1 /* $NetBSD: n_erf.c,v 1.4 1999/07/02 15:37:36 simonb Exp $ */ 2 /*- 3 * Copyright (c) 1992, 1993 4 * The Regents of the University of California. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 1. Redistributions of source code must retain the above copyright 10 * notice, this list of conditions and the following disclaimer. 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 3. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 #if 0 37 static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93"; 38 #endif 39 #endif /* not lint */ 40 41 #include "mathimpl.h" 42 43 /* Modified Nov 30, 1992 P. McILROY: 44 * Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp) 45 * Replaced even+odd with direct calculation for x < .84375, 46 * to avoid destructive cancellation. 47 * 48 * Performance of erfc(x): 49 * In 300000 trials in the range [.83, .84375] the 50 * maximum observed error was 3.6ulp. 51 * 52 * In [.84735,1.25] the maximum observed error was <2.5ulp in 53 * 100000 runs in the range [1.2, 1.25]. 54 * 55 * In [1.25,26] (Not including subnormal results) 56 * the error is < 1.7ulp. 57 */ 58 59 /* double erf(double x) 60 * double erfc(double x) 61 * x 62 * 2 |\ 63 * erf(x) = --------- | exp(-t*t)dt 64 * sqrt(pi) \| 65 * 0 66 * 67 * erfc(x) = 1-erf(x) 68 * 69 * Method: 70 * 1. Reduce x to |x| by erf(-x) = -erf(x) 71 * 2. For x in [0, 0.84375] 72 * erf(x) = x + x*P(x^2) 73 * erfc(x) = 1 - erf(x) if x<=0.25 74 * = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375] 75 * where 76 * 2 2 4 20 77 * P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x ) 78 * is an approximation to (erf(x)-x)/x with precision 79 * 80 * -56.45 81 * | P - (erf(x)-x)/x | <= 2 82 * 83 * 84 * Remark. The formula is derived by noting 85 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 86 * and that 87 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 88 * is close to one. The interval is chosen because the fixed 89 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 90 * near 0.6174), and by some experiment, 0.84375 is chosen to 91 * guarantee the error is less than one ulp for erf. 92 * 93 * 3. For x in [0.84375,1.25], let s = x - 1, and 94 * c = 0.84506291151 rounded to single (24 bits) 95 * erf(x) = c + P1(s)/Q1(s) 96 * erfc(x) = (1-c) - P1(s)/Q1(s) 97 * |P1/Q1 - (erf(x)-c)| <= 2**-59.06 98 * Remark: here we use the taylor series expansion at x=1. 99 * erf(1+s) = erf(1) + s*Poly(s) 100 * = 0.845.. + P1(s)/Q1(s) 101 * That is, we use rational approximation to approximate 102 * erf(1+s) - (c = (single)0.84506291151) 103 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 104 * where 105 * P1(s) = degree 6 poly in s 106 * Q1(s) = degree 6 poly in s 107 * 108 * 4. For x in [1.25, 2]; [2, 4] 109 * erf(x) = 1.0 - tiny 110 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z)) 111 * 112 * Where z = 1/(x*x), R is degree 9, and S is degree 3; 113 * 114 * 5. For x in [4,28] 115 * erf(x) = 1.0 - tiny 116 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z)) 117 * 118 * Where P is degree 14 polynomial in 1/(x*x). 119 * 120 * Notes: 121 * Here 4 and 5 make use of the asymptotic series 122 * exp(-x*x) 123 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ); 124 * x*sqrt(pi) 125 * 126 * where for z = 1/(x*x) 127 * P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...)))) 128 * 129 * Thus we use rational approximation to approximate 130 * erfc*x*exp(x*x) ~ 1/sqrt(pi); 131 * 132 * The error bound for the target function, G(z) for 133 * the interval 134 * [4, 28]: 135 * |eps + 1/(z)P(z) - G(z)| < 2**(-56.61) 136 * for [2, 4]: 137 * |R(z)/S(z) - G(z)| < 2**(-58.24) 138 * for [1.25, 2]: 139 * |R(z)/S(z) - G(z)| < 2**(-58.12) 140 * 141 * 6. For inf > x >= 28 142 * erf(x) = 1 - tiny (raise inexact) 143 * erfc(x) = tiny*tiny (raise underflow) 144 * 145 * 7. Special cases: 146 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 147 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 148 * erfc/erf(NaN) is NaN 149 */ 150 151 #if defined(__vax__) || defined(tahoe) 152 #define _IEEE 0 153 #define TRUNC(x) (double)(x) = (float)(x) 154 #else 155 #define _IEEE 1 156 #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000 157 #define infnan(x) 0.0 158 #endif 159 160 #ifdef _IEEE_LIBM 161 /* 162 * redefining "___function" to "function" in _IEEE_LIBM mode 163 */ 164 #include "ieee_libm.h" 165 #endif 166 167 static double 168 tiny = 1e-300, 169 half = 0.5, 170 one = 1.0, 171 two = 2.0, 172 c = 8.45062911510467529297e-01, /* (float)0.84506291151 */ 173 /* 174 * Coefficients for approximation to erf in [0,0.84375] 175 */ 176 p0t8 = 1.02703333676410051049867154944018394163280, 177 p0 = 1.283791670955125638123339436800229927041e-0001, 178 p1 = -3.761263890318340796574473028946097022260e-0001, 179 p2 = 1.128379167093567004871858633779992337238e-0001, 180 p3 = -2.686617064084433642889526516177508374437e-0002, 181 p4 = 5.223977576966219409445780927846432273191e-0003, 182 p5 = -8.548323822001639515038738961618255438422e-0004, 183 p6 = 1.205520092530505090384383082516403772317e-0004, 184 p7 = -1.492214100762529635365672665955239554276e-0005, 185 p8 = 1.640186161764254363152286358441771740838e-0006, 186 p9 = -1.571599331700515057841960987689515895479e-0007, 187 p10= 1.073087585213621540635426191486561494058e-0008; 188 /* 189 * Coefficients for approximation to erf in [0.84375,1.25] 190 */ 191 static double 192 pa0 = -2.362118560752659485957248365514511540287e-0003, 193 pa1 = 4.148561186837483359654781492060070469522e-0001, 194 pa2 = -3.722078760357013107593507594535478633044e-0001, 195 pa3 = 3.183466199011617316853636418691420262160e-0001, 196 pa4 = -1.108946942823966771253985510891237782544e-0001, 197 pa5 = 3.547830432561823343969797140537411825179e-0002, 198 pa6 = -2.166375594868790886906539848893221184820e-0003, 199 qa1 = 1.064208804008442270765369280952419863524e-0001, 200 qa2 = 5.403979177021710663441167681878575087235e-0001, 201 qa3 = 7.182865441419627066207655332170665812023e-0002, 202 qa4 = 1.261712198087616469108438860983447773726e-0001, 203 qa5 = 1.363708391202905087876983523620537833157e-0002, 204 qa6 = 1.198449984679910764099772682882189711364e-0002; 205 /* 206 * log(sqrt(pi)) for large x expansions. 207 * The tail (lsqrtPI_lo) is included in the rational 208 * approximations. 209 */ 210 static double 211 lsqrtPI_hi = .5723649429247000819387380943226; 212 /* 213 * lsqrtPI_lo = .000000000000000005132975581353913; 214 * 215 * Coefficients for approximation to erfc in [2, 4] 216 */ 217 static double 218 rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */ 219 rb1 = 2.15592846101742183841910806188e-008, 220 rb2 = 6.24998557732436510470108714799e-001, 221 rb3 = 8.24849222231141787631258921465e+000, 222 rb4 = 2.63974967372233173534823436057e+001, 223 rb5 = 9.86383092541570505318304640241e+000, 224 rb6 = -7.28024154841991322228977878694e+000, 225 rb7 = 5.96303287280680116566600190708e+000, 226 rb8 = -4.40070358507372993983608466806e+000, 227 rb9 = 2.39923700182518073731330332521e+000, 228 rb10 = -6.89257464785841156285073338950e-001, 229 sb1 = 1.56641558965626774835300238919e+001, 230 sb2 = 7.20522741000949622502957936376e+001, 231 sb3 = 9.60121069770492994166488642804e+001; 232 /* 233 * Coefficients for approximation to erfc in [1.25, 2] 234 */ 235 static double 236 rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */ 237 rc1 = 1.28735722546372485255126993930e-005, 238 rc2 = 6.24664954087883916855616917019e-001, 239 rc3 = 4.69798884785807402408863708843e+000, 240 rc4 = 7.61618295853929705430118701770e+000, 241 rc5 = 9.15640208659364240872946538730e-001, 242 rc6 = -3.59753040425048631334448145935e-001, 243 rc7 = 1.42862267989304403403849619281e-001, 244 rc8 = -4.74392758811439801958087514322e-002, 245 rc9 = 1.09964787987580810135757047874e-002, 246 rc10 = -1.28856240494889325194638463046e-003, 247 sc1 = 9.97395106984001955652274773456e+000, 248 sc2 = 2.80952153365721279953959310660e+001, 249 sc3 = 2.19826478142545234106819407316e+001; 250 /* 251 * Coefficients for approximation to erfc in [4,28] 252 */ 253 static double 254 rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */ 255 rd1 = -4.99999999999640086151350330820e-001, 256 rd2 = 6.24999999772906433825880867516e-001, 257 rd3 = -1.54166659428052432723177389562e+000, 258 rd4 = 5.51561147405411844601985649206e+000, 259 rd5 = -2.55046307982949826964613748714e+001, 260 rd6 = 1.43631424382843846387913799845e+002, 261 rd7 = -9.45789244999420134263345971704e+002, 262 rd8 = 6.94834146607051206956384703517e+003, 263 rd9 = -5.27176414235983393155038356781e+004, 264 rd10 = 3.68530281128672766499221324921e+005, 265 rd11 = -2.06466642800404317677021026611e+006, 266 rd12 = 7.78293889471135381609201431274e+006, 267 rd13 = -1.42821001129434127360582351685e+007; 268 269 double erf(x) 270 double x; 271 { 272 double R,S,P,Q,ax,s,y,z,r; 273 if(!finite(x)) { /* erf(nan)=nan */ 274 if (isnan(x)) 275 return(x); 276 return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */ 277 } 278 if ((ax = x) < 0) 279 ax = - ax; 280 if (ax < .84375) { 281 if (ax < 3.7e-09) { 282 if (ax < 1.0e-308) 283 return 0.125*(8.0*x+p0t8*x); /*avoid underflow */ 284 return x + p0*x; 285 } 286 y = x*x; 287 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 288 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 289 return x + x*(p0+r); 290 } 291 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 292 s = fabs(x)-one; 293 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 294 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 295 if (x>=0) 296 return (c + P/Q); 297 else 298 return (-c - P/Q); 299 } 300 if (ax >= 6.0) { /* inf>|x|>=6 */ 301 if (x >= 0.0) 302 return (one-tiny); 303 else 304 return (tiny-one); 305 } 306 /* 1.25 <= |x| < 6 */ 307 z = -ax*ax; 308 s = -one/z; 309 if (ax < 2.0) { 310 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 311 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 312 S = one+s*(sc1+s*(sc2+s*sc3)); 313 } else { 314 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 315 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 316 S = one+s*(sb1+s*(sb2+s*sb3)); 317 } 318 y = (R/S -.5*s) - lsqrtPI_hi; 319 z += y; 320 z = exp(z)/ax; 321 if (x >= 0) 322 return (one-z); 323 else 324 return (z-one); 325 } 326 327 double erfc(x) 328 double x; 329 { 330 double R,S,P,Q,s,ax,y,z,r; 331 if (!finite(x)) { 332 if (isnan(x)) /* erfc(NaN) = NaN */ 333 return(x); 334 else if (x > 0) /* erfc(+-inf)=0,2 */ 335 return 0.0; 336 else 337 return 2.0; 338 } 339 if ((ax = x) < 0) 340 ax = -ax; 341 if (ax < .84375) { /* |x|<0.84375 */ 342 if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */ 343 return one-x; 344 y = x*x; 345 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 346 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 347 if (ax < .0625) { /* |x|<2**-4 */ 348 return (one-(x+x*(p0+r))); 349 } else { 350 r = x*(p0+r); 351 r += (x-half); 352 return (half - r); 353 } 354 } 355 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 356 s = ax-one; 357 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 358 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 359 if (x>=0) { 360 z = one-c; return z - P/Q; 361 } else { 362 z = c+P/Q; return one+z; 363 } 364 } 365 if (ax >= 28) { /* Out of range */ 366 if (x>0) 367 return (tiny*tiny); 368 else 369 return (two-tiny); 370 } 371 z = ax; 372 TRUNC(z); 373 y = z - ax; y *= (ax+z); 374 z *= -z; /* Here z + y = -x^2 */ 375 s = one/(-z-y); /* 1/(x*x) */ 376 if (ax >= 4) { /* 6 <= ax */ 377 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+ 378 s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10 379 +s*(rd11+s*(rd12+s*rd13)))))))))))); 380 y += rd0; 381 } else if (ax >= 2) { 382 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 383 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 384 S = one+s*(sb1+s*(sb2+s*sb3)); 385 y += R/S; 386 R = -.5*s; 387 } else { 388 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 389 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 390 S = one+s*(sc1+s*(sc2+s*sc3)); 391 y += R/S; 392 R = -.5*s; 393 } 394 /* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */ 395 s = ((R + y) - lsqrtPI_hi) + z; 396 y = (((z-s) - lsqrtPI_hi) + R) + y; 397 r = __exp__D(s, y)/x; 398 if (x>0) 399 return r; 400 else 401 return two-r; 402 } 403