1/*							cbrtq.c
2 *
3 *	Cube root, __float128 precision
4 *
5 *
6 *
7 * SYNOPSIS:
8 *
9 * __float128 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
57#include "quadmath-imp.h"
58
59static const long double CBRT2 = 1.259921049894873164767210607278228350570251Q;
60static const long double CBRT4 = 1.587401051968199474751705639272308260391493Q;
61static const long double CBRT2I = 0.7937005259840997373758528196361541301957467Q;
62static const long double CBRT4I = 0.6299605249474365823836053036391141752851257Q;
63
64
65__float128
66cbrtq ( __float128 x)
67{
68  int e, rem, sign;
69  __float128 z;
70
71  if (!finiteq (x))
72    return x + x;
73
74  if (x == 0)
75    return (x);
76
77  if (x > 0)
78    sign = 1;
79  else
80    {
81      sign = -1;
82      x = -x;
83    }
84
85  z = x;
86 /* extract power of 2, leaving mantissa between 0.5 and 1  */
87  x = frexpq (x, &e);
88
89  /* Approximate cube root of number between .5 and 1,
90     peak relative error = 1.2e-6  */
91  x = ((((1.3584464340920900529734e-1Q * x
92	  - 6.3986917220457538402318e-1Q) * x
93	 + 1.2875551670318751538055e0Q) * x
94	- 1.4897083391357284957891e0Q) * x
95       + 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q;
96
97  /* exponent divided by 3 */
98  if (e >= 0)
99    {
100      rem = e;
101      e /= 3;
102      rem -= 3 * e;
103      if (rem == 1)
104	x *= CBRT2;
105      else if (rem == 2)
106	x *= CBRT4;
107    }
108  else
109    {				/* argument less than 1 */
110      e = -e;
111      rem = e;
112      e /= 3;
113      rem -= 3 * e;
114      if (rem == 1)
115	x *= CBRT2I;
116      else if (rem == 2)
117	x *= CBRT4I;
118      e = -e;
119    }
120
121  /* multiply by power of 2 */
122  x = ldexpq (x, e);
123
124  /* Newton iteration */
125  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
126  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
127  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
128
129  if (sign < 0)
130    x = -x;
131  return (x);
132}
133