1 2/* @(#)s_erf.c 1.3 95/01/18 */ 3/* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunSoft, a Sun Microsystems, Inc. business. 8 * Permission to use, copy, modify, and distribute this 9 * software is freely granted, provided that this notice 10 * is preserved. 11 * ==================================================== 12 */ 13 14/* double ieee_erf(double x) 15 * double ieee_erfc(double x) 16 * x 17 * 2 |\ 18 * ieee_erf(x) = --------- | ieee_exp(-t*t)dt 19 * ieee_sqrt(pi) \| 20 * 0 21 * 22 * ieee_erfc(x) = 1-ieee_erf(x) 23 * Note that 24 * erf(-x) = -ieee_erf(x) 25 * erfc(-x) = 2 - ieee_erfc(x) 26 * 27 * Method: 28 * 1. For |x| in [0, 0.84375] 29 * ieee_erf(x) = x + x*R(x^2) 30 * ieee_erfc(x) = 1 - ieee_erf(x) if x in [-.84375,0.25] 31 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 32 * where R = P/Q where P is an odd poly of degree 8 and 33 * Q is an odd poly of degree 10. 34 * -57.90 35 * | R - (ieee_erf(x)-x)/x | <= 2 36 * 37 * 38 * Remark. The formula is derived by noting 39 * ieee_erf(x) = (2/ieee_sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 40 * and that 41 * 2/ieee_sqrt(pi) = 1.128379167095512573896158903121545171688 42 * is close to one. The interval is chosen because the fix 43 * point of ieee_erf(x) is near 0.6174 (i.e., ieee_erf(x)=x when x is 44 * near 0.6174), and by some experiment, 0.84375 is chosen to 45 * guarantee the error is less than one ulp for erf. 46 * 47 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 48 * c = 0.84506291151 rounded to single (24 bits) 49 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 50 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 51 * 1+(c+P1(s)/Q1(s)) if x < 0 52 * |P1/Q1 - (ieee_erf(|x|)-c)| <= 2**-59.06 53 * Remark: here we use the taylor series expansion at x=1. 54 * erf(1+s) = ieee_erf(1) + s*Poly(s) 55 * = 0.845.. + P1(s)/Q1(s) 56 * That is, we use rational approximation to approximate 57 * erf(1+s) - (c = (single)0.84506291151) 58 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 59 * where 60 * P1(s) = degree 6 poly in s 61 * Q1(s) = degree 6 poly in s 62 * 63 * 3. For x in [1.25,1/0.35(~2.857143)], 64 * erfc(x) = (1/x)*ieee_exp(-x*x-0.5625+R1/S1) 65 * erf(x) = 1 - ieee_erfc(x) 66 * where 67 * R1(z) = degree 7 poly in z, (z=1/x^2) 68 * S1(z) = degree 8 poly in z 69 * 70 * 4. For x in [1/0.35,28] 71 * erfc(x) = (1/x)*ieee_exp(-x*x-0.5625+R2/S2) if x > 0 72 * = 2.0 - (1/x)*ieee_exp(-x*x-0.5625+R2/S2) if -6<x<0 73 * = 2.0 - tiny (if x <= -6) 74 * erf(x) = sign(x)*(1.0 - ieee_erfc(x)) if x < 6, else 75 * erf(x) = sign(x)*(1.0 - tiny) 76 * where 77 * R2(z) = degree 6 poly in z, (z=1/x^2) 78 * S2(z) = degree 7 poly in z 79 * 80 * Note1: 81 * To compute ieee_exp(-x*x-0.5625+R/S), let s be a single 82 * precision number and s := x; then 83 * -x*x = -s*s + (s-x)*(s+x) 84 * ieee_exp(-x*x-0.5626+R/S) = 85 * exp(-s*s-0.5625)*ieee_exp((s-x)*(s+x)+R/S); 86 * Note2: 87 * Here 4 and 5 make use of the asymptotic series 88 * ieee_exp(-x*x) 89 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 90 * x*ieee_sqrt(pi) 91 * We use rational approximation to approximate 92 * g(s)=f(1/x^2) = ieee_log(ieee_erfc(x)*x) - x*x + 0.5625 93 * Here is the error bound for R1/S1 and R2/S2 94 * |R1/S1 - f(x)| < 2**(-62.57) 95 * |R2/S2 - f(x)| < 2**(-61.52) 96 * 97 * 5. For inf > x >= 28 98 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 99 * erfc(x) = tiny*tiny (raise underflow) if x > 0 100 * = 2 - tiny if x<0 101 * 102 * 7. Special case: 103 * erf(0) = 0, ieee_erf(inf) = 1, ieee_erf(-inf) = -1, 104 * erfc(0) = 1, ieee_erfc(inf) = 0, ieee_erfc(-inf) = 2, 105 * erfc/ieee_erf(NaN) is NaN 106 */ 107 108 109#include "fdlibm.h" 110 111#ifdef __STDC__ 112static const double 113#else 114static double 115#endif 116tiny = 1e-300, 117half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 118one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 119two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 120 /* c = (float)0.84506291151 */ 121erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 122/* 123 * Coefficients for approximation to erf on [0,0.84375] 124 */ 125efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 126efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 127pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 128pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 129pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 130pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 131pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 132qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 133qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 134qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 135qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 136qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 137/* 138 * Coefficients for approximation to erf in [0.84375,1.25] 139 */ 140pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 141pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 142pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 143pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 144pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 145pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 146pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 147qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 148qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 149qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 150qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 151qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 152qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 153/* 154 * Coefficients for approximation to erfc in [1.25,1/0.35] 155 */ 156ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 157ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 158ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 159ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 160ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 161ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 162ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 163ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 164sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 165sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 166sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 167sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 168sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 169sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 170sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 171sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 172/* 173 * Coefficients for approximation to erfc in [1/.35,28] 174 */ 175rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 176rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 177rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 178rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 179rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 180rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 181rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 182sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 183sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 184sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 185sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 186sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 187sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 188sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 189 190#ifdef __STDC__ 191 double ieee_erf(double x) 192#else 193 double ieee_erf(x) 194 double x; 195#endif 196{ 197 int hx,ix,i; 198 double R,S,P,Q,s,y,z,r; 199 hx = __HI(x); 200 ix = hx&0x7fffffff; 201 if(ix>=0x7ff00000) { /* ieee_erf(nan)=nan */ 202 i = ((unsigned)hx>>31)<<1; 203 return (double)(1-i)+one/x; /* ieee_erf(+-inf)=+-1 */ 204 } 205 206 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 207 if(ix < 0x3e300000) { /* |x|<2**-28 */ 208 if (ix < 0x00800000) 209 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 210 return x + efx*x; 211 } 212 z = x*x; 213 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 214 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 215 y = r/s; 216 return x + x*y; 217 } 218 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 219 s = ieee_fabs(x)-one; 220 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 221 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 222 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 223 } 224 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 225 if(hx>=0) return one-tiny; else return tiny-one; 226 } 227 x = ieee_fabs(x); 228 s = one/(x*x); 229 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 230 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 231 ra5+s*(ra6+s*ra7)))))); 232 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 233 sa5+s*(sa6+s*(sa7+s*sa8))))))); 234 } else { /* |x| >= 1/0.35 */ 235 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 236 rb5+s*rb6))))); 237 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 238 sb5+s*(sb6+s*sb7)))))); 239 } 240 z = x; 241 __LO(z) = 0; 242 r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 243 if(hx>=0) return one-r/x; else return r/x-one; 244} 245 246#ifdef __STDC__ 247 double ieee_erfc(double x) 248#else 249 double ieee_erfc(x) 250 double x; 251#endif 252{ 253 int hx,ix; 254 double R,S,P,Q,s,y,z,r; 255 hx = __HI(x); 256 ix = hx&0x7fffffff; 257 if(ix>=0x7ff00000) { /* ieee_erfc(nan)=nan */ 258 /* ieee_erfc(+-inf)=0,2 */ 259 return (double)(((unsigned)hx>>31)<<1)+one/x; 260 } 261 262 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 263 if(ix < 0x3c700000) /* |x|<2**-56 */ 264 return one-x; 265 z = x*x; 266 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 267 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 268 y = r/s; 269 if(hx < 0x3fd00000) { /* x<1/4 */ 270 return one-(x+x*y); 271 } else { 272 r = x*y; 273 r += (x-half); 274 return half - r ; 275 } 276 } 277 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 278 s = ieee_fabs(x)-one; 279 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 280 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 281 if(hx>=0) { 282 z = one-erx; return z - P/Q; 283 } else { 284 z = erx+P/Q; return one+z; 285 } 286 } 287 if (ix < 0x403c0000) { /* |x|<28 */ 288 x = ieee_fabs(x); 289 s = one/(x*x); 290 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 291 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 292 ra5+s*(ra6+s*ra7)))))); 293 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 294 sa5+s*(sa6+s*(sa7+s*sa8))))))); 295 } else { /* |x| >= 1/.35 ~ 2.857143 */ 296 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 297 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 298 rb5+s*rb6))))); 299 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 300 sb5+s*(sb6+s*sb7)))))); 301 } 302 z = x; 303 __LO(z) = 0; 304 r = __ieee754_exp(-z*z-0.5625)* 305 __ieee754_exp((z-x)*(z+x)+R/S); 306 if(hx>0) return r/x; else return two-r/x; 307 } else { 308 if(hx>0) return tiny*tiny; else return two-tiny; 309 } 310} 311