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