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