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 /*
38 Cephes Math Library Release 2.2: January, 1991
39 Copyright 1984, 1991 by Stephen L. Moshier
40 Adapted 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
59 static const long double CBRT2 = 1.259921049894873164767210607278228350570251Q;
60 static const long double CBRT4 = 1.587401051968199474751705639272308260391493Q;
61 static const long double CBRT2I = 0.7937005259840997373758528196361541301957467Q;
62 static const long double CBRT4I = 0.6299605249474365823836053036391141752851257Q;
63
64
65 __float128
cbrtq(__float128 x)66 cbrtq ( __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