nsz Git - libm/blob - src/math/erf.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* double erf(double x)
  13  * double erfc(double x)
  14  *                           x
  15  *                    2      |\
  16  *     erf(x)  =  ---------  | exp(-t*t)dt
  17  *                 sqrt(pi) \|
  18  *                           0
  19  *
  20  *     erfc(x) =  1-erf(x)
  21  *  Note that
  22  *              erf(-x) = -erf(x)
  23  *              erfc(-x) = 2 - erfc(x)
  24  *
  25  * Method:
  26  *      1. For |x| in [0, 0.84375]
  27  *          erf(x)  = x + x*R(x^2)
  28  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
  29  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
  30  *         where R = P/Q where P is an odd poly of degree 8 and
  31  *         Q is an odd poly of degree 10.
  32  *                                               -57.90
  33  *                      | R - (erf(x)-x)/x | <= 2
  34  *
  35  *
  36  *         Remark. The formula is derived by noting
  37  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
  38  *         and that
  39  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
  40  *         is close to one. The interval is chosen because the fix
  41  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
  42  *         near 0.6174), and by some experiment, 0.84375 is chosen to
  43  *         guarantee the error is less than one ulp for erf.
  44  *
  45  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
  46  *         c = 0.84506291151 rounded to single (24 bits)
  47  *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
  48  *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
  49  *                        1+(c+P1(s)/Q1(s))    if x < 0
  50  *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
  51  *         Remark: here we use the taylor series expansion at x=1.
  52  *              erf(1+s) = erf(1) + s*Poly(s)
  53  *                       = 0.845.. + P1(s)/Q1(s)
  54  *         That is, we use rational approximation to approximate
  55  *                      erf(1+s) - (c = (single)0.84506291151)
  56  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
  57  *         where
  58  *              P1(s) = degree 6 poly in s
  59  *              Q1(s) = degree 6 poly in s
  60  *
  61  *      3. For x in [1.25,1/0.35(~2.857143)],
  62  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
  63  *              erf(x)  = 1 - erfc(x)
  64  *         where
  65  *              R1(z) = degree 7 poly in z, (z=1/x^2)
  66  *              S1(z) = degree 8 poly in z
  67  *
  68  *      4. For x in [1/0.35,28]
  69  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
  70  *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
  71  *                      = 2.0 - tiny            (if x <= -6)
  72  *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
  73  *              erf(x)  = sign(x)*(1.0 - tiny)
  74  *         where
  75  *              R2(z) = degree 6 poly in z, (z=1/x^2)
  76  *              S2(z) = degree 7 poly in z
  77  *
  78  *      Note1:
  79  *         To compute exp(-x*x-0.5625+R/S), let s be a single
  80  *         precision number and s := x; then
  81  *              -x*x = -s*s + (s-x)*(s+x)
  82  *              exp(-x*x-0.5626+R/S) =
  83  *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
  84  *      Note2:
  85  *         Here 4 and 5 make use of the asymptotic series
  86  *                        exp(-x*x)
  87  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
  88  *                        x*sqrt(pi)
  89  *         We use rational approximation to approximate
  90  *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
  91  *         Here is the error bound for R1/S1 and R2/S2
  92  *              |R1/S1 - f(x)|  < 2**(-62.57)
  93  *              |R2/S2 - f(x)|  < 2**(-61.52)
  94  *
  95  *      5. For inf > x >= 28
  96  *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
  97  *              erfc(x) = tiny*tiny (raise underflow) if x > 0
  98  *                      = 2 - tiny if x<0
  99  *
 100  *      7. Special case:
 101  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
 102  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
 103  *              erfc/erf(NaN) is NaN
 104  */
 105
 106 #include "libm.h"
 107
 108 static const double
 109 tiny = 1e-300,
 110 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 111 one  = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 112 two  = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
 113 /* c = (float)0.84506291151 */
 114 erx  = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 115 /*
 116  * Coefficients for approximation to  erf on [0,0.84375]
 117  */
 118 efx  =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 119 efx8 =  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
 120 pp0  =  1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 121 pp1  = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
 122 pp2  = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
 123 pp3  = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
 124 pp4  = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
 125 qq1  =  3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
 126 qq2  =  6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
 127 qq3  =  5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
 128 qq4  =  1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
 129 qq5  = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
 130 /*
 131  * Coefficients for approximation to  erf  in [0.84375,1.25]
 132  */
 133 pa0  = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
 134 pa1  =  4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
 135 pa2  = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
 136 pa3  =  3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
 137 pa4  = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
 138 pa5  =  3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
 139 pa6  = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
 140 qa1  =  1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
 141 qa2  =  5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
 142 qa3  =  7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
 143 qa4  =  1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
 144 qa5  =  1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
 145 qa6  =  1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
 146 /*
 147  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 148  */
 149 ra0  = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
 150 ra1  = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
 151 ra2  = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
 152 ra3  = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
 153 ra4  = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
 154 ra5  = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
 155 ra6  = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
 156 ra7  = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
 157 sa1  =  1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
 158 sa2  =  1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
 159 sa3  =  4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
 160 sa4  =  6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
 161 sa5  =  4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
 162 sa6  =  1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
 163 sa7  =  6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
 164 sa8  = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
 165 /*
 166  * Coefficients for approximation to  erfc in [1/.35,28]
 167  */
 168 rb0  = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
 169 rb1  = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
 170 rb2  = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
 171 rb3  = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
 172 rb4  = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
 173 rb5  = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
 174 rb6  = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
 175 sb1  =  3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
 176 sb2  =  3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
 177 sb3  =  1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
 178 sb4  =  3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
 179 sb5  =  2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
 180 sb6  =  4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
 181 sb7  = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
 182
 183 double erf(double x)
 184 {
 185         int32_t hx,ix,i;
 186         double R,S,P,Q,s,y,z,r;
 187
 188         GET_HIGH_WORD(hx, x);
 189         ix = hx & 0x7fffffff;
 190         if (ix >= 0x7ff00000) {
 191                 /* erf(nan)=nan, erf(+-inf)=+-1 */
 192                 i = ((uint32_t)hx>>31)<<1;
 193                 return (double)(1-i) + one/x;
 194         }
 195         if (ix < 0x3feb0000) {  /* |x|<0.84375 */
 196                 if (ix < 0x3e300000) {  /* |x|<2**-28 */
 197                         if (ix < 0x00800000)
 198                                 /* avoid underflow */
 199                                 return 0.125*(8.0*x + efx8*x);
 200                         return x + efx*x;
 201                 }
 202                 z = x*x;
 203                 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 204                 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 205                 y = r/s;
 206                 return x + x*y;
 207         }
 208         if (ix < 0x3ff40000) {  /* 0.84375 <= |x| < 1.25 */
 209                 s = fabs(x)-one;
 210                 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 211                 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 212                 if (hx >= 0)
 213                         return erx + P/Q;
 214                 return -erx - P/Q;
 215         }
 216         if (ix >= 0x40180000) {  /* inf > |x| >= 6 */
 217                 if (hx >= 0)
 218                         return one-tiny;
 219                 return tiny-one;
 220         }
 221         x = fabs(x);
 222         s = one/(x*x);
 223         if (ix < 0x4006DB6E) {  /* |x| < 1/0.35 */
 224                 R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 225                      ra5+s*(ra6+s*ra7))))));
 226                 S = one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 227                      sa5+s*(sa6+s*(sa7+s*sa8)))))));
 228         } else {                /* |x| >= 1/0.35 */
 229                 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 230                      rb5+s*rb6)))));
 231                 S = one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 232                      sb5+s*(sb6+s*sb7))))));
 233         }
 234         z = x;
 235         SET_LOW_WORD(z,0);
 236         r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
 237         if (hx >= 0)
 238                 return one-r/x;
 239         return r/x-one;
 240 }
 241
 242 double erfc(double x)
 243 {
 244         int32_t hx,ix;
 245         double R,S,P,Q,s,y,z,r;
 246
 247         GET_HIGH_WORD(hx, x);
 248         ix = hx & 0x7fffffff;
 249         if (ix >= 0x7ff00000) {
 250                 /* erfc(nan)=nan, erfc(+-inf)=0,2 */
 251                 return (double)(((uint32_t)hx>>31)<<1) + one/x;
 252         }
 253         if (ix < 0x3feb0000) {  /* |x| < 0.84375 */
 254                 if (ix < 0x3c700000)  /* |x| < 2**-56 */
 255                         return one - x;
 256                 z = x*x;
 257                 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 258                 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 259                 y = r/s;
 260                 if (hx < 0x3fd00000) {  /* x < 1/4 */
 261                         return one - (x+x*y);
 262                 } else {
 263                         r = x*y;
 264                         r += x-half;
 265                         return half - r ;
 266                 }
 267         }
 268         if (ix < 0x3ff40000) {  /* 0.84375 <= |x| < 1.25 */
 269                 s = fabs(x)-one;
 270                 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 271                 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 272                 if (hx >= 0) {
 273                         z = one-erx;
 274                         return z - P/Q;
 275                 } else {
 276                         z = erx+P/Q;
 277                         return one+z;
 278                 }
 279         }
 280         if (ix < 0x403c0000) {  /* |x| < 28 */
 281                 x = fabs(x);
 282                 s = one/(x*x);
 283                 if (ix < 0x4006DB6D) {  /* |x| < 1/.35 ~ 2.857143*/
 284                         R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 285                              ra5+s*(ra6+s*ra7))))));
 286                         S = one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 287                              sa5+s*(sa6+s*(sa7+s*sa8)))))));
 288                 } else {                /* |x| >= 1/.35 ~ 2.857143 */
 289                         if (hx < 0 && ix >= 0x40180000)  /* x < -6 */
 290                                 return two-tiny;
 291                         R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 292                              rb5+s*rb6)))));
 293                         S = one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 294                              sb5+s*(sb6+s*sb7))))));
 295                 }
 296                 z = x;
 297                 SET_LOW_WORD(z, 0);
 298                 r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S);
 299                 if (hx > 0)
 300                         return r/x;
 301                 return two-r/x;
 302         }
 303         if (hx > 0)
 304                 return tiny*tiny;
 305         return two-tiny;
 306 }