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_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 + 306 z *(qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); 307 y = r / s; 308 return (x + x * y); 309 } 310 if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 311 s = fabs(x) - one; 312 P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + 313 s * (pa5 + s * pa6))))); 314 Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + 315 s * (qa5 + s * qa6))))); 316 if (hx >= 0) 317 return (erx + P / Q); 318 else 319 return (-erx - P / Q); 320 } 321 if (ix >= 0x40180000) { /* inf > |x| >= 6 */ 322 if (hx >= 0) 323 return (one - tiny); 324 else 325 return (tiny - one); 326 } 327 x = fabs(x); 328 s = one / (x * x); 329 if (ix < 0x4006DB6E) { /* |x| < 1/0.35 */ 330 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + 331 s * (ra5 + s * (ra6 + s * ra7)))))); 332 S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + 333 s * (sa5 + s * (sa6 + s * (sa7 + s * sa8))))))); 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 + 371 z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); 372 y = r / s; 373 if (hx < 0x3fd00000) { /* x < 1/4 */ 374 return (one - (x + x * y)); 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 } else { 391 z = erx + P / Q; 392 return (one + z); 393 } 394 } 395 if (ix < 0x403c0000) { /* |x|<28 */ 396 x = fabs(x); 397 s = one / (x * x); 398 if (ix < 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143 */ 399 R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + 400 s * (ra5 + s * (ra6 + s * ra7)))))); 401 S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + 402 s * (sa5 + s * (sa6 + s * (sa7 + s * sa8))))))); 403 } else { 404 /* |x| >= 1/.35 ~ 2.857143 */ 405 if (hx < 0 && ix >= 0x40180000) 406 return (two - tiny); /* x < -6 */ 407 408 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 } else { 421 if (hx > 0) 422 return (tiny * tiny); 423 else 424 return (two - tiny); 425 } 426 } 427