1 /*- 2 * Copyright (c) 1992, 1993 3 * The Regents of the University of California. All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 06/04/93"; 10 #endif /* not lint */ 11 12 /* Modified Nov 30, 1992 P. McILROY: 13 * Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp) 14 * Replaced even+odd with direct calculation for x < .84375, 15 * to avoid destructive cancellation. 16 * 17 * Performance of erfc(x): 18 * In 300000 trials in the range [.83, .84375] the 19 * maximum observed error was 3.6ulp. 20 * 21 * In [.84735,1.25] the maximum observed error was <2.5ulp in 22 * 100000 runs in the range [1.2, 1.25]. 23 * 24 * In [1.25,26] (Not including subnormal results) 25 * the error is < 1.7ulp. 26 */ 27 28 /* double erf(double x) 29 * double erfc(double x) 30 * x 31 * 2 |\ 32 * erf(x) = --------- | exp(-t*t)dt 33 * sqrt(pi) \| 34 * 0 35 * 36 * erfc(x) = 1-erf(x) 37 * 38 * Method: 39 * 1. Reduce x to |x| by erf(-x) = -erf(x) 40 * 2. For x in [0, 0.84375] 41 * erf(x) = x + x*P(x^2) 42 * erfc(x) = 1 - erf(x) if x<=0.25 43 * = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375] 44 * where 45 * 2 2 4 20 46 * P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x ) 47 * is an approximation to (erf(x)-x)/x with precision 48 * 49 * -56.45 50 * | P - (erf(x)-x)/x | <= 2 51 * 52 * 53 * Remark. The formula is derived by noting 54 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 55 * and that 56 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 57 * is close to one. The interval is chosen because the fixed 58 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 59 * near 0.6174), and by some experiment, 0.84375 is chosen to 60 * guarantee the error is less than one ulp for erf. 61 * 62 * 3. For x in [0.84375,1.25], let s = x - 1, and 63 * c = 0.84506291151 rounded to single (24 bits) 64 * erf(x) = c + P1(s)/Q1(s) 65 * erfc(x) = (1-c) - P1(s)/Q1(s) 66 * |P1/Q1 - (erf(x)-c)| <= 2**-59.06 67 * Remark: here we use the taylor series expansion at x=1. 68 * erf(1+s) = erf(1) + s*Poly(s) 69 * = 0.845.. + P1(s)/Q1(s) 70 * That is, we use rational approximation to approximate 71 * erf(1+s) - (c = (single)0.84506291151) 72 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 73 * where 74 * P1(s) = degree 6 poly in s 75 * Q1(s) = degree 6 poly in s 76 * 77 * 4. For x in [1.25, 2]; [2, 4] 78 * erf(x) = 1.0 - tiny 79 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z)) 80 * 81 * Where z = 1/(x*x), R is degree 9, and S is degree 3; 82 * 83 * 5. For x in [4,28] 84 * erf(x) = 1.0 - tiny 85 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z)) 86 * 87 * Where P is degree 14 polynomial in 1/(x*x). 88 * 89 * Notes: 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 * 95 * where for z = 1/(x*x) 96 * P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...)))) 97 * 98 * Thus we use rational approximation to approximate 99 * erfc*x*exp(x*x) ~ 1/sqrt(pi); 100 * 101 * The error bound for the target function, G(z) for 102 * the interval 103 * [4, 28]: 104 * |eps + 1/(z)P(z) - G(z)| < 2**(-56.61) 105 * for [2, 4]: 106 * |R(z)/S(z) - G(z)| < 2**(-58.24) 107 * for [1.25, 2]: 108 * |R(z)/S(z) - G(z)| < 2**(-58.12) 109 * 110 * 6. For inf > x >= 28 111 * erf(x) = 1 - tiny (raise inexact) 112 * erfc(x) = tiny*tiny (raise underflow) 113 * 114 * 7. Special cases: 115 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 116 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 117 * erfc/erf(NaN) is NaN 118 */ 119 120 #if defined(vax) || defined(tahoe) 121 #define _IEEE 0 122 #define TRUNC(x) (double) (float) (x) 123 #else 124 #define _IEEE 1 125 #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000 126 #define infnan(x) 0.0 127 #endif 128 129 #ifdef _IEEE_LIBM 130 /* 131 * redefining "___function" to "function" in _IEEE_LIBM mode 132 */ 133 #include "ieee_libm.h" 134 #endif 135 136 static double 137 tiny = 1e-300, 138 half = 0.5, 139 one = 1.0, 140 two = 2.0, 141 c = 8.45062911510467529297e-01, /* (float)0.84506291151 */ 142 /* 143 * Coefficients for approximation to erf in [0,0.84375] 144 */ 145 p0t8 = 1.02703333676410051049867154944018394163280, 146 p0 = 1.283791670955125638123339436800229927041e-0001, 147 p1 = -3.761263890318340796574473028946097022260e-0001, 148 p2 = 1.128379167093567004871858633779992337238e-0001, 149 p3 = -2.686617064084433642889526516177508374437e-0002, 150 p4 = 5.223977576966219409445780927846432273191e-0003, 151 p5 = -8.548323822001639515038738961618255438422e-0004, 152 p6 = 1.205520092530505090384383082516403772317e-0004, 153 p7 = -1.492214100762529635365672665955239554276e-0005, 154 p8 = 1.640186161764254363152286358441771740838e-0006, 155 p9 = -1.571599331700515057841960987689515895479e-0007, 156 p10= 1.073087585213621540635426191486561494058e-0008; 157 /* 158 * Coefficients for approximation to erf in [0.84375,1.25] 159 */ 160 static double 161 pa0 = -2.362118560752659485957248365514511540287e-0003, 162 pa1 = 4.148561186837483359654781492060070469522e-0001, 163 pa2 = -3.722078760357013107593507594535478633044e-0001, 164 pa3 = 3.183466199011617316853636418691420262160e-0001, 165 pa4 = -1.108946942823966771253985510891237782544e-0001, 166 pa5 = 3.547830432561823343969797140537411825179e-0002, 167 pa6 = -2.166375594868790886906539848893221184820e-0003, 168 qa1 = 1.064208804008442270765369280952419863524e-0001, 169 qa2 = 5.403979177021710663441167681878575087235e-0001, 170 qa3 = 7.182865441419627066207655332170665812023e-0002, 171 qa4 = 1.261712198087616469108438860983447773726e-0001, 172 qa5 = 1.363708391202905087876983523620537833157e-0002, 173 qa6 = 1.198449984679910764099772682882189711364e-0002; 174 /* 175 * log(sqrt(pi)) for large x expansions. 176 * The tail (lsqrtPI_lo) is included in the rational 177 * approximations. 178 */ 179 static double 180 lsqrtPI_hi = .5723649429247000819387380943226; 181 /* 182 * lsqrtPI_lo = .000000000000000005132975581353913; 183 * 184 * Coefficients for approximation to erfc in [2, 4] 185 */ 186 static double 187 rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */ 188 rb1 = 2.15592846101742183841910806188e-008, 189 rb2 = 6.24998557732436510470108714799e-001, 190 rb3 = 8.24849222231141787631258921465e+000, 191 rb4 = 2.63974967372233173534823436057e+001, 192 rb5 = 9.86383092541570505318304640241e+000, 193 rb6 = -7.28024154841991322228977878694e+000, 194 rb7 = 5.96303287280680116566600190708e+000, 195 rb8 = -4.40070358507372993983608466806e+000, 196 rb9 = 2.39923700182518073731330332521e+000, 197 rb10 = -6.89257464785841156285073338950e-001, 198 sb1 = 1.56641558965626774835300238919e+001, 199 sb2 = 7.20522741000949622502957936376e+001, 200 sb3 = 9.60121069770492994166488642804e+001; 201 /* 202 * Coefficients for approximation to erfc in [1.25, 2] 203 */ 204 static double 205 rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */ 206 rc1 = 1.28735722546372485255126993930e-005, 207 rc2 = 6.24664954087883916855616917019e-001, 208 rc3 = 4.69798884785807402408863708843e+000, 209 rc4 = 7.61618295853929705430118701770e+000, 210 rc5 = 9.15640208659364240872946538730e-001, 211 rc6 = -3.59753040425048631334448145935e-001, 212 rc7 = 1.42862267989304403403849619281e-001, 213 rc8 = -4.74392758811439801958087514322e-002, 214 rc9 = 1.09964787987580810135757047874e-002, 215 rc10 = -1.28856240494889325194638463046e-003, 216 sc1 = 9.97395106984001955652274773456e+000, 217 sc2 = 2.80952153365721279953959310660e+001, 218 sc3 = 2.19826478142545234106819407316e+001; 219 /* 220 * Coefficients for approximation to erfc in [4,28] 221 */ 222 static double 223 rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */ 224 rd1 = -4.99999999999640086151350330820e-001, 225 rd2 = 6.24999999772906433825880867516e-001, 226 rd3 = -1.54166659428052432723177389562e+000, 227 rd4 = 5.51561147405411844601985649206e+000, 228 rd5 = -2.55046307982949826964613748714e+001, 229 rd6 = 1.43631424382843846387913799845e+002, 230 rd7 = -9.45789244999420134263345971704e+002, 231 rd8 = 6.94834146607051206956384703517e+003, 232 rd9 = -5.27176414235983393155038356781e+004, 233 rd10 = 3.68530281128672766499221324921e+005, 234 rd11 = -2.06466642800404317677021026611e+006, 235 rd12 = 7.78293889471135381609201431274e+006, 236 rd13 = -1.42821001129434127360582351685e+007; 237 238 double erf(x) 239 double x; 240 { 241 double R,S,P,Q,ax,s,y,z,r,fabs(),exp(); 242 if(!finite(x)) { /* erf(nan)=nan */ 243 if (isnan(x)) 244 return(x); 245 return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */ 246 } 247 if ((ax = x) < 0) 248 ax = - ax; 249 if (ax < .84375) { 250 if (ax < 3.7e-09) { 251 if (ax < 1.0e-308) 252 return 0.125*(8.0*x+p0t8*x); /*avoid underflow */ 253 return x + p0*x; 254 } 255 y = x*x; 256 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 257 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 258 return x + x*(p0+r); 259 } 260 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 261 s = fabs(x)-one; 262 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 263 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 264 if (x>=0) 265 return (c + P/Q); 266 else 267 return (-c - P/Q); 268 } 269 if (ax >= 6.0) { /* inf>|x|>=6 */ 270 if (x >= 0.0) 271 return (one-tiny); 272 else 273 return (tiny-one); 274 } 275 /* 1.25 <= |x| < 6 */ 276 z = -ax*ax; 277 s = -one/z; 278 if (ax < 2.0) { 279 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 280 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 281 S = one+s*(sc1+s*(sc2+s*sc3)); 282 } else { 283 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 284 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 285 S = one+s*(sb1+s*(sb2+s*sb3)); 286 } 287 y = (R/S -.5*s) - lsqrtPI_hi; 288 z += y; 289 z = exp(z)/ax; 290 if (x >= 0) 291 return (one-z); 292 else 293 return (z-one); 294 } 295 296 double erfc(x) 297 double x; 298 { 299 double R,S,P,Q,s,ax,y,z,r,fabs(),__exp__D(); 300 if (!finite(x)) { 301 if (isnan(x)) /* erfc(NaN) = NaN */ 302 return(x); 303 else if (x > 0) /* erfc(+-inf)=0,2 */ 304 return 0.0; 305 else 306 return 2.0; 307 } 308 if ((ax = x) < 0) 309 ax = -ax; 310 if (ax < .84375) { /* |x|<0.84375 */ 311 if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */ 312 return one-x; 313 y = x*x; 314 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 315 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 316 if (ax < .0625) { /* |x|<2**-4 */ 317 return (one-(x+x*(p0+r))); 318 } else { 319 r = x*(p0+r); 320 r += (x-half); 321 return (half - r); 322 } 323 } 324 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 325 s = ax-one; 326 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 327 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 328 if (x>=0) { 329 z = one-c; return z - P/Q; 330 } else { 331 z = c+P/Q; return one+z; 332 } 333 } 334 if (ax >= 28) /* Out of range */ 335 if (x>0) 336 return (tiny*tiny); 337 else 338 return (two-tiny); 339 z = ax; 340 TRUNC(z); 341 y = z - ax; y *= (ax+z); 342 z *= -z; /* Here z + y = -x^2 */ 343 s = one/(-z-y); /* 1/(x*x) */ 344 if (ax >= 4) { /* 6 <= ax */ 345 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+ 346 s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10 347 +s*(rd11+s*(rd12+s*rd13)))))))))))); 348 y += rd0; 349 } else if (ax >= 2) { 350 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 351 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 352 S = one+s*(sb1+s*(sb2+s*sb3)); 353 y += R/S; 354 R = -.5*s; 355 } else { 356 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 357 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 358 S = one+s*(sc1+s*(sc2+s*sc3)); 359 y += R/S; 360 R = -.5*s; 361 } 362 /* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */ 363 s = ((R + y) - lsqrtPI_hi) + z; 364 y = (((z-s) - lsqrtPI_hi) + R) + y; 365 r = __exp__D(s, y)/x; 366 if (x>0) 367 return r; 368 else 369 return two-r; 370 } 371