js/src/fdlibm/s_erf.c

   1 /* -*- Mode: C; tab-width: 8; indent-tabs-mode: nil; c-basic-offset: 4 -*-
   2  *
   3  * ***** BEGIN LICENSE BLOCK *****
   4  * Version: MPL 1.1/GPL 2.0/LGPL 2.1
   5  *
   6  * The contents of this file are subject to the Mozilla Public License Version
   7  * 1.1 (the "License"); you may not use this file except in compliance with
   8  * the License. You may obtain a copy of the License at
   9  * http://www.mozilla.org/MPL/
  10  *
  11  * Software distributed under the License is distributed on an "AS IS" basis,
  12  * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
  13  * for the specific language governing rights and limitations under the
  14  * License.
  15  *
  16  * The Original Code is Mozilla Communicator client code, released
  17  * March 31, 1998.
  18  *
  19  * The Initial Developer of the Original Code is
  20  * Sun Microsystems, Inc.
  21  * Portions created by the Initial Developer are Copyright (C) 1998
  22  * the Initial Developer. All Rights Reserved.
  23  *
  24  * Contributor(s):
  25  *
  26  * Alternatively, the contents of this file may be used under the terms of
  27  * either of the GNU General Public License Version 2 or later (the "GPL"),
  28  * or the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
  29  * in which case the provisions of the GPL or the LGPL are applicable instead
  30  * of those above. If you wish to allow use of your version of this file only
  31  * under the terms of either the GPL or the LGPL, and not to allow others to
  32  * use your version of this file under the terms of the MPL, indicate your
  33  * decision by deleting the provisions above and replace them with the notice
  34  * and other provisions required by the GPL or the LGPL. If you do not delete
  35  * the provisions above, a recipient may use your version of this file under
  36  * the terms of any one of the MPL, the GPL or the LGPL.
  37  *
  38  * ***** END LICENSE BLOCK ***** */
  39
  40 /* @(#)s_erf.c 1.3 95/01/18 */
  41 /*
  42  * ====================================================
  43  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  44  *
  45  * Developed at SunSoft, a Sun Microsystems, Inc. business.
  46  * Permission to use, copy, modify, and distribute this
  47  * software is freely granted, provided that this notice
  48  * is preserved.
  49  * ====================================================
  50  */
  51
  52 /* double erf(double x)
  53  * double erfc(double x)
  54  *                           x
  55  *                    2      |\
  56  *     erf(x)  =  ---------  | exp(-t*t)dt
  57  *                 sqrt(pi) \|
  58  *                           0
  59  *
  60  *     erfc(x) =  1-erf(x)
  61  *  Note that
  62  *              erf(-x) = -erf(x)
  63  *              erfc(-x) = 2 - erfc(x)
  64  *
  65  * Method:
  66  *      1. For |x| in [0, 0.84375]
  67  *          erf(x)  = x + x*R(x^2)
  68  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
  69  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
  70  *         where R = P/Q where P is an odd poly of degree 8 and
  71  *         Q is an odd poly of degree 10.
  72  *                                               -57.90
  73  *                      | R - (erf(x)-x)/x | <= 2
  74  *
  75  *
  76  *         Remark. The formula is derived by noting
  77  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
  78  *         and that
  79  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
  80  *         is close to one. The interval is chosen because the fix
  81  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
  82  *         near 0.6174), and by some experiment, 0.84375 is chosen to
  83  *         guarantee the error is less than one ulp for erf.
  84  *
  85  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
  86  *         c = 0.84506291151 rounded to single (24 bits)
  87  *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
  88  *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
  89  *                        1+(c+P1(s)/Q1(s))    if x < 0
  90  *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
  91  *         Remark: here we use the taylor series expansion at x=1.
  92  *              erf(1+s) = erf(1) + s*Poly(s)
  93  *                       = 0.845.. + P1(s)/Q1(s)
  94  *         That is, we use rational approximation to approximate
  95  *                      erf(1+s) - (c = (single)0.84506291151)
  96  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
  97  *         where
  98  *              P1(s) = degree 6 poly in s
  99  *              Q1(s) = degree 6 poly in s
 100  *
 101  *      3. For x in [1.25,1/0.35(~2.857143)],
 102  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
 103  *              erf(x)  = 1 - erfc(x)
 104  *         where
 105  *              R1(z) = degree 7 poly in z, (z=1/x^2)
 106  *              S1(z) = degree 8 poly in z
 107  *
 108  *      4. For x in [1/0.35,28]
 109  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
 110  *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
 111  *                      = 2.0 - tiny            (if x <= -6)
 112  *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
 113  *              erf(x)  = sign(x)*(1.0 - tiny)
 114  *         where
 115  *              R2(z) = degree 6 poly in z, (z=1/x^2)
 116  *              S2(z) = degree 7 poly in z
 117  *
 118  *      Note1:
 119  *         To compute exp(-x*x-0.5625+R/S), let s be a single
 120  *         precision number and s := x; then
 121  *              -x*x = -s*s + (s-x)*(s+x)
 122  *              exp(-x*x-0.5626+R/S) =
 123  *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
 124  *      Note2:
 125  *         Here 4 and 5 make use of the asymptotic series
 126  *                        exp(-x*x)
 127  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
 128  *                        x*sqrt(pi)
 129  *         We use rational approximation to approximate
 130  *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
 131  *         Here is the error bound for R1/S1 and R2/S2
 132  *              |R1/S1 - f(x)|  < 2**(-62.57)
 133  *              |R2/S2 - f(x)|  < 2**(-61.52)
 134  *
 135  *      5. For inf > x >= 28
 136  *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
 137  *              erfc(x) = tiny*tiny (raise underflow) if x > 0
 138  *                      = 2 - tiny if x<0
 139  *
 140  *      7. Special case:
 141  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
 142  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
 143  *              erfc/erf(NaN) is NaN
 144  */
 145
 146
 147 #include "fdlibm.h"
 148
 149 #ifdef __STDC__
 150 static const double
 151 #else
 152 static double
 153 #endif
 154 tiny        = 1e-300,
 155 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 156 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 157 two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
 158         /* c = (float)0.84506291151 */
 159 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 160 /*
 161  * Coefficients for approximation to  erf on [0,0.84375]
 162  */
 163 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 164 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
 165 pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 166 pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
 167 pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
 168 pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
 169 pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
 170 qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
 171 qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
 172 qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
 173 qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
 174 qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
 175 /*
 176  * Coefficients for approximation to  erf  in [0.84375,1.25]
 177  */
 178 pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
 179 pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
 180 pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
 181 pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
 182 pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
 183 pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
 184 pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
 185 qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
 186 qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
 187 qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
 188 qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
 189 qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
 190 qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
 191 /*
 192  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 193  */
 194 ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
 195 ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
 196 ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
 197 ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
 198 ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
 199 ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
 200 ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
 201 ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
 202 sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
 203 sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
 204 sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
 205 sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
 206 sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
 207 sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
 208 sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
 209 sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
 210 /*
 211  * Coefficients for approximation to  erfc in [1/.35,28]
 212  */
 213 rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
 214 rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
 215 rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
 216 rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
 217 rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
 218 rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
 219 rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
 220 sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
 221 sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
 222 sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
 223 sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
 224 sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
 225 sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
 226 sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
 227
 228 #ifdef __STDC__
 229         double fd_erf(double x)
 230 #else
 231         double fd_erf(x)
 232         double x;
 233 #endif
 234 {
 235         fd_twoints u;
 236         int hx,ix,i;
 237         double R,S,P,Q,s,y,z,r;
 238         u.d = x;
 239         hx = __HI(u);
 240         ix = hx&0x7fffffff;
 241         if(ix>=0x7ff00000) {            /* erf(nan)=nan */
 242             i = ((unsigned)hx>>31)<<1;
 243             return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
 244         }
 245
 246         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
 247             if(ix < 0x3e300000) {       /* |x|<2**-28 */
 248                 if (ix < 0x00800000)
 249                     return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
 250                 return x + efx*x;
 251             }
 252             z = x*x;
 253             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 254             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 255             y = r/s;
 256             return x + x*y;
 257         }
 258         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
 259             s = fd_fabs(x)-one;
 260             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 261             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 262             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 263         }
 264         if (ix >= 0x40180000) {         /* inf>|x|>=6 */
 265             if(hx>=0) return one-tiny; else return tiny-one;
 266         }
 267         x = fd_fabs(x);
 268         s = one/(x*x);
 269         if(ix< 0x4006DB6E) {    /* |x| < 1/0.35 */
 270             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 271                                 ra5+s*(ra6+s*ra7))))));
 272             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 273                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
 274         } else {        /* |x| >= 1/0.35 */
 275             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 276                                 rb5+s*rb6)))));
 277             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 278                                 sb5+s*(sb6+s*sb7))))));
 279         }
 280         z  = x;
 281         u.d = z;
 282         __LO(u) = 0;
 283         z = u.d;
 284         r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
 285         if(hx>=0) return one-r/x; else return  r/x-one;
 286 }
 287
 288 #ifdef __STDC__
 289         double erfc(double x)
 290 #else
 291         double erfc(x)
 292         double x;
 293 #endif
 294 {
 295         fd_twoints u;
 296         int hx,ix;
 297         double R,S,P,Q,s,y,z,r;
 298         u.d = x;
 299         hx = __HI(u);
 300         ix = hx&0x7fffffff;
 301         if(ix>=0x7ff00000) {                    /* erfc(nan)=nan */
 302                                                 /* erfc(+-inf)=0,2 */
 303             return (double)(((unsigned)hx>>31)<<1)+one/x;
 304         }
 305
 306         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
 307             if(ix < 0x3c700000)         /* |x|<2**-56 */
 308                 return one-x;
 309             z = x*x;
 310             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 311             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 312             y = r/s;
 313             if(hx < 0x3fd00000) {       /* x<1/4 */
 314                 return one-(x+x*y);
 315             } else {
 316                 r = x*y;
 317                 r += (x-half);
 318                 return half - r ;
 319             }
 320         }
 321         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
 322             s = fd_fabs(x)-one;
 323             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 324             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 325             if(hx>=0) {
 326                 z  = one-erx; return z - P/Q;
 327             } else {
 328                 z = erx+P/Q; return one+z;
 329             }
 330         }
 331         if (ix < 0x403c0000) {          /* |x|<28 */
 332             x = fd_fabs(x);
 333             s = one/(x*x);
 334             if(ix< 0x4006DB6D) {        /* |x| < 1/.35 ~ 2.857143*/
 335                 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 336                                 ra5+s*(ra6+s*ra7))))));
 337                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 338                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
 339             } else {                    /* |x| >= 1/.35 ~ 2.857143 */
 340                 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
 341                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 342                                 rb5+s*rb6)))));
 343                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 344                                 sb5+s*(sb6+s*sb7))))));
 345             }
 346             z  = x;
 347             u.d = z;
 348             __LO(u)  = 0;
 349             z = u.d;
 350             r  =  __ieee754_exp(-z*z-0.5625)*
 351                         __ieee754_exp((z-x)*(z+x)+R/S);
 352             if(hx>0) return r/x; else return two-r/x;
 353         } else {
 354             if(hx>0) return tiny*tiny; else return two-tiny;
 355         }
 356 }