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 254969 2013-08-27 19:46:56Z kargl $
  16  */
  17
  18
  19 #include "math.h"
  20 #include "math_private.h"
  21
  22 static const float
  23 tiny        = 1e-30,
  24 half=  5.0000000000e-01, /* 0x3F000000 */
  25 one =  1.0000000000e+00, /* 0x3F800000 */
  26 two =  2.0000000000e+00, /* 0x40000000 */
  27 /*
  28  * Coefficients for approximation to erf on [0,0.84375]
  29  */
  30 efx =  1.2837916613e-01, /* 0x3e0375d4 */
  31 efx8=  1.0270333290e+00, /* 0x3f8375d4 */
  32 /*
  33  *  Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]:
  34  *  |(erf(x) - x)/x - p(x)/q(x)| < 2**-31.
  35  */
  36 pp0  =  1.28379166e-01F, /*  0x1.06eba8p-3 */
  37 pp1  = -3.36030394e-01F, /* -0x1.58185ap-2 */
  38 pp2  = -1.86260219e-03F, /* -0x1.e8451ep-10 */
  39 qq1  =  3.12324286e-01F, /*  0x1.3fd1f0p-2 */
  40 qq2  =  2.16070302e-02F, /*  0x1.620274p-6 */
  41 qq3  = -1.98859419e-03F, /* -0x1.04a626p-9 */
  42 /*
  43  * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]:
  44  * |(erf(x) - erx) - p(x)/q(x)| < 2**-36.
  45  */
  46 erx  =  8.42697144e-01F, /*  0x1.af7600p-1.  erf(1) rounded to 16 bits. */
  47 pa0  =  3.64939137e-06F, /*  0x1.e9d022p-19 */
  48 pa1  =  4.15109694e-01F, /*  0x1.a91284p-2 */
  49 pa2  = -1.65179938e-01F, /* -0x1.5249dcp-3 */
  50 pa3  =  1.10914491e-01F, /*  0x1.c64e46p-4 */
  51 qa1  =  6.02074385e-01F, /*  0x1.344318p-1 */
  52 qa2  =  5.35934687e-01F, /*  0x1.126608p-1 */
  53 qa3  =  1.68576106e-01F, /*  0x1.593e6ep-3 */
  54 qa4  =  5.62181212e-02F, /*  0x1.cc89f2p-5 */
  55 /*
  56  * Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]:
  57  * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30
  58  */
  59 ra0  = -9.87132732e-03F, /* -0x1.4376b2p-7 */
  60 ra1  = -5.53605914e-01F, /* -0x1.1b723cp-1 */
  61 ra2  = -2.17589188e+00F, /* -0x1.1683a0p+1 */
  62 ra3  = -1.43268085e+00F, /* -0x1.6ec42cp+0 */
  63 sa1  =  5.45995426e+00F, /*  0x1.5d6fe4p+2 */
  64 sa2  =  6.69798088e+00F, /*  0x1.acabb8p+2 */
  65 sa3  =  1.43113089e+00F, /*  0x1.6e5e98p+0 */
  66 sa4  = -5.77397496e-02F, /* -0x1.d90108p-5 */
  67 /*
  68  * Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]:
  69  * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42
  70  */
  71 rb0  = -9.86494310e-03F, /* -0x1.434124p-7 */
  72 rb1  = -6.25171244e-01F, /* -0x1.401672p-1 */
  73 rb2  = -6.16498327e+00F, /* -0x1.8a8f16p+2 */
  74 rb3  = -1.66696873e+01F, /* -0x1.0ab70ap+4 */
  75 rb4  = -9.53764343e+00F, /* -0x1.313460p+3 */
  76 sb1  =  1.26884899e+01F, /*  0x1.96081cp+3 */
  77 sb2  =  4.51839523e+01F, /*  0x1.6978bcp+5 */
  78 sb3  =  4.72810211e+01F, /*  0x1.7a3f88p+5 */
  79 sb4  =  8.93033314e+00F; /*  0x1.1dc54ap+3 */
  80
  81 float
  82 erff(float x)
  83 {
  84         int32_t hx,ix,i;
  85         float R,S,P,Q,s,y,z,r;
  86         GET_FLOAT_WORD(hx,x);
  87         ix = hx&0x7fffffff;
  88         if(ix>=0x7f800000) {            /* erf(nan)=nan */
  89             i = ((u_int32_t)hx>>31)<<1;
  90             return (float)(1-i)+one/x;  /* erf(+-inf)=+-1 */
  91         }
  92
  93         if(ix < 0x3f580000) {           /* |x|<0.84375 */
  94             if(ix < 0x38800000) {       /* |x|<2**-14 */
  95                 if (ix < 0x04000000)    /* |x|<0x1p-119 */
  96                     return (8*x+efx8*x)/8;      /* avoid spurious underflow */
  97                 return x + efx*x;
  98             }
  99             z = x*x;
 100             r = pp0+z*(pp1+z*pp2);
 101             s = one+z*(qq1+z*(qq2+z*qq3));
 102             y = r/s;
 103             return x + x*y;
 104         }
 105         if(ix < 0x3fa00000) {           /* 0.84375 <= |x| < 1.25 */
 106             s = fabsf(x)-one;
 107             P = pa0+s*(pa1+s*(pa2+s*pa3));
 108             Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
 109             if(hx>=0) return erx + P/Q; else return -erx - P/Q;
 110         }
 111         if (ix >= 0x40800000) {         /* inf>|x|>=4 */
 112             if(hx>=0) return one-tiny; else return tiny-one;
 113         }
 114         x = fabsf(x);
 115         s = one/(x*x);
 116         if(ix< 0x4036DB6E) {    /* |x| < 1/0.35 */
 117             R=ra0+s*(ra1+s*(ra2+s*ra3));
 118             S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
 119         } else {        /* |x| >= 1/0.35 */
 120             R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
 121             S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
 122         }
 123         SET_FLOAT_WORD(z,hx&0xffffe000);
 124         r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
 125         if(hx>=0) return one-r/x; else return  r/x-one;
 126 }
 127
 128 float
 129 erfcf(float x)
 130 {
 131         int32_t hx,ix;
 132         float R,S,P,Q,s,y,z,r;
 133         GET_FLOAT_WORD(hx,x);
 134         ix = hx&0x7fffffff;
 135         if(ix>=0x7f800000) {                    /* erfc(nan)=nan */
 136                                                 /* erfc(+-inf)=0,2 */
 137             return (float)(((u_int32_t)hx>>31)<<1)+one/x;
 138         }
 139
 140         if(ix < 0x3f580000) {           /* |x|<0.84375 */
 141             if(ix < 0x33800000)         /* |x|<2**-24 */
 142                 return one-x;
 143             z = x*x;
 144             r = pp0+z*(pp1+z*pp2);
 145             s = one+z*(qq1+z*(qq2+z*qq3));
 146             y = r/s;
 147             if(hx < 0x3e800000) {       /* x<1/4 */
 148                 return one-(x+x*y);
 149             } else {
 150                 r = x*y;
 151                 r += (x-half);
 152                 return half - r ;
 153             }
 154         }
 155         if(ix < 0x3fa00000) {           /* 0.84375 <= |x| < 1.25 */
 156             s = fabsf(x)-one;
 157             P = pa0+s*(pa1+s*(pa2+s*pa3));
 158             Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4)));
 159             if(hx>=0) {
 160                 z  = one-erx; return z - P/Q;
 161             } else {
 162                 z = erx+P/Q; return one+z;
 163             }
 164         }
 165         if (ix < 0x41300000) {          /* |x|<11 */
 166             x = fabsf(x);
 167             s = one/(x*x);
 168             if(ix< 0x4036DB6D) {        /* |x| < 1/.35 ~ 2.857143*/
 169                 R=ra0+s*(ra1+s*(ra2+s*ra3));
 170                 S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4)));
 171             } else {                    /* |x| >= 1/.35 ~ 2.857143 */
 172                 if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */
 173                 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4)));
 174                 S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4)));
 175             }
 176             SET_FLOAT_WORD(z,hx&0xffffe000);
 177             r  = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S);
 178             if(hx>0) return r/x; else return two-r/x;
 179         } else {
 180             if(hx>0) return tiny*tiny; else return two-tiny;
 181         }
 182 }