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 #include "libm_macros.h" 31 32 /* INDENT OFF */ 33 34 /* 35 * cbrt: double precision cube root 36 * 37 * Algorithm: bit hacking, table lookup, and polynomial approximation 38 * 39 * For normal x, write x = s*2^(3j)*z where s = +/-1, j is an integer, 40 * and 1 <= z < 8. Let y := s*2^j. From a table, find u such that 41 * u^3 is computable exactly and |(z-u^3)/u^3| <~ 2^-8. We construct 42 * y, z, and the table index from x by a few integer operations. 43 * 44 * Now cbrt(x) = y*u*(1+t)^(1/3) where t = (z-u^3)/u^3. We approximate 45 * (1+t)^(1/3) by a polynomial 1+p(t), where p(t) := t*(p1+t*(p2+...+ 46 * (p5+t*p6))). By computing the result as y*(u+u*p(t)), we can bound 47 * the worst case error by .51 ulp. 48 * 49 * Notes: 50 * 51 * 1. For subnormal x, we scale x by 2^54, compute the cube root, and 52 * scale the result by 2^-18. 53 * 54 * 2. cbrt(+/-inf) = +/-inf and cbrt(NaN) is NaN. 55 */ 56 57 /* 58 * for i = 0, ..., 385 59 * form x(i) with high word 0x3ff00000 + (i << 13) and low word 0; 60 * then TBL[i] = cbrt(x(i)) rounded to 17 significant bits 61 */ 62 static const double __libm_TBL_cbrt[] = { 63 1.00000000000000000e+00, 1.00259399414062500e+00, 1.00518798828125000e+00, 64 1.00775146484375000e+00, 1.01031494140625000e+00, 1.01284790039062500e+00, 65 1.01538085937500000e+00, 1.01791381835937500e+00, 1.02041625976562500e+00, 66 1.02290344238281250e+00, 1.02539062500000000e+00, 1.02786254882812500e+00, 67 1.03031921386718750e+00, 1.03277587890625000e+00, 1.03520202636718750e+00, 68 1.03762817382812500e+00, 1.04003906250000000e+00, 1.04244995117187500e+00, 69 1.04483032226562500e+00, 1.04721069335937500e+00, 1.04959106445312500e+00, 70 1.05194091796875000e+00, 1.05429077148437500e+00, 1.05662536621093750e+00, 71 1.05895996093750000e+00, 1.06127929687500000e+00, 1.06358337402343750e+00, 72 1.06587219238281250e+00, 1.06816101074218750e+00, 1.07044982910156250e+00, 73 1.07270812988281250e+00, 1.07496643066406250e+00, 1.07722473144531250e+00, 74 1.07945251464843750e+00, 1.08168029785156250e+00, 1.08390808105468750e+00, 75 1.08612060546875000e+00, 1.08831787109375000e+00, 1.09051513671875000e+00, 76 1.09269714355468750e+00, 1.09487915039062500e+00, 1.09704589843750000e+00, 77 1.09921264648437500e+00, 1.10136413574218750e+00, 1.10350036621093750e+00, 78 1.10563659667968750e+00, 1.10775756835937500e+00, 1.10987854003906250e+00, 79 1.11198425292968750e+00, 1.11408996582031250e+00, 1.11618041992187500e+00, 80 1.11827087402343750e+00, 1.12034606933593750e+00, 1.12242126464843750e+00, 81 1.12448120117187500e+00, 1.12654113769531250e+00, 1.12858581542968750e+00, 82 1.13063049316406250e+00, 1.13265991210937500e+00, 1.13468933105468750e+00, 83 1.13670349121093750e+00, 1.13871765136718750e+00, 1.14073181152343750e+00, 84 1.14273071289062500e+00, 1.14471435546875000e+00, 1.14669799804687500e+00, 85 1.14868164062500000e+00, 1.15065002441406250e+00, 1.15260314941406250e+00, 86 1.15457153320312500e+00, 1.15650939941406250e+00, 1.15846252441406250e+00, 87 1.16040039062500000e+00, 1.16232299804687500e+00, 1.16424560546875000e+00, 88 1.16616821289062500e+00, 1.16807556152343750e+00, 1.16998291015625000e+00, 89 1.17189025878906250e+00, 1.17378234863281250e+00, 1.17567443847656250e+00, 90 1.17755126953125000e+00, 1.17942810058593750e+00, 1.18128967285156250e+00, 91 1.18315124511718750e+00, 1.18501281738281250e+00, 1.18685913085937500e+00, 92 1.18870544433593750e+00, 1.19055175781250000e+00, 1.19238281250000000e+00, 93 1.19421386718750000e+00, 1.19602966308593750e+00, 1.19786071777343750e+00, 94 1.19966125488281250e+00, 1.20147705078125000e+00, 1.20327758789062500e+00, 95 1.20507812500000000e+00, 1.20686340332031250e+00, 1.20864868164062500e+00, 96 1.21043395996093750e+00, 1.21220397949218750e+00, 1.21397399902343750e+00, 97 1.21572875976562500e+00, 1.21749877929687500e+00, 1.21925354003906250e+00, 98 1.22099304199218750e+00, 1.22274780273437500e+00, 1.22448730468750000e+00, 99 1.22621154785156250e+00, 1.22795104980468750e+00, 1.22967529296875000e+00, 100 1.23138427734375000e+00, 1.23310852050781250e+00, 1.23481750488281250e+00, 101 1.23652648925781250e+00, 1.23822021484375000e+00, 1.23991394042968750e+00, 102 1.24160766601562500e+00, 1.24330139160156250e+00, 1.24497985839843750e+00, 103 1.24665832519531250e+00, 1.24833679199218750e+00, 1.25000000000000000e+00, 104 1.25166320800781250e+00, 1.25332641601562500e+00, 1.25497436523437500e+00, 105 1.25663757324218750e+00, 1.25828552246093750e+00, 1.25991821289062500e+00, 106 1.26319885253906250e+00, 1.26644897460937500e+00, 1.26968383789062500e+00, 107 1.27290344238281250e+00, 1.27612304687500000e+00, 1.27931213378906250e+00, 108 1.28248596191406250e+00, 1.28564453125000000e+00, 1.28878784179687500e+00, 109 1.29191589355468750e+00, 1.29502868652343750e+00, 1.29812622070312500e+00, 110 1.30120849609375000e+00, 1.30427551269531250e+00, 1.30732727050781250e+00, 111 1.31036376953125000e+00, 1.31340026855468750e+00, 1.31640625000000000e+00, 112 1.31941223144531250e+00, 1.32238769531250000e+00, 1.32536315917968750e+00, 113 1.32832336425781250e+00, 1.33126831054687500e+00, 1.33419799804687500e+00, 114 1.33712768554687500e+00, 1.34002685546875000e+00, 1.34292602539062500e+00, 115 1.34580993652343750e+00, 1.34867858886718750e+00, 1.35153198242187500e+00, 116 1.35437011718750000e+00, 1.35720825195312500e+00, 1.36003112792968750e+00, 117 1.36283874511718750e+00, 1.36564636230468750e+00, 1.36842346191406250e+00, 118 1.37120056152343750e+00, 1.37396240234375000e+00, 1.37672424316406250e+00, 119 1.37945556640625000e+00, 1.38218688964843750e+00, 1.38491821289062500e+00, 120 1.38761901855468750e+00, 1.39031982421875000e+00, 1.39302062988281250e+00, 121 1.39569091796875000e+00, 1.39836120605468750e+00, 1.40101623535156250e+00, 122 1.40367126464843750e+00, 1.40631103515625000e+00, 1.40893554687500000e+00, 123 1.41156005859375000e+00, 1.41416931152343750e+00, 1.41676330566406250e+00, 124 1.41935729980468750e+00, 1.42193603515625000e+00, 1.42449951171875000e+00, 125 1.42706298828125000e+00, 1.42962646484375000e+00, 1.43215942382812500e+00, 126 1.43469238281250000e+00, 1.43722534179687500e+00, 1.43974304199218750e+00, 127 1.44224548339843750e+00, 1.44474792480468750e+00, 1.44723510742187500e+00, 128 1.44972229003906250e+00, 1.45219421386718750e+00, 1.45466613769531250e+00, 129 1.45712280273437500e+00, 1.45956420898437500e+00, 1.46200561523437500e+00, 130 1.46444702148437500e+00, 1.46687316894531250e+00, 1.46928405761718750e+00, 131 1.47169494628906250e+00, 1.47409057617187500e+00, 1.47648620605468750e+00, 132 1.47886657714843750e+00, 1.48124694824218750e+00, 1.48361206054687500e+00, 133 1.48597717285156250e+00, 1.48834228515625000e+00, 1.49067687988281250e+00, 134 1.49302673339843750e+00, 1.49536132812500000e+00, 1.49768066406250000e+00, 135 1.50000000000000000e+00, 1.50230407714843750e+00, 1.50460815429687500e+00, 136 1.50691223144531250e+00, 1.50920104980468750e+00, 1.51148986816406250e+00, 137 1.51376342773437500e+00, 1.51603698730468750e+00, 1.51829528808593750e+00, 138 1.52055358886718750e+00, 1.52279663085937500e+00, 1.52503967285156250e+00, 139 1.52728271484375000e+00, 1.52951049804687500e+00, 1.53173828125000000e+00, 140 1.53395080566406250e+00, 1.53616333007812500e+00, 1.53836059570312500e+00, 141 1.54055786132812500e+00, 1.54275512695312500e+00, 1.54493713378906250e+00, 142 1.54711914062500000e+00, 1.54928588867187500e+00, 1.55145263671875000e+00, 143 1.55361938476562500e+00, 1.55577087402343750e+00, 1.55792236328125000e+00, 144 1.56005859375000000e+00, 1.56219482421875000e+00, 1.56433105468750000e+00, 145 1.56645202636718750e+00, 1.56857299804687500e+00, 1.57069396972656250e+00, 146 1.57279968261718750e+00, 1.57490539550781250e+00, 1.57699584960937500e+00, 147 1.57908630371093750e+00, 1.58117675781250000e+00, 1.58325195312500000e+00, 148 1.58532714843750000e+00, 1.58740234375000000e+00, 1.59152221679687500e+00, 149 1.59562683105468750e+00, 1.59970092773437500e+00, 1.60375976562500000e+00, 150 1.60780334472656250e+00, 1.61183166503906250e+00, 1.61582946777343750e+00, 151 1.61981201171875000e+00, 1.62376403808593750e+00, 1.62770080566406250e+00, 152 1.63162231445312500e+00, 1.63552856445312500e+00, 1.63941955566406250e+00, 153 1.64328002929687500e+00, 1.64714050292968750e+00, 1.65097045898437500e+00, 154 1.65476989746093750e+00, 1.65856933593750000e+00, 1.66235351562500000e+00, 155 1.66610717773437500e+00, 1.66986083984375000e+00, 1.67358398437500000e+00, 156 1.67729187011718750e+00, 1.68098449707031250e+00, 1.68466186523437500e+00, 157 1.68832397460937500e+00, 1.69197082519531250e+00, 1.69560241699218750e+00, 158 1.69921875000000000e+00, 1.70281982421875000e+00, 1.70640563964843750e+00, 159 1.70997619628906250e+00, 1.71353149414062500e+00, 1.71707153320312500e+00, 160 1.72059631347656250e+00, 1.72410583496093750e+00, 1.72760009765625000e+00, 161 1.73109436035156250e+00, 1.73455810546875000e+00, 1.73800659179687500e+00, 162 1.74145507812500000e+00, 1.74488830566406250e+00, 1.74829101562500000e+00, 163 1.75169372558593750e+00, 1.75508117675781250e+00, 1.75846862792968750e+00, 164 1.76182556152343750e+00, 1.76516723632812500e+00, 1.76850891113281250e+00, 165 1.77183532714843750e+00, 1.77514648437500000e+00, 1.77844238281250000e+00, 166 1.78173828125000000e+00, 1.78500366210937500e+00, 1.78826904296875000e+00, 167 1.79151916503906250e+00, 1.79476928710937500e+00, 1.79798889160156250e+00, 168 1.80120849609375000e+00, 1.80441284179687500e+00, 1.80760192871093750e+00, 169 1.81079101562500000e+00, 1.81396484375000000e+00, 1.81712341308593750e+00, 170 1.82026672363281250e+00, 1.82341003417968750e+00, 1.82653808593750000e+00, 171 1.82965087890625000e+00, 1.83276367187500000e+00, 1.83586120605468750e+00, 172 1.83894348144531250e+00, 1.84201049804687500e+00, 1.84507751464843750e+00, 173 1.84812927246093750e+00, 1.85118103027343750e+00, 1.85421752929687500e+00, 174 1.85723876953125000e+00, 1.86026000976562500e+00, 1.86326599121093750e+00, 175 1.86625671386718750e+00, 1.86924743652343750e+00, 1.87222290039062500e+00, 176 1.87518310546875000e+00, 1.87814331054687500e+00, 1.88108825683593750e+00, 177 1.88403320312500000e+00, 1.88696289062500000e+00, 1.88987731933593750e+00, 178 1.89279174804687500e+00, 1.89569091796875000e+00, 1.89859008789062500e+00, 179 1.90147399902343750e+00, 1.90435791015625000e+00, 1.90722656250000000e+00, 180 1.91007995605468750e+00, 1.91293334960937500e+00, 1.91577148437500000e+00, 181 1.91860961914062500e+00, 1.92143249511718750e+00, 1.92425537109375000e+00, 182 1.92706298828125000e+00, 1.92985534667968750e+00, 1.93264770507812500e+00, 183 1.93544006347656250e+00, 1.93821716308593750e+00, 1.94097900390625000e+00, 184 1.94374084472656250e+00, 1.94650268554687500e+00, 1.94924926757812500e+00, 185 1.95198059082031250e+00, 1.95471191406250000e+00, 1.95742797851562500e+00, 186 1.96014404296875000e+00, 1.96286010742187500e+00, 1.96556091308593750e+00, 187 1.96824645996093750e+00, 1.97093200683593750e+00, 1.97361755371093750e+00, 188 1.97628784179687500e+00, 1.97894287109375000e+00, 1.98159790039062500e+00, 189 1.98425292968750000e+00, 1.98689270019531250e+00, 1.98953247070312500e+00, 190 1.99215698242187500e+00, 1.99478149414062500e+00, 1.99739074707031250e+00, 191 2.00000000000000000e+00, 192 }; 193 194 /* 195 * The polynomial p(x) := p1*x + p2*x^2 + ... + p6*x^6 satisfies 196 * 197 * |(1+x)^(1/3) - 1 - p(x)| < 2^-63 for |x| < 0.003914 198 */ 199 static const double C[] = { 200 3.33333333333333340735623180707664400321413178600e-0001, 201 -1.11111111111111111992797989129069515334791432304e-0001, 202 6.17283950578506695710302115234720605072083379082e-0002, 203 -4.11522633731005164138964638666647311514892319010e-0002, 204 3.01788343105268728151735586597807324859173704847e-0002, 205 -2.34723340038386971009665073968507263074215090751e-0002, 206 18014398509481984.0 207 }; 208 209 #define p1 C[0] 210 #define p2 C[1] 211 #define p3 C[2] 212 #define p4 C[3] 213 #define p5 C[4] 214 #define p6 C[5] 215 #define two54 C[6] 216 217 /* INDENT ON */ 218 219 #pragma weak cbrt = __cbrt 220 221 double __cbrt(double x) 222 { 223 union { 224 unsigned int i[2]; 225 double d; 226 } xx, yy; 227 double t, u, w; 228 unsigned int hx, sx, ex, j, offset; 229 230 xx.d = x; 231 hx = xx.i[HIWORD] & ~0x80000000; 232 sx = xx.i[HIWORD] & 0x80000000; 233 234 /* handle special cases */ 235 if (hx >= 0x7ff00000) /* x is inf or nan */ 236 #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN) 237 return hx >= 0x7ff80000 ? x : x + x; 238 /* assumes sparc-like QNaN */ 239 #else 240 return x + x; 241 #endif 242 243 if (hx < 0x00100000) { /* x is subnormal or zero */ 244 if ((hx | xx.i[LOWORD]) == 0) 245 return x; 246 247 /* scale x to normal range */ 248 xx.d = x * two54; 249 hx = xx.i[HIWORD] & ~0x80000000; 250 offset = 0x29800000; 251 } 252 else 253 offset = 0x2aa00000; 254 255 ex = hx & 0x7ff00000; 256 j = (ex >> 2) + (ex >> 4) + (ex >> 6); 257 j = j + (j >> 6); 258 j = 0x7ff00000 & (j + 0x2aa00); /* j is ex/3 */ 259 hx -= (j + j + j); 260 xx.i[HIWORD] = 0x3ff00000 + hx; 261 262 u = __libm_TBL_cbrt[(hx + 0x1000) >> 13]; 263 w = u * u * u; 264 t = (xx.d - w) / w; 265 266 yy.i[HIWORD] = sx | (j + offset); 267 yy.i[LOWORD] = 0; 268 269 w = t * t; 270 return yy.d * (u + u * (t * (p1 + t * p2 + w * p3) + 271 (w * w) * (p4 + t * p5 + w * p6))); 272 } 273