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 16 #include "cdefs-compat.h" 17 //__FBSDID("$FreeBSD: src/lib/msun/src/s_erff.c,v 1.8 2008/02/22 02:30:35 das Exp $"); 18 19 #include <openlibm_math.h> 20 21 #include "math_private.h" 22 23 static const float 24 tiny = 1e-30, 25 half= 5.0000000000e-01, /* 0x3F000000 */ 26 one = 1.0000000000e+00, /* 0x3F800000 */ 27 two = 2.0000000000e+00, /* 0x40000000 */ 28 /* 29 * Coefficients for approximation to erf on [0,0.84375] 30 */ 31 efx = 1.2837916613e-01, /* 0x3e0375d4 */ 32 efx8= 1.0270333290e+00, /* 0x3f8375d4 */ 33 /* 34 * Domain [0, 0.84375], range ~[-5.4446e-10,5.5197e-10]: 35 * |(erf(x) - x)/x - p(x)/q(x)| < 2**-31. 36 */ 37 pp0 = 1.28379166e-01F, /* 0x1.06eba8p-3 */ 38 pp1 = -3.36030394e-01F, /* -0x1.58185ap-2 */ 39 pp2 = -1.86260219e-03F, /* -0x1.e8451ep-10 */ 40 qq1 = 3.12324286e-01F, /* 0x1.3fd1f0p-2 */ 41 qq2 = 2.16070302e-02F, /* 0x1.620274p-6 */ 42 qq3 = -1.98859419e-03F, /* -0x1.04a626p-9 */ 43 /* 44 * Domain [0.84375, 1.25], range ~[-1.953e-11,1.940e-11]: 45 * |(erf(x) - erx) - p(x)/q(x)| < 2**-36. 46 */ 47 erx = 8.42697144e-01F, /* 0x1.af7600p-1. erf(1) rounded to 16 bits. */ 48 pa0 = 3.64939137e-06F, /* 0x1.e9d022p-19 */ 49 pa1 = 4.15109694e-01F, /* 0x1.a91284p-2 */ 50 pa2 = -1.65179938e-01F, /* -0x1.5249dcp-3 */ 51 pa3 = 1.10914491e-01F, /* 0x1.c64e46p-4 */ 52 qa1 = 6.02074385e-01F, /* 0x1.344318p-1 */ 53 qa2 = 5.35934687e-01F, /* 0x1.126608p-1 */ 54 qa3 = 1.68576106e-01F, /* 0x1.593e6ep-3 */ 55 qa4 = 5.62181212e-02F, /* 0x1.cc89f2p-5 */ 56 /* 57 * Domain [1.25,1/0.35], range ~[-7.043e-10,7.457e-10]: 58 * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-30 59 */ 60 ra0 = -9.87132732e-03F, /* -0x1.4376b2p-7 */ 61 ra1 = -5.53605914e-01F, /* -0x1.1b723cp-1 */ 62 ra2 = -2.17589188e+00F, /* -0x1.1683a0p+1 */ 63 ra3 = -1.43268085e+00F, /* -0x1.6ec42cp+0 */ 64 sa1 = 5.45995426e+00F, /* 0x1.5d6fe4p+2 */ 65 sa2 = 6.69798088e+00F, /* 0x1.acabb8p+2 */ 66 sa3 = 1.43113089e+00F, /* 0x1.6e5e98p+0 */ 67 sa4 = -5.77397496e-02F, /* -0x1.d90108p-5 */ 68 /* 69 * Domain [1/0.35, 11], range ~[-2.264e-13,2.336e-13]: 70 * |log(x*erfc(x)) + x**2 + 0.5625 - r(x)/s(x)| < 2**-42 71 */ 72 rb0 = -9.86494310e-03F, /* -0x1.434124p-7 */ 73 rb1 = -6.25171244e-01F, /* -0x1.401672p-1 */ 74 rb2 = -6.16498327e+00F, /* -0x1.8a8f16p+2 */ 75 rb3 = -1.66696873e+01F, /* -0x1.0ab70ap+4 */ 76 rb4 = -9.53764343e+00F, /* -0x1.313460p+3 */ 77 sb1 = 1.26884899e+01F, /* 0x1.96081cp+3 */ 78 sb2 = 4.51839523e+01F, /* 0x1.6978bcp+5 */ 79 sb3 = 4.72810211e+01F, /* 0x1.7a3f88p+5 */ 80 sb4 = 8.93033314e+00F; /* 0x1.1dc54ap+3 */ 81 82 83 OLM_DLLEXPORT float 84 erff(float x) 85 { 86 int32_t hx,ix,i; 87 float R,S,P,Q,s,y,z,r; 88 GET_FLOAT_WORD(hx,x); 89 ix = hx&0x7fffffff; 90 if(ix>=0x7f800000) { /* erf(nan)=nan */ 91 i = ((u_int32_t)hx>>31)<<1; 92 return (float)(1-i)+one/x; /* erf(+-inf)=+-1 */ 93 } 94 95 if(ix < 0x3f580000) { /* |x|<0.84375 */ 96 if(ix < 0x38800000) { /* |x|<2**-14 */ 97 if (ix < 0x04000000) /* |x|<0x1p-119 */ 98 return (8*x+efx8*x)/8; /* avoid spurious underflow */ 99 return x + efx*x; 100 } 101 z = x*x; 102 r = pp0+z*(pp1+z*pp2); 103 s = one+z*(qq1+z*(qq2+z*qq3)); 104 y = r/s; 105 return x + x*y; 106 } 107 if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ 108 s = fabsf(x)-one; 109 P = pa0+s*(pa1+s*(pa2+s*pa3)); 110 Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4))); 111 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 112 } 113 if (ix >= 0x40800000) { /* inf>|x|>=4 */ 114 if(hx>=0) return one-tiny; else return tiny-one; 115 } 116 x = fabsf(x); 117 s = one/(x*x); 118 if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */ 119 R=ra0+s*(ra1+s*(ra2+s*ra3)); 120 S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4))); 121 } else { /* |x| >= 1/0.35 */ 122 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4))); 123 S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4))); 124 } 125 SET_FLOAT_WORD(z,hx&0xffffe000); 126 r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S); 127 if(hx>=0) return one-r/x; else return r/x-one; 128 } 129 130 OLM_DLLEXPORT float 131 erfcf(float x) 132 { 133 int32_t hx,ix; 134 float R,S,P,Q,s,y,z,r; 135 GET_FLOAT_WORD(hx,x); 136 ix = hx&0x7fffffff; 137 if(ix>=0x7f800000) { /* erfc(nan)=nan */ 138 /* erfc(+-inf)=0,2 */ 139 return (float)(((u_int32_t)hx>>31)<<1)+one/x; 140 } 141 142 if(ix < 0x3f580000) { /* |x|<0.84375 */ 143 if(ix < 0x33800000) /* |x|<2**-56 */ 144 return one-x; 145 z = x*x; 146 r = pp0+z*(pp1+z*pp2); 147 s = one+z*(qq1+z*(qq2+z*qq3)); 148 y = r/s; 149 if(hx < 0x3e800000) { /* x<1/4 */ 150 return one-(x+x*y); 151 } else { 152 r = x*y; 153 r += (x-half); 154 return half - r ; 155 } 156 } 157 if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ 158 s = fabsf(x)-one; 159 P = pa0+s*(pa1+s*(pa2+s*pa3)); 160 Q = one+s*(qa1+s*(qa2+s*(qa3+s*qa4))); 161 if(hx>=0) { 162 z = one-erx; return z - P/Q; 163 } else { 164 z = erx+P/Q; return one+z; 165 } 166 } 167 if (ix < 0x41300000) { /* |x|<28 */ 168 x = fabsf(x); 169 s = one/(x*x); 170 if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 171 R=ra0+s*(ra1+s*(ra2+s*ra3)); 172 S=one+s*(sa1+s*(sa2+s*(sa3+s*sa4))); 173 } else { /* |x| >= 1/.35 ~ 2.857143 */ 174 if(hx<0&&ix>=0x40a00000) return two-tiny;/* x < -5 */ 175 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*rb4))); 176 S=one+s*(sb1+s*(sb2+s*(sb3+s*sb4))); 177 } 178 SET_FLOAT_WORD(z,hx&0xffffe000); 179 r = expf(-z*z-0.5625F)*expf((z-x)*(z+x)+R/S); 180 if(hx>0) return r/x; else return two-r/x; 181 } else { 182 if(hx>0) return tiny*tiny; else return two-tiny; 183 } 184 } 185