usr/src/lib/libm/common/LD/erfl.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21
  22 /*
  23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  24  */
  25 /*
  26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  27  * Use is subject to license terms.
  28  */
  29
  30 /* long double function erf,erfc (long double x)
  31  * K.C. Ng, September, 1989.
  32  *                           x
  33  *                    2      |\
  34  *     erf(x)  =  ---------  | exp(-t*t)dt
  35  *                 sqrt(pi) \|
  36  *                           0
  37  *
  38  *     erfc(x) =  1-erf(x)
  39  *
  40  * method:
  41  *      Since erf(-x) = -erf(x), we assume x>=0.
  42  *      For x near 0, we have the expansion
  43  *
  44  *          erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....).
  45  *
  46  *      Since 2/sqrt(pi) = 1.128379167095512573896158903121545171688,
  47  *      we use x + x*P(x^2) to approximate erf(x). This formula will
  48  *      guarantee the error less than one ulp where x is not too far
  49  *      away from 0. We note that erf(x)=x at x = 0.6174...... After
  50  *      some experiment, we choose the following approximation on
  51  *      interval [0,0.84375].
  52  *
  53  *      For x in [0,0.84375]
  54  *                 2                2        4               40
  55  *         P =  P(x ) = (p0 + p1 * x + p2 * x + ... + p20 * x  )
  56  *
  57  *         erf(x)  = x + x*P
  58  *         erfc(x) = 1 - erf(x)           if x<=0.25
  59  *                 = 0.5 + ((0.5-x)-x*P)  if x in [0.25,0.84375]
  60  *      precision: |P(x^2)-(erf(x)-x)/x| <= 2**-122.50
  61  *
  62  *      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  *      precision: |P1/Q1 - (erf(x)-c)| <= 2**-118.41
  67  *
  68  *
  69  *      For x in [1.25,1.75], let s = x - 1.5, and
  70  *      c = 0.95478588343 rounded to single (24 bits)
  71  *         erf(x)  = c  + P2(s)/Q2(s)
  72  *         erfc(x) = (1-c)  - P2(s)/Q2(s)
  73  *      precision: |P1/Q1 - (erf(x)-c)| <= 2**-123.83
  74  *
  75  *
  76  *      For x in [1.75,16/3]
  77  *         erfc(x) = exp(-x*x)*(1/x)*R1(1/x)/S1(1/x)
  78  *         erf(x)  = 1 - erfc(x)
  79  *      precision: absolute error of R1/S1 is bounded by 2**-124.03
  80  *
  81  *      For x in [16/3,107]
  82  *         erfc(x) = exp(-x*x)*(1/x)*R2(1/x)/S2(1/x)
  83  *         erf(x)  = 1 - erfc(x) (if x>=9 simple return erf(x)=1 with inexact)
  84  *      precision: absolute error of R2/S2 is bounded by 2**-120.07
  85  *
  86  *      Else if inf > x >= 107
  87  *         erf(x)  = 1 with inexact
  88  *         erfc(x) = 0 with underflow
  89  *
  90  *      Special case:
  91  *         erf(inf)  = 1
  92  *         erfc(inf) = 0
  93  */
  94
  95 #pragma weak __erfl = erfl
  96 #pragma weak __erfcl = erfcl
  97
  98 #include "libm.h"
  99 #include "longdouble.h"
 100
 101 static long double
 102 tiny        = 1e-40L,
 103 nearunfl    = 1e-4000L,
 104 half        = 0.5L,
 105 one         = 1.0L,
 106 onehalf     = 1.5L,
 107 L16_3       = 16.0L/3.0L;
 108 /*
 109  * Coefficients for even polynomial P for erf(x)=x+x*P(x^2) on [0,0.84375]
 110  */
 111 static long double P[] = {      /* 21 coeffs */
 112    1.283791670955125738961589031215451715556e-0001L,
 113   -3.761263890318375246320529677071815594603e-0001L,
 114    1.128379167095512573896158903121205899135e-0001L,
 115   -2.686617064513125175943235483344625046092e-0002L,
 116    5.223977625442187842111846652980454568389e-0003L,
 117   -8.548327023450852832546626271083862724358e-0004L,
 118    1.205533298178966425102164715902231976672e-0004L,
 119   -1.492565035840625097674944905027897838996e-0005L,
 120    1.646211436588924733604648849172936692024e-0006L,
 121   -1.636584469123491976815834704799733514987e-0007L,
 122    1.480719281587897445302529007144770739305e-0008L,
 123   -1.229055530170782843046467986464722047175e-0009L,
 124    9.422759064320307357553954945760654341633e-0011L,
 125   -6.711366846653439036162105104991433380926e-0012L,
 126    4.463224090341893165100275380693843116240e-0013L,
 127   -2.783513452582658245422635662559779162312e-0014L,
 128    1.634227412586960195251346878863754661546e-0015L,
 129   -9.060782672889577722765711455623117802795e-0017L,
 130    4.741341801266246873412159213893613602354e-0018L,
 131   -2.272417596497826188374846636534317381203e-0019L,
 132    8.069088733716068462496835658928566920933e-0021L,
 133 };
 134
 135 /*
 136  * Rational erf(x) = ((float)0.84506291151) + P1(x-1)/Q1(x-1) on [0.84375,1.25]
 137  */
 138 static long double C1   = (long double)((float)0.84506291151);
 139 static long double P1[] = {     /*  12 top coeffs */
 140   -2.362118560752659955654364917390741930316e-0003L,
 141    4.129623379624420034078926610650759979146e-0001L,
 142   -3.973857505403547283109417923182669976904e-0002L,
 143    4.357503184084022439763567513078036755183e-0002L,
 144    8.015593623388421371247676683754171456950e-0002L,
 145   -1.034459310403352486685467221776778474602e-0002L,
 146    5.671850295381046679675355719017720821383e-0003L,
 147    1.219262563232763998351452194968781174318e-0003L,
 148    5.390833481581033423020320734201065475098e-0004L,
 149   -1.978853912815115495053119023517805528300e-0004L,
 150    6.184234513953600118335017885706420552487e-0005L,
 151   -5.331802711697810861017518515816271808286e-0006L,
 152 };
 153 static long double Q1[] = {     /*  12 bottom coeffs with leading 1.0 hidden */
 154    9.081506296064882195280178373107623196655e-0001L,
 155    6.821049531968204097604392183650687642520e-0001L,
 156    4.067869178233539502315055970743271822838e-0001L,
 157    1.702332233546316765818144723063881095577e-0001L,
 158    7.498098377690553934266423088708614219356e-0002L,
 159    2.050154396918178697056927234366372760310e-0002L,
 160    7.012988534031999899054782333851905939379e-0003L,
 161    1.149904787014400354649843451234570731076e-0003L,
 162    3.185620255011299476196039491205159718620e-0004L,
 163    1.273405072153008775426376193374105840517e-0005L,
 164    4.753866999959432971956781228148402971454e-0006L,
 165   -1.002287602111660026053981728549540200683e-0006L,
 166 };
 167 /*
 168  * Rational erf(x) = ((float)0.95478588343) + P2(x-1.5)/Q2(x-1.5)
 169  * on [1.25,1.75]
 170  */
 171 static long double C2   = (long double)((float)0.95478588343);
 172 static long double P2[] = {     /*  12 top coeffs */
 173    1.131926304864446730135126164594785863512e-0002L,
 174    1.273617996967754151544330055186210322832e-0001L,
 175   -8.169980734667512519897816907190281143423e-0002L,
 176    9.512267486090321197833634271787944271746e-0002L,
 177   -2.394251569804872160005274999735914368170e-0002L,
 178    1.108768660227528667525252333184520222905e-0002L,
 179    3.527435492933902414662043314373277494221e-0004L,
 180    4.946116273341953463584319006669474625971e-0004L,
 181   -4.289851942513144714600285769022420962418e-0005L,
 182    8.304719841341952705874781636002085119978e-0005L,
 183   -1.040460226177309338781902252282849903189e-0005L,
 184    2.122913331584921470381327583672044434087e-0006L,
 185 };
 186 static long double Q2[] = {     /*  13 bottom coeffs with leading 1.0 hidden */
 187    7.448815737306992749168727691042003832150e-0001L,
 188    7.161813850236008294484744312430122188043e-0001L,
 189    3.603134756584225766144922727405641236121e-0001L,
 190    1.955811609133766478080550795194535852653e-0001L,
 191    7.253059963716225972479693813787810711233e-0002L,
 192    2.752391253757421424212770221541238324978e-0002L,
 193    7.677654852085240257439050673446546828005e-0003L,
 194    2.141102244555509687346497060326630061069e-0003L,
 195    4.342123013830957093949563339130674364271e-0004L,
 196    8.664587895570043348530991997272212150316e-0005L,
 197    1.109201582511752087060167429397033701988e-0005L,
 198    1.357834375781831062713347000030984364311e-0006L,
 199    4.957746280594384997273090385060680016451e-0008L,
 200 };
 201 /*
 202  * erfc(x) = exp(-x*x)/x * R1(1/x)/S1(1/x) on [1.75, 16/3]
 203  */
 204 static long double R1[] = {     /*  14 top coeffs */
 205    4.630195122654315016370705767621550602948e+0006L,
 206    1.257949521746494830700654204488675713628e+0007L,
 207    1.704153822720260272814743497376181625707e+0007L,
 208    1.502600568706061872381577539537315739943e+0007L,
 209    9.543710793431995284827024445387333922861e+0006L,
 210    4.589344808584091011652238164935949522427e+0006L,
 211    1.714660662941745791190907071920671844289e+0006L,
 212    5.034802147768798894307672256192466283867e+0005L,
 213    1.162286400443554670553152110447126850725e+0005L,
 214    2.086643834548901681362757308058660399137e+0004L,
 215    2.839793161868140305907004392890348777338e+0003L,
 216    2.786687241658423601778258694498655680778e+0002L,
 217    1.779177837102695602425897452623985786464e+0001L,
 218    5.641895835477470769043614623819144434731e-0001L,
 219 };
 220 static long double S1[] = {     /* 15 bottom coeffs with leading 1.0 hidden */
 221    4.630195122654331529595606896287596843110e+0006L,
 222    1.780411093345512024324781084220509055058e+0007L,
 223    3.250113097051800703707108623715776848283e+0007L,
 224    3.737857099176755050912193712123489115755e+0007L,
 225    3.029787497516578821459174055870781168593e+0007L,
 226    1.833850619965384765005769632103205777227e+0007L,
 227    8.562719999736915722210391222639186586498e+0006L,
 228    3.139684562074658971315545539760008136973e+0006L,
 229    9.106421313731384880027703627454366930945e+0005L,
 230    2.085108342384266508613267136003194920001e+0005L,
 231    3.723126272693120340730491416449539290600e+0004L,
 232    5.049169878567344046145695360784436929802e+0003L,
 233    4.944274532748010767670150730035392093899e+0002L,
 234    3.153510608818213929982940249162268971412e+0001L,
 235    1.0e00L,
 236 };
 237
 238 /*
 239  * erfc(x) = exp(-x*x)/x * R2(1/x)/S2(1/x) on [16/3, 107]
 240  */
 241 static long double R2[] = {     /*  15 top coeffs in reverse order!!*/
 242    2.447288012254302966796326587537136931669e+0005L,
 243    8.768592567189861896653369912716538739016e+0005L,
 244    1.552293152581780065761497908005779524953e+0006L,
 245    1.792075924835942935864231657504259926729e+0006L,
 246    1.504001463155897344947500222052694835875e+0006L,
 247    9.699485556326891411801230186016013019935e+0005L,
 248    4.961449933661807969863435013364796037700e+0005L,
 249    2.048726544693474028061176764716228273791e+0005L,
 250    6.891532964330949722479061090551896886635e+0004L,
 251    1.888014709010307507771964047905823237985e+0004L,
 252    4.189692064988957745054734809642495644502e+0003L,
 253    7.362346487427048068212968889642741734621e+0002L,
 254    9.980359714211411423007641056580813116207e+0001L,
 255    9.426910895135379181107191962193485174159e+0000L,
 256    5.641895835477562869480794515623601280429e-0001L,
 257 };
 258 static long double S2[] = {     /* 16 coefficients */
 259    2.447282203601902971246004716790604686880e+0005L,
 260    1.153009852759385309367759460934808489833e+0006L,
 261    2.608580649612639131548966265078663384849e+0006L,
 262    3.766673917346623308850202792390569025740e+0006L,
 263    3.890566255138383910789924920541335370691e+0006L,
 264    3.052882073900746207613166259994150527732e+0006L,
 265    1.885574519970380988460241047248519418407e+0006L,
 266    9.369722034759943185851450846811445012922e+0005L,
 267    3.792278350536686111444869752624492443659e+0005L,
 268    1.257750606950115799965366001773094058720e+0005L,
 269    3.410830600242369370645608634643620355058e+0004L,
 270    7.513984469742343134851326863175067271240e+0003L,
 271    1.313296320593190002554779998138695507840e+0003L,
 272    1.773972700887629157006326333696896516769e+0002L,
 273    1.670876451822586800422009013880457094162e+0001L,
 274    1.000L,
 275 };
 276
 277 long double erfl(x)
 278 long double x;
 279 {
 280         long double erfcl(long double),s,y,t;
 281
 282         if (!finitel(x)) {
 283             if (x != x) return x+x;     /* NaN */
 284             return copysignl(one,x);    /* return +-1.0 is x=Inf */
 285         }
 286
 287         y = fabsl(x);
 288         if (y <= 0.84375L) {
 289             if (y<=tiny) return x+P[0]*x;
 290             s = y*y;
 291             t = __poly_libmq(s,21,P);
 292             return  x+x*t;
 293         }
 294         if (y<=1.25L) {
 295             s = y-one;
 296             t = C1+__poly_libmq(s,12,P1)/(one+s*__poly_libmq(s,12,Q1));
 297             return (signbitl(x))? -t: t;
 298         } else if (y<=1.75L) {
 299             s = y-onehalf;
 300             t = C2+__poly_libmq(s,12,P2)/(one+s*__poly_libmq(s,13,Q2));
 301             return (signbitl(x))? -t: t;
 302         }
 303         if (y<=9.0L) t = erfcl(y); else t = tiny;
 304         return (signbitl(x))? t-one: one-t;
 305 }
 306
 307 long double erfcl(x)
 308 long double x;
 309 {
 310         long double erfl(long double),s,y,t;
 311
 312         if (!finitel(x)) {
 313             if (x != x) return x+x;     /* NaN */
 314                                         /* return 2.0 if x= -inf
 315                                                   0.0 if x= +inf */
 316             if (x<0.0L) return 2.0L; else return 0.0L;
 317         }
 318
 319         if (x <= 0.84375L) {
 320             if (x<=0.25) return one-erfl(x);
 321             s = x*x;
 322             t = half-x;
 323             t = t - x*__poly_libmq(s,21,P);
 324             return  half+t;
 325         }
 326         if (x<=1.25L) {
 327             s = x-one;
 328             t = one-C1;
 329             return t - __poly_libmq(s,12,P1)/(one+s*__poly_libmq(s,12,Q1));
 330         } else if (x<=1.75L) {
 331             s = x-onehalf;
 332             t = one-C2;
 333             return t - __poly_libmq(s,12,P2)/(one+s*__poly_libmq(s,13,Q2));
 334         }
 335         if (x>=107.0L) return nearunfl*nearunfl;                /* underflow */
 336         else if (x >= L16_3) {
 337             y = __poly_libmq(x,15,R2);
 338             t = y/__poly_libmq(x,16,S2);
 339         } else {
 340             y = __poly_libmq(x,14,R1);
 341             t = y/__poly_libmq(x,15,S1);
 342         }
 343         /* see comment in ../Q/erfl.c */
 344         y = x;
 345         *(int*)&y = 0;
 346         t *= expl(-y*y)*expl(-(x-y)*(x+y));
 347         return  t;
 348 }