/* mpfr_root -- kth root.

Copyright 2005-2017 Free Software Foundation, Inc.

Contributed by the AriC and Caramba 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, except for large

    values of k, for which this would be inefficient or yield internal

    integer overflows:

    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 + t where 0 <= t 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.

 */

static int

mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k,

               mpfr_rnd_t rnd_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, f;

  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 == 0)

        {

          MPFR_SET_NAN (y);

          MPFR_RET_NAN;

        }

      else /* y = x^(1/1) = x */

        return mpfr_set (y, x, rnd_mode);

    }

  /* Singular values */

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))

    {

      if (MPFR_IS_NAN (x))

        {

          MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */

          MPFR_RET_NAN;

        }

      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);

        }

      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 */

  if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && (k % 2 == 0)))

    {

      MPFR_SET_NAN (y);

      MPFR_RET_NAN;

    }

  /* Special case |x| = 1. Note that if x = -1, then k is odd

     (NaN results have already been filtered), so that y = -1. */

  if (mpfr_cmpabs (x, __gmpfr_one) == 0)

    return mpfr_set (y, x, rnd_mode);

  /* General case */

  /* For large k, use exp(log(x)/k). The threshold of 100 seems to be quite

     good when the precision goes to infinity. */

  if (k > 100)

    return mpfr_root_aux (y, x, k, rnd_mode);

  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 <= r < k */

  MPFR_ASSERTD (0 <= r && r < k);

  /* 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^sh so that root(m,k) will give

     exactly n bits: we want k*(n-1)+1 <= size_m + sh <= k*n

     i.e. sh = k*f + r with f = max(floor((k*n-size_m-r)/k),0) */

  if ((mpfr_exp_t) size_m + r >= k * (mpfr_exp_t) n)

    f = 0; /* we already have too many bits */

  else

    f = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k;

  sh = k * f + r;

  mpz_mul_2exp (m, m, sh);

  e = e - sh;

  /* 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);

}

/* Compute y <- x^(1/k) using exp(log(x)/k).

   Assume all special cases have been eliminated before.

   In the extended exponent range, overflows/underflows are not possible.

   Assume x > 0, or x < 0 and k odd.

   Also assume |x| <> 1 because log(1) = 0, which does not have an exponent

   and would yield a failure in the error bound computation. A priori, this

   constraint is quite artificial because if |x| is close enough to 1, then

   the exponent of log|x| does not need to be used (in the code, err would

   be 1 in such a domain). So this constraint |x| <> 1 could be avoided in

   the code. However, this is an exact case easy to detect, so that such a

   change would be useless. Values very close to 1 are not an issue, since

   an underflow is not possible before the MPFR_GET_EXP.

*/

static int

mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)

{

  int inexact, exact_root = 0;

  mpfr_prec_t w; /* working precision */

  mpfr_t absx, t;

  MPFR_GROUP_DECL(group);

  MPFR_TMP_DECL(marker);

  MPFR_ZIV_DECL(loop);

  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_TMP_INIT_ABS (absx, x);

  MPFR_TMP_MARK(marker);

  w = MPFR_PREC(y) + 10;

  /* Take some guard bits to prepare for the 'expt' lost bits below.

     If |x| < 2^k, then log|x| < k, thus taking log2(k) bits should be fine. */

  if (MPFR_GET_EXP(x) > 0)

    w += MPFR_INT_CEIL_LOG2 (MPFR_GET_EXP(x));

  MPFR_GROUP_INIT_1(group, w, t);

  MPFR_SAVE_EXPO_MARK (expo);

  MPFR_ZIV_INIT (loop, w);

  for (;;)

    {

      mpfr_exp_t expt;

      unsigned int err;

      mpfr_log (t, absx, MPFR_RNDN);

      /* t = log|x| * (1 + theta) with |theta| <= 2^(-w) */

      mpfr_div_ui (t, t, k, MPFR_RNDN);

      /* No possible underflow in mpfr_log and mpfr_div_ui. */

      expt = MPFR_GET_EXP (t);  /* assumes t <> 0 */

      /* t = log|x|/k * (1 + theta) + eps with |theta| <= 2^(-w)

         and |eps| <= 1/2 ulp(t), thus the total error is bounded

         by 1.5 * 2^(expt - w) */

      mpfr_exp (t, t, MPFR_RNDN);

      /* t = |x|^(1/k) * exp(tau) * (1 + theta1) with

         |tau| <= 1.5 * 2^(expt - w) and |theta1| <= 2^(-w).

         For |tau| <= 0.5 we have |exp(tau)-1| < 4/3*tau, thus

         for w >= expt + 2 we have:

         t = |x|^(1/k) * (1 + 2^(expt+2)*theta2) * (1 + theta1) with

         |theta1|, |theta2| <= 2^(-w).

         If expt+2 > 0, as long as w >= 1, we have:

         t = |x|^(1/k) * (1 + 2^(expt+3)*theta3) with |theta3| < 2^(-w).

         For expt+2 = 0, we have:

         t = |x|^(1/k) * (1 + 2^2*theta3) with |theta3| < 2^(-w).

         Finally for expt+2 < 0 we have:

         t = |x|^(1/k) * (1 + 2*theta3) with |theta3| < 2^(-w).

      */

      err = (expt + 2 > 0) ? expt + 3

        : (expt + 2 == 0) ? 2 : 1;

      /* now t = |x|^(1/k) * (1 + 2^(err-w)) thus the error is at most

         2^(EXP(t) - w + err) */

      if (MPFR_LIKELY (MPFR_CAN_ROUND(t, w - err, MPFR_PREC(y), rnd_mode)))

        break;

      /* If we fail to round correctly, check for an exact result or a

         midpoint result with MPFR_RNDN (regarded as hard-to-round in

         all precisions in order to determine the ternary value). */

      {

        mpfr_t z, zk;

        mpfr_init2 (z, MPFR_PREC(y) + (rnd_mode == MPFR_RNDN));

        mpfr_init2 (zk, MPFR_PREC(x));

        mpfr_set (z, t, MPFR_RNDN);

        inexact = mpfr_pow_ui (zk, z, k, MPFR_RNDN);

        exact_root = !inexact && mpfr_equal_p (zk, absx);

        if (exact_root) /* z is the exact root, thus round z directly */

          inexact = mpfr_set4 (y, z, rnd_mode, MPFR_SIGN (x));

        mpfr_clear (zk);

        mpfr_clear (z);

        if (exact_root)

          break;

      }

      MPFR_ZIV_NEXT (loop, w);

      MPFR_GROUP_REPREC_1(group, w, t);

    }

  MPFR_ZIV_FREE (loop);

  if (!exact_root)

    inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (x));

  MPFR_GROUP_CLEAR(group);

  MPFR_TMP_FREE(marker);

  MPFR_SAVE_EXPO_FREE (expo);

  return mpfr_check_range (y, inexact, rnd_mode);

}