1/*							cbrtq.c
2 *
3 *	Cube root, long double precision
4 *
5 *
6 *
7 * SYNOPSIS:
8 *
9 * long double x, y, cbrtq();
10 *
11 * y = cbrtq( x );
12 *
13 *
14 *
15 * DESCRIPTION:
16 *
17 * Returns the cube root of the argument, which may be negative.
18 *
19 * Range reduction involves determining the power of 2 of
20 * the argument.  A polynomial of degree 2 applied to the
21 * mantissa, and multiplication by the cube root of 1, 2, or 4
22 * approximates the root to within about 0.1%.  Then Newton's
23 * iteration is used three times to converge to an accurate
24 * result.
25 *
26 *
27 *
28 * ACCURACY:
29 *
30 *                      Relative error:
31 * arithmetic   domain     # trials      peak         rms
32 *    IEEE       -8,8       100000      1.3e-34     3.9e-35
33 *    IEEE    exp(+-707)    100000      1.3e-34     4.3e-35
34 *
35 */
36
37/*
38Cephes Math Library Release 2.2: January, 1991
39Copyright 1984, 1991 by Stephen L. Moshier
40Adapted for glibc October, 2001.
41
42    This library is free software; you can redistribute it and/or
43    modify it under the terms of the GNU Lesser General Public
44    License as published by the Free Software Foundation; either
45    version 2.1 of the License, or (at your option) any later version.
46
47    This library is distributed in the hope that it will be useful,
48    but WITHOUT ANY WARRANTY; without even the implied warranty of
49    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
50    Lesser General Public License for more details.
51
52    You should have received a copy of the GNU Lesser General Public
53    License along with this library; if not, see
54    <http://www.gnu.org/licenses/>.  */
55
56#include "quadmath-imp.h"
57
58static const __float128 CBRT2 = 1.259921049894873164767210607278228350570251Q;
59static const __float128 CBRT4 = 1.587401051968199474751705639272308260391493Q;
60static const __float128 CBRT2I = 0.7937005259840997373758528196361541301957467Q;
61static const __float128 CBRT4I = 0.6299605249474365823836053036391141752851257Q;
62
63
64__float128
65cbrtq (__float128 x)
66{
67  int e, rem, sign;
68  __float128 z;
69
70  if (!finiteq (x))
71    return x + x;
72
73  if (x == 0)
74    return (x);
75
76  if (x > 0)
77    sign = 1;
78  else
79    {
80      sign = -1;
81      x = -x;
82    }
83
84  z = x;
85 /* extract power of 2, leaving mantissa between 0.5 and 1  */
86  x = frexpq (x, &e);
87
88  /* Approximate cube root of number between .5 and 1,
89     peak relative error = 1.2e-6  */
90  x = ((((1.3584464340920900529734e-1Q * x
91	  - 6.3986917220457538402318e-1Q) * x
92	 + 1.2875551670318751538055e0Q) * x
93	- 1.4897083391357284957891e0Q) * x
94       + 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q;
95
96  /* exponent divided by 3 */
97  if (e >= 0)
98    {
99      rem = e;
100      e /= 3;
101      rem -= 3 * e;
102      if (rem == 1)
103	x *= CBRT2;
104      else if (rem == 2)
105	x *= CBRT4;
106    }
107  else
108    {				/* argument less than 1 */
109      e = -e;
110      rem = e;
111      e /= 3;
112      rem -= 3 * e;
113      if (rem == 1)
114	x *= CBRT2I;
115      else if (rem == 2)
116	x *= CBRT4I;
117      e = -e;
118    }
119
120  /* multiply by power of 2 */
121  x = ldexpq (x, e);
122
123  /* Newton iteration */
124  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
125  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
126  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
127
128  if (sign < 0)
129    x = -x;
130  return (x);
131}
132