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 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