1 /* mpfr_root -- kth root.
3 Copyright 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
4 Contributed by the Arenaire and Caramel projects, INRIA.
6 This file is part of the GNU MPFR Library.
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.
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.
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. */
23 #define MPFR_NEED_LONGLONG_H
24 #include "mpfr-impl.h"
26 /* The computation of y = x^(1/k) is done as follows:
28 Let x = sign * m * 2^(k*e) where m is an integer
30 with 2^(k*(n-1)) <= m < 2^(k*n) where n = PREC(y)
32 and m = s^k + r where 0 <= r and m < (s+1)^k
34 we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(k*(n-1))
35 i.e. m must have at least k*(n-1)+1 bits
37 then, not taking into account the sign, the result will be
38 x^(1/k) = s * 2^e or (s+1) * 2^e according to the rounding mode.
42 mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
46 mpfr_prec_t n, size_m, tmp;
47 int inexact, negative;
48 MPFR_SAVE_EXPO_DECL (expo);
51 (("x[%Pu]=%.*Rg k=%lu rnd=%d",
52 mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
53 ("y[%Pu]=%.*Rg inexact=%d",
54 mpfr_get_prec (y), mpfr_log_prec, y, inexact));
56 if (MPFR_UNLIKELY (k <= 1))
58 if (k < 1) /* k==0 => y=x^(1/0)=x^(+Inf) */
60 /* For 0 <= x < 1 => +0.
66 if (MPFR_IS_NEG (x) && !MPFR_IS_ZERO (x))
71 inexact = mpfr_cmp (x, __gmpfr_one);
73 return mpfr_set_ui (y, 1, rnd_mode); /* 1 may be Out of Range */
75 return mpfr_set_ui (y, 0, rnd_mode); /* 0+ */
87 else /* y =x^(1/1)=x */
88 return mpfr_set (y, x, rnd_mode);
92 else if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
96 MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
99 else if (MPFR_IS_INF (x)) /* +Inf^(1/k) = +Inf
100 -Inf^(1/k) = -Inf if k odd
101 -Inf^(1/k) = NaN if k even */
103 if (MPFR_IS_NEG(x) && (k % 2 == 0))
109 MPFR_SET_SAME_SIGN (y, x);
112 else /* x is necessarily 0: (+0)^(1/k) = +0
115 MPFR_ASSERTD (MPFR_IS_ZERO (x));
117 MPFR_SET_SAME_SIGN (y, x);
122 /* Returns NAN for x < 0 and k even */
123 else if (MPFR_IS_NEG (x) && (k % 2 == 0))
130 MPFR_SAVE_EXPO_MARK (expo);
133 e = mpfr_get_z_2exp (m, x); /* x = m * 2^e */
134 if ((negative = MPFR_IS_NEG(x)))
136 r = e % (mpfr_exp_t) k;
138 r += k; /* now r = e (mod k) with 0 <= e < r */
139 /* x = (m*2^r) * 2^(e-r) where e-r is a multiple of k */
141 MPFR_MPZ_SIZEINBASE2 (size_m, m);
142 /* for rounding to nearest, we want the round bit to be in the root */
143 n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN);
145 /* we now multiply m by 2^(r+k*sh) so that root(m,k) will give
146 exactly n bits: we want k*(n-1)+1 <= size_m + k*sh + r <= k*n
147 i.e. sh = floor ((kn-size_m-r)/k) */
148 if ((mpfr_exp_t) size_m + r > k * (mpfr_exp_t) n)
149 sh = 0; /* we already have too many bits */
151 sh = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k;
155 mpz_mul_2exp (m, m, sh);
160 mpz_mul_2exp (m, m, r);
164 /* invariant: x = m*2^e, with e divisible by k */
166 /* we reuse the variable m to store the kth root, since it is not needed
167 any more: we just need to know if the root is exact */
168 inexact = mpz_root (m, m, k) == 0;
170 MPFR_MPZ_SIZEINBASE2 (tmp, m);
172 if (sh > 0) /* we have to flush to 0 the last sh bits from m */
174 inexact = inexact || ((mpfr_exp_t) mpz_scan1 (m, 0) < sh);
175 mpz_fdiv_q_2exp (m, m, sh);
182 rnd_mode = MPFR_INVERT_RND (rnd_mode);
183 if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA
184 || (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0)))
185 inexact = 1, mpz_add_ui (m, m, 1);
190 /* either inexact is not zero, and the conversion is exact, i.e. inexact
191 is not changed; or inexact=0, and inexact is set only when
192 rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */
193 inexact += mpfr_set_z (y, m, MPFR_RNDN);
194 MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mpfr_exp_t) k);
198 MPFR_CHANGE_SIGN (y);
203 MPFR_SAVE_EXPO_FREE (expo);
204 return mpfr_check_range (y, inexact, rnd_mode);