sysdeps/ieee754/dbl-64/s_erf.c

   1 /* @(#)s_erf.c 5.1 93/09/24 */
   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 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $";
  18 #endif
  19
  20 /* double erf(double x)
  21  * double erfc(double x)
  22  *                           x
  23  *                    2      |\
  24  *     erf(x)  =  ---------  | exp(-t*t)dt
  25  *                 sqrt(pi) \|
  26  *                           0
  27  *
  28  *     erfc(x) =  1-erf(x)
  29  *  Note that
  30  *              erf(-x) = -erf(x)
  31  *              erfc(-x) = 2 - erfc(x)
  32  *
  33  * Method:
  34  *      1. For |x| in [0, 0.84375]
  35  *          erf(x)  = x + x*R(x^2)
  36  *          erfc(x) = 1 - erf(x)           if x in [-.84375,0.25]
  37  *                  = 0.5 + ((0.5-x)-x*R)  if x in [0.25,0.84375]
  38  *         where R = P/Q where P is an odd poly of degree 8 and
  39  *         Q is an odd poly of degree 10.
  40  *                                               -57.90
  41  *                      | R - (erf(x)-x)/x | <= 2
  42  *
  43  *
  44  *         Remark. The formula is derived by noting
  45  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
  46  *         and that
  47  *          2/sqrt(pi) = 1.128379167095512573896158903121545171688
  48  *         is close to one. The interval is chosen because the fix
  49  *         point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
  50  *         near 0.6174), and by some experiment, 0.84375 is chosen to
  51  *         guarantee the error is less than one ulp for erf.
  52  *
  53  *      2. For |x| in [0.84375,1.25], let s = |x| - 1, and
  54  *         c = 0.84506291151 rounded to single (24 bits)
  55  *              erf(x)  = sign(x) * (c  + P1(s)/Q1(s))
  56  *              erfc(x) = (1-c)  - P1(s)/Q1(s) if x > 0
  57  *                        1+(c+P1(s)/Q1(s))    if x < 0
  58  *              |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
  59  *         Remark: here we use the taylor series expansion at x=1.
  60  *              erf(1+s) = erf(1) + s*Poly(s)
  61  *                       = 0.845.. + P1(s)/Q1(s)
  62  *         That is, we use rational approximation to approximate
  63  *                      erf(1+s) - (c = (single)0.84506291151)
  64  *         Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
  65  *         where
  66  *              P1(s) = degree 6 poly in s
  67  *              Q1(s) = degree 6 poly in s
  68  *
  69  *      3. For x in [1.25,1/0.35(~2.857143)],
  70  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
  71  *              erf(x)  = 1 - erfc(x)
  72  *         where
  73  *              R1(z) = degree 7 poly in z, (z=1/x^2)
  74  *              S1(z) = degree 8 poly in z
  75  *
  76  *      4. For x in [1/0.35,28]
  77  *              erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
  78  *                      = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
  79  *                      = 2.0 - tiny            (if x <= -6)
  80  *              erf(x)  = sign(x)*(1.0 - erfc(x)) if x < 6, else
  81  *              erf(x)  = sign(x)*(1.0 - tiny)
  82  *         where
  83  *              R2(z) = degree 6 poly in z, (z=1/x^2)
  84  *              S2(z) = degree 7 poly in z
  85  *
  86  *      Note1:
  87  *         To compute exp(-x*x-0.5625+R/S), let s be a single
  88  *         precision number and s := x; then
  89  *              -x*x = -s*s + (s-x)*(s+x)
  90  *              exp(-x*x-0.5626+R/S) =
  91  *                      exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
  92  *      Note2:
  93  *         Here 4 and 5 make use of the asymptotic series
  94  *                        exp(-x*x)
  95  *              erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
  96  *                        x*sqrt(pi)
  97  *         We use rational approximation to approximate
  98  *              g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
  99  *         Here is the error bound for R1/S1 and R2/S2
 100  *              |R1/S1 - f(x)|  < 2**(-62.57)
 101  *              |R2/S2 - f(x)|  < 2**(-61.52)
 102  *
 103  *      5. For inf > x >= 28
 104  *              erf(x)  = sign(x) *(1 - tiny)  (raise inexact)
 105  *              erfc(x) = tiny*tiny (raise underflow) if x > 0
 106  *                      = 2 - tiny if x<0
 107  *
 108  *      7. Special case:
 109  *              erf(0)  = 0, erf(inf)  = 1, erf(-inf) = -1,
 110  *              erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
 111  *              erfc/erf(NaN) is NaN
 112  */
 113
 114
 115 #include <math.h>
 116 #include <math_private.h>
 117
 118 static const double
 119 tiny        = 1e-300,
 120 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 121 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 122 two =  2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
 123         /* c = (float)0.84506291151 */
 124 erx =  8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 125 /*
 126  * Coefficients for approximation to  erf on [0,0.84375]
 127  */
 128 efx =  1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 129 efx8=  1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
 130 pp[]  =  {1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 131  -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
 132  -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
 133  -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
 134  -2.37630166566501626084e-05}, /* 0xBEF8EAD6, 0x120016AC */
 135 qq[]  =  {0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
 136   6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
 137   5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
 138   1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
 139  -3.96022827877536812320e-06}, /* 0xBED09C43, 0x42A26120 */
 140 /*
 141  * Coefficients for approximation to  erf  in [0.84375,1.25]
 142  */
 143 pa[]  = {-2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
 144   4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
 145  -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
 146   3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
 147  -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
 148   3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
 149  -2.16637559486879084300e-03}, /* 0xBF61BF38, 0x0A96073F */
 150 qa[]  =  {0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
 151   5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
 152   7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
 153   1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
 154   1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
 155   1.19844998467991074170e-02}, /* 0x3F888B54, 0x5735151D */
 156 /*
 157  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 158  */
 159 ra[]  = {-9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
 160  -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
 161  -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
 162  -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
 163  -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
 164  -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
 165  -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
 166  -9.81432934416914548592e+00}, /* 0xC023A0EF, 0xC69AC25C */
 167 sa[]  =  {0.0,1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
 168   1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
 169   4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
 170   6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
 171   4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
 172   1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
 173   6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
 174  -6.04244152148580987438e-02}, /* 0xBFAEEFF2, 0xEE749A62 */
 175 /*
 176  * Coefficients for approximation to  erfc in [1/.35,28]
 177  */
 178 rb[]  = {-9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
 179  -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
 180  -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
 181  -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
 182  -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
 183  -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
 184  -4.83519191608651397019e+02}, /* 0xC07E384E, 0x9BDC383F */
 185 sb[]  =  {0.0,3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
 186   3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
 187   1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
 188   3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
 189   2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
 190   4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
 191  -2.24409524465858183362e+01}; /* 0xC03670E2, 0x42712D62 */
 192
 193 double __erf(double x)
 194 {
 195         int32_t hx,ix,i;
 196         double R,S,P,Q,s,y,z,r;
 197         GET_HIGH_WORD(hx,x);
 198         ix = hx&0x7fffffff;
 199         if(ix>=0x7ff00000) {            /* erf(nan)=nan */
 200             i = ((u_int32_t)hx>>31)<<1;
 201             return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */
 202         }
 203
 204         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
 205             double r1,r2,s1,s2,s3,z2,z4;
 206             if(ix < 0x3e300000) {       /* |x|<2**-28 */
 207                 if (ix < 0x00800000)
 208                     return 0.125*(8.0*x+efx8*x);  /*avoid underflow */
 209                 return x + efx*x;
 210             }
 211             z = x*x;
 212 #ifdef DO_NOT_USE_THIS
 213             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 214             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 215 #else
 216             r1 = pp[0]+z*pp[1]; z2=z*z;
 217             r2 = pp[2]+z*pp[3]; z4=z2*z2;
 218             s1 = one+z*qq[1];
 219             s2 = qq[2]+z*qq[3];
 220             s3 = qq[4]+z*qq[5];
 221             r = r1 + z2*r2 + z4*pp[4];
 222             s  = s1 + z2*s2 + z4*s3;
 223 #endif
 224             y = r/s;
 225             return x + x*y;
 226         }
 227         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
 228             double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
 229             s = fabs(x)-one;
 230 #ifdef DO_NOT_USE_THIS
 231             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 232             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 233 #else
 234             P1 = pa[0]+s*pa[1]; s2=s*s;
 235             Q1 = one+s*qa[1];   s4=s2*s2;
 236             P2 = pa[2]+s*pa[3]; s6=s4*s2;
 237             Q2 = qa[2]+s*qa[3];
 238             P3 = pa[4]+s*pa[5];
 239             Q3 = qa[4]+s*qa[5];
 240             P4 = pa[6];
 241             Q4 = qa[6];
 242             P = P1 + s2*P2 + s4*P3 + s6*P4;
 243             Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
 244 #endif
 245             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 246         }
 247         if (ix >= 0x40180000) {         /* inf>|x|>=6 */
 248             if(hx>=0) return one-tiny; else return tiny-one;
 249         }
 250         x = fabs(x);
 251         s = one/(x*x);
 252         if(ix< 0x4006DB6E) {    /* |x| < 1/0.35 */
 253 #ifdef DO_NOT_USE_THIS
 254             R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 255                                 ra5+s*(ra6+s*ra7))))));
 256             S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 257                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
 258 #else
 259             double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
 260             R1 = ra[0]+s*ra[1];s2 = s*s;
 261             S1 = one+s*sa[1];  s4 = s2*s2;
 262             R2 = ra[2]+s*ra[3];s6 = s4*s2;
 263             S2 = sa[2]+s*sa[3];s8 = s4*s4;
 264             R3 = ra[4]+s*ra[5];
 265             S3 = sa[4]+s*sa[5];
 266             R4 = ra[6]+s*ra[7];
 267             S4 = sa[6]+s*sa[7];
 268             R = R1 + s2*R2 + s4*R3 + s6*R4;
 269             S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
 270 #endif
 271         } else {        /* |x| >= 1/0.35 */
 272 #ifdef DO_NOT_USE_THIS
 273             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 274                                 rb5+s*rb6)))));
 275             S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 276                                 sb5+s*(sb6+s*sb7))))));
 277 #else
 278             double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
 279             R1 = rb[0]+s*rb[1];s2 = s*s;
 280             S1 = one+s*sb[1];  s4 = s2*s2;
 281             R2 = rb[2]+s*rb[3];s6 = s4*s2;
 282             S2 = sb[2]+s*sb[3];
 283             R3 = rb[4]+s*rb[5];
 284             S3 = sb[4]+s*sb[5];
 285             S4 = sb[6]+s*sb[7];
 286             R = R1 + s2*R2 + s4*R3 + s6*rb[6];
 287             S = S1 + s2*S2 + s4*S3 + s6*S4;
 288 #endif
 289         }
 290         z  = x;
 291         SET_LOW_WORD(z,0);
 292         r  =  __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S);
 293         if(hx>=0) return one-r/x; else return  r/x-one;
 294 }
 295 weak_alias (__erf, erf)
 296 #ifdef NO_LONG_DOUBLE
 297 strong_alias (__erf, __erfl)
 298 weak_alias (__erf, erfl)
 299 #endif
 300
 301 double __erfc(double x)
 302 {
 303         int32_t hx,ix;
 304         double R,S,P,Q,s,y,z,r;
 305         GET_HIGH_WORD(hx,x);
 306         ix = hx&0x7fffffff;
 307         if(ix>=0x7ff00000) {                    /* erfc(nan)=nan */
 308                                                 /* erfc(+-inf)=0,2 */
 309             return (double)(((u_int32_t)hx>>31)<<1)+one/x;
 310         }
 311
 312         if(ix < 0x3feb0000) {           /* |x|<0.84375 */
 313             double r1,r2,s1,s2,s3,z2,z4;
 314             if(ix < 0x3c700000)         /* |x|<2**-56 */
 315                 return one-x;
 316             z = x*x;
 317 #ifdef DO_NOT_USE_THIS
 318             r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
 319             s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
 320 #else
 321             r1 = pp[0]+z*pp[1]; z2=z*z;
 322             r2 = pp[2]+z*pp[3]; z4=z2*z2;
 323             s1 = one+z*qq[1];
 324             s2 = qq[2]+z*qq[3];
 325             s3 = qq[4]+z*qq[5];
 326             r = r1 + z2*r2 + z4*pp[4];
 327             s  = s1 + z2*s2 + z4*s3;
 328 #endif
 329             y = r/s;
 330             if(hx < 0x3fd00000) {       /* x<1/4 */
 331                 return one-(x+x*y);
 332             } else {
 333                 r = x*y;
 334                 r += (x-half);
 335                 return half - r ;
 336             }
 337         }
 338         if(ix < 0x3ff40000) {           /* 0.84375 <= |x| < 1.25 */
 339             double s2,s4,s6,P1,P2,P3,P4,Q1,Q2,Q3,Q4;
 340             s = fabs(x)-one;
 341 #ifdef DO_NOT_USE_THIS
 342             P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
 343             Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
 344 #else
 345             P1 = pa[0]+s*pa[1]; s2=s*s;
 346             Q1 = one+s*qa[1];   s4=s2*s2;
 347             P2 = pa[2]+s*pa[3]; s6=s4*s2;
 348             Q2 = qa[2]+s*qa[3];
 349             P3 = pa[4]+s*pa[5];
 350             Q3 = qa[4]+s*qa[5];
 351             P4 = pa[6];
 352             Q4 = qa[6];
 353             P = P1 + s2*P2 + s4*P3 + s6*P4;
 354             Q = Q1 + s2*Q2 + s4*Q3 + s6*Q4;
 355 #endif
 356             if(hx>=0) {
 357                 z  = one-erx; return z - P/Q;
 358             } else {
 359                 z = erx+P/Q; return one+z;
 360             }
 361         }
 362         if (ix < 0x403c0000) {          /* |x|<28 */
 363             x = fabs(x);
 364             s = one/(x*x);
 365             if(ix< 0x4006DB6D) {        /* |x| < 1/.35 ~ 2.857143*/
 366 #ifdef DO_NOT_USE_THIS
 367                 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
 368                                 ra5+s*(ra6+s*ra7))))));
 369                 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
 370                                 sa5+s*(sa6+s*(sa7+s*sa8)))))));
 371 #else
 372                 double R1,R2,R3,R4,S1,S2,S3,S4,s2,s4,s6,s8;
 373             R1 = ra[0]+s*ra[1];s2 = s*s;
 374             S1 = one+s*sa[1];  s4 = s2*s2;
 375             R2 = ra[2]+s*ra[3];s6 = s4*s2;
 376             S2 = sa[2]+s*sa[3];s8 = s4*s4;
 377             R3 = ra[4]+s*ra[5];
 378             S3 = sa[4]+s*sa[5];
 379             R4 = ra[6]+s*ra[7];
 380             S4 = sa[6]+s*sa[7];
 381             R = R1 + s2*R2 + s4*R3 + s6*R4;
 382             S = S1 + s2*S2 + s4*S3 + s6*S4 + s8*sa[8];
 383 #endif
 384             } else {                    /* |x| >= 1/.35 ~ 2.857143 */
 385                 double R1,R2,R3,S1,S2,S3,S4,s2,s4,s6;
 386                 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */
 387 #ifdef DO_NOT_USE_THIS
 388                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
 389                                 rb5+s*rb6)))));
 390                 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
 391                                 sb5+s*(sb6+s*sb7))))));
 392 #else
 393                 R1 = rb[0]+s*rb[1];s2 = s*s;
 394                 S1 = one+s*sb[1];  s4 = s2*s2;
 395                 R2 = rb[2]+s*rb[3];s6 = s4*s2;
 396                 S2 = sb[2]+s*sb[3];
 397                 R3 = rb[4]+s*rb[5];
 398                 S3 = sb[4]+s*sb[5];
 399                 S4 = sb[6]+s*sb[7];
 400                 R = R1 + s2*R2 + s4*R3 + s6*rb[6];
 401                 S = S1 + s2*S2 + s4*S3 + s6*S4;
 402 #endif
 403             }
 404             z  = x;
 405             SET_LOW_WORD(z,0);
 406             r  =  __ieee754_exp(-z*z-0.5625)*
 407                         __ieee754_exp((z-x)*(z+x)+R/S);
 408             if(hx>0) return r/x; else return two-r/x;
 409         } else {
 410             if(hx>0) return tiny*tiny; else return two-tiny;
 411         }
 412 }
 413 weak_alias (__erfc, erfc)
 414 #ifdef NO_LONG_DOUBLE
 415 strong_alias (__erfc, __erfcl)
 416 weak_alias (__erfc, erfcl)
 417 #endif