1/* $NetBSD: n_erf.c,v 1.7 2005/05/03 04:18:32 matt 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. Neither the name of the University nor the names of its contributors 15 * may be used to endorse or promote products derived from this software 16 * without specific prior written permission. 17 * 18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 28 * SUCH DAMAGE. 29 */ 30 31#ifndef lint 32#if 0 33static char sccsid[] = "@(#)erf.c 8.1 (Berkeley) 6/4/93"; 34#endif 35#endif /* not lint */ 36 37#include "mathimpl.h" 38 39/* Modified Nov 30, 1992 P. McILROY: 40 * Replaced expansions for x >= 1.25 (error 1.7ulp vs ~6ulp) 41 * Replaced even+odd with direct calculation for x < .84375, 42 * to avoid destructive cancellation. 43 * 44 * Performance of erfc(x): 45 * In 300000 trials in the range [.83, .84375] the 46 * maximum observed error was 3.6ulp. 47 * 48 * In [.84735,1.25] the maximum observed error was <2.5ulp in 49 * 100000 runs in the range [1.2, 1.25]. 50 * 51 * In [1.25,26] (Not including subnormal results) 52 * the error is < 1.7ulp. 53 */ 54 55/* double erf(double x) 56 * double erfc(double x) 57 * x 58 * 2 |\ 59 * erf(x) = --------- | exp(-t*t)dt 60 * sqrt(pi) \| 61 * 0 62 * 63 * erfc(x) = 1-erf(x) 64 * 65 * Method: 66 * 1. Reduce x to |x| by erf(-x) = -erf(x) 67 * 2. For x in [0, 0.84375] 68 * erf(x) = x + x*P(x^2) 69 * erfc(x) = 1 - erf(x) if x<=0.25 70 * = 0.5 + ((0.5-x)-x*P) if x in [0.25,0.84375] 71 * where 72 * 2 2 4 20 73 * P = P(x ) = (p0 + p1 * x + p2 * x + ... + p10 * x ) 74 * is an approximation to (erf(x)-x)/x with precision 75 * 76 * -56.45 77 * | P - (erf(x)-x)/x | <= 2 78 * 79 * 80 * Remark. The formula is derived by noting 81 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 82 * and that 83 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 84 * is close to one. The interval is chosen because the fixed 85 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 86 * near 0.6174), and by some experiment, 0.84375 is chosen to 87 * guarantee the error is less than one ulp for erf. 88 * 89 * 3. For x in [0.84375,1.25], let s = x - 1, and 90 * c = 0.84506291151 rounded to single (24 bits) 91 * erf(x) = c + P1(s)/Q1(s) 92 * erfc(x) = (1-c) - P1(s)/Q1(s) 93 * |P1/Q1 - (erf(x)-c)| <= 2**-59.06 94 * Remark: here we use the taylor series expansion at x=1. 95 * erf(1+s) = erf(1) + s*Poly(s) 96 * = 0.845.. + P1(s)/Q1(s) 97 * That is, we use rational approximation to approximate 98 * erf(1+s) - (c = (single)0.84506291151) 99 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 100 * where 101 * P1(s) = degree 6 poly in s 102 * Q1(s) = degree 6 poly in s 103 * 104 * 4. For x in [1.25, 2]; [2, 4] 105 * erf(x) = 1.0 - tiny 106 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi) -.5z + R(z)/S(z)) 107 * 108 * Where z = 1/(x*x), R is degree 9, and S is degree 3; 109 * 110 * 5. For x in [4,28] 111 * erf(x) = 1.0 - tiny 112 * erfc(x) = (1/x)exp(-x*x-(.5*log(pi)+eps + zP(z)) 113 * 114 * Where P is degree 14 polynomial in 1/(x*x). 115 * 116 * Notes: 117 * Here 4 and 5 make use of the asymptotic series 118 * exp(-x*x) 119 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ); 120 * x*sqrt(pi) 121 * 122 * where for z = 1/(x*x) 123 * P(z) ~ z/2*(-1 + z*3/2*(1 + z*5/2*(-1 + z*7/2*(1 +...)))) 124 * 125 * Thus we use rational approximation to approximate 126 * erfc*x*exp(x*x) ~ 1/sqrt(pi); 127 * 128 * The error bound for the target function, G(z) for 129 * the interval 130 * [4, 28]: 131 * |eps + 1/(z)P(z) - G(z)| < 2**(-56.61) 132 * for [2, 4]: 133 * |R(z)/S(z) - G(z)| < 2**(-58.24) 134 * for [1.25, 2]: 135 * |R(z)/S(z) - G(z)| < 2**(-58.12) 136 * 137 * 6. For inf > x >= 28 138 * erf(x) = 1 - tiny (raise inexact) 139 * erfc(x) = tiny*tiny (raise underflow) 140 * 141 * 7. Special cases: 142 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 143 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 144 * erfc/erf(NaN) is NaN 145 */ 146 147#if defined(__vax__) || defined(tahoe) 148#define _IEEE 0 149#define TRUNC(x) (x) = (float)(x) 150#else 151#define _IEEE 1 152#define TRUNC(x) *(((int *) &x) + 1) &= 0xf8000000 153#define infnan(x) 0.0 154#endif 155 156#ifdef _IEEE_LIBM 157/* 158 * redefining "___function" to "function" in _IEEE_LIBM mode 159 */ 160#include "ieee_libm.h" 161#endif 162 163static const double 164tiny = _TINY, 165half = 0.5, 166one = 1.0, 167two = 2.0, 168c = 8.45062911510467529297e-01, /* (float)0.84506291151 */ 169/* 170 * Coefficients for approximation to erf in [0,0.84375] 171 */ 172p0t8 = 1.02703333676410051049867154944018394163280, 173p0 = 1.283791670955125638123339436800229927041e-0001, 174p1 = -3.761263890318340796574473028946097022260e-0001, 175p2 = 1.128379167093567004871858633779992337238e-0001, 176p3 = -2.686617064084433642889526516177508374437e-0002, 177p4 = 5.223977576966219409445780927846432273191e-0003, 178p5 = -8.548323822001639515038738961618255438422e-0004, 179p6 = 1.205520092530505090384383082516403772317e-0004, 180p7 = -1.492214100762529635365672665955239554276e-0005, 181p8 = 1.640186161764254363152286358441771740838e-0006, 182p9 = -1.571599331700515057841960987689515895479e-0007, 183p10= 1.073087585213621540635426191486561494058e-0008; 184/* 185 * Coefficients for approximation to erf in [0.84375,1.25] 186 */ 187static const double 188pa0 = -2.362118560752659485957248365514511540287e-0003, 189pa1 = 4.148561186837483359654781492060070469522e-0001, 190pa2 = -3.722078760357013107593507594535478633044e-0001, 191pa3 = 3.183466199011617316853636418691420262160e-0001, 192pa4 = -1.108946942823966771253985510891237782544e-0001, 193pa5 = 3.547830432561823343969797140537411825179e-0002, 194pa6 = -2.166375594868790886906539848893221184820e-0003, 195qa1 = 1.064208804008442270765369280952419863524e-0001, 196qa2 = 5.403979177021710663441167681878575087235e-0001, 197qa3 = 7.182865441419627066207655332170665812023e-0002, 198qa4 = 1.261712198087616469108438860983447773726e-0001, 199qa5 = 1.363708391202905087876983523620537833157e-0002, 200qa6 = 1.198449984679910764099772682882189711364e-0002; 201/* 202 * log(sqrt(pi)) for large x expansions. 203 * The tail (lsqrtPI_lo) is included in the rational 204 * approximations. 205*/ 206static const double 207 lsqrtPI_hi = .5723649429247000819387380943226; 208/* 209 * lsqrtPI_lo = .000000000000000005132975581353913; 210 * 211 * Coefficients for approximation to erfc in [2, 4] 212*/ 213static const double 214rb0 = -1.5306508387410807582e-010, /* includes lsqrtPI_lo */ 215rb1 = 2.15592846101742183841910806188e-008, 216rb2 = 6.24998557732436510470108714799e-001, 217rb3 = 8.24849222231141787631258921465e+000, 218rb4 = 2.63974967372233173534823436057e+001, 219rb5 = 9.86383092541570505318304640241e+000, 220rb6 = -7.28024154841991322228977878694e+000, 221rb7 = 5.96303287280680116566600190708e+000, 222rb8 = -4.40070358507372993983608466806e+000, 223rb9 = 2.39923700182518073731330332521e+000, 224rb10 = -6.89257464785841156285073338950e-001, 225sb1 = 1.56641558965626774835300238919e+001, 226sb2 = 7.20522741000949622502957936376e+001, 227sb3 = 9.60121069770492994166488642804e+001; 228/* 229 * Coefficients for approximation to erfc in [1.25, 2] 230*/ 231static const double 232rc0 = -2.47925334685189288817e-007, /* includes lsqrtPI_lo */ 233rc1 = 1.28735722546372485255126993930e-005, 234rc2 = 6.24664954087883916855616917019e-001, 235rc3 = 4.69798884785807402408863708843e+000, 236rc4 = 7.61618295853929705430118701770e+000, 237rc5 = 9.15640208659364240872946538730e-001, 238rc6 = -3.59753040425048631334448145935e-001, 239rc7 = 1.42862267989304403403849619281e-001, 240rc8 = -4.74392758811439801958087514322e-002, 241rc9 = 1.09964787987580810135757047874e-002, 242rc10 = -1.28856240494889325194638463046e-003, 243sc1 = 9.97395106984001955652274773456e+000, 244sc2 = 2.80952153365721279953959310660e+001, 245sc3 = 2.19826478142545234106819407316e+001; 246/* 247 * Coefficients for approximation to erfc in [4,28] 248 */ 249static const double 250rd0 = -2.1491361969012978677e-016, /* includes lsqrtPI_lo */ 251rd1 = -4.99999999999640086151350330820e-001, 252rd2 = 6.24999999772906433825880867516e-001, 253rd3 = -1.54166659428052432723177389562e+000, 254rd4 = 5.51561147405411844601985649206e+000, 255rd5 = -2.55046307982949826964613748714e+001, 256rd6 = 1.43631424382843846387913799845e+002, 257rd7 = -9.45789244999420134263345971704e+002, 258rd8 = 6.94834146607051206956384703517e+003, 259rd9 = -5.27176414235983393155038356781e+004, 260rd10 = 3.68530281128672766499221324921e+005, 261rd11 = -2.06466642800404317677021026611e+006, 262rd12 = 7.78293889471135381609201431274e+006, 263rd13 = -1.42821001129434127360582351685e+007; 264 265double 266erf(double x) 267{ 268 double R,S,P,Q,ax,s,y,z,r; 269 if(!finite(x)) { /* erf(nan)=nan */ 270 if (isnan(x)) 271 return(x); 272 return (x > 0 ? one : -one); /* erf(+/-inf)= +/-1 */ 273 } 274 if ((ax = x) < 0) 275 ax = - ax; 276 if (ax < .84375) { 277 if (ax < 3.7e-09) { 278 if (ax < _TINYER) 279 return 0.125*(8.0*x+p0t8*x); /*avoid underflow */ 280 return x + p0*x; 281 } 282 y = x*x; 283 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 284 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 285 return x + x*(p0+r); 286 } 287 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 288 s = fabs(x)-one; 289 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 290 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 291 if (x>=0) 292 return (c + P/Q); 293 else 294 return (-c - P/Q); 295 } 296 if (ax >= 6.0) { /* inf>|x|>=6 */ 297 if (x >= 0.0) 298 return (one-tiny); 299 else 300 return (tiny-one); 301 } 302 /* 1.25 <= |x| < 6 */ 303 z = -ax*ax; 304 s = -one/z; 305 if (ax < 2.0) { 306 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 307 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 308 S = one+s*(sc1+s*(sc2+s*sc3)); 309 } else { 310 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 311 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 312 S = one+s*(sb1+s*(sb2+s*sb3)); 313 } 314 y = (R/S -.5*s) - lsqrtPI_hi; 315 z += y; 316 z = exp(z)/ax; 317 if (x >= 0) 318 return (one-z); 319 else 320 return (z-one); 321} 322 323double 324erfc(double x) 325{ 326 double R,S,P,Q,s,ax,y,z,r; 327 if (!finite(x)) { 328 if (isnan(x)) /* erfc(NaN) = NaN */ 329 return(x); 330 else if (x > 0) /* erfc(+-inf)=0,2 */ 331 return 0.0; 332 else 333 return 2.0; 334 } 335 if ((ax = x) < 0) 336 ax = -ax; 337 if (ax < .84375) { /* |x|<0.84375 */ 338 if (ax < 1.38777878078144568e-17) /* |x|<2**-56 */ 339 return one-x; 340 y = x*x; 341 r = y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+ 342 y*(p6+y*(p7+y*(p8+y*(p9+y*p10))))))))); 343 if (ax < .0625) { /* |x|<2**-4 */ 344 return (one-(x+x*(p0+r))); 345 } else { 346 r = x*(p0+r); 347 r += (x-half); 348 return (half - r); 349 } 350 } 351 if (ax < 1.25) { /* 0.84375 <= |x| < 1.25 */ 352 s = ax-one; 353 P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 354 Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 355 if (x>=0) { 356 z = one-c; return z - P/Q; 357 } else { 358 z = c+P/Q; return one+z; 359 } 360 } 361 if (ax >= 28) { /* Out of range */ 362 if (x>0) 363 return (tiny*tiny); 364 else 365 return (two-tiny); 366 } 367 z = ax; 368 TRUNC(z); 369 y = z - ax; y *= (ax+z); 370 z *= -z; /* Here z + y = -x^2 */ 371 s = one/(-z-y); /* 1/(x*x) */ 372 if (ax >= 4) { /* 6 <= ax */ 373 R = s*(rd1+s*(rd2+s*(rd3+s*(rd4+s*(rd5+ 374 s*(rd6+s*(rd7+s*(rd8+s*(rd9+s*(rd10 375 +s*(rd11+s*(rd12+s*rd13)))))))))))); 376 y += rd0; 377 } else if (ax >= 2) { 378 R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+ 379 s*(rb6+s*(rb7+s*(rb8+s*(rb9+s*rb10))))))))); 380 S = one+s*(sb1+s*(sb2+s*sb3)); 381 y += R/S; 382 R = -.5*s; 383 } else { 384 R = rc0+s*(rc1+s*(rc2+s*(rc3+s*(rc4+s*(rc5+ 385 s*(rc6+s*(rc7+s*(rc8+s*(rc9+s*rc10))))))))); 386 S = one+s*(sc1+s*(sc2+s*sc3)); 387 y += R/S; 388 R = -.5*s; 389 } 390 /* return exp(-x^2 - lsqrtPI_hi + R + y)/x; */ 391 s = ((R + y) - lsqrtPI_hi) + z; 392 y = (((z-s) - lsqrtPI_hi) + R) + y; 393 r = __exp__D(s, y)/x; 394 if (x>0) 395 return r; 396 else 397 return two-r; 398} 399