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 <https://www.gnu.org/licenses/>. */
55
56
57#include <math.h>
58#include <math_private.h>
59#include <libm-alias-ldouble.h>
60
61static const _Float128 CBRT2 = L(1.259921049894873164767210607278228350570251);
62static const _Float128 CBRT4 = L(1.587401051968199474751705639272308260391493);
63static const _Float128 CBRT2I = L(0.7937005259840997373758528196361541301957467);
64static const _Float128 CBRT4I = L(0.6299605249474365823836053036391141752851257);
65
66
67_Float128
68__cbrtl (_Float128 x)
69{
70 int e, rem, sign;
71 _Float128 z;
72
73 if (!isfinite (x))
74 return x + x;
75
76 if (x == 0)
77 return (x);
78
79 if (x > 0)
80 sign = 1;
81 else
82 {
83 sign = -1;
84 x = -x;
85 }
86
87 z = x;
88 /* extract power of 2, leaving mantissa between 0.5 and 1 */
89 x = __frexpl (x, &e);
90
91 /* Approximate cube root of number between .5 and 1,
92 peak relative error = 1.2e-6 */
93 x = ((((L(1.3584464340920900529734e-1) * x
94 - L(6.3986917220457538402318e-1)) * x
95 + L(1.2875551670318751538055e0)) * x
96 - L(1.4897083391357284957891e0)) * x
97 + L(1.3304961236013647092521e0)) * x + L(3.7568280825958912391243e-1);
98
99 /* exponent divided by 3 */
100 if (e >= 0)
101 {
102 rem = e;
103 e /= 3;
104 rem -= 3 * e;
105 if (rem == 1)
106 x *= CBRT2;
107 else if (rem == 2)
108 x *= CBRT4;
109 }
110 else
111 { /* argument less than 1 */
112 e = -e;
113 rem = e;
114 e /= 3;
115 rem -= 3 * e;
116 if (rem == 1)
117 x *= CBRT2I;
118 else if (rem == 2)
119 x *= CBRT4I;
120 e = -e;
121 }
122
123 /* multiply by power of 2 */
124 x = __ldexpl (x, e);
125
126 /* Newton iteration */
127 x -= (x - (z / (x * x))) * L(0.3333333333333333333333333333333333333333);
128 x -= (x - (z / (x * x))) * L(0.3333333333333333333333333333333333333333);
129 x -= (x - (z / (x * x))) * L(0.3333333333333333333333333333333333333333);
130
131 if (sign < 0)
132 x = -x;
133 return (x);
134}
135
136libm_alias_ldouble (__cbrt, cbrt)
137