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 /* INDENT OFF */ 31 /* 32 * long double __k_cexpl(long double x, int *n); 33 * Returns the exponential of x in the form of 2**n * y, y=__k_cexpl(x,&n). 34 * 35 * 1. Argument Reduction: given the input x, find r and integer k 36 * and j such that 37 * x = (32k+j)*ln2 + r, |r| <= (1/64)*ln2 . 38 * 39 * 2. expl(x) = 2^k * (2^(j/32) + 2^(j/32)*expm1(r)) 40 * Note: 41 * a. expm1(r) = (2r)/(2-R), R = r - r^2*(t1 + t2*r^2) 42 * b. 2^(j/32) is represented as 43 * exp2_32_hi[j]+exp2_32_lo[j] 44 * where 45 * exp2_32_hi[j] = 2^(j/32) rounded 46 * exp2_32_lo[j] = 2^(j/32) - exp2_32_hi[j]. 47 * 48 * Special cases: 49 * expl(INF) is INF, expl(NaN) is NaN; 50 * expl(-INF)= 0; 51 * for finite argument, only expl(0)=1 is exact. 52 * 53 * Accuracy: 54 * according to an error analysis, the error is always less than 55 * an ulp (unit in the last place). 56 * 57 * Misc. info. 58 * When |x| is really big, say |x| > 1000000, the accuracy 59 * is not important because the ultimate result will over or under 60 * flow. So we will simply replace n = 1000000 and r = 0.0. For 61 * moderate size x, according to an error analysis, the error is 62 * always less than 1 ulp (unit in the last place). 63 * 64 * Constants: 65 * Only decimal values are given. We assume that the compiler will convert 66 * from decimal to binary accurately enough to produce the correct 67 * hexadecimal values. 68 */ 69 /* INDENT ON */ 70 71 #include "libm.h" /* __k_cexpl */ 72 #include "complex_wrapper.h" /* HI_XWORD */ 73 74 /* INDENT OFF */ 75 /* ln2/32 = 0.0216608493924982909192885037955680177523593791987579766912713 */ 76 #if defined(__x86) 77 static const long double 78 /* 43 significant bits, 21 trailing zeros */ 79 ln2_32hi = 2.166084939249657281834515742957592010498046875e-2L, 80 ln2_32lo = 1.7181009433463659920976473789104487579766912713e-15L; 81 static const long double exp2_32_hi[] = { /* exp2_32[j] = 2^(j/32) */ 82 1.0000000000000000000000000e+00L, 83 1.0218971486541166782081522e+00L, 84 1.0442737824274138402382006e+00L, 85 1.0671404006768236181297224e+00L, 86 1.0905077326652576591003302e+00L, 87 1.1143867425958925362894369e+00L, 88 1.1387886347566916536971221e+00L, 89 1.1637248587775775137938619e+00L, 90 1.1892071150027210666875674e+00L, 91 1.2152473599804688780476325e+00L, 92 1.2418578120734840485256747e+00L, 93 1.2690509571917332224885722e+00L, 94 1.2968395546510096659215822e+00L, 95 1.3252366431597412945939118e+00L, 96 1.3542555469368927282668852e+00L, 97 1.3839098819638319548151403e+00L, 98 1.4142135623730950487637881e+00L, 99 1.4451808069770466200253470e+00L, 100 1.4768261459394993113155431e+00L, 101 1.5091644275934227397133885e+00L, 102 1.5422108254079408235859630e+00L, 103 1.5759808451078864864006862e+00L, 104 1.6104903319492543080837174e+00L, 105 1.6457554781539648445110730e+00L, 106 1.6817928305074290860378350e+00L, 107 1.7186192981224779156032914e+00L, 108 1.7562521603732994831094730e+00L, 109 1.7947090750031071864148413e+00L, 110 1.8340080864093424633989166e+00L, 111 1.8741676341102999013002103e+00L, 112 1.9152065613971472938202589e+00L, 113 1.9571441241754002689657438e+00L, 114 }; 115 static const long double exp2_32_lo[] = { 116 0.0000000000000000000000000e+00L, 117 2.6327965667180882569382524e-20L, 118 8.3765863521895191129661899e-20L, 119 3.9798705777454504249209575e-20L, 120 1.0668046596651558640993042e-19L, 121 1.9376009847285360448117114e-20L, 122 6.7081819456112953751277576e-21L, 123 1.9711680502629186462729727e-20L, 124 2.9932584438449523689104569e-20L, 125 6.8887754153039109411061914e-20L, 126 6.8002718741225378942847820e-20L, 127 6.5846917376975403439742349e-20L, 128 1.2171958727511372194876001e-20L, 129 3.5625253228704087115438260e-20L, 130 3.1129551559077560956309179e-20L, 131 5.7519192396164779846216492e-20L, 132 3.7900651177865141593101239e-20L, 133 1.1659262405698741798080115e-20L, 134 7.1364385105284695967172478e-20L, 135 5.2631003710812203588788949e-20L, 136 2.6328853788732632868460580e-20L, 137 5.4583950085438242788190141e-20L, 138 9.5803254376938269960718656e-20L, 139 7.6837733983874245823512279e-21L, 140 2.4415965910835093824202087e-20L, 141 2.6052966871016580981769728e-20L, 142 2.6876456344632553875309579e-21L, 143 1.2861930155613700201703279e-20L, 144 8.8166633394037485606572294e-20L, 145 2.9788615389580190940837037e-20L, 146 5.2352341619805098677422139e-20L, 147 5.2578463064010463732242363e-20L, 148 }; 149 #else /* sparc */ 150 static const long double 151 /* 0x3FF962E4 2FEFA39E F35793C7 00000000 */ 152 ln2_32hi = 2.166084939249829091928849858592451515688e-2L, 153 ln2_32lo = 5.209643502595475652782654157501186731779e-27L; 154 static const long double exp2_32_hi[] = { /* exp2_32[j] = 2^(j/32) */ 155 1.000000000000000000000000000000000000000e+0000L, 156 1.021897148654116678234480134783299439782e+0000L, 157 1.044273782427413840321966478739929008785e+0000L, 158 1.067140400676823618169521120992809162607e+0000L, 159 1.090507732665257659207010655760707978993e+0000L, 160 1.114386742595892536308812956919603067800e+0000L, 161 1.138788634756691653703830283841511254720e+0000L, 162 1.163724858777577513813573599092185312343e+0000L, 163 1.189207115002721066717499970560475915293e+0000L, 164 1.215247359980468878116520251338798457624e+0000L, 165 1.241857812073484048593677468726595605511e+0000L, 166 1.269050957191733222554419081032338004715e+0000L, 167 1.296839554651009665933754117792451159835e+0000L, 168 1.325236643159741294629537095498721674113e+0000L, 169 1.354255546936892728298014740140702804343e+0000L, 170 1.383909881963831954872659527265192818002e+0000L, 171 1.414213562373095048801688724209698078570e+0000L, 172 1.445180806977046620037006241471670905678e+0000L, 173 1.476826145939499311386907480374049923924e+0000L, 174 1.509164427593422739766019551033193531420e+0000L, 175 1.542210825407940823612291862090734841307e+0000L, 176 1.575980845107886486455270160181905008906e+0000L, 177 1.610490331949254308179520667357400583459e+0000L, 178 1.645755478153964844518756724725822445667e+0000L, 179 1.681792830507429086062250952466429790080e+0000L, 180 1.718619298122477915629344376456312504516e+0000L, 181 1.756252160373299483112160619375313221294e+0000L, 182 1.794709075003107186427703242127781814354e+0000L, 183 1.834008086409342463487083189588288856077e+0000L, 184 1.874167634110299901329998949954446534439e+0000L, 185 1.915206561397147293872611270295830887850e+0000L, 186 1.957144124175400269018322251626871491190e+0000L, 187 }; 188 189 static const long double exp2_32_lo[] = { 190 +0.000000000000000000000000000000000000000e+0000L, 191 +1.805067874203309547455733330545737864651e-0035L, 192 -9.374520292280427421957567419730832143843e-0035L, 193 -1.596968447292758770712909630231499971233e-0035L, 194 +9.112493410125022978511686101672486662119e-0035L, 195 -6.504228206978548287230374775259388710985e-0035L, 196 -8.148468844525851137325691767488155323605e-0035L, 197 -5.066214576721800313372330745142903350963e-0035L, 198 -1.359830974688816973749875638245919118924e-0035L, 199 +9.497427635563196470307710566433246597109e-0035L, 200 -3.283170523176998601615065965333915261932e-0036L, 201 -5.017235709387190410290186530458428950862e-0035L, 202 -2.391474797689109171622834301602640139258e-0035L, 203 -8.350571357633908815298890737944083853080e-0036L, 204 +7.036756889073265042421737190671876440729e-0035L, 205 -5.182484853064646457536893018566956189817e-0035L, 206 +9.422242548621832065692116736394064879758e-0035L, 207 -3.967500825398862309167306130216418281103e-0035L, 208 +7.143528991563300614523273615092767243521e-0035L, 209 +1.159871252867985124246517834100444327747e-0035L, 210 +4.696933478358115495309739213201874466685e-0035L, 211 -3.386513175995004710799241984999819165197e-0035L, 212 -8.587318774298247068868655935103874453522e-0035L, 213 -9.605951548749350503185499362246069088835e-0035L, 214 +9.609733932128012784507558697141785813655e-0035L, 215 +6.378397921440028439244761449780848545957e-0035L, 216 +7.792430785695864249456461125169277701177e-0035L, 217 +7.361337767588456524131930836633932195088e-0035L, 218 -6.472995147913347230035214575612170525266e-0035L, 219 +8.587474417953698694278798062295229624207e-0035L, 220 +2.371815422825174835691651228302690977951e-0035L, 221 -3.026891682096118773004597373421900314256e-0037L, 222 }; 223 #endif 224 225 static const long double 226 one = 1.0L, 227 two = 2.0L, 228 ln2_64 = 1.083042469624914545964425189778400898568e-2L, 229 invln2_32 = 4.616624130844682903551758979206054839765e+1L; 230 231 /* rational approximation coeffs for [-(ln2)/64,(ln2)/64] */ 232 static const long double 233 t1 = 1.666666666666666666666666666660876387437e-1L, 234 t2 = -2.777777777777777777777707812093173478756e-3L, 235 t3 = 6.613756613756613482074280932874221202424e-5L, 236 t4 = -1.653439153392139954169609822742235851120e-6L, 237 t5 = 4.175314851769539751387852116610973796053e-8L; 238 /* INDENT ON */ 239 240 long double 241 __k_cexpl(long double x, int *n) { 242 int hx, ix, j, k; 243 long double t, r; 244 245 *n = 0; 246 hx = HI_XWORD(x); 247 ix = hx & 0x7fffffff; 248 if (hx >= 0x7fff0000) 249 return (x + x); /* NaN of +inf */ 250 if (((unsigned) hx) >= 0xffff0000) 251 return (-one / x); /* NaN or -inf */ 252 if (ix < 0x3fc30000) 253 return (one + x); /* |x|<2^-60 */ 254 if (hx > 0) { 255 if (hx > 0x401086a0) { /* x > 200000 */ 256 *n = 200000; 257 return (one); 258 } 259 k = (int) (invln2_32 * (x + ln2_64)); 260 } else { 261 if (ix > 0x401086a0) { /* x < -200000 */ 262 *n = -200000; 263 return (one); 264 } 265 k = (int) (invln2_32 * (x - ln2_64)); 266 } 267 j = k & 0x1f; 268 *n = k >> 5; 269 t = (long double) k; 270 x = (x - t * ln2_32hi) - t * ln2_32lo; 271 t = x * x; 272 r = (x - t * (t1 + t * (t2 + t * (t3 + t * (t4 + t * t5))))) - two; 273 x = exp2_32_hi[j] - ((exp2_32_hi[j] * (x + x)) / r - exp2_32_lo[j]); 274 k >>= 5; 275 if (k > 240) { 276 XFSCALE(x, 240); 277 *n -= 240; 278 } else if (k > 0) { 279 XFSCALE(x, k); 280 *n = 0; 281 } 282 return (x); 283 } 284