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 <errno.h>
 116 #include <float.h>
 117 #include <math.h>
 118 #include <math-narrow-eval.h>
 119 #include <math_private.h>
 120 #include <libm-alias-double.h>
 121 #include <fix-int-fp-convert-zero.h>
 122
 123 static const double
 124   tiny = 1e-300,
 125   half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 126   one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 127   two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
 128 /* c = (float)0.84506291151 */
 129   erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
 130 /*
 131  * Coefficients for approximation to  erf on [0,0.84375]
 132  */
 133   efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */
 134   pp[] = { 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
 135            -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
 136            -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
 137            -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
 138            -2.37630166566501626084e-05 }, /* 0xBEF8EAD6, 0x120016AC */
 139   qq[] = { 0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
 140            6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
 141            5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
 142            1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
 143            -3.96022827877536812320e-06 }, /* 0xBED09C43, 0x42A26120 */
 144 /*
 145  * Coefficients for approximation to  erf  in [0.84375,1.25]
 146  */
 147   pa[] = { -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
 148            4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
 149            -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
 150            3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
 151            -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
 152            3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
 153            -2.16637559486879084300e-03 }, /* 0xBF61BF38, 0x0A96073F */
 154   qa[] = { 0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
 155            5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
 156            7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
 157            1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
 158            1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
 159            1.19844998467991074170e-02 }, /* 0x3F888B54, 0x5735151D */
 160 /*
 161  * Coefficients for approximation to  erfc in [1.25,1/0.35]
 162  */
 163   ra[] = { -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
 164            -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
 165            -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
 166            -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
 167            -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
 168            -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
 169            -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
 170            -9.81432934416914548592e+00 }, /* 0xC023A0EF, 0xC69AC25C */
 171   sa[] = { 0.0, 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
 172            1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
 173            4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
 174            6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
 175            4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
 176            1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
 177            6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
 178            -6.04244152148580987438e-02 }, /* 0xBFAEEFF2, 0xEE749A62 */
 179 /*
 180  * Coefficients for approximation to  erfc in [1/.35,28]
 181  */
 182   rb[] = { -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
 183            -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
 184            -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
 185            -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
 186            -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
 187            -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
 188            -4.83519191608651397019e+02 }, /* 0xC07E384E, 0x9BDC383F */
 189   sb[] = { 0.0, 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
 190            3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
 191            1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
 192            3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
 193            2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
 194            4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
 195            -2.24409524465858183362e+01 }; /* 0xC03670E2, 0x42712D62 */
 196
 197 double
 198 __erf (double x)
 199 {
 200   int32_t hx, ix, i;
 201   double R, S, P, Q, s, y, z, r;
 202   GET_HIGH_WORD (hx, x);
 203   ix = hx & 0x7fffffff;
 204   if (ix >= 0x7ff00000)                 /* erf(nan)=nan */
 205     {
 206       i = ((uint32_t) hx >> 31) << 1;
 207       return (double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
 208     }
 209
 210   if (ix < 0x3feb0000)                  /* |x|<0.84375 */
 211     {
 212       double r1, r2, s1, s2, s3, z2, z4;
 213       if (ix < 0x3e300000)              /* |x|<2**-28 */
 214         {
 215           if (ix < 0x00800000)
 216             {
 217               /* Avoid spurious underflow.  */
 218               double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
 219               math_check_force_underflow (ret);
 220               return ret;
 221             }
 222           return x + efx * x;
 223         }
 224       z = x * x;
 225       r1 = pp[0] + z * pp[1]; z2 = z * z;
 226       r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
 227       s1 = one + z * qq[1];
 228       s2 = qq[2] + z * qq[3];
 229       s3 = qq[4] + z * qq[5];
 230       r = r1 + z2 * r2 + z4 * pp[4];
 231       s = s1 + z2 * s2 + z4 * s3;
 232       y = r / s;
 233       return x + x * y;
 234     }
 235   if (ix < 0x3ff40000)                  /* 0.84375 <= |x| < 1.25 */
 236     {
 237       double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
 238       s = fabs (x) - one;
 239       P1 = pa[0] + s * pa[1]; s2 = s * s;
 240       Q1 = one + s * qa[1];   s4 = s2 * s2;
 241       P2 = pa[2] + s * pa[3]; s6 = s4 * s2;
 242       Q2 = qa[2] + s * qa[3];
 243       P3 = pa[4] + s * pa[5];
 244       Q3 = qa[4] + s * qa[5];
 245       P4 = pa[6];
 246       Q4 = qa[6];
 247       P = P1 + s2 * P2 + s4 * P3 + s6 * P4;
 248       Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4;
 249       if (hx >= 0)
 250         return erx + P / Q;
 251       else
 252         return -erx - P / Q;
 253     }
 254   if (ix >= 0x40180000)                 /* inf>|x|>=6 */
 255     {
 256       if (hx >= 0)
 257         return one - tiny;
 258       else
 259         return tiny - one;
 260     }
 261   x = fabs (x);
 262   s = one / (x * x);
 263   if (ix < 0x4006DB6E)          /* |x| < 1/0.35 */
 264     {
 265       double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
 266       R1 = ra[0] + s * ra[1]; s2 = s * s;
 267       S1 = one + s * sa[1];  s4 = s2 * s2;
 268       R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
 269       S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
 270       R3 = ra[4] + s * ra[5];
 271       S3 = sa[4] + s * sa[5];
 272       R4 = ra[6] + s * ra[7];
 273       S4 = sa[6] + s * sa[7];
 274       R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
 275       S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
 276     }
 277   else                  /* |x| >= 1/0.35 */
 278     {
 279       double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
 280       R1 = rb[0] + s * rb[1]; s2 = s * s;
 281       S1 = one + s * sb[1];  s4 = s2 * s2;
 282       R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
 283       S2 = sb[2] + s * sb[3];
 284       R3 = rb[4] + s * rb[5];
 285       S3 = sb[4] + s * sb[5];
 286       S4 = sb[6] + s * sb[7];
 287       R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
 288       S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
 289     }
 290   z = x;
 291   SET_LOW_WORD (z, 0);
 292   r = __ieee754_exp (-z * z - 0.5625) *
 293       __ieee754_exp ((z - x) * (z + x) + R / S);
 294   if (hx >= 0)
 295     return one - r / x;
 296   else
 297     return r / x - one;
 298 }
 299 libm_alias_double (__erf, erf)
 300
 301 double
 302 __erfc (double x)
 303 {
 304   int32_t hx, ix;
 305   double R, S, P, Q, s, y, z, r;
 306   GET_HIGH_WORD (hx, x);
 307   ix = hx & 0x7fffffff;
 308   if (ix >= 0x7ff00000)                         /* erfc(nan)=nan */
 309     {                                           /* erfc(+-inf)=0,2 */
 310       double ret = (double) (((uint32_t) hx >> 31) << 1) + one / x;
 311       if (FIX_INT_FP_CONVERT_ZERO && ret == 0.0)
 312         return 0.0;
 313       return ret;
 314     }
 315
 316   if (ix < 0x3feb0000)                  /* |x|<0.84375 */
 317     {
 318       double r1, r2, s1, s2, s3, z2, z4;
 319       if (ix < 0x3c700000)              /* |x|<2**-56 */
 320         return one - x;
 321       z = x * x;
 322       r1 = pp[0] + z * pp[1]; z2 = z * z;
 323       r2 = pp[2] + z * pp[3]; z4 = z2 * z2;
 324       s1 = one + z * qq[1];
 325       s2 = qq[2] + z * qq[3];
 326       s3 = qq[4] + z * qq[5];
 327       r = r1 + z2 * r2 + z4 * pp[4];
 328       s = s1 + z2 * s2 + z4 * s3;
 329       y = r / s;
 330       if (hx < 0x3fd00000)              /* x<1/4 */
 331         {
 332           return one - (x + x * y);
 333         }
 334       else
 335         {
 336           r = x * y;
 337           r += (x - half);
 338           return half - r;
 339         }
 340     }
 341   if (ix < 0x3ff40000)                  /* 0.84375 <= |x| < 1.25 */
 342     {
 343       double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4;
 344       s = fabs (x) - one;
 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       if (hx >= 0)
 356         {
 357           z = one - erx; return z - P / Q;
 358         }
 359       else
 360         {
 361           z = erx + P / Q; return one + z;
 362         }
 363     }
 364   if (ix < 0x403c0000)                  /* |x|<28 */
 365     {
 366       x = fabs (x);
 367       s = one / (x * x);
 368       if (ix < 0x4006DB6D)              /* |x| < 1/.35 ~ 2.857143*/
 369         {
 370           double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8;
 371           R1 = ra[0] + s * ra[1]; s2 = s * s;
 372           S1 = one + s * sa[1];  s4 = s2 * s2;
 373           R2 = ra[2] + s * ra[3]; s6 = s4 * s2;
 374           S2 = sa[2] + s * sa[3]; s8 = s4 * s4;
 375           R3 = ra[4] + s * ra[5];
 376           S3 = sa[4] + s * sa[5];
 377           R4 = ra[6] + s * ra[7];
 378           S4 = sa[6] + s * sa[7];
 379           R = R1 + s2 * R2 + s4 * R3 + s6 * R4;
 380           S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8];
 381         }
 382       else                              /* |x| >= 1/.35 ~ 2.857143 */
 383         {
 384           double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6;
 385           if (hx < 0 && ix >= 0x40180000)
 386             return two - tiny;                           /* x < -6 */
 387           R1 = rb[0] + s * rb[1]; s2 = s * s;
 388           S1 = one + s * sb[1];  s4 = s2 * s2;
 389           R2 = rb[2] + s * rb[3]; s6 = s4 * s2;
 390           S2 = sb[2] + s * sb[3];
 391           R3 = rb[4] + s * rb[5];
 392           S3 = sb[4] + s * sb[5];
 393           S4 = sb[6] + s * sb[7];
 394           R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6];
 395           S = S1 + s2 * S2 + s4 * S3 + s6 * S4;
 396         }
 397       z = x;
 398       SET_LOW_WORD (z, 0);
 399       r = __ieee754_exp (-z * z - 0.5625) *
 400           __ieee754_exp ((z - x) * (z + x) + R / S);
 401       if (hx > 0)
 402         {
 403           double ret = math_narrow_eval (r / x);
 404           if (ret == 0)
 405             __set_errno (ERANGE);
 406           return ret;
 407         }
 408       else
 409         return two - r / x;
 410     }
 411   else
 412     {
 413       if (hx > 0)
 414         {
 415           __set_errno (ERANGE);
 416           return tiny * tiny;
 417         }
 418       else
 419         return two - tiny;
 420     }
 421 }
 422 libm_alias_double (__erfc, erfc)