1 /* 2 * This file is part of the MicroPython project, http://micropython.org/ 3 * 4 * These math functions are taken from newlib-nano-2, the newlib/libm/math 5 * directory, available from https://github.com/32bitmicro/newlib-nano-2. 6 * 7 * Appropriate copyright headers are reproduced below. 8 */ 9 10 /* sf_erf.c -- float version of s_erf.c. 11 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. 12 */ 13 14 /* 15 * ==================================================== 16 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 17 * 18 * Developed at SunPro, a Sun Microsystems, Inc. business. 19 * Permission to use, copy, modify, and distribute this 20 * software is freely granted, provided that this notice 21 * is preserved. 22 * ==================================================== 23 */ 24 25 #include "fdlibm.h" 26 27 #define __ieee754_expf expf 28 29 #ifdef __v810__ 30 #define const 31 #endif 32 33 #ifdef __STDC__ 34 static const float 35 #else 36 static float 37 #endif 38 tiny = 1e-30f, 39 half= 5.0000000000e-01f, /* 0x3F000000 */ 40 one = 1.0000000000e+00f, /* 0x3F800000 */ 41 two = 2.0000000000e+00f, /* 0x40000000 */ 42 /* c = (subfloat)0.84506291151 */ 43 erx = 8.4506291151e-01f, /* 0x3f58560b */ 44 /* 45 * Coefficients for approximation to erf on [0,0.84375] 46 */ 47 efx = 1.2837916613e-01f, /* 0x3e0375d4 */ 48 efx8= 1.0270333290e+00f, /* 0x3f8375d4 */ 49 pp0 = 1.2837916613e-01f, /* 0x3e0375d4 */ 50 pp1 = -3.2504209876e-01f, /* 0xbea66beb */ 51 pp2 = -2.8481749818e-02f, /* 0xbce9528f */ 52 pp3 = -5.7702702470e-03f, /* 0xbbbd1489 */ 53 pp4 = -2.3763017452e-05f, /* 0xb7c756b1 */ 54 qq1 = 3.9791721106e-01f, /* 0x3ecbbbce */ 55 qq2 = 6.5022252500e-02f, /* 0x3d852a63 */ 56 qq3 = 5.0813062117e-03f, /* 0x3ba68116 */ 57 qq4 = 1.3249473704e-04f, /* 0x390aee49 */ 58 qq5 = -3.9602282413e-06f, /* 0xb684e21a */ 59 /* 60 * Coefficients for approximation to erf in [0.84375,1.25] 61 */ 62 pa0 = -2.3621185683e-03f, /* 0xbb1acdc6 */ 63 pa1 = 4.1485610604e-01f, /* 0x3ed46805 */ 64 pa2 = -3.7220788002e-01f, /* 0xbebe9208 */ 65 pa3 = 3.1834661961e-01f, /* 0x3ea2fe54 */ 66 pa4 = -1.1089469492e-01f, /* 0xbde31cc2 */ 67 pa5 = 3.5478305072e-02f, /* 0x3d1151b3 */ 68 pa6 = -2.1663755178e-03f, /* 0xbb0df9c0 */ 69 qa1 = 1.0642088205e-01f, /* 0x3dd9f331 */ 70 qa2 = 5.4039794207e-01f, /* 0x3f0a5785 */ 71 qa3 = 7.1828655899e-02f, /* 0x3d931ae7 */ 72 qa4 = 1.2617121637e-01f, /* 0x3e013307 */ 73 qa5 = 1.3637083583e-02f, /* 0x3c5f6e13 */ 74 qa6 = 1.1984500103e-02f, /* 0x3c445aa3 */ 75 /* 76 * Coefficients for approximation to erfc in [1.25,1/0.35] 77 */ 78 ra0 = -9.8649440333e-03f, /* 0xbc21a093 */ 79 ra1 = -6.9385856390e-01f, /* 0xbf31a0b7 */ 80 ra2 = -1.0558626175e+01f, /* 0xc128f022 */ 81 ra3 = -6.2375331879e+01f, /* 0xc2798057 */ 82 ra4 = -1.6239666748e+02f, /* 0xc322658c */ 83 ra5 = -1.8460508728e+02f, /* 0xc3389ae7 */ 84 ra6 = -8.1287437439e+01f, /* 0xc2a2932b */ 85 ra7 = -9.8143291473e+00f, /* 0xc11d077e */ 86 sa1 = 1.9651271820e+01f, /* 0x419d35ce */ 87 sa2 = 1.3765776062e+02f, /* 0x4309a863 */ 88 sa3 = 4.3456588745e+02f, /* 0x43d9486f */ 89 sa4 = 6.4538726807e+02f, /* 0x442158c9 */ 90 sa5 = 4.2900814819e+02f, /* 0x43d6810b */ 91 sa6 = 1.0863500214e+02f, /* 0x42d9451f */ 92 sa7 = 6.5702495575e+00f, /* 0x40d23f7c */ 93 sa8 = -6.0424413532e-02f, /* 0xbd777f97 */ 94 /* 95 * Coefficients for approximation to erfc in [1/.35,28] 96 */ 97 rb0 = -9.8649431020e-03f, /* 0xbc21a092 */ 98 rb1 = -7.9928326607e-01f, /* 0xbf4c9dd4 */ 99 rb2 = -1.7757955551e+01f, /* 0xc18e104b */ 100 rb3 = -1.6063638306e+02f, /* 0xc320a2ea */ 101 rb4 = -6.3756646729e+02f, /* 0xc41f6441 */ 102 rb5 = -1.0250950928e+03f, /* 0xc480230b */ 103 rb6 = -4.8351919556e+02f, /* 0xc3f1c275 */ 104 sb1 = 3.0338060379e+01f, /* 0x41f2b459 */ 105 sb2 = 3.2579251099e+02f, /* 0x43a2e571 */ 106 sb3 = 1.5367296143e+03f, /* 0x44c01759 */ 107 sb4 = 3.1998581543e+03f, /* 0x4547fdbb */ 108 sb5 = 2.5530502930e+03f, /* 0x451f90ce */ 109 sb6 = 4.7452853394e+02f, /* 0x43ed43a7 */ 110 sb7 = -2.2440952301e+01f; /* 0xc1b38712 */ 111 112 #ifdef __STDC__ erff(float x)113 float erff(float x) 114 #else 115 float erff(x) 116 float x; 117 #endif 118 { 119 __int32_t hx,ix,i; 120 float R,S,P,Q,s,y,z,r; 121 GET_FLOAT_WORD(hx,x); 122 ix = hx&0x7fffffff; 123 if(!FLT_UWORD_IS_FINITE(ix)) { /* erf(nan)=nan */ 124 i = ((__uint32_t)hx>>31)<<1; 125 return (float)(1-i)+one/x; /* erf(+-inf)=+-1 */ 126 } 127 128 if(ix < 0x3f580000) { /* |x|<0.84375 */ 129 if(ix < 0x31800000) { /* |x|<2**-28 */ 130 if (ix < 0x04000000) 131 /*avoid underflow */ 132 return (float)0.125*((float)8.0*x+efx8*x); 133 return x + efx*x; 134 } 135 z = x*x; 136 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 137 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 138 y = r/s; 139 return x + x*y; 140 } 141 if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ 142 s = fabsf(x)-one; 143 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 144 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 145 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 146 } 147 if (ix >= 0x40c00000) { /* inf>|x|>=6 */ 148 if(hx>=0) return one-tiny; else return tiny-one; 149 } 150 x = fabsf(x); 151 s = one/(x*x); 152 if(ix< 0x4036DB6E) { /* |x| < 1/0.35 */ 153 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 154 ra5+s*(ra6+s*ra7)))))); 155 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 156 sa5+s*(sa6+s*(sa7+s*sa8))))))); 157 } else { /* |x| >= 1/0.35 */ 158 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 159 rb5+s*rb6))))); 160 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 161 sb5+s*(sb6+s*sb7)))))); 162 } 163 GET_FLOAT_WORD(ix,x); 164 SET_FLOAT_WORD(z,ix&0xfffff000); 165 r = __ieee754_expf(-z*z-(float)0.5625)*__ieee754_expf((z-x)*(z+x)+R/S); 166 if(hx>=0) return one-r/x; else return r/x-one; 167 } 168 169 #ifdef __STDC__ erfcf(float x)170 float erfcf(float x) 171 #else 172 float erfcf(x) 173 float x; 174 #endif 175 { 176 __int32_t hx,ix; 177 float R,S,P,Q,s,y,z,r; 178 GET_FLOAT_WORD(hx,x); 179 ix = hx&0x7fffffff; 180 if(!FLT_UWORD_IS_FINITE(ix)) { /* erfc(nan)=nan */ 181 /* erfc(+-inf)=0,2 */ 182 return (float)(((__uint32_t)hx>>31)<<1)+one/x; 183 } 184 185 if(ix < 0x3f580000) { /* |x|<0.84375 */ 186 if(ix < 0x23800000) /* |x|<2**-56 */ 187 return one-x; 188 z = x*x; 189 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 190 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 191 y = r/s; 192 if(hx < 0x3e800000) { /* x<1/4 */ 193 return one-(x+x*y); 194 } else { 195 r = x*y; 196 r += (x-half); 197 return half - r ; 198 } 199 } 200 if(ix < 0x3fa00000) { /* 0.84375 <= |x| < 1.25 */ 201 s = fabsf(x)-one; 202 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 203 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 204 if(hx>=0) { 205 z = one-erx; return z - P/Q; 206 } else { 207 z = erx+P/Q; return one+z; 208 } 209 } 210 if (ix < 0x41e00000) { /* |x|<28 */ 211 x = fabsf(x); 212 s = one/(x*x); 213 if(ix< 0x4036DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 214 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 215 ra5+s*(ra6+s*ra7)))))); 216 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 217 sa5+s*(sa6+s*(sa7+s*sa8))))))); 218 } else { /* |x| >= 1/.35 ~ 2.857143 */ 219 if(hx<0&&ix>=0x40c00000) return two-tiny;/* x < -6 */ 220 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 221 rb5+s*rb6))))); 222 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 223 sb5+s*(sb6+s*sb7)))))); 224 } 225 GET_FLOAT_WORD(ix,x); 226 SET_FLOAT_WORD(z,ix&0xfffff000); 227 r = __ieee754_expf(-z*z-(float)0.5625)* 228 __ieee754_expf((z-x)*(z+x)+R/S); 229 if(hx>0) return r/x; else return two-r/x; 230 } else { 231 if(hx>0) return tiny*tiny; else return two-tiny; 232 } 233 } 234 235 #ifdef _DOUBLE_IS_32BITS 236 237 #ifdef __STDC__ erf(double x)238 double erf(double x) 239 #else 240 double erf(x) 241 double x; 242 #endif 243 { 244 return (double) erff((float) x); 245 } 246 247 #ifdef __STDC__ erfc(double x)248 double erfc(double x) 249 #else 250 double erfc(x) 251 double x; 252 #endif 253 { 254 return (double) erfcf((float) x); 255 } 256 257 #endif /* defined(_DOUBLE_IS_32BITS) */ 258