 335 R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 +
337 336 s * (rb5 + s * rb6)))));
338 337 S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 +
339 338 s * (sb5 + s * (sb6 + s * sb7))))));
340 339 }
341 340 z = x;
342 341 ((int *) &z)[LOWORD] = 0;
343 342 r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
344 343 if (hx >= 0)
345 344 return (one - r / x);
346 345 else
347 346 return (r / x - one);
348 347 }
349 348
350 349 double
351 350 erfc(double x) {
352 351 int hx, ix;
353 352 double R, S, P, Q, s, y, z, r;
354 353
355 354 hx = ((int *) &x)[HIWORD];
356 355 ix = hx & 0x7fffffff;
357 356 if (ix >= 0x7ff00000) { /* erfc(nan)=nan */
358 357 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
359 358 if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */
360 359 return (x);
361 360 #endif
362 361 /* erfc(+-inf)=0,2 */
363 362 return ((double) (((unsigned) hx >> 31) << 1) + one / x);
364 363 }
365 364
366 365 if (ix < 0x3feb0000) { /* |x| < 0.84375 */
367 366 if (ix < 0x3c700000) /* |x| < 2**-56 */
368 367 return (one - x);
369 368 z = x * x;
370 369 r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
371 370 s = one +
372 371 z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
373 372 y = r / s;
374 373 if (hx < 0x3fd00000) { /* x < 1/4 */
375 374 return (one - (x + x * y));
376 375 } else {
377 376 r = x * y;
378 377 r += (x - half);
379 378 return (half - r);
380 379 }
381 380 }
382 381 if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
383 382 s = fabs(x) - one;
384 383 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 +
385 384 s * (pa5 + s * pa6)))));
386 385 Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 +
387 386 s * (qa5 + s * qa6)))));
388 387 if (hx >= 0) {
389 388 z = one - erx;
390 389 return (z - P / Q);
391 390 } else {
392 391 z = erx + P / Q;
393 392 return (one + z);
394 393 }
395 394 }
396 395 if (ix < 0x403c0000) { /* |x|<28 */
397 396 x = fabs(x);
398 397 s = one / (x * x);
399 398 if (ix < 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143 */
400 399 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 +
401 400 s * (ra5 + s * (ra6 + s * ra7))))));
402 401 S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 +
403 402 s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
404 403 } else {
405 404 /* |x| >= 1/.35 ~ 2.857143 */
406 405 if (hx < 0 && ix >= 0x40180000)
407 406 return (two - tiny); /* x < -6 */
408 407
409 408 R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 +
410 409 s * (rb5 + s * rb6)))));
411 410 S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 +
412 411 s * (sb5 + s * (sb6 + s * sb7))))));
413 412 }
414 413 z = x;
415 414 ((int *) &z)[LOWORD] = 0;
416 415 r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
417 416 if (hx > 0)
418 417 return (r / x);
419 418 else
420 419 return (two - r / x);
421 420 } else {
422 421 if (hx > 0)
423 422 return (tiny * tiny);
424 423 else
425 424 return (two - tiny);
426 425 }
427 426 }
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