n_erf.c revision 1.1
1 /* $NetBSD: n_erf.c,v 1.1 1995/10/10 23:36:43 ragge 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 static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93"; 37 #endif /* not lint */ 38 39 #include "mathimpl.h" 40 41 /* Modified Nov 30, 1992 P. McILROY: 42 * Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp) 43 * Replaced even+odd with direct calculation for x < .84375, 44 * to avoid destructive cancellation. 45 * 46 * Performance of erfc(x): 47 * In 300000 trials in the range [.83, .84375] the 48 * maximum observed error was 3.6ulp. 49 * 50 * In [.84735,1.25] the maximum observed error was <2.5ulp in 51 * 100000 runs in the range [1.2, 1.25]. 52 * 53 * In [1.25,26] (Not including subnormal results) 54 * the error is < 1.7ulp. 55 */ 56 57 /* double erf(double x) 58 * double erfc(double x) 59 * x 60 * 2 |\ 61 * erf(x) = --------- | exp(-t*t)dt 62 * sqrt(pi) \| 63 * 0 64 * 65 * erfc(x) = 1-erf(x) 66 * 67 * Method: 68 * 1. Reduce x to |x| by erf(-x) = -erf(x) 69 * 2. For x in [0, 0.84375] 70 * erf(x) = x + x*P(x^2) 71 * erfc(x) = 1 - erf(x) if x<=0.25 72 * = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375] 73 * where 74 * 2 2 4 20 75 * P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x ) 76 * is an approximation to (erf(x)-x)/x with precision 77 * 78 * -56.45 79 * | P - (erf(x)-x)/x | <= 2 80 * 81 * 82 * Remark. The formula is derived by noting 83 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 84 * and that 85 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 86 * is close to one. The interval is chosen because the fixed 87 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 88 * near 0.6174), and by some experiment, 0.84375 is chosen to 89 * guarantee the error is less than one ulp for erf. 90 * 91 * 3. For x in [0.84375,1.25], let s = x - 1, and 92 * c = 0.84506291151 rounded to single (24 bits) 93 * erf(x) = c + P1(s)/Q1(s) 94 * erfc(x) = (1-c) - P1(s)/Q1(s) 95 * |P1/Q1 - (erf(x)-c)| <= 2**-59.06 96 * Remark: here we use the taylor series expansion at x=1. 97 * erf(1+s) = erf(1) + s*Poly(s) 98 * = 0.845.. + P1(s)/Q1(s) 99 * That is, we use rational approximation to approximate 100 * erf(1+s) - (c = (single)0.84506291151) 101 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 102 * where 103 * P1(s) = degree 6 poly in s 104 * Q1(s) = degree 6 poly in s 105 * 106 * 4. For x in [1.25, 2]; [2, 4] 107 * erf(x) = 1.0 - tiny 108 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z)) 109 * 110 * Where z = 1/(x*x), R is degree 9, and S is degree 3; 111 * 112 * 5. For x in [4,28] 113 * erf(x) = 1.0 - tiny 114 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z)) 115 * 116 * Where P is degree 14 polynomial in 1/(x*x). 117 * 118 * Notes: 119 * Here 4 and 5 make use of the asymptotic series 120 * exp(-x*x) 121 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ); 122 * x*sqrt(pi) 123 * 124 * where for z = 1/(x*x) 125 * P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...)))) 126 * 127 * Thus we use rational approximation to approximate 128 * erfc*x*exp(x*x) ~ 1/sqrt(pi); 129 * 130 * The error bound for the target function, G(z) for 131 * the interval 132 * [4, 28]: 133 * |eps + 1/(z)P(z) - G(z)| < 2**(-56.61) 134 * for [2, 4]: 135 * |R(z)/S(z) - G(z)| < 2**(-58.24) 136 * for [1.25, 2]: 137 * |R(z)/S(z) - G(z)| < 2**(-58.12) 138 * 139 * 6. For inf > x >= 28 140 * erf(x) = 1 - tiny (raise inexact) 141 * erfc(x) = tiny*tiny (raise underflow) 142 * 143 * 7. Special cases: 144 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 145 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 146 * erfc/erf(NaN) is NaN 147 */ 148 149 #if defined(vax) || defined(tahoe) 150 #define _IEEE 0 151 #define TRUNC(x) (double) (float) (x) 152 #else 153 #define _IEEE 1 154 #define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000 155 #define infnan(x) 0.0 156 #endif 157 158 #ifdef _IEEE_LIBM 159 /* 160 * redefining "___function" to "function" in _IEEE_LIBM mode 161 */ 162 #include "ieee_libm.h" 163 #endif 164 165 static double 166 tiny = 1e-300, 167 half = 0.5, 168 one = 1.0, 169 two = 2.0, 170 c = 8.45062911510467529297e-01, /* (float)0.84506291151 */ 171 /* 172 * Coefficients for approximation to erf in [0,0.84375] 173 */ 174 p0t8 = 1.02703333676410051049867154944018394163280, 175 p0 = 1.283791670955125638123339436800229927041e-0001, 176 p1 = -3.761263890318340796574473028946097022260e-0001, 177 p2 = 1.128379167093567004871858633779992337238e-0001, 178 p3 = -2.686617064084433642889526516177508374437e-0002, 179 p4 = 5.223977576966219409445780927846432273191e-0003, 180 p5 = -8.548323822001639515038738961618255438422e-0004, 181 p6 = 1.205520092530505090384383082516403772317e-0004, 182 p7 = -1.492214100762529635365672665955239554276e-0005, 183 p8 = 1.640186161764254363152286358441771740838e-0006, 184 p9 = -1.571599331700515057841960987689515895479e-0007, 185 p10= 1.073087585213621540635426191486561494058e-0008; 186 /* 187 * Coefficients for approximation to erf in [0.84375,1.25] 188 */ 189 static double 190 pa0 = -2.362118560752659485957248365514511540287e-0003, 191 pa1 = 4.148561186837483359654781492060070469522e-0001, 192 pa2 = -3.722078760357013107593507594535478633044e-0001, 193 pa3 = 3.183466199011617316853636418691420262160e-0001, 194 pa4 = -1.108946942823966771253985510891237782544e-0001, 195 pa5 = 3.547830432561823343969797140537411825179e-0002, 196 pa6 = -2.166375594868790886906539848893221184820e-0003, 197 qa1 = 1.064208804008442270765369280952419863524e-0001, 198 qa2 = 5.403979177021710663441167681878575087235e-0001, 199 qa3 = 7.182865441419627066207655332170665812023e-0002, 200 qa4 = 1.261712198087616469108438860983447773726e-0001, 201 qa5 = 1.363708391202905087876983523620537833157e-0002, 202 qa6 = 1.198449984679910764099772682882189711364e-0002; 203 /* 204 * log(sqrt(pi)) for large x expansions. 205 * The tail (lsqrtPI_lo) is included in the rational 206 * approximations. 207 */ 208 static double 209 lsqrtPI_hi = .5723649429247000819387380943226; 210 /* 211 * lsqrtPI_lo = .000000000000000005132975581353913; 212 * 213 * Coefficients for approximation to erfc in [2, 4] 214 */ 215 static double 216 rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */ 217 rb1 = 2.15592846101742183841910806188e-008, 218 rb2 = 6.24998557732436510470108714799e-001, 219 rb3 = 8.24849222231141787631258921465e+000, 220 rb4 = 2.63974967372233173534823436057e+001, 221 rb5 = 9.86383092541570505318304640241e+000, 222 rb6 = -7.28024154841991322228977878694e+000, 223 rb7 = 5.96303287280680116566600190708e+000, 224 rb8 = -4.40070358507372993983608466806e+000, 225 rb9 = 2.39923700182518073731330332521e+000, 226 rb10 = -6.89257464785841156285073338950e-001, 227 sb1 = 1.56641558965626774835300238919e+001, 228 sb2 = 7.20522741000949622502957936376e+001, 229 sb3 = 9.60121069770492994166488642804e+001; 230 /* 231 * Coefficients for approximation to erfc in [1.25, 2] 232 */ 233 static double 234 rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */ 235 rc1 = 1.28735722546372485255126993930e-005, 236 rc2 = 6.24664954087883916855616917019e-001, 237 rc3 = 4.69798884785807402408863708843e+000, 238 rc4 = 7.61618295853929705430118701770e+000, 239 rc5 = 9.15640208659364240872946538730e-001, 240 rc6 = -3.59753040425048631334448145935e-001, 241 rc7 = 1.42862267989304403403849619281e-001, 242 rc8 = -4.74392758811439801958087514322e-002, 243 rc9 = 1.09964787987580810135757047874e-002, 244 rc10 = -1.28856240494889325194638463046e-003, 245 sc1 = 9.97395106984001955652274773456e+000, 246 sc2 = 2.80952153365721279953959310660e+001, 247 sc3 = 2.19826478142545234106819407316e+001; 248 /* 249 * Coefficients for approximation to erfc in [4,28] 250 */ 251 static double 252 rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */ 253 rd1 = -4.99999999999640086151350330820e-001, 254 rd2 = 6.24999999772906433825880867516e-001, 255 rd3 = -1.54166659428052432723177389562e+000, 256 rd4 = 5.51561147405411844601985649206e+000, 257 rd5 = -2.55046307982949826964613748714e+001, 258 rd6 = 1.43631424382843846387913799845e+002, 259 rd7 = -9.45789244999420134263345971704e+002, 260 rd8 = 6.94834146607051206956384703517e+003, 261 rd9 = -5.27176414235983393155038356781e+004, 262 rd10 = 3.68530281128672766499221324921e+005, 263 rd11 = -2.06466642800404317677021026611e+006, 264 rd12 = 7.78293889471135381609201431274e+006, 265 rd13 = -1.42821001129434127360582351685e+007; 266 267 double erf(x) 268 double x; 269 { 270 double R,S,P,Q,ax,s,y,z,r,fabs(),exp(); 271 if(!finite(x)) { /* erf(nan)=nan */ 272 if (isnan(x)) 273 return(x); 274 return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */ 275 } 276 if ((ax = x) < 0) 277 ax = - ax; 278 if (ax < .84375) { 279 if (ax < 3.7e-09) { 280 if (ax < 1.0e-308) 281 return 0.125*(8.0*x+p0t8*x); /*avoid underflow */ 282 return x + p0*x; 283 } 284 y = x*x; 285 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 286 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 287 return x + x*(p0+r); 288 } 289 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 290 s = fabs(x)-one; 291 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 292 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 293 if (x>=0) 294 return (c + P/Q); 295 else 296 return (-c - P/Q); 297 } 298 if (ax >= 6.0) { /* inf>|x|>=6 */ 299 if (x >= 0.0) 300 return (one-tiny); 301 else 302 return (tiny-one); 303 } 304 /* 1.25 <= |x| < 6 */ 305 z = -ax*ax; 306 s = -one/z; 307 if (ax < 2.0) { 308 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 309 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 310 S = one+s*(sc1+s*(sc2+s*sc3)); 311 } else { 312 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 313 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 314 S = one+s*(sb1+s*(sb2+s*sb3)); 315 } 316 y = (R/S -.5*s) - lsqrtPI_hi; 317 z += y; 318 z = exp(z)/ax; 319 if (x >= 0) 320 return (one-z); 321 else 322 return (z-one); 323 } 324 325 double erfc(x) 326 double x; 327 { 328 double R,S,P,Q,s,ax,y,z,r,fabs(),__exp__D(); 329 if (!finite(x)) { 330 if (isnan(x)) /* erfc(NaN) = NaN */ 331 return(x); 332 else if (x > 0) /* erfc(+-inf)=0,2 */ 333 return 0.0; 334 else 335 return 2.0; 336 } 337 if ((ax = x) < 0) 338 ax = -ax; 339 if (ax < .84375) { /* |x|<0.84375 */ 340 if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */ 341 return one-x; 342 y = x*x; 343 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 344 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 345 if (ax < .0625) { /* |x|<2**-4 */ 346 return (one-(x+x*(p0+r))); 347 } else { 348 r = x*(p0+r); 349 r += (x-half); 350 return (half - r); 351 } 352 } 353 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 354 s = ax-one; 355 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 356 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 357 if (x>=0) { 358 z = one-c; return z - P/Q; 359 } else { 360 z = c+P/Q; return one+z; 361 } 362 } 363 if (ax >= 28) /* Out of range */ 364 if (x>0) 365 return (tiny*tiny); 366 else 367 return (two-tiny); 368 z = ax; 369 TRUNC(z); 370 y = z - ax; y *= (ax+z); 371 z *= -z; /* Here z + y = -x^2 */ 372 s = one/(-z-y); /* 1/(x*x) */ 373 if (ax >= 4) { /* 6 <= ax */ 374 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+ 375 s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10 376 +s*(rd11+s*(rd12+s*rd13)))))))))))); 377 y += rd0; 378 } else if (ax >= 2) { 379 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 380 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 381 S = one+s*(sb1+s*(sb2+s*sb3)); 382 y += R/S; 383 R = -.5*s; 384 } else { 385 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 386 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 387 S = one+s*(sc1+s*(sc2+s*sc3)); 388 y += R/S; 389 R = -.5*s; 390 } 391 /* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */ 392 s = ((R + y) - lsqrtPI_hi) + z; 393 y = (((z-s) - lsqrtPI_hi) + R) + y; 394 r = __exp__D(s, y)/x; 395 if (x>0) 396 return r; 397 else 398 return two-r; 399 } 400
Indexes created Tue Sep 23 07:09:52 GMT 2025