path: root/pow_z.c

/* mpfr_pow_z -- power function x^z with z a MPZ

Copyright 2005, 2006, 2007, 2008 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao projects, INRIA.

This file is part of the MPFR Library.

The 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 2.1 of the License, or (at your
option) any later version.

The 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 MPFR Library; see the file COPYING.LIB.  If not, 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"

/* y <- x^|z| with z != 0
   if cr=1: ensures correct rounding of y
   if cr=0: does not ensure correct rounding, and uses the precision of y
   as working precision (warning, y and x might be the same variable). */
static int
mpfr_pow_pos_z (mpfr_ptr y, mpfr_srcptr x, mpz_srcptr z, mp_rnd_t rnd, int cr)
{
  mpfr_t res;
  mp_prec_t prec, err;
  int inexact;
  mp_rnd_t rnd1, rnd2;
  mpz_t absz;
  mp_size_t size_z;
  MPFR_ZIV_DECL (loop);
  MPFR_BLOCK_DECL (flags);

  MPFR_ASSERTD (mpz_sgn (z) != 0);

  if (MPFR_UNLIKELY (mpz_cmpabs_ui (z, 1) == 0))
    return mpfr_set (y, x, rnd);

  absz[0] = z[0];
  SIZ (absz) = ABS(SIZ(absz)); /* Hack to get abs(z) */
  MPFR_MPZ_SIZEINBASE2 (size_z, z);

  /* round towards 1 (or -1) to avoid spurious overflow/underflow,
     i.e. if an overflow or underflow occurs, it is a real exception
     and is not just due to the rounding error. */
  rnd1 = (MPFR_EXP(x) >= 1) ? GMP_RNDZ
    : (MPFR_IS_POS(x) ? GMP_RNDU : GMP_RNDD);
  rnd2 = (MPFR_EXP(x) >= 1) ? GMP_RNDD : GMP_RNDU;

  if (cr != 0)
    prec = MPFR_PREC (y) + 3 + size_z + MPFR_INT_CEIL_LOG2 (MPFR_PREC (y));
  else
    prec = MPFR_PREC (y);
  mpfr_init2 (res, prec);

  MPFR_ZIV_INIT (loop, prec);
  for (;;)
    {
      unsigned int inexmul;  /* will be non-zero if res may be inexact */
      mp_size_t i = size_z;

      /* now 2^(i-1) <= z < 2^i */
      /* see below (case z < 0) for the error analysis, which is identical,
         except if z=n, the maximal relative error is here 2(n-1)2^(-prec)
         instead of 2(2n-1)2^(-prec) for z<0. */
      MPFR_ASSERTD (prec > (mpfr_prec_t) i);
      err = prec - 1 - (mpfr_prec_t) i;

      MPFR_BLOCK (flags,
                  inexmul = mpfr_mul (res, x, x, rnd2);
                  MPFR_ASSERTD (i >= 2);
                  if (mpz_tstbit (absz, i - 2))
                    inexmul |= mpfr_mul (res, res, x, rnd1);
                  for (i -= 3; i >= 0 && !MPFR_BLOCK_EXCEP; i--)
                    {
                      inexmul |= mpfr_mul (res, res, res, rnd2);
                      if (mpz_tstbit (absz, i))
                        inexmul |= mpfr_mul (res, res, x, rnd1);
                    });
      if (MPFR_LIKELY (inexmul == 0 || cr == 0
                       || MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags)
                       || MPFR_CAN_ROUND (res, err, MPFR_PREC (y), rnd)))
        break;
      /* Can't decide correct rounding, increase the precision */
      MPFR_ZIV_NEXT (loop, prec);
      mpfr_set_prec (res, prec);
    }
  MPFR_ZIV_FREE (loop);

  /* Check Overflow */
  if (MPFR_OVERFLOW (flags))
    inexact = mpfr_overflow (y, rnd, mpz_odd_p (absz) ?
                             MPFR_SIGN (x) : MPFR_SIGN_POS);
  /* Check Underflow */
  else if (MPFR_UNDERFLOW (flags))
    inexact = mpfr_underflow (y, rnd == GMP_RNDN ? GMP_RNDZ : rnd,
                              mpz_odd_p (absz) ? MPFR_SIGN (x) :
                              MPFR_SIGN_POS);
  else
    inexact = mpfr_set (y, res, rnd);

  mpfr_clear (res);
  return inexact;
}

/* The computation of y = pow(x,z) is done by
 *    y = set_ui(1)      if z = 0
 *    y = pow_ui(x,z)    if z > 0
 *    y = pow_ui(1/x,-z) if z < 0
 *
 * Note: in case z < 0, we could also compute 1/pow_ui(x,-z). However, in
 * case MAX < 1/MIN, where MAX is the largest positive value, i.e.,
 * MAX = nextbelow(+Inf), and MIN is the smallest positive value, i.e.,
 * MIN = nextabove(+0), then x^(-z) might produce an overflow, whereas
 * x^z is representable.
 */

int
mpfr_pow_z (mpfr_ptr y, mpfr_srcptr x, mpz_srcptr z, mp_rnd_t rnd)
{
  int   inexact;
  mpz_t tmp;
  MPFR_SAVE_EXPO_DECL (expo);

  /* x^0 = 1 for any x, even a NaN */
  if (MPFR_UNLIKELY (mpz_sgn (z) == 0))
    return mpfr_set_ui (y, 1, rnd);

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (x))
        {
          /* Inf^n = Inf, (-Inf)^n = Inf for n even, -Inf for n odd */
          /* Inf ^(-n) = 0, sign = + if x>0 or z even */
          if (mpz_sgn (z) > 0)
            MPFR_SET_INF (y);
          else
            MPFR_SET_ZERO (y);
          if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && mpz_odd_p (z)))
            MPFR_SET_NEG (y);
          else
            MPFR_SET_POS (y);
          MPFR_RET (0);
        }
      else /* x is zero */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO(x));
          if (mpz_sgn (z) > 0)
            /* 0^n = +/-0 for any n */
            MPFR_SET_ZERO (y);
          else
            /* 0^(-n) if +/- INF */
            MPFR_SET_INF (y);
          if (MPFR_LIKELY (MPFR_IS_POS (x) || mpz_even_p (z)))
            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
     Do it if n > 0 too (faster). */
  if (MPFR_UNLIKELY (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
                                       MPFR_EXP (x) - 1) == 0))
    {
      mp_exp_t expx = MPFR_EXP (x); /* warning: x and y may be the same
                                       variable */
      mpfr_set_si (y, mpz_odd_p (z) ? MPFR_INT_SIGN(x) : 1, rnd);
      MPFR_ASSERTD (MPFR_IS_FP (y));
      mpz_init (tmp);
      mpz_mul_si (tmp, z, expx-1);
      MPFR_ASSERTD (MPFR_GET_EXP (y) == 1);
      mpz_add_ui (tmp, tmp, 1);
      inexact = 0;
      if (MPFR_UNLIKELY (mpz_cmp_si (tmp, __gmpfr_emin) < 0))
        {
          /* The following test is necessary because in the rounding to the
           * nearest mode, mpfr_underflow always rounds away from 0. In
           * this rounding mode, we need to round to 0 if:
           *   _ |y| < 2^(emin-2), or
           *   _ |y| = 2^(emin-2) and the absolute value of the exact
           *     result is <= 2^(emin-2).
           * NOTE: y is a power of 2 and inexact = 0!
           */
          if (rnd == GMP_RNDN && mpz_cmp_si (tmp, __gmpfr_emin-1) < 0)
            rnd = GMP_RNDZ;
          inexact = mpfr_underflow (y, rnd, MPFR_SIGN (y));
        }
      else if (MPFR_UNLIKELY (mpz_cmp_si (tmp, __gmpfr_emax) > 0))
        inexact = mpfr_overflow (y, rnd, MPFR_SIGN (y));
      else
        MPFR_SET_EXP (y, mpz_get_si (tmp));
      mpz_clear (tmp);
      MPFR_RET (inexact);
    }

  MPFR_SAVE_EXPO_MARK (expo);

  if (mpz_sgn (z) > 0)
    inexact = mpfr_pow_pos_z (y, x, z, rnd, 1);
  else
    {
      /* Declaration of the intermediary variable */
      mpfr_t t;
      mp_prec_t Nt;   /* Precision of the intermediary variable */
      mp_rnd_t rnd1;
      mp_size_t size_z;
      MPFR_ZIV_DECL (loop);

      MPFR_MPZ_SIZEINBASE2 (size_z, z);

      /* initial working precision */
      Nt = MPFR_PREC (y);
      Nt = Nt + size_z + 3 + MPFR_INT_CEIL_LOG2 (Nt);
      /* ensures Nt >= bits(z)+2 */

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

      /* we choose a rounding towards 1, to avoid overflow or underflow */
      rnd1 = (MPFR_EXP(x) >= 1) ? GMP_RNDZ :
        ((MPFR_SIGN(x) > 0) ? GMP_RNDU : GMP_RNDZ);

      MPFR_ZIV_INIT (loop, Nt);
      for (;;)
        {
          /* compute (1/x)^(-z), -z>0 */
          mpfr_ui_div (t, 1, x, rnd1); /* t = (1/x)*(1+theta) where
                                          |theta| <= 2^(-Nt) */
          mpfr_pow_pos_z (t, t, z, rnd1, 0);
          /* Now if z=-n, t = x^z*(1+theta)^(2n-1) where |theta| <= 2^(-Nt),
             with theta maybe different from above. If (2n-1)*2^(-Nt) <= 1/2,
             which is satisfied as soon as Nt >= bits(z)+2, then we can use
             Lemma \ref{lemma_graillat} from algorithms.tex, which yields
             t = x^z*(1+theta) with |theta| <= 2(2n-1)*2^(-Nt), thus the
             error is bounded by 2(2n-1) ulps <= 2^(bits(z)+2) ulps. */
          if (MPFR_UNLIKELY (MPFR_IS_ZERO (t)))
            {
              MPFR_ZIV_FREE (loop);
              mpfr_clear (t);
              MPFR_SAVE_EXPO_FREE (expo);
              return mpfr_underflow (y, rnd == GMP_RNDN ? GMP_RNDZ : rnd,
                                     mpz_odd_p (z) ? MPFR_SIGN (x) :
                                     MPFR_SIGN_POS);
            }
          if (MPFR_UNLIKELY (MPFR_IS_INF (t)))
            {
              MPFR_ZIV_FREE (loop);
              mpfr_clear (t);
              MPFR_SAVE_EXPO_FREE (expo);
              return mpfr_overflow (y, rnd,
                                    mpz_odd_p (z) ? MPFR_SIGN (x) :
                                    MPFR_SIGN_POS);
            }
          if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - size_z - 2, MPFR_PREC (y),
                                           rnd)))
            break;
          /* actualisation 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);
    }

  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inexact, rnd);
}