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comparison libquadmath/math/cbrtq.c @ 111:04ced10e8804

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gcc 7
author kono
date Fri, 27 Oct 2017 22:46:09 +0900
parents 561a7518be6b
children 1830386684a0
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68:561a7518be6b 111:04ced10e8804
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
1 #include "quadmath-imp.h" 57 #include "quadmath-imp.h"
2 #include <math.h> 58
3 #include <float.h> 59 static const __float128 CBRT2 = 1.259921049894873164767210607278228350570251Q;
60 static const __float128 CBRT4 = 1.587401051968199474751705639272308260391493Q;
61 static const __float128 CBRT2I = 0.7937005259840997373758528196361541301957467Q;
62 static const __float128 CBRT4I = 0.6299605249474365823836053036391141752851257Q;
63
4 64
5 __float128 65 __float128
6 cbrtq (const __float128 x) 66 cbrtq ( __float128 x)
7 { 67 {
8 __float128 y; 68 int e, rem, sign;
9 int exp, i; 69 __float128 z;
70
71 if (!finiteq (x))
72 return x + x;
10 73
11 if (x == 0) 74 if (x == 0)
12 return x; 75 return (x);
13 76
14 if (isnanq (x)) 77 if (x > 0)
15 return x; 78 sign = 1;
79 else
80 {
81 sign = -1;
82 x = -x;
83 }
16 84
17 if (x <= DBL_MAX && x >= DBL_MIN) 85 z = x;
18 { 86 /* extract power of 2, leaving mantissa between 0.5 and 1 */
19 /* Use double result as starting point. */ 87 x = frexpq (x, &e);
20 y = cbrt ((double) x);
21 88
22 /* Two Newton iterations. */ 89 /* Approximate cube root of number between .5 and 1,
23 y -= 0.333333333333333333333333333333333333333333333333333Q 90 peak relative error = 1.2e-6 */
24 * (y - x / (y * y)); 91 x = ((((1.3584464340920900529734e-1Q * x
25 y -= 0.333333333333333333333333333333333333333333333333333Q 92 - 6.3986917220457538402318e-1Q) * x
26 * (y - x / (y * y)); 93 + 1.2875551670318751538055e0Q) * x
27 return y; 94 - 1.4897083391357284957891e0Q) * x
28 } 95 + 1.3304961236013647092521e0Q) * x + 3.7568280825958912391243e-1Q;
29 96
30 #ifdef HAVE_CBRTL 97 /* exponent divided by 3 */
31 if (x <= LDBL_MAX && x >= LDBL_MIN) 98 if (e >= 0)
32 { 99 {
33 /* Use long double result as starting point. */ 100 rem = e;
34 y = cbrtl ((long double) x); 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 }
35 120
36 /* One Newton iteration. */ 121 /* multiply by power of 2 */
37 y -= 0.333333333333333333333333333333333333333333333333333Q 122 x = ldexpq (x, e);
38 * (y - x / (y * y));
39 return y;
40 }
41 #endif
42 123
43 /* If we're outside of the range of C types, we have to compute 124 /* Newton iteration */
44 the initial guess the hard way. */ 125 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
45 y = frexpq (x, &exp); 126 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
127 x -= (x - (z / (x * x))) * 0.3333333333333333333333333333333333333333Q;
46 128
47 i = exp % 3; 129 if (sign < 0)
48 y = (i >= 0 ? i : -i); 130 x = -x;
49 if (i == 1) 131 return (x);
50 y *= 2, exp--;
51 else if (i == 2)
52 y *= 4, exp -= 2;
53
54 y = cbrt (y);
55 y = scalbnq (y, exp / 3);
56
57 /* Two Newton iterations. */
58 y -= 0.333333333333333333333333333333333333333333333333333Q
59 * (y - x / (y * y));
60 y -= 0.333333333333333333333333333333333333333333333333333Q
61 * (y - x / (y * y));
62 return y;
63 } 132 }
64