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