1 /* origin: FreeBSD /usr/src/lib/msun/src/s_erff.c */ 2 /* 3 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 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 "libm.h" 17 18 static const float 19 erx = 8.4506291151e-01, /* 0x3f58560b */ 20 /* 21 * Coefficients for approximation to erf on [0,0.84375] 22 */ 23 efx8 = 1.0270333290e+00, /* 0x3f8375d4 */ 24 pp0 = 1.2837916613e-01, /* 0x3e0375d4 */ 25 pp1 = -3.2504209876e-01, /* 0xbea66beb */ 26 pp2 = -2.8481749818e-02, /* 0xbce9528f */ 27 pp3 = -5.7702702470e-03, /* 0xbbbd1489 */ 28 pp4 = -2.3763017452e-05, /* 0xb7c756b1 */ 29 qq1 = 3.9791721106e-01, /* 0x3ecbbbce */ 30 qq2 = 6.5022252500e-02, /* 0x3d852a63 */ 31 qq3 = 5.0813062117e-03, /* 0x3ba68116 */ 32 qq4 = 1.3249473704e-04, /* 0x390aee49 */ 33 qq5 = -3.9602282413e-06, /* 0xb684e21a */ 34 /* 35 * Coefficients for approximation to erf in [0.84375,1.25] 36 */ 37 pa0 = -2.3621185683e-03, /* 0xbb1acdc6 */ 38 pa1 = 4.1485610604e-01, /* 0x3ed46805 */ 39 pa2 = -3.7220788002e-01, /* 0xbebe9208 */ 40 pa3 = 3.1834661961e-01, /* 0x3ea2fe54 */ 41 pa4 = -1.1089469492e-01, /* 0xbde31cc2 */ 42 pa5 = 3.5478305072e-02, /* 0x3d1151b3 */ 43 pa6 = -2.1663755178e-03, /* 0xbb0df9c0 */ 44 qa1 = 1.0642088205e-01, /* 0x3dd9f331 */ 45 qa2 = 5.4039794207e-01, /* 0x3f0a5785 */ 46 qa3 = 7.1828655899e-02, /* 0x3d931ae7 */ 47 qa4 = 1.2617121637e-01, /* 0x3e013307 */ 48 qa5 = 1.3637083583e-02, /* 0x3c5f6e13 */ 49 qa6 = 1.1984500103e-02, /* 0x3c445aa3 */ 50 /* 51 * Coefficients for approximation to erfc in [1.25,1/0.35] 52 */ 53 ra0 = -9.8649440333e-03, /* 0xbc21a093 */ 54 ra1 = -6.9385856390e-01, /* 0xbf31a0b7 */ 55 ra2 = -1.0558626175e+01, /* 0xc128f022 */ 56 ra3 = -6.2375331879e+01, /* 0xc2798057 */ 57 ra4 = -1.6239666748e+02, /* 0xc322658c */ 58 ra5 = -1.8460508728e+02, /* 0xc3389ae7 */ 59 ra6 = -8.1287437439e+01, /* 0xc2a2932b */ 60 ra7 = -9.8143291473e+00, /* 0xc11d077e */ 61 sa1 = 1.9651271820e+01, /* 0x419d35ce */ 62 sa2 = 1.3765776062e+02, /* 0x4309a863 */ 63 sa3 = 4.3456588745e+02, /* 0x43d9486f */ 64 sa4 = 6.4538726807e+02, /* 0x442158c9 */ 65 sa5 = 4.2900814819e+02, /* 0x43d6810b */ 66 sa6 = 1.0863500214e+02, /* 0x42d9451f */ 67 sa7 = 6.5702495575e+00, /* 0x40d23f7c */ 68 sa8 = -6.0424413532e-02, /* 0xbd777f97 */ 69 /* 70 * Coefficients for approximation to erfc in [1/.35,28] 71 */ 72 rb0 = -9.8649431020e-03, /* 0xbc21a092 */ 73 rb1 = -7.9928326607e-01, /* 0xbf4c9dd4 */ 74 rb2 = -1.7757955551e+01, /* 0xc18e104b */ 75 rb3 = -1.6063638306e+02, /* 0xc320a2ea */ 76 rb4 = -6.3756646729e+02, /* 0xc41f6441 */ 77 rb5 = -1.0250950928e+03, /* 0xc480230b */ 78 rb6 = -4.8351919556e+02, /* 0xc3f1c275 */ 79 sb1 = 3.0338060379e+01, /* 0x41f2b459 */ 80 sb2 = 3.2579251099e+02, /* 0x43a2e571 */ 81 sb3 = 1.5367296143e+03, /* 0x44c01759 */ 82 sb4 = 3.1998581543e+03, /* 0x4547fdbb */ 83 sb5 = 2.5530502930e+03, /* 0x451f90ce */ 84 sb6 = 4.7452853394e+02, /* 0x43ed43a7 */ 85 sb7 = -2.2440952301e+01; /* 0xc1b38712 */ 86 87 static float erfc1(float x) 88 { 89 float_t s,P,Q; 90 91 s = fabsf(x) - 1; 92 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 93 Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 94 return 1 - erx - P/Q; 95 } 96 97 static float erfc2(uint32_t ix, float x) 98 { 99 float_t s,R,S; 100 float z; 101 102 if (ix < 0x3fa00000) /* |x| < 1.25 */ 103 return erfc1(x); 104 105 x = fabsf(x); 106 s = 1/(x*x); 107 if (ix < 0x4036db6d) { /* |x| < 1/0.35 */ 108 R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 109 ra5+s*(ra6+s*ra7)))))); 110 S = 1.0f+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 111 sa5+s*(sa6+s*(sa7+s*sa8))))))); 112 } else { /* |x| >= 1/0.35 */ 113 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 114 rb5+s*rb6))))); 115 S = 1.0f+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 116 sb5+s*(sb6+s*sb7)))))); 117 } 118 GET_FLOAT_WORD(ix, x); 119 SET_FLOAT_WORD(z, ix&0xffffe000); 120 return expf(-z*z - 0.5625f) * expf((z-x)*(z+x) + R/S)/x; 121 } 122 123 float erff(float x) 124 { 125 float r,s,z,y; 126 uint32_t ix; 127 int sign; 128 129 GET_FLOAT_WORD(ix, x); 130 sign = ix>>31; 131 ix &= 0x7fffffff; 132 if (ix >= 0x7f800000) { 133 /* erf(nan)=nan, erf(+-inf)=+-1 */ 134 return 1-2*sign + 1/x; 135 } 136 if (ix < 0x3f580000) { /* |x| < 0.84375 */ 137 if (ix < 0x31800000) { /* |x| < 2**-28 */ 138 /*avoid underflow */ 139 return 0.125f*(8*x + efx8*x); 140 } 141 z = x*x; 142 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 143 s = 1+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 144 y = r/s; 145 return x + x*y; 146 } 147 if (ix < 0x40c00000) /* |x| < 6 */ 148 y = 1 - erfc2(ix,x); 149 else 150 y = 1 - 0x1p-120f; 151 return sign ? -y : y; 152 } 153 154 float erfcf(float x) 155 { 156 float r,s,z,y; 157 uint32_t ix; 158 int sign; 159 160 GET_FLOAT_WORD(ix, x); 161 sign = ix>>31; 162 ix &= 0x7fffffff; 163 if (ix >= 0x7f800000) { 164 /* erfc(nan)=nan, erfc(+-inf)=0,2 */ 165 return 2*sign + 1/x; 166 } 167 168 if (ix < 0x3f580000) { /* |x| < 0.84375 */ 169 if (ix < 0x23800000) /* |x| < 2**-56 */ 170 return 1.0f - x; 171 z = x*x; 172 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 173 s = 1.0f+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 174 y = r/s; 175 if (sign || ix < 0x3e800000) /* x < 1/4 */ 176 return 1.0f - (x+x*y); 177 return 0.5f - (x - 0.5f + x*y); 178 } 179 if (ix < 0x41e00000) { /* |x| < 28 */ 180 return sign ? 2 - erfc2(ix,x) : erfc2(ix,x); 181 } 182 return sign ? 2 - 0x1p-120f : 0x1p-120f*0x1p-120f; 183 } 184