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 }