lib/libm/noieee_src/n_erf.c

   1 /*      $NetBSD: n_erf.c,v 1.6 2003/08/07 16:44:50 agc Exp $    */
   2 /*-
   3  * Copyright (c) 1992, 1993
   4  *      The Regents of the University of California.  All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  * 3. Neither the name of the University nor the names of its contributors
  15  *    may be used to endorse or promote products derived from this software
  16  *    without specific prior written permission.
  17  *
  18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  28  * SUCH DAMAGE.
  29  */
  30
  31 #ifndef lint
  32 #if 0
  33 static char sccsid[] = "@(#)erf.c       8.1 (Berkeley) 6/4/93";
  34 #endif
  35 #endif /* not lint */
  36
  37 #include "mathimpl.h"
  38
  39 /* Modified Nov 30, 1992 P. McILROY:
  40  *      Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp)
  41  * Replaced even+odd with direct calculation for x < .84375,
  42  * to avoid destructive cancellation.
  43  *
  44  * Performance of erfc(x):
  45  * In 300000 trials in the range [.83, .84375] the
  46  * maximum observed error was 3.6ulp.
  47  *
  48  * In [.84735,1.25] the maximum observed error was <2.5ulp in
  49  * 100000 runs in the range [1.2, 1.25].
  50  *
  51  * In [1.25,26] (Not including subnormal results)
  52  * the error is < 1.7ulp.
  53  */
  54
  55 /* double erf(double x)
  56  * double erfc(double x)
  57  *                           x
  58  *                    2      |\
  59  *     erf(x)  =  ---------  | exp(-t*t)dt
  60  *                 sqrt(pi) \|
  61  *                           0
  62  *
  63  *     erfc(x) =  1-erf(x)
  64  *
  65  * Method:
  66  *      1. Reduce x to |x| by erf(-x) = -erf(x)
  67  *      2. For x in [0, 0.84375]
  68  *          erf(x)  = x + x*P(x^2)
  69  *          erfc(x) = 1 - erf(x)           if x<=0.25
  70  *                  = 0.5 + ((0.5-x)-x*P)  if x in [0.25,0.84375]
  71  *         where
  72  *                      2                2        4               20
  73  *              P =  P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x  )
  74  *         is an approximation to (erf(x)-x)/x with precision
  75  *
  76  *                                               -56.45
  77  *                      | P - (erf(x)-x)/x | <= 2
  78  *
  79  *
  80  *         Remark. The formula is derived by noting
  81  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
  82  *         and that
  83  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
  84  *         is close to one. The interval is chosen because the fixed
  85  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
  86  *         near 0.6174), and by some experiment, 0.84375 is chosen to
  87  *         guarantee the error is less than one ulp for erf.
  88  *
  89  *      3. For x in [0.84375,1.25], let s = x - 1, and
  90  *         c = 0.84506291151 rounded to single (24 bits)
  91  *              erf(x)  = c  + P1(s)/Q1(s)
  92  *              erfc(x) = (1-c)  - P1(s)/Q1(s)
  93  *              |P1/Q1 - (erf(x)-c)| <= 2**-59.06
  94  *         Remark: here we use the taylor series expansion at x=1.
  95  *              erf(1+s) = erf(1) + s*Poly(s)
  96  *                       = 0.845.. + P1(s)/Q1(s)
  97  *         That is, we use rational approximation to approximate
  98  *                      erf(1+s) - (c = (single)0.84506291151)
  99  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
 100  *         where
 101  *              P1(s) = degree 6 poly in s
 102  *              Q1(s) = degree 6 poly in s
 103  *
 104  *      4. For x in [1.25, 2]; [2, 4]
 105  *              erf(x)  = 1.0 - tiny
 106  *              erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z))
 107  *
 108  *      Where z = 1/(x*x), R is degree 9, and S is degree 3;
 109  *
 110  *      5. For x in [4,28]
 111  *              erf(x)  = 1.0 - tiny
 112  *              erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z))
 113  *
 114  *      Where P is degree 14 polynomial in 1/(x*x).
 115  *
 116  *      Notes:
 117  *         Here 4 and 5 make use of the asymptotic series
 118  *                        exp(-x*x)
 119  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) );
 120  *                        x*sqrt(pi)
 121  *
 122  *              where for z = 1/(x*x)
 123  *              P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...))))
 124  *
 125  *         Thus we use rational approximation to approximate
 126  *              erfc*x*exp(x*x) ~ 1/sqrt(pi);
 127  *
 128  *              The error bound for the target function, G(z) for
 129  *              the interval
 130  *              [4, 28]:
 131  *              |eps + 1/(z)P(z) - G(z)| < 2**(-56.61)
 132  *              for [2, 4]:
 133  *              |R(z)/S(z) - G(z)|       < 2**(-58.24)
 134  *              for [1.25, 2]:
 135  *              |R(z)/S(z) - G(z)|       < 2**(-58.12)
 136  *
 137  *      6. For inf > x >= 28
 138  *              erf(x)  = 1 - tiny  (raise inexact)
 139  *              erfc(x) = tiny*tiny (raise underflow)
 140  *
 141  *      7. Special cases:
 142  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
 143  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
 144  *              erfc/erf(NaN) is NaN
 145  */
 146
 147 #if defined(__vax__) || defined(tahoe)
 148 #define _IEEE   0
 149 #define TRUNC(x) (x) = (float)(x)
 150 #else
 151 #define _IEEE   1
 152 #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000
 153 #define infnan(x) 0.0
 154 #endif
 155
 156 #ifdef _IEEE_LIBM
 157 /*
 158  * redefining "___function" to "function" in _IEEE_LIBM mode
 159  */
 160 #include "ieee_libm.h"
 161 #endif
 162
 163 static const double
 164 tiny        = 1e-300,
 165 half        = 0.5,
 166 one         = 1.0,
 167 two         = 2.0,
 168 c           = 8.45062911510467529297e-01, /* (float)0.84506291151 */
 169 /*
 170  * Coefficients for approximation to erf in [0,0.84375]
 171  */
 172 p0t8 = 1.02703333676410051049867154944018394163280,
 173 p0 =   1.283791670955125638123339436800229927041e-0001,
 174 p1 =  -3.761263890318340796574473028946097022260e-0001,
 175 p2 =   1.128379167093567004871858633779992337238e-0001,
 176 p3 =  -2.686617064084433642889526516177508374437e-0002,
 177 p4 =   5.223977576966219409445780927846432273191e-0003,
 178 p5 =  -8.548323822001639515038738961618255438422e-0004,
 179 p6 =   1.205520092530505090384383082516403772317e-0004,
 180 p7 =  -1.492214100762529635365672665955239554276e-0005,
 181 p8 =   1.640186161764254363152286358441771740838e-0006,
 182 p9 =  -1.571599331700515057841960987689515895479e-0007,
 183 p10=   1.073087585213621540635426191486561494058e-0008;
 184 /*
 185  * Coefficients for approximation to erf in [0.84375,1.25]
 186  */
 187 static const double
 188 pa0 =  -2.362118560752659485957248365514511540287e-0003,
 189 pa1 =   4.148561186837483359654781492060070469522e-0001,
 190 pa2 =  -3.722078760357013107593507594535478633044e-0001,
 191 pa3 =   3.183466199011617316853636418691420262160e-0001,
 192 pa4 =  -1.108946942823966771253985510891237782544e-0001,
 193 pa5 =   3.547830432561823343969797140537411825179e-0002,
 194 pa6 =  -2.166375594868790886906539848893221184820e-0003,
 195 qa1 =   1.064208804008442270765369280952419863524e-0001,
 196 qa2 =   5.403979177021710663441167681878575087235e-0001,
 197 qa3 =   7.182865441419627066207655332170665812023e-0002,
 198 qa4 =   1.261712198087616469108438860983447773726e-0001,
 199 qa5 =   1.363708391202905087876983523620537833157e-0002,
 200 qa6 =   1.198449984679910764099772682882189711364e-0002;
 201 /*
 202  * log(sqrt(pi)) for large x expansions.
 203  * The tail (lsqrtPI_lo) is included in the rational
 204  * approximations.
 205 */
 206 static const double
 207    lsqrtPI_hi = .5723649429247000819387380943226;
 208 /*
 209  * lsqrtPI_lo = .000000000000000005132975581353913;
 210  *
 211  * Coefficients for approximation to erfc in [2, 4]
 212 */
 213 static const double
 214 rb0  =  -1.5306508387410807582e-010,    /* includes lsqrtPI_lo */
 215 rb1  =   2.15592846101742183841910806188e-008,
 216 rb2  =   6.24998557732436510470108714799e-001,
 217 rb3  =   8.24849222231141787631258921465e+000,
 218 rb4  =   2.63974967372233173534823436057e+001,
 219 rb5  =   9.86383092541570505318304640241e+000,
 220 rb6  =  -7.28024154841991322228977878694e+000,
 221 rb7  =   5.96303287280680116566600190708e+000,
 222 rb8  =  -4.40070358507372993983608466806e+000,
 223 rb9  =   2.39923700182518073731330332521e+000,
 224 rb10 =  -6.89257464785841156285073338950e-001,
 225 sb1  =   1.56641558965626774835300238919e+001,
 226 sb2  =   7.20522741000949622502957936376e+001,
 227 sb3  =   9.60121069770492994166488642804e+001;
 228 /*
 229  * Coefficients for approximation to erfc in [1.25, 2]
 230 */
 231 static const double
 232 rc0  =  -2.47925334685189288817e-007,   /* includes lsqrtPI_lo */
 233 rc1  =   1.28735722546372485255126993930e-005,
 234 rc2  =   6.24664954087883916855616917019e-001,
 235 rc3  =   4.69798884785807402408863708843e+000,
 236 rc4  =   7.61618295853929705430118701770e+000,
 237 rc5  =   9.15640208659364240872946538730e-001,
 238 rc6  =  -3.59753040425048631334448145935e-001,
 239 rc7  =   1.42862267989304403403849619281e-001,
 240 rc8  =  -4.74392758811439801958087514322e-002,
 241 rc9  =   1.09964787987580810135757047874e-002,
 242 rc10 =  -1.28856240494889325194638463046e-003,
 243 sc1  =   9.97395106984001955652274773456e+000,
 244 sc2  =   2.80952153365721279953959310660e+001,
 245 sc3  =   2.19826478142545234106819407316e+001;
 246 /*
 247  * Coefficients for approximation to  erfc in [4,28]
 248  */
 249 static const double
 250 rd0  =  -2.1491361969012978677e-016,    /* includes lsqrtPI_lo */
 251 rd1  =  -4.99999999999640086151350330820e-001,
 252 rd2  =   6.24999999772906433825880867516e-001,
 253 rd3  =  -1.54166659428052432723177389562e+000,
 254 rd4  =   5.51561147405411844601985649206e+000,
 255 rd5  =  -2.55046307982949826964613748714e+001,
 256 rd6  =   1.43631424382843846387913799845e+002,
 257 rd7  =  -9.45789244999420134263345971704e+002,
 258 rd8  =   6.94834146607051206956384703517e+003,
 259 rd9  =  -5.27176414235983393155038356781e+004,
 260 rd10 =   3.68530281128672766499221324921e+005,
 261 rd11 =  -2.06466642800404317677021026611e+006,
 262 rd12 =   7.78293889471135381609201431274e+006,
 263 rd13 =  -1.42821001129434127360582351685e+007;
 264
 265 double
 266 erf(double x)
 267 {
 268         double R,S,P,Q,ax,s,y,z,r;
 269         if(!finite(x)) {                /* erf(nan)=nan */
 270             if (isnan(x))
 271                 return(x);
 272             return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */
 273         }
 274         if ((ax = x) < 0)
 275                 ax = - ax;
 276         if (ax < .84375) {
 277             if (ax < 3.7e-09) {
 278                 if (ax < 1.0e-308)
 279                     return 0.125*(8.0*x+p0t8*x);  /*avoid underflow */
 280                 return x + p0*x;
 281             }
 282             y = x*x;
 283             r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
 284                         y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
 285             return x + x*(p0+r);
 286         }
 287         if (ax < 1.25) {                /* 0.84375 <= |x| < 1.25 */
 288             s = fabs(x)-one;
 289             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 290             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 291             if (x>=0)
 292                 return (c + P/Q);
 293             else
 294                 return (-c - P/Q);
 295         }
 296         if (ax >= 6.0) {                /* inf>|x|>=6 */
 297             if (x >= 0.0)
 298                 return (one-tiny);
 299             else
 300                 return (tiny-one);
 301         }
 302     /* 1.25 <= |x| < 6 */
 303         z = -ax*ax;
 304         s = -one/z;
 305         if (ax < 2.0) {
 306                 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
 307                         s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
 308                 S = one+s*(sc1+s*(sc2+s*sc3));
 309         } else {
 310                 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
 311                         s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
 312                 S = one+s*(sb1+s*(sb2+s*sb3));
 313         }
 314         y = (R/S -.5*s) - lsqrtPI_hi;
 315         z += y;
 316         z = exp(z)/ax;
 317         if (x >= 0)
 318                 return (one-z);
 319         else
 320                 return (z-one);
 321 }
 322
 323 double
 324 erfc(double x)
 325 {
 326         double R,S,P,Q,s,ax,y,z,r;
 327         if (!finite(x)) {
 328                 if (isnan(x))           /* erfc(NaN) = NaN */
 329                         return(x);
 330                 else if (x > 0)         /* erfc(+-inf)=0,2 */
 331                         return 0.0;
 332                 else
 333                         return 2.0;
 334         }
 335         if ((ax = x) < 0)
 336                 ax = -ax;
 337         if (ax < .84375) {                      /* |x|<0.84375 */
 338             if (ax < 1.38777878078144568e-17)   /* |x|<2**-56 */
 339                 return one-x;
 340             y = x*x;
 341             r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+
 342                         y*(p6+y*(p7+y*(p8+y*(p9+y*p10)))))))));
 343             if (ax < .0625) {   /* |x|<2**-4 */
 344                 return (one-(x+x*(p0+r)));
 345             } else {
 346                 r = x*(p0+r);
 347                 r += (x-half);
 348                 return (half - r);
 349             }
 350         }
 351         if (ax < 1.25) {                /* 0.84375 <= |x| < 1.25 */
 352             s = ax-one;
 353             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 354             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 355             if (x>=0) {
 356                 z  = one-c; return z - P/Q;
 357             } else {
 358                 z = c+P/Q; return one+z;
 359             }
 360         }
 361         if (ax >= 28) { /* Out of range */
 362                 if (x>0)
 363                         return (tiny*tiny);
 364                 else
 365                         return (two-tiny);
 366         }
 367         z = ax;
 368         TRUNC(z);
 369         y = z - ax; y *= (ax+z);
 370         z *= -z;                        /* Here z + y = -x^2 */
 371                 s = one/(-z-y);         /* 1/(x*x) */
 372         if (ax >= 4) {                  /* 6 <= ax */
 373                 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+
 374                         s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10
 375                         +s*(rd11+s*(rd12+s*rd13))))))))))));
 376                 y += rd0;
 377         } else if (ax >= 2) {
 378                 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+
 379                         s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10)))))))));
 380                 S = one+s*(sb1+s*(sb2+s*sb3));
 381                 y += R/S;
 382                 R = -.5*s;
 383         } else {
 384                 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+
 385                         s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10)))))))));
 386                 S = one+s*(sc1+s*(sc2+s*sc3));
 387                 y += R/S;
 388                 R = -.5*s;
 389         }
 390         /* return exp(-x^2 - lsqrtPI_hi + R + y)/x;     */
 391         s = ((R + y) - lsqrtPI_hi) + z;
 392         y = (((z-s) - lsqrtPI_hi) + R) + y;
 393         r = __exp__D(s, y)/x;
 394         if (x>0)
 395                 return r;
 396         else
 397                 return two-r;
 398 }