1 /*- 2 * Copyright (c) 1992 The Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 */ 7 8 #ifndef lint 9 static char sccsid[] = "@(#)erf.c 5.6 (Berkeley) 12/14/92"; 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 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 58 * near 0.6174), and by some experiment, 0.84375 is chosen to 59 * guarantee the error is less than one ulp for erf. 60 * 61 * 3. For x in [0.84375,1.25], let s = x - 1, and 62 * c = 0.84506291151 rounded to single (24 bits) 63 * erf(x) = c + P1(s)/Q1(s) 64 * erfc(x) = (1-c) - P1(s)/Q1(s) 65 * |P1/Q1 - (erf(x)-c)| <= 2**-59.06 66 * Remark: here we use the taylor series expansion at x=1. 67 * erf(1+s) = erf(1) + s*Poly(s) 68 * = 0.845.. + P1(s)/Q1(s) 69 * That is, we use rational approximation to approximate 70 * erf(1+s) - (c = (single)0.84506291151) 71 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 72 * where 73 * P1(s) = degree 7 poly in s 74 * 75 * 4. For x in [1.25, 2]; [2, 4] 76 * erf(x) = 1.0 - tiny 77 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z)) 78 * 79 * Where z = 1/(x*x), R is degree 9, and S is degree 3; 80 * 81 * 5. For x in [4,28] 82 * erf(x) = 1.0 - tiny 83 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z)) 84 * 85 * Where P is degree 14 polynomial in 1/(x*x). 86 * 87 * Notes: 88 * Here 4 and 5 make use of the asymptotic series 89 * exp(-x*x) 90 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ); 91 * x*sqrt(pi) 92 * 93 * where for z = 1/(x*x) 94 * P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...)))) 95 * 96 * Thus we use rational approximation to approximate 97 * erfc*x*exp(x*x) ~ 1/sqrt(pi); 98 * 99 * The error bound for the target function, G(z) for 100 * the interval 101 * [4, 28]: 102 * |eps + 1/(z)P(z) - G(z)| < 2**(-56.61) 103 * for [2, 4]: 104 * |R(z)/S(z) - G(z)| < 2**(-58.24) 105 * for [1.25, 2]: 106 * |R(z)/S(z) - G(z)| < 2**(-58.12) 107 * 108 * 6. For inf > x >= 28 109 * erf(x) = 1 - tiny (raise inexact) 110 * erfc(x) = tiny*tiny (raise underflow) 111 * 112 * 7. Special cases: 113 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 114 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 115 * erfc/erf(NaN) is NaN 116 */ 117 118 #if defined(vax) || defined(tahoe) 119 #define _IEEE 0 120 #define TRUNC(x) (double) (float) (x) 121 #else 122 #define _IEEE 1 123 #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000 124 #define infnan(x) 0.0 125 #endif 126 127 #ifdef _IEEE_LIBM 128 /* 129 * redefining "___function" to "function" in _IEEE_LIBM mode 130 */ 131 #include "ieee_libm.h" 132 #endif 133 134 static double 135 tiny = 1e-300, 136 half = 0.5, 137 one = 1.0, 138 two = 2.0, 139 c = 8.45062911510467529297e-01, /* (float)0.84506291151 */ 140 /* 141 * Coefficients for approximation to erf in [0,0.84375] 142 */ 143 p0t8 = 1.02703333676410051049867154944018394163280, 144 p0 = 1.283791670955125638123339436800229927041e-0001, 145 p1 = -3.761263890318340796574473028946097022260e-0001, 146 p2 = 1.128379167093567004871858633779992337238e-0001, 147 p3 = -2.686617064084433642889526516177508374437e-0002, 148 p4 = 5.223977576966219409445780927846432273191e-0003, 149 p5 = -8.548323822001639515038738961618255438422e-0004, 150 p6 = 1.205520092530505090384383082516403772317e-0004, 151 p7 = -1.492214100762529635365672665955239554276e-0005, 152 p8 = 1.640186161764254363152286358441771740838e-0006, 153 p9 = -1.571599331700515057841960987689515895479e-0007, 154 p10= 1.073087585213621540635426191486561494058e-0008; 155 /* 156 * Coefficients for approximation to erf in [0.84375,1.25] 157 */ 158 static double 159 pa0 = -2.362118560752659485957248365514511540287e-0003, 160 pa1 = 4.148561186837483359654781492060070469522e-0001, 161 pa2 = -3.722078760357013107593507594535478633044e-0001, 162 pa3 = 3.183466199011617316853636418691420262160e-0001, 163 pa4 = -1.108946942823966771253985510891237782544e-0001, 164 pa5 = 3.547830432561823343969797140537411825179e-0002, 165 pa6 = -2.166375594868790886906539848893221184820e-0003, 166 qa1 = 1.064208804008442270765369280952419863524e-0001, 167 qa2 = 5.403979177021710663441167681878575087235e-0001, 168 qa3 = 7.182865441419627066207655332170665812023e-0002, 169 qa4 = 1.261712198087616469108438860983447773726e-0001, 170 qa5 = 1.363708391202905087876983523620537833157e-0002, 171 qa6 = 1.198449984679910764099772682882189711364e-0002; 172 /* 173 * log(sqrt(pi)) for large x expansions. 174 * The tail (lsqrtPI_lo) is included in the rational 175 * approximations. 176 */ 177 static double 178 lsqrtPI_hi = .5723649429247000819387380943226; 179 /* 180 * lsqrtPI_lo = .000000000000000005132975581353913; 181 * 182 * Coefficients for approximation to erfc in [2, 4] 183 */ 184 static double 185 rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */ 186 rb1 = 2.15592846101742183841910806188e-008, 187 rb2 = 6.24998557732436510470108714799e-001, 188 rb3 = 8.24849222231141787631258921465e+000, 189 rb4 = 2.63974967372233173534823436057e+001, 190 rb5 = 9.86383092541570505318304640241e+000, 191 rb6 = -7.28024154841991322228977878694e+000, 192 rb7 = 5.96303287280680116566600190708e+000, 193 rb8 = -4.40070358507372993983608466806e+000, 194 rb9 = 2.39923700182518073731330332521e+000, 195 rb10 = -6.89257464785841156285073338950e-001, 196 sb1 = 1.56641558965626774835300238919e+001, 197 sb2 = 7.20522741000949622502957936376e+001, 198 sb3 = 9.60121069770492994166488642804e+001; 199 /* 200 * Coefficients for approximation to erfc in [1.25, 2] 201 */ 202 static double 203 rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */ 204 rc1 = 1.28735722546372485255126993930e-005, 205 rc2 = 6.24664954087883916855616917019e-001, 206 rc3 = 4.69798884785807402408863708843e+000, 207 rc4 = 7.61618295853929705430118701770e+000, 208 rc5 = 9.15640208659364240872946538730e-001, 209 rc6 = -3.59753040425048631334448145935e-001, 210 rc7 = 1.42862267989304403403849619281e-001, 211 rc8 = -4.74392758811439801958087514322e-002, 212 rc9 = 1.09964787987580810135757047874e-002, 213 rc10 = -1.28856240494889325194638463046e-003, 214 sc1 = 9.97395106984001955652274773456e+000, 215 sc2 = 2.80952153365721279953959310660e+001, 216 sc3 = 2.19826478142545234106819407316e+001; 217 /* 218 * Coefficients for approximation to erfc in [4,28] 219 */ 220 static double 221 rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */ 222 rd1 = -4.99999999999640086151350330820e-001, 223 rd2 = 6.24999999772906433825880867516e-001, 224 rd3 = -1.54166659428052432723177389562e+000, 225 rd4 = 5.51561147405411844601985649206e+000, 226 rd5 = -2.55046307982949826964613748714e+001, 227 rd6 = 1.43631424382843846387913799845e+002, 228 rd7 = -9.45789244999420134263345971704e+002, 229 rd8 = 6.94834146607051206956384703517e+003, 230 rd9 = -5.27176414235983393155038356781e+004, 231 rd10 = 3.68530281128672766499221324921e+005, 232 rd11 = -2.06466642800404317677021026611e+006, 233 rd12 = 7.78293889471135381609201431274e+006, 234 rd13 = -1.42821001129434127360582351685e+007; 235 236 double erf(x) 237 double x; 238 { 239 double R,S,P,Q,ax,s,y,z,r,fabs(),exp(); 240 if(!finite(x)) { /* erf(nan)=nan */ 241 if (isnan(x)) 242 return(x); 243 return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */ 244 } 245 if ((ax = x) < 0) 246 ax = - ax; 247 if (ax < .84375) { 248 if (ax < 3.7e-09) { 249 if (ax < 1.0e-308) 250 return 0.125*(8.0*x+p0t8*x); /*avoid underflow */ 251 return x + p0*x; 252 } 253 y = x*x; 254 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 255 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 256 return x + x*(p0+r); 257 } 258 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 259 s = fabs(x)-one; 260 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 261 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 262 if (x>=0) 263 return (c + P/Q); 264 else 265 return (-c - P/Q); 266 } 267 if (ax >= 6.0) { /* inf>|x|>=6 */ 268 if (x >= 0.0) 269 return (one-tiny); 270 else 271 return (tiny-one); 272 } 273 /* 1.25 <= |x| < 6 */ 274 z = -ax*ax; 275 s = -one/z; 276 if (ax < 2.0) { 277 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 278 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 279 S = one+s*(sc1+s*(sc2+s*sc3)); 280 } else { 281 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 282 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 283 S = one+s*(sb1+s*(sb2+s*sb3)); 284 } 285 y = (R/S -.5*s) - lsqrtPI_hi; 286 z += y; 287 z = exp(z)/ax; 288 if (x >= 0) 289 return (one-z); 290 else 291 return (z-one); 292 } 293 294 double erfc(x) 295 double x; 296 { 297 double R,S,P,Q,s,ax,y,z,r,fabs(),exp__D(); 298 if (!finite(x)) { 299 if (isnan(x)) /* erfc(NaN) = NaN */ 300 return(x); 301 else if (x > 0) /* erfc(+-inf)=0,2 */ 302 return 0.0; 303 else 304 return 2.0; 305 } 306 if ((ax = x) < 0) 307 ax = -ax; 308 if (ax < .84375) { /* |x|<0.84375 */ 309 if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */ 310 return one-x; 311 y = x*x; 312 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 313 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 314 if (ax < .0625) { /* |x|<2**-4 */ 315 return (one-(x+x*(p0+r))); 316 } else { 317 r = x*(p0+r); 318 r += (x-half); 319 return (half - r); 320 } 321 } 322 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 323 s = ax-one; 324 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 325 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 326 if (x>=0) { 327 z = one-c; return z - P/Q; 328 } else { 329 z = c+P/Q; return one+z; 330 } 331 } 332 if (ax >= 28) /* Out of range */ 333 if (x>0) 334 return (tiny*tiny); 335 else 336 return (two-tiny); 337 z = ax; 338 TRUNC(z); 339 y = z - ax; y *= (ax+z); 340 z *= -z; /* Here z + y = -x^2 */ 341 s = one/(-z-y); /* 1/(x*x) */ 342 if (ax >= 4) { /* 6 <= ax */ 343 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+ 344 s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10 345 +s*(rd11+s*(rd12+s*rd13)))))))))))); 346 y += rd0; 347 } else if (ax >= 2) { 348 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 349 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 350 S = one+s*(sb1+s*(sb2+s*sb3)); 351 y += R/S; 352 R = -.5*s; 353 } else { 354 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 355 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 356 S = one+s*(sc1+s*(sc2+s*sc3)); 357 y += R/S; 358 R = -.5*s; 359 } 360 /* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */ 361 s = ((R + y) - lsqrtPI_hi) + z; 362 y = (((z-s) - lsqrtPI_hi) + R) + y; 363 r = exp__D(s, y)/x; 364 if (x>0) 365 return r; 366 else 367 return two-r; 368 } 369