contrib/mpfr/cbrt.c

   1 /* mpfr_cbrt -- cube root function.
   2
   3 Copyright 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
   4 Contributed by the Arenaire and Cacao 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 2.1 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.LIB.  If not, write to
  20 the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
  21 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, mp_rnd_t rnd_mode)
  46 {
  47   mpz_t m;
  48   mp_exp_t e, r, sh;
  49   mp_prec_t n, size_m, tmp;
  50   int inexact, negative;
  51   MPFR_SAVE_EXPO_DECL (expo);
  52
  53   /* special values */
  54   if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
  55     {
  56       if (MPFR_IS_NAN (x))
  57         {
  58           MPFR_SET_NAN (y);
  59           MPFR_RET_NAN;
  60         }
  61       else if (MPFR_IS_INF (x))
  62         {
  63           MPFR_SET_INF (y);
  64           MPFR_SET_SAME_SIGN (y, x);
  65           MPFR_RET (0);
  66         }
  67       /* case 0: cbrt(+/- 0) = +/- 0 */
  68       else /* x is necessarily 0 */
  69         {
  70           MPFR_ASSERTD (MPFR_IS_ZERO (x));
  71           MPFR_SET_ZERO (y);
  72           MPFR_SET_SAME_SIGN (y, x);
  73           MPFR_RET (0);
  74         }
  75     }
  76
  77   /* General case */
  78   MPFR_SAVE_EXPO_MARK (expo);
  79   mpz_init (m);
  80
  81   e = mpfr_get_z_exp (m, x);                /* x = m * 2^e */
  82   if ((negative = MPFR_IS_NEG(x)))
  83     mpz_neg (m, m);
  84   r = e % 3;
  85   if (r < 0)
  86     r += 3;
  87   /* x = (m*2^r) * 2^(e-r) = (m*2^r) * 2^(3*q) */
  88
  89   MPFR_MPZ_SIZEINBASE2 (size_m, m);
  90   n = MPFR_PREC (y) + (rnd_mode == GMP_RNDN);
  91
  92   /* we want 3*n-2 <= size_m + 3*sh + r <= 3*n
  93      i.e. 3*sh + size_m + r <= 3*n */
  94   sh = (3 * (mp_exp_t) n - (mp_exp_t) size_m - r) / 3;
  95   sh = 3 * sh + r;
  96   if (sh >= 0)
  97     {
  98       mpz_mul_2exp (m, m, sh);
  99       e = e - sh;
 100     }
 101   else if (r > 0)
 102     {
 103       mpz_mul_2exp (m, m, r);
 104       e = e - r;
 105     }
 106
 107   /* invariant: x = m*2^e, with e divisible by 3 */
 108
 109   /* we reuse the variable m to store the cube root, since it is not needed
 110      any more: we just need to know if the root is exact */
 111   inexact = mpz_root (m, m, 3) == 0;
 112
 113   MPFR_MPZ_SIZEINBASE2 (tmp, m);
 114   sh = tmp - n;
 115   if (sh > 0) /* we have to flush to 0 the last sh bits from m */
 116     {
 117       inexact = inexact || ((mp_exp_t) mpz_scan1 (m, 0) < sh);
 118       mpz_div_2exp (m, m, sh);
 119       e += 3 * sh;
 120     }
 121
 122   if (inexact)
 123     {
 124       if (negative)
 125         rnd_mode = MPFR_INVERT_RND (rnd_mode);
 126       if (rnd_mode == GMP_RNDU
 127           || (rnd_mode == GMP_RNDN && mpz_tstbit (m, 0)))
 128         inexact = 1, mpz_add_ui (m, m, 1);
 129       else
 130         inexact = -1;
 131     }
 132
 133   /* either inexact is not zero, and the conversion is exact, i.e. inexact
 134      is not changed; or inexact=0, and inexact is set only when
 135      rnd_mode=GMP_RNDN and bit (n+1) from m is 1 */
 136   inexact += mpfr_set_z (y, m, GMP_RNDN);
 137   MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / 3);
 138
 139   if (negative)
 140     {
 141       MPFR_CHANGE_SIGN (y);
 142       inexact = -inexact;
 143     }
 144
 145   mpz_clear (m);
 146   MPFR_SAVE_EXPO_FREE (expo);
 147   return mpfr_check_range (y, inexact, rnd_mode);
 148 }