lib/libm/src/s_erff.c

   1 /* s_erff.c -- float version of s_erf.c.
   2  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
   3  */
   4
   5 /*
   6  * ====================================================
   7  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   8  *
   9  * Developed at SunPro, a Sun Microsystems, Inc. business.
  10  * Permission to use, copy, modify, and distribute this
  11  * software is freely granted, provided that this notice
  12  * is preserved.
  13  * ====================================================
  14  *
  15  * $FreeBSD: head/lib/msun/src/s_erff.c 268590 2014-07-13 16:24:16Z kargl $
  16  */
  17
  18
  19 #include "math.h"
  20 #include "math_private.h"
  21
  22 /* XXX Prevent compilers from erroneously constant folding: */
  23 static const volatile float tiny = 1e-30;
  24
  25 static const float
  26 half= 0.5,
  27 one = 1,
  28 two = 2,
  29 erx = 8.42697144e-01,                   /* 0x3f57bb00 */
  30 /*
  31  * In the domain [0, 2**-14], only the first term in the power series
  32  * expansion of erf(x) is used.  The magnitude of the first neglected
  33  * terms is less than 2**-42.
  34  */
  35 efx = 1.28379166e-01, /* 0x3e0375d4 */
  36 efx8= 1.02703333e+00, /* 0x3f8375d4 */
  37 /*
  38  * Domain [0, 0.84375], range ~[-5.4419e-10, 5.5179e-10]:
  39  * |(erf(x) - x)/x - pp(x)/qq(x)| < 2**-31
  40  */
  41 pp0  =  1.28379166e-01, /* 0x3e0375d4 */
  42 pp1  = -3.36030394e-01, /* 0xbeac0c2d */
  43 pp2  = -1.86261395e-03, /* 0xbaf422f4 */
  44 qq1  =  3.12324315e-01, /* 0x3e9fe8f9 */
  45 qq2  =  2.16070414e-02, /* 0x3cb10140 */
  46 qq3  = -1.98859372e-03, /* 0xbb025311 */
  47 /*
  48  * Domain [0.84375, 1.25], range ~[-1.023e-9, 1.023e-9]:
  49  * |(erf(x) - erx) - pa(x)/qa(x)| < 2**-31
  50  */
  51 pa0  =  3.65041046e-06, /* 0x3674f993 */
  52 pa1  =  4.15109307e-01, /* 0x3ed48935 */
  53 pa2  = -2.09395722e-01, /* 0xbe566bd5 */
  54 pa3  =  8.67677554e-02, /* 0x3db1b34b */
  55 qa1  =  4.95560974e-01, /* 0x3efdba2b */
  56 qa2  =  3.71248513e-01, /* 0x3ebe1449 */
  57 qa3  =  3.92478965e-02, /* 0x3d20c267 */
  58 /*
  59  * Domain [1.25,1/0.35], range ~[-4.821e-9, 4.927e-9]:
  60  * |log(x*erfc(x)) + x**2 + 0.5625 - ra(x)/sa(x)| < 2**-28
  61  */
  62 ra0  = -9.88156721e-03, /* 0xbc21e64c */
  63 ra1  = -5.43658376e-01, /* 0xbf0b2d32 */
  64 ra2  = -1.66828310e+00, /* 0xbfd58a4d */
  65 ra3  = -6.91554189e-01, /* 0xbf3109b2 */
  66 sa1  =  4.48581553e+00, /* 0x408f8bcd */
  67 sa2  =  4.10799170e+00, /* 0x408374ab */
  68 sa3  =  5.53855181e-01, /* 0x3f0dc974 */
  69 /*
  70  * Domain [2.85715, 11], range ~[-1.484e-9, 1.505e-9]:
  71  * |log(x*erfc(x)) + x**2 + 0.5625 - rb(x)/sb(x)| < 2**-30
  72  */
  73 rb0  = -9.86496918e-03, /* 0xbc21a0ae */
  74 rb1  = -5.48049808e-01, /* 0xbf0c4cfe */
  75 rb2  = -1.84115684e+00, /* 0xbfebab07 */
  76 sb1  =  4.87132740e+00, /* 0x409be1ea */
  77 sb2  =  3.04982710e+00, /* 0x4043305e */
  78 sb3  = -7.61900663e-01; /* 0xbf430bec */
  79
  80 float
  81 erff(float x)
  82 {
  83         int32_t hx,ix,i;
  84         float R,S,P,Q,s,y,z,r;
  85         GET_FLOAT_WORD(hx,x);
  86         ix = hx&0x7fffffff;
  87         if(ix>=0x7f800000) {            /* erff(nan)=nan */
  88             i = ((u_int32_t)hx>>31)<<1;
  89             return (float)(1-i)+one/x;  /* erff(+-inf)=+-1 */
  90         }
  91
  92         if(ix < 0x3f580000) {           /* |x|<0.84375 */
  93             if(ix < 0x38800000) {       /* |x|<2**-14 */
  94                 if (ix < 0x04000000)    /* |x|<0x1p-119 */
  95                     return (8*x+efx8*x)/8;      /* avoid spurious underflow */
  96                 return x + efx*x;
  97             }
  98             z = x*x;
  99             r = pp0+z*(pp1+z*pp2);
 100             s = one+z*(qq1+z*(qq2+z*qq3));
 101             y = r/s;
 102             return x + x*y;
 103         }
 104         if(ix < 0x3fa00000) {           /* 0.84375 <= |x| < 1.25 */
 105             s = fabsf(x)-one;
 106             P = pa0+s*(pa1+s*(pa2+s*pa3));
 107             Q = one+s*(qa1+s*(qa2+s*qa3));
 108             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 109         }
 110         if (ix >= 0x40800000) {         /* inf>|x|>=4 */
 111             if(hx>=0) return one-tiny; else return tiny-one;
 112         }
 113         x = fabsf(x);
 114         s = one/(x*x);
 115         if(ix< 0x4036db8c) {    /* |x| < 2.85715 ~ 1/0.35 */
 116             R=ra0+s*(ra1+s*(ra2+s*ra3));
 117             S=one+s*(sa1+s*(sa2+s*sa3));
 118         } else {        /* |x| >= 2.85715 ~ 1/0.35 */
 119             R=rb0+s*(rb1+s*rb2);
 120             S=one+s*(sb1+s*(sb2+s*sb3));
 121         }
 122         SET_FLOAT_WORD(z,hx&0xffffe000);
 123         r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
 124         if(hx>=0) return one-r/x; else return  r/x-one;
 125 }
 126
 127 float
 128 erfcf(float x)
 129 {
 130         int32_t hx,ix;
 131         float R,S,P,Q,s,y,z,r;
 132         GET_FLOAT_WORD(hx,x);
 133         ix = hx&0x7fffffff;
 134         if(ix>=0x7f800000) {                    /* erfcf(nan)=nan */
 135                                                 /* erfcf(+-inf)=0,2 */
 136             return (float)(((u_int32_t)hx>>31)<<1)+one/x;
 137         }
 138
 139         if(ix < 0x3f580000) {           /* |x|<0.84375 */
 140             if(ix < 0x33800000)         /* |x|<2**-24 */
 141                 return one-x;
 142             z = x*x;
 143             r = pp0+z*(pp1+z*pp2);
 144             s = one+z*(qq1+z*(qq2+z*qq3));
 145             y = r/s;
 146             if(hx < 0x3e800000) {       /* x<1/4 */
 147                 return one-(x+x*y);
 148             } else {
 149                 r = x*y;
 150                 r += (x-half);
 151                 return half - r ;
 152             }
 153         }
 154         if(ix < 0x3fa00000) {           /* 0.84375 <= |x| < 1.25 */
 155             s = fabsf(x)-one;
 156             P = pa0+s*(pa1+s*(pa2+s*pa3));
 157             Q = one+s*(qa1+s*(qa2+s*qa3));
 158             if(hx>=0) {
 159                 z  = one-erx; return z - P/Q;
 160             } else {
 161                 z = erx+P/Q; return one+z;
 162             }
 163         }
 164         if (ix < 0x41300000) {          /* |x|<11 */
 165             x = fabsf(x);
 166             s = one/(x*x);
 167             if(ix< 0x4036db8c) {        /* |x| < 2.85715 ~ 1/.35 */
 168                 R=ra0+s*(ra1+s*(ra2+s*ra3));
 169                 S=one+s*(sa1+s*(sa2+s*sa3));
 170             } else {                    /* |x| >= 2.85715 ~ 1/.35 */
 171                 if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
 172                 R=rb0+s*(rb1+s*rb2);
 173                 S=one+s*(sb1+s*(sb2+s*sb3));
 174             }
 175             SET_FLOAT_WORD(z,hx&0xffffe000);
 176             r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
 177             if(hx>0) return r/x; else return two-r/x;
 178         } else {
 179             if(hx>0) return tiny*tiny; else return two-tiny;
 180         }
 181 }