1 2 /* @(#)s_erf.c 5.1 93/09/24 */ 3 /* 4 * ==================================================== 5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 6 * 7 * Developed at SunPro, 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 /* 15 FUNCTION 16 <<erf>>, <<erff>>, <<erfc>>, <<erfcf>>---error function 17 INDEX 18 erf 19 INDEX 20 erff 21 INDEX 22 erfc 23 INDEX 24 erfcf 25 26 ANSI_SYNOPSIS 27 #include <math.h> 28 double erf(double <[x]>); 29 float erff(float <[x]>); 30 double erfc(double <[x]>); 31 float erfcf(float <[x]>); 32 TRAD_SYNOPSIS 33 #include <math.h> 34 35 double erf(<[x]>) 36 double <[x]>; 37 38 float erff(<[x]>) 39 float <[x]>; 40 41 double erfc(<[x]>) 42 double <[x]>; 43 44 float erfcf(<[x]>) 45 float <[x]>; 46 47 DESCRIPTION 48 <<erf>> calculates an approximation to the ``error function'', 49 which estimates the probability that an observation will fall within 50 <[x]> standard deviations of the mean (assuming a normal 51 distribution). 52 @tex 53 The error function is defined as 54 $${2\over\sqrt\pi}\times\int_0^x e^{-t^2}dt$$ 55 @end tex 56 57 <<erfc>> calculates the complementary probability; that is, 58 <<erfc(<[x]>)>> is <<1 - erf(<[x]>)>>. <<erfc>> is computed directly, 59 so that you can use it to avoid the loss of precision that would 60 result from subtracting large probabilities (on large <[x]>) from 1. 61 62 <<erff>> and <<erfcf>> differ from <<erf>> and <<erfc>> only in the 63 argument and result types. 64 65 RETURNS 66 For positive arguments, <<erf>> and all its variants return a 67 probability---a number between 0 and 1. 68 69 PORTABILITY 70 None of the variants of <<erf>> are ANSI C. 71 */ 72 73 /* double erf(double x) 74 * double erfc(double x) 75 * x 76 * 2 |\ 77 * erf(x) = --------- | exp(-t*t)dt 78 * sqrt(pi) \| 79 * 0 80 * 81 * erfc(x) = 1-erf(x) 82 * Note that 83 * erf(-x) = -erf(x) 84 * erfc(-x) = 2 - erfc(x) 85 * 86 * Method: 87 * 1. For |x| in [0, 0.84375] 88 * erf(x) = x + x*R(x^2) 89 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 90 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 91 * where R = P/Q where P is an odd poly of degree 8 and 92 * Q is an odd poly of degree 10. 93 * -57.90 94 * | R - (erf(x)-x)/x | <= 2 95 * 96 * 97 * Remark. The formula is derived by noting 98 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 99 * and that 100 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 101 * is close to one. The interval is chosen because the fix 102 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 103 * near 0.6174), and by some experiment, 0.84375 is chosen to 104 * guarantee the error is less than one ulp for erf. 105 * 106 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 107 * c = 0.84506291151 rounded to single (24 bits) 108 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 109 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 110 * 1+(c+P1(s)/Q1(s)) if x < 0 111 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 112 * Remark: here we use the taylor series expansion at x=1. 113 * erf(1+s) = erf(1) + s*Poly(s) 114 * = 0.845.. + P1(s)/Q1(s) 115 * That is, we use rational approximation to approximate 116 * erf(1+s) - (c = (single)0.84506291151) 117 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 118 * where 119 * P1(s) = degree 6 poly in s 120 * Q1(s) = degree 6 poly in s 121 * 122 * 3. For x in [1.25,1/0.35(~2.857143)], 123 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 124 * erf(x) = 1 - erfc(x) 125 * where 126 * R1(z) = degree 7 poly in z, (z=1/x^2) 127 * S1(z) = degree 8 poly in z 128 * 129 * 4. For x in [1/0.35,28] 130 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 131 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 132 * = 2.0 - tiny (if x <= -6) 133 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 134 * erf(x) = sign(x)*(1.0 - tiny) 135 * where 136 * R2(z) = degree 6 poly in z, (z=1/x^2) 137 * S2(z) = degree 7 poly in z 138 * 139 * Note1: 140 * To compute exp(-x*x-0.5625+R/S), let s be a single 141 * precision number and s := x; then 142 * -x*x = -s*s + (s-x)*(s+x) 143 * exp(-x*x-0.5626+R/S) = 144 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 145 * Note2: 146 * Here 4 and 5 make use of the asymptotic series 147 * exp(-x*x) 148 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 149 * x*sqrt(pi) 150 * We use rational approximation to approximate 151 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 152 * Here is the error bound for R1/S1 and R2/S2 153 * |R1/S1 - f(x)| < 2**(-62.57) 154 * |R2/S2 - f(x)| < 2**(-61.52) 155 * 156 * 5. For inf > x >= 28 157 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 158 * erfc(x) = tiny*tiny (raise underflow) if x > 0 159 * = 2 - tiny if x<0 160 * 161 * 7. Special case: 162 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 163 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 164 * erfc/erf(NaN) is NaN 165 */ 166 167 168 #include "fdlibm.h" 169 170 #ifndef _DOUBLE_IS_32BITS 171 172 #ifdef __STDC__ 173 static const double 174 #else 175 static double 176 #endif 177 tiny = 1e-300, 178 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 179 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 180 two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 181 /* c = (float)0.84506291151 */ 182 erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 183 /* 184 * Coefficients for approximation to erf on [0,0.84375] 185 */ 186 efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 187 efx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 188 pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 189 pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 190 pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 191 pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 192 pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 193 qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 194 qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 195 qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 196 qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 197 qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 198 /* 199 * Coefficients for approximation to erf in [0.84375,1.25] 200 */ 201 pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 202 pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 203 pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 204 pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 205 pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 206 pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 207 pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 208 qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 209 qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 210 qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 211 qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 212 qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 213 qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 214 /* 215 * Coefficients for approximation to erfc in [1.25,1/0.35] 216 */ 217 ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 218 ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 219 ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 220 ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 221 ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 222 ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 223 ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 224 ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 225 sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 226 sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 227 sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 228 sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 229 sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 230 sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 231 sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 232 sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 233 /* 234 * Coefficients for approximation to erfc in [1/.35,28] 235 */ 236 rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 237 rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 238 rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 239 rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 240 rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 241 rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 242 rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 243 sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 244 sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 245 sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 246 sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 247 sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 248 sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 249 sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 250 251 #ifdef __STDC__ erf(double x)252 double erf(double x) 253 #else 254 double erf(x) 255 double x; 256 #endif 257 { 258 __int32_t hx,ix,i; 259 double R,S,P,Q,s,y,z,r; 260 GET_HIGH_WORD(hx,x); 261 ix = hx&0x7fffffff; 262 if(ix>=0x7ff00000) { /* erf(nan)=nan */ 263 i = ((__uint32_t)hx>>31)<<1; 264 return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 265 } 266 267 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 268 if(ix < 0x3e300000) { /* |x|<2**-28 */ 269 if (ix < 0x00800000) 270 return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 271 return x + efx*x; 272 } 273 z = x*x; 274 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 275 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 276 y = r/s; 277 return x + x*y; 278 } 279 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 280 s = fabs(x)-one; 281 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 282 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 283 if(hx>=0) return erx + P/Q; else return -erx - P/Q; 284 } 285 if (ix >= 0x40180000) { /* inf>|x|>=6 */ 286 if(hx>=0) return one-tiny; else return tiny-one; 287 } 288 x = fabs(x); 289 s = one/(x*x); 290 if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 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| >= 1/0.35 */ 296 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 297 rb5+s*rb6))))); 298 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 299 sb5+s*(sb6+s*sb7)))))); 300 } 301 z = x; 302 SET_LOW_WORD(z,0); 303 r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S); 304 if(hx>=0) return one-r/x; else return r/x-one; 305 } 306 307 #ifdef __STDC__ erfc(double x)308 double erfc(double x) 309 #else 310 double erfc(x) 311 double x; 312 #endif 313 { 314 __int32_t hx,ix; 315 double R,S,P,Q,s,y,z,r; 316 GET_HIGH_WORD(hx,x); 317 ix = hx&0x7fffffff; 318 if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 319 /* erfc(+-inf)=0,2 */ 320 return (double)(((__uint32_t)hx>>31)<<1)+one/x; 321 } 322 323 if(ix < 0x3feb0000) { /* |x|<0.84375 */ 324 if(ix < 0x3c700000) /* |x|<2**-56 */ 325 return one-x; 326 z = x*x; 327 r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 328 s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 329 y = r/s; 330 if(hx < 0x3fd00000) { /* x<1/4 */ 331 return one-(x+x*y); 332 } else { 333 r = x*y; 334 r += (x-half); 335 return half - r ; 336 } 337 } 338 if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 339 s = fabs(x)-one; 340 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 341 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 342 if(hx>=0) { 343 z = one-erx; return z - P/Q; 344 } else { 345 z = erx+P/Q; return one+z; 346 } 347 } 348 if (ix < 0x403c0000) { /* |x|<28 */ 349 x = fabs(x); 350 s = one/(x*x); 351 if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 352 R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 353 ra5+s*(ra6+s*ra7)))))); 354 S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 355 sa5+s*(sa6+s*(sa7+s*sa8))))))); 356 } else { /* |x| >= 1/.35 ~ 2.857143 */ 357 if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 358 R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 359 rb5+s*rb6))))); 360 S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 361 sb5+s*(sb6+s*sb7)))))); 362 } 363 z = x; 364 SET_LOW_WORD(z,0); 365 r = exp(-z*z-0.5625)* 366 exp((z-x)*(z+x)+R/S); 367 if(hx>0) return r/x; else return two-r/x; 368 } else { 369 if(hx>0) return tiny*tiny; else return two-tiny; 370 } 371 } 372 373 #endif /* _DOUBLE_IS_32BITS */ 374