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 }