1 /* @(#)s_erf.c 5.1 93/09/24 */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 10 * ==================================================== 11 */ 12 13 #ifndef lint 14 static char rcsid[] = "$Id: s_erf.c,v 1.4 1994/03/03 17:04:31 jtc Exp $"; 15 #endif 16 17 /* double erf(double x) 18 * double erfc(double x) 19 * x 20 * 2 |\ 21 * erf(x) = --------- | exp(-t*t)dt 22 * sqrt(pi) \| 23 * 0 24 * 25 * erfc(x) = 1-erf(x) 26 * Note that 27 * erf(-x) = -erf(x) 28 * erfc(-x) = 2 - erfc(x) 29 * 30 * Method: 31 * 1. For |x| in [0, 0.84375] 32 * erf(x) = x + x*R(x^2) 33 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 34 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 35 * where R = P/Q where P is an odd poly of degree 8 and 36 * Q is an odd poly of degree 10. 37 * -57.90 38 * | R - (erf(x)-x)/x | <= 2 39 * 40 * 41 * Remark. The formula is derived by noting 42 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 43 * and that 44 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 45 * is close to one. The interval is chosen because the fix 46 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 47 * near 0.6174), and by some experiment, 0.84375 is chosen to 48 * guarantee the error is less than one ulp for erf. 49 * 50 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 51 * c = 0.84506291151 rounded to single (24 bits) 52 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 53 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 54 * 1+(c+P1(s)/Q1(s)) if x < 0 55 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 56 * Remark: here we use the taylor series expansion at x=1. 57 * erf(1+s) = erf(1) + s*Poly(s) 58 * = 0.845.. + P1(s)/Q1(s) 59 * That is, we use rational approximation to approximate 60 * erf(1+s) - (c = (single)0.84506291151) 61 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 62 * where 63 * P1(s) = degree 6 poly in s 64 * Q1(s) = degree 6 poly in s 65 * 66 * 3. For x in [1.25,1/0.35(~2.857143)], 67 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 68 * erf(x) = 1 - erfc(x) 69 * where 70 * R1(z) = degree 7 poly in z, (z=1/x^2) 71 * S1(z) = degree 8 poly in z 72 * 73 * 4. For x in [1/0.35,28] 74 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 75 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 76 * = 2.0 - tiny (if x <= -6) 77 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 78 * erf(x) = sign(x)*(1.0 - tiny) 79 * where 80 * R2(z) = degree 6 poly in z, (z=1/x^2) 81 * S2(z) = degree 7 poly in z 82 * 83 * Note1: 84 * To compute exp(-x*x-0.5625+R/S), let s be a single 85 * precision number and s := x; then 86 * -x*x = -s*s + (s-x)*(s+x) 87 * exp(-x*x-0.5626+R/S) = 88 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 89 * Note2: 90 * Here 4 and 5 make use of the asymptotic series 91 * exp(-x*x) 92 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 93 * x*sqrt(pi) 94 * We use rational approximation to approximate 95 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 96 * Here is the error bound for R1/S1 and R2/S2 97 * |R1/S1 - f(x)| < 2**(-62.57) 98 * |R2/S2 - f(x)| < 2**(-61.52) 99 * 100 * 5. For inf > x >= 28 101 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 102 * erfc(x) = tiny*tiny (raise underflow) if x > 0 103 * = 2 - tiny if x<0 104 * 105 * 7. Special case: 106 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 107 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 108 * erfc/erf(NaN) is NaN 109 */ 110 111 112 #include <math.h> 113 #include <machine/endian.h> 114 115 #if BYTE_ORDER == LITTLE_ENDIAN 116 #define n0 1 117 #else 118 #define n0 0 119 #endif 120 121 #ifdef __STDC__ 122 static const double 123 #else 124 static double 125 #endif 126 tiny = 1e-300, 127 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 128 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 129 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 130 /* c = (float)0.84506291151 */ 131 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 132 /* 133 * Coefficients for approximation to erf on [0,0.84375] 134 */ 135 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 136 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 137 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 138 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 139 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 140 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 141 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 142 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 143 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 144 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 145 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 146 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 147 /* 148 * Coefficients for approximation to erf in [0.84375,1.25] 149 */ 150 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 151 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 152 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 153 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 154 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 155 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 156 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 157 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 158 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 159 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 160 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 161 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 162 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 163 /* 164 * Coefficients for approximation to erfc in [1.25,1/0.35] 165 */ 166 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 167 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 168 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 169 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 170 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 171 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 172 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 173 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 174 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 175 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 176 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 177 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 178 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 179 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 180 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 181 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 182 /* 183 * Coefficients for approximation to erfc in [1/.35,28] 184 */ 185 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 186 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 187 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 188 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 189 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 190 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 191 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 192 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 193 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 194 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 195 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 196 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 197 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 198 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 199 200 #ifdef __STDC__ 201 double erf(double x) 202 #else 203 double erf(x) 204 double x; 205 #endif 206 { 207 int hx,ix,i; 208 double R,S,P,Q,s,y,z,r; 209 hx = *(n0+(int*)&x); 210 ix = hx&0x7fffffff; 211 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 212 i = ((unsigned)hx>>31)<<1; 213 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 214 } 215 216 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 217 if(ix < 0x3e300000) { /* |x|<2**-28 */ 218 if (ix < 0x00800000) 219 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 220 return x + efx*x; 221 } 222 z = x*x; 223 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 224 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 225 y = r/s; 226 return x + x*y; 227 } 228 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 229 s = fabs(x)-one; 230 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 231 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 232 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 233 } 234 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 235 if(hx>=0) return one-tiny; else return tiny-one; 236 } 237 x = fabs(x); 238 s = one/(x*x); 239 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 240 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 241 ra5+s*(ra6+s*ra7)))))); 242 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 243 sa5+s*(sa6+s*(sa7+s*sa8))))))); 244 } else { /* |x| >= 1/0.35 */ 245 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 246 rb5+s*rb6))))); 247 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 248 sb5+s*(sb6+s*sb7)))))); 249 } 250 z = x; 251 *(1-n0+(int*)&z) = 0; 252 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 253 if(hx>=0) return one-r/x; else return r/x-one; 254 } 255 256 #ifdef __STDC__ 257 double erfc(double x) 258 #else 259 double erfc(x) 260 double x; 261 #endif 262 { 263 int hx,ix; 264 double R,S,P,Q,s,y,z,r; 265 hx = *(n0+(int*)&x); 266 ix = hx&0x7fffffff; 267 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 268 /* erfc(+-inf)=0,2 */ 269 return (double)(((unsigned)hx>>31)<<1)+one/x; 270 } 271 272 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 273 if(ix < 0x3c700000) /* |x|<2**-56 */ 274 return one-x; 275 z = x*x; 276 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 277 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 278 y = r/s; 279 if(hx < 0x3fd00000) { /* x<1/4 */ 280 return one-(x+x*y); 281 } else { 282 r = x*y; 283 r += (x-half); 284 return half - r ; 285 } 286 } 287 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 288 s = fabs(x)-one; 289 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 290 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 291 if(hx>=0) { 292 z = one-erx; return z - P/Q; 293 } else { 294 z = erx+P/Q; return one+z; 295 } 296 } 297 if (ix < 0x403c0000) { /* |x|<28 */ 298 x = fabs(x); 299 s = one/(x*x); 300 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 301 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 302 ra5+s*(ra6+s*ra7)))))); 303 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 304 sa5+s*(sa6+s*(sa7+s*sa8))))))); 305 } else { /* |x| >= 1/.35 ~ 2.857143 */ 306 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 307 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 308 rb5+s*rb6))))); 309 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 310 sb5+s*(sb6+s*sb7)))))); 311 } 312 z = x; 313 *(1-n0+(int*)&z) = 0; 314 r = __ieee754_exp(-z*z-0.5625)* 315 __ieee754_exp((z-x)*(z+x)+R/S); 316 if(hx>0) return r/x; else return two-r/x; 317 } else { 318 if(hx>0) return tiny*tiny; else return two-tiny; 319 } 320 } 321