/* mpfr_pow_si -- power function x^y with y a signed int

Copyright 2001-2020 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
https://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  /* for MPFR_INT_CEIL_LOG2 */
#include "mpfr-impl.h"

/* The computation of y = pow_si(x,n) is done by
 *    y = pow_ui(x,n)       if n >= 0
 *    y = 1 / pow_ui(x,-n)  if n < 0
 */

int
mpfr_pow_si (mpfr_ptr y, mpfr_srcptr x, long int n, mpfr_rnd_t rnd)
{
  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg n=%ld rnd=%d",
      mpfr_get_prec (x), mpfr_log_prec, x, n, rnd),
     ("y[%Pu]=%.*Rg", mpfr_get_prec (y), mpfr_log_prec, y));

  if (n >= 0)
    return mpfr_pow_ui (y, x, n, rnd);
  else
    {
      if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
        {
          if (MPFR_IS_NAN (x))
            {
              MPFR_SET_NAN (y);
              MPFR_RET_NAN;
            }
          else
            {
              int positive = MPFR_IS_POS (x) || ((unsigned long) n & 1) == 0;
              if (MPFR_IS_INF (x))
                MPFR_SET_ZERO (y);
              else /* x is zero */
                {
                  MPFR_ASSERTD (MPFR_IS_ZERO (x));
                  MPFR_SET_INF (y);
                  MPFR_SET_DIVBY0 ();
                }
              if (positive)
                MPFR_SET_POS (y);
              else
                MPFR_SET_NEG (y);
              MPFR_RET (0);
            }
        }

      /* detect exact powers: x^(-n) is exact iff x is a power of 2 */
      if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), MPFR_EXP(x) - 1) == 0)
        {
          mpfr_exp_t expx = MPFR_EXP (x) - 1, expy;
          MPFR_ASSERTD (n < 0);
          /* Warning: n * expx may overflow!
           *
           * Some systems (apparently alpha-freebsd) abort with
           * LONG_MIN / 1, and LONG_MIN / -1 is undefined.
           * http://www.freebsd.org/cgi/query-pr.cgi?pr=72024
           *
           * Proof of the overflow checking. The expressions below are
           * assumed to be on the rational numbers, but the word "overflow"
           * still has its own meaning in the C context. / still denotes
           * the integer (truncated) division, and // denotes the exact
           * division.
           * - First, (__gmpfr_emin - 1) / n and (__gmpfr_emax - 1) / n
           *   cannot overflow due to the constraints on the exponents of
           *   MPFR numbers.
           * - If n = -1, then n * expx = - expx, which is representable
           *   because of the constraints on the exponents of MPFR numbers.
           * - If expx = 0, then n * expx = 0, which is representable.
           * - If n < -1 and expx > 0:
           *   + If expx > (__gmpfr_emin - 1) / n, then
           *           expx >= (__gmpfr_emin - 1) / n + 1
           *                > (__gmpfr_emin - 1) // n,
           *     and
           *           n * expx < __gmpfr_emin - 1,
           *     i.e.
           *           n * expx <= __gmpfr_emin - 2.
           *     This corresponds to an underflow, with a null result in
           *     the rounding-to-nearest mode.
           *   + If expx <= (__gmpfr_emin - 1) / n, then n * expx cannot
           *     overflow since 0 < expx <= (__gmpfr_emin - 1) / n and
           *           0 > n * expx >= n * ((__gmpfr_emin - 1) / n)
           *                        >= __gmpfr_emin - 1.
           * - If n < -1 and expx < 0:
           *   + If expx < (__gmpfr_emax - 1) / n, then
           *           expx <= (__gmpfr_emax - 1) / n - 1
           *                < (__gmpfr_emax - 1) // n,
           *     and
           *           n * expx > __gmpfr_emax - 1,
           *     i.e.
           *           n * expx >= __gmpfr_emax.
           *     This corresponds to an overflow (2^(n * expx) has an
           *     exponent > __gmpfr_emax).
           *   + If expx >= (__gmpfr_emax - 1) / n, then n * expx cannot
           *     overflow since 0 > expx >= (__gmpfr_emax - 1) / n and
           *           0 < n * expx <= n * ((__gmpfr_emax - 1) / n)
           *                        <= __gmpfr_emax - 1.
           * Note: one could use expx bounds based on MPFR_EXP_MIN and
           * MPFR_EXP_MAX instead of __gmpfr_emin and __gmpfr_emax. The
           * current bounds do not lead to noticeably slower code and
           * allow us to avoid a bug in Sun's compiler for Solaris/x86
           * (when optimizations are enabled); known affected versions:
           *   cc: Sun C 5.8 2005/10/13
           *   cc: Sun C 5.8 Patch 121016-02 2006/03/31
           *   cc: Sun C 5.8 Patch 121016-04 2006/10/18
           */
          expy =
            n != -1 && expx > 0 && expx > (__gmpfr_emin - 1) / n ?
            MPFR_EMIN_MIN - 2 /* Underflow */ :
            n != -1 && expx < 0 && expx < (__gmpfr_emax - 1) / n ?
            MPFR_EMAX_MAX /* Overflow */ : n * expx;
          return mpfr_set_si_2exp (y, n % 2 ? MPFR_INT_SIGN (x) : 1,
                                   expy, rnd);
        }

      /* General case */
      {
        /* Declaration of the intermediary variable */
        mpfr_t t;
        /* Declaration of the size variable */
        mpfr_prec_t Ny;                              /* target precision */
        mpfr_prec_t Nt;                              /* working precision */
        mpfr_rnd_t rnd1;
        int size_n;
        int inexact;
        unsigned long abs_n;
        MPFR_SAVE_EXPO_DECL (expo);
        MPFR_ZIV_DECL (loop);

        abs_n = - (unsigned long) n;
        size_n = mpfr_nbits_ulong (abs_n);

        /* initial working precision */
        Ny = MPFR_PREC (y);
        Nt = Ny + size_n + 3 + MPFR_INT_CEIL_LOG2 (Ny);

        MPFR_SAVE_EXPO_MARK (expo);

        /* initialize of intermediary   variable */
        mpfr_init2 (t, Nt);

        /* We will compute rnd(rnd1(1/x) ^ |n|), where rnd1 is the rounding
           toward sign(x), to avoid spurious overflow or underflow, as in
           mpfr_pow_z. */
        rnd1 = MPFR_EXP (x) < 1 ? MPFR_RNDZ :
          (MPFR_IS_POS (x) ? MPFR_RNDU : MPFR_RNDD);

        MPFR_ZIV_INIT (loop, Nt);
        for (;;)
          {
            MPFR_BLOCK_DECL (flags);

            /* compute (1/x)^|n| */
            MPFR_BLOCK (flags, mpfr_ui_div (t, 1, x, rnd1));
            MPFR_ASSERTD (! MPFR_UNDERFLOW (flags));
            /* t = (1/x)*(1+theta) where |theta| <= 2^(-Nt) */
            if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
              goto overflow;
            MPFR_BLOCK (flags, mpfr_pow_ui (t, t, abs_n, rnd));
            /* t = (1/x)^|n|*(1+theta')^(|n|+1) where |theta'| <= 2^(-Nt).
               If (|n|+1)*2^(-Nt) <= 1/2, which is satisfied as soon as
               Nt >= bits(n)+2, then we can use Lemma \ref{lemma_graillat}
               from algorithms.tex, which yields x^n*(1+theta) with
               |theta| <= 2(|n|+1)*2^(-Nt), thus the error is bounded by
               2(|n|+1) ulps <= 2^(bits(n)+2) ulps. */
            if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
              {
              overflow:
                MPFR_ZIV_FREE (loop);
                mpfr_clear (t);
                MPFR_SAVE_EXPO_FREE (expo);
                MPFR_LOG_MSG (("overflow\n", 0));
                return mpfr_overflow (y, rnd, abs_n & 1 ?
                                      MPFR_SIGN (x) : MPFR_SIGN_POS);
              }
            if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
              {
                MPFR_ZIV_FREE (loop);
                mpfr_clear (t);
                MPFR_LOG_MSG (("underflow\n", 0));
                if (rnd == MPFR_RNDN)
                  {
                    mpfr_t y2, nn;

                    /* We cannot decide now whether the result should be
                       rounded toward zero or away from zero. So, like
                       in mpfr_pow_pos_z, let's use the general case of
                       mpfr_pow in precision 2. */
                    MPFR_ASSERTD (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
                                                    MPFR_EXP (x) - 1) != 0);
                    mpfr_init2 (y2, 2);
                    mpfr_init2 (nn, sizeof (long) * CHAR_BIT);
                    inexact = mpfr_set_si (nn, n, MPFR_RNDN);
                    MPFR_ASSERTN (inexact == 0);
                    inexact = mpfr_pow_general (y2, x, nn, rnd, 1,
                                                (mpfr_save_expo_t *) NULL);
                    mpfr_clear (nn);
                    mpfr_set (y, y2, MPFR_RNDN);
                    mpfr_clear (y2);
                    MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
                    goto end;
                  }
                else
                  {
                    MPFR_SAVE_EXPO_FREE (expo);
                    return mpfr_underflow (y, rnd, abs_n & 1 ?
                                           MPFR_SIGN (x) : MPFR_SIGN_POS);
                  }
              }
            /* error estimate -- see pow function in algorithms.ps */
            if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - size_n - 2, Ny, rnd)))
              break;

            /* actualization of the precision */
            MPFR_ZIV_NEXT (loop, Nt);
            mpfr_set_prec (t, Nt);
          }
        MPFR_ZIV_FREE (loop);

        inexact = mpfr_set (y, t, rnd);
        mpfr_clear (t);

      end:
        MPFR_SAVE_EXPO_FREE (expo);
        return mpfr_check_range (y, inexact, rnd);
      }
    }
}
