about summary refs log tree commit diff
path: root/sysdeps/ieee754/ldbl-128/s_cbrtl.c
blob: d93684e50e863698e2cdc69fd94010d1df6d93c5 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
/*							cbrtl.c
 *
 *	Cube root, long double precision
 *
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, cbrtl();
 *
 * y = cbrtl( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns the cube root of the argument, which may be negative.
 *
 * Range reduction involves determining the power of 2 of
 * the argument.  A polynomial of degree 2 applied to the
 * mantissa, and multiplication by the cube root of 1, 2, or 4
 * approximates the root to within about 0.1%.  Then Newton's
 * iteration is used three times to converge to an accurate
 * result.
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE       -8,8       100000      1.3e-34     3.9e-35
 *    IEEE    exp(+-707)    100000      1.3e-34     4.3e-35
 *
 */

/*
Cephes Math Library Release 2.2: January, 1991
Copyright 1984, 1991 by Stephen L. Moshier
Adapted for glibc October, 2001.

    This library is free software; you can redistribute it and/or
    modify it under the terms of the GNU Lesser General Public
    License as published by the Free Software Foundation; either
    version 2.1 of the License, or (at your option) any later version.

    This library is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
    Lesser General Public License for more details.

    You should have received a copy of the GNU Lesser General Public
    License along with this library; if not, see
    <http://www.gnu.org/licenses/>.  */


#include <math.h>
#include <math_private.h>

static const _Float128 CBRT2 = 1.259921049894873164767210607278228350570251L;
static const _Float128 CBRT4 = 1.587401051968199474751705639272308260391493L;
static const _Float128 CBRT2I = 0.7937005259840997373758528196361541301957467L;
static const _Float128 CBRT4I = 0.6299605249474365823836053036391141752851257L;


_Float128
__cbrtl (_Float128 x)
{
  int e, rem, sign;
  _Float128 z;

  if (!isfinite (x))
    return x + x;

  if (x == 0)
    return (x);

  if (x > 0)
    sign = 1;
  else
    {
      sign = -1;
      x = -x;
    }

  z = x;
 /* extract power of 2, leaving mantissa between 0.5 and 1  */
  x = __frexpl (x, &e);

  /* Approximate cube root of number between .5 and 1,
     peak relative error = 1.2e-6  */
  x = ((((1.3584464340920900529734e-1L * x
	  - 6.3986917220457538402318e-1L) * x
	 + 1.2875551670318751538055e0L) * x
	- 1.4897083391357284957891e0L) * x
       + 1.3304961236013647092521e0L) * x + 3.7568280825958912391243e-1L;

  /* exponent divided by 3 */
  if (e >= 0)
    {
      rem = e;
      e /= 3;
      rem -= 3 * e;
      if (rem == 1)
	x *= CBRT2;
      else if (rem == 2)
	x *= CBRT4;
    }
  else
    {				/* argument less than 1 */
      e = -e;
      rem = e;
      e /= 3;
      rem -= 3 * e;
      if (rem == 1)
	x *= CBRT2I;
      else if (rem == 2)
	x *= CBRT4I;
      e = -e;
    }

  /* multiply by power of 2 */
  x = __ldexpl (x, e);

  /* Newton iteration */
  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;
  x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333L;

  if (sign < 0)
    x = -x;
  return (x);
}

weak_alias (__cbrtl, cbrtl)