root.c

/* mpfr_root -- kth root.

Copyright 2005-2015 Free Software Foundation, Inc.
Contributed by the AriC and Caramel projects, INRIA.

This file is part of the GNU MPFR Library.

The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */

#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"

 /* The computation of y = x^(1/k) is done as follows:

    Let x = sign * m * 2^(k*e) where m is an integer

    with 2^(k*(n-1)) <= m < 2^(k*n) where n = PREC(y)

    and m = s^k + r where 0 <= r and m < (s+1)^k

    we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(k*(n-1))
    i.e. m must have at least k*(n-1)+1 bits

    then, not taking into account the sign, the result will be
    x^(1/k) = s * 2^e or (s+1) * 2^e according to the rounding mode.
 */

int
mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
{
  mpz_t m;
  mpfr_exp_t e, r, sh;
  mpfr_prec_t n, size_m, tmp;
  int inexact, negative;
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg k=%lu rnd=%d",
      mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
     ("y[%Pu]=%.*Rg inexact=%d",
      mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  if (MPFR_UNLIKELY (k <= 1))
    {
      if (k < 1) /* k==0 => y=x^(1/0)=x^(+Inf) */
#if 0
        /* For 0 <= x < 1 => +0.
           For x = 1      => 1.
           For x > 1,     => +Inf.
           For x < 0      => NaN.
        */
        {
          if (MPFR_IS_NEG (x) && !MPFR_IS_ZERO (x))
            {
              MPFR_SET_NAN (y);
              MPFR_RET_NAN;
            }
          inexact = mpfr_cmp (x, __gmpfr_one);
          if (inexact == 0)
            return mpfr_set_ui (y, 1, rnd_mode); /* 1 may be Out of Range */
          else if (inexact < 0)
            return mpfr_set_ui (y, 0, rnd_mode); /* 0+ */
          else
            {
              mpfr_set_inf (y, 1);
              return 0;
            }
        }
#endif
      {
        MPFR_SET_NAN (y);
        MPFR_RET_NAN;
      }
      else /* y =x^(1/1)=x */
        return mpfr_set (y, x, rnd_mode);
    }

  /* Singular values */
  else if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (x)) /* +Inf^(1/k) = +Inf
                                   -Inf^(1/k) = -Inf if k odd
                                   -Inf^(1/k) = NaN if k even */
        {
          if (MPFR_IS_NEG(x) && (k % 2 == 0))
            {
              MPFR_SET_NAN (y);
              MPFR_RET_NAN;
            }
          MPFR_SET_INF (y);
          MPFR_SET_SAME_SIGN (y, x);
          MPFR_RET (0);
        }
      else /* x is necessarily 0: (+0)^(1/k) = +0
                                  (-0)^(1/k) = -0 */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          MPFR_SET_ZERO (y);
          MPFR_SET_SAME_SIGN (y, x);
          MPFR_RET (0);
        }
    }

  /* Returns NAN for x < 0 and k even */
  else if (MPFR_IS_NEG (x) && (k % 2 == 0))
    {
      MPFR_SET_NAN (y);
      MPFR_RET_NAN;
    }

  /* General case */
  MPFR_SAVE_EXPO_MARK (expo);
  mpz_init (m);

  e = mpfr_get_z_2exp (m, x);                /* x = m * 2^e */
  if ((negative = MPFR_IS_NEG(x)))
    mpz_neg (m, m);
  r = e % (mpfr_exp_t) k;
  if (r < 0)
    r += k; /* now r = e (mod k) with 0 <= e < r */
  /* x = (m*2^r) * 2^(e-r) where e-r is a multiple of k */

  MPFR_MPZ_SIZEINBASE2 (size_m, m);
  /* for rounding to nearest, we want the round bit to be in the root */
  n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN);

  /* we now multiply m by 2^(r+k*sh) so that root(m,k) will give
     exactly n bits: we want k*(n-1)+1 <= size_m + k*sh + r <= k*n
     i.e. sh = floor ((kn-size_m-r)/k) */
  if ((mpfr_exp_t) size_m + r > k * (mpfr_exp_t) n)
    sh = 0; /* we already have too many bits */
  else
    sh = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k;
  sh = k * sh + r;
  if (sh >= 0)
    {
      mpz_mul_2exp (m, m, sh);
      e = e - sh;
    }
  else if (r > 0)
    {
      mpz_mul_2exp (m, m, r);
      e = e - r;
    }

  /* invariant: x = m*2^e, with e divisible by k */

  /* we reuse the variable m to store the kth root, since it is not needed
     any more: we just need to know if the root is exact */
  inexact = mpz_root (m, m, k) == 0;

  MPFR_MPZ_SIZEINBASE2 (tmp, m);
  sh = tmp - n;
  if (sh > 0) /* we have to flush to 0 the last sh bits from m */
    {
      inexact = inexact || ((mpfr_exp_t) mpz_scan1 (m, 0) < sh);
      mpz_fdiv_q_2exp (m, m, sh);
      e += k * sh;
    }

  if (inexact)
    {
      if (negative)
        rnd_mode = MPFR_INVERT_RND (rnd_mode);
      if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA
          || (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0)))
        inexact = 1, mpz_add_ui (m, m, 1);
      else
        inexact = -1;
    }

  /* either inexact is not zero, and the conversion is exact, i.e. inexact
     is not changed; or inexact=0, and inexact is set only when
     rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */
  inexact += mpfr_set_z (y, m, MPFR_RNDN);
  MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mpfr_exp_t) k);

  if (negative)
    {
      MPFR_CHANGE_SIGN (y);
      inexact = -inexact;
    }

  mpz_clear (m);
  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inexact, rnd_mode);
}