1/* cbrtl.c
2 *
3 * Cube root, long double precision
4 *
5 *
6 *
7 * SYNOPSIS:
8 *
9 * long double x, y, cbrtl();
10 *
11 * y = cbrtl( 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 <math.h>
58#include <math_private.h>
59
60static const _Float128 CBRT2 = L(1.259921049894873164767210607278228350570251);
61static const _Float128 CBRT4 = L(1.587401051968199474751705639272308260391493);
62static const _Float128 CBRT2I = L(0.7937005259840997373758528196361541301957467);
63static const _Float128 CBRT4I = L(0.6299605249474365823836053036391141752851257);
64
65
66_Float128
67__cbrtl (_Float128 x)
68{
69 int e, rem, sign;
70 _Float128 z;
71
72 if (!isfinite (x))
73 return x + x;
74
75 if (x == 0)
76 return (x);
77
78 if (x > 0)
79 sign = 1;
80 else
81 {
82 sign = -1;
83 x = -x;
84 }
85
86 z = x;
87 /* extract power of 2, leaving mantissa between 0.5 and 1 */
88 x = __frexpl (x, &e);
89
90 /* Approximate cube root of number between .5 and 1,
91 peak relative error = 1.2e-6 */
92 x = ((((L(1.3584464340920900529734e-1) * x
93 - L(6.3986917220457538402318e-1)) * x
94 + L(1.2875551670318751538055e0)) * x
95 - L(1.4897083391357284957891e0)) * x
96 + L(1.3304961236013647092521e0)) * x + L(3.7568280825958912391243e-1);
97
98 /* exponent divided by 3 */
99 if (e >= 0)
100 {
101 rem = e;
102 e /= 3;
103 rem -= 3 * e;
104 if (rem == 1)
105 x *= CBRT2;
106 else if (rem == 2)
107 x *= CBRT4;
108 }
109 else
110 { /* argument less than 1 */
111 e = -e;
112 rem = e;
113 e /= 3;
114 rem -= 3 * e;
115 if (rem == 1)
116 x *= CBRT2I;
117 else if (rem == 2)
118 x *= CBRT4I;
119 e = -e;
120 }
121
122 /* multiply by power of 2 */
123 x = __ldexpl (x, e);
124
125 /* Newton iteration */
126 x -= (x - (z / (x * x))) * L(0.3333333333333333333333333333333333333333);
127 x -= (x - (z / (x * x))) * L(0.3333333333333333333333333333333333333333);
128 x -= (x - (z / (x * x))) * L(0.3333333333333333333333333333333333333333);
129
130 if (sign < 0)
131 x = -x;
132 return (x);
133}
134
135weak_alias (__cbrtl, cbrtl)
136