1 /* mpfr_cbrt -- cube root function.
2
3 Copyright 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
4 Contributed by the AriC and Caramel projects, INRIA.
5
6 This file is part of the GNU MPFR Library.
7
8 The GNU MPFR Library is free software; you can redistribute it and/or modify
9 it under the terms of the GNU Lesser General Public License as published by
10 the Free Software Foundation; either version 3 of the License, or (at your
11 option) any later version.
12
13 The GNU MPFR Library is distributed in the hope that it will be useful, but
14 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
15 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
16 License for more details.
17
18 You should have received a copy of the GNU Lesser General Public License
19 along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
20 http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
21 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
22
23 #define MPFR_NEED_LONGLONG_H
24 #include "mpfr-impl.h"
25
26 /* The computation of y = x^(1/3) is done as follows:
27
28 Let x = sign * m * 2^(3*e) where m is an integer
29
30 with 2^(3n-3) <= m < 2^(3n) where n = PREC(y)
31
32 and m = s^3 + r where 0 <= r and m < (s+1)^3
33
34 we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(3n-3)
35 i.e. m must have at least 3n-2 bits
36
37 then x^(1/3) = s * 2^e if r=0
38 x^(1/3) = (s+1) * 2^e if round up
39 x^(1/3) = (s-1) * 2^e if round down
40 x^(1/3) = s * 2^e if nearest and r < 3/2*s^2+3/4*s+1/8
41 (s+1) * 2^e otherwise
42 */
43
44 int
mpfr_cbrt(mpfr_ptr y,mpfr_srcptr x,mpfr_rnd_t rnd_mode)45 mpfr_cbrt (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
46 {
47 mpz_t m;
48 mpfr_exp_t e, r, sh;
49 mpfr_prec_t n, size_m, tmp;
50 int inexact, negative;
51 MPFR_SAVE_EXPO_DECL (expo);
52
53 MPFR_LOG_FUNC (
54 ("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
55 ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
56 inexact));
57
58 /* special values */
59 if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
60 {
61 if (MPFR_IS_NAN (x))
62 {
63 MPFR_SET_NAN (y);
64 MPFR_RET_NAN;
65 }
66 else if (MPFR_IS_INF (x))
67 {
68 MPFR_SET_INF (y);
69 MPFR_SET_SAME_SIGN (y, x);
70 MPFR_RET (0);
71 }
72 /* case 0: cbrt(+/- 0) = +/- 0 */
73 else /* x is necessarily 0 */
74 {
75 MPFR_ASSERTD (MPFR_IS_ZERO (x));
76 MPFR_SET_ZERO (y);
77 MPFR_SET_SAME_SIGN (y, x);
78 MPFR_RET (0);
79 }
80 }
81
82 /* General case */
83 MPFR_SAVE_EXPO_MARK (expo);
84 mpz_init (m);
85
86 e = mpfr_get_z_2exp (m, x); /* x = m * 2^e */
87 if ((negative = MPFR_IS_NEG(x)))
88 mpz_neg (m, m);
89 r = e % 3;
90 if (r < 0)
91 r += 3;
92 /* x = (m*2^r) * 2^(e-r) = (m*2^r) * 2^(3*q) */
93
94 MPFR_MPZ_SIZEINBASE2 (size_m, m);
95 n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN);
96
97 /* we want 3*n-2 <= size_m + 3*sh + r <= 3*n
98 i.e. 3*sh + size_m + r <= 3*n */
99 sh = (3 * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / 3;
100 sh = 3 * sh + r;
101 if (sh >= 0)
102 {
103 mpz_mul_2exp (m, m, sh);
104 e = e - sh;
105 }
106 else if (r > 0)
107 {
108 mpz_mul_2exp (m, m, r);
109 e = e - r;
110 }
111
112 /* invariant: x = m*2^e, with e divisible by 3 */
113
114 /* we reuse the variable m to store the cube root, since it is not needed
115 any more: we just need to know if the root is exact */
116 inexact = mpz_root (m, m, 3) == 0;
117
118 MPFR_MPZ_SIZEINBASE2 (tmp, m);
119 sh = tmp - n;
120 if (sh > 0) /* we have to flush to 0 the last sh bits from m */
121 {
122 inexact = inexact || ((mpfr_exp_t) mpz_scan1 (m, 0) < sh);
123 mpz_fdiv_q_2exp (m, m, sh);
124 e += 3 * sh;
125 }
126
127 if (inexact)
128 {
129 if (negative)
130 rnd_mode = MPFR_INVERT_RND (rnd_mode);
131 if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA
132 || (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0)))
133 inexact = 1, mpz_add_ui (m, m, 1);
134 else
135 inexact = -1;
136 }
137
138 /* either inexact is not zero, and the conversion is exact, i.e. inexact
139 is not changed; or inexact=0, and inexact is set only when
140 rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */
141 inexact += mpfr_set_z (y, m, MPFR_RNDN);
142 MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / 3);
143
144 if (negative)
145 {
146 MPFR_CHANGE_SIGN (y);
147 inexact = -inexact;
148 }
149
150 mpz_clear (m);
151 MPFR_SAVE_EXPO_FREE (expo);
152 return mpfr_check_range (y, inexact, rnd_mode);
153 }
154