src/compound.c - master - githubmirror/mpfr

src/compound.c
702 строки26 KB
Vincent Lefevre
Updated the "Contributed" lines.
02 фев 2025, 02:39
02 фев 2025, 02:39a774f60
/* mpfr_compound, mpfr_compound_si --- compound(x,y) = (1+x)^y
   The case of integer y, that is, compound(x,n) = (1+x)^n with n in Z,
   is specified by the IEEE 754 standard. The case of non-integer y is
   a trivial extension based on the rules of mpfr_pow for some special
   values.

Copyright 2021-2025 Free Software Foundation, Inc.
Contributed by the Pascaline 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/>. */

#define MPFR_NEED_LONGLONG_H /* needed for MPFR_INT_CEIL_LOG2 */
#include "mpfr-impl.h"

/* assuming |(1+x)^n - 1| < 1/4*ulp(1), return correct rounding,
   where s is the sign of n*log2(1+x) */
static int
mpfr_compound_near_one (mpfr_ptr y, int s, mpfr_rnd_t rnd_mode)
{
  mpfr_set_ui (y, 1, rnd_mode); /* exact */
  if (rnd_mode == MPFR_RNDN || rnd_mode == MPFR_RNDF
      || (s > 0 && (rnd_mode == MPFR_RNDZ || rnd_mode == MPFR_RNDD))
      || (s < 0 && (rnd_mode == MPFR_RNDA || rnd_mode == MPFR_RNDU)))
    {
      /* round toward 1 */
      return -s;
    }
  else if (s > 0) /* necessarily RNDA or RNDU */
    {
      /* round toward +Inf */
      mpfr_nextabove (y);
      return +1;
    }
  else /* necessarily s < 0 and RNDZ or RNDD */
    {
      /* round toward 0 */
      mpfr_nextbelow (y);
      return -1;
    }
}

/* put in y the correctly rounded value of (1+x)^n */
int
mpfr_compound_si (mpfr_ptr y, mpfr_srcptr x, long n, mpfr_rnd_t rnd_mode)
{
  int inexact, compared, k, nloop;
  mpfr_t t, u;
  mpfr_prec_t py, prec, extra;
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("x[%Pd]=%.*Rg n=%ld rnd=%d",
      mpfr_get_prec(x), mpfr_log_prec, x, n, rnd_mode),
     ("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  /* Special cases */
  if (MPFR_IS_SINGULAR (x))
    {
      if (MPFR_IS_INF (x) && MPFR_IS_NEG (x))
        {
          /* compound(-Inf,n) is NaN */
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (n == 0 || MPFR_IS_ZERO (x))
        {
          /* compound(x,0) = 1 for x >= -1 or NaN (the only special value
             of x that is not concerned is -Inf, already handled);
             compound(0,n) = 1 */
          return mpfr_set_ui (y, 1, rnd_mode);
        }
      else if (MPFR_IS_NAN (x))
        {
          /* compound(NaN,n) is NaN, except for n = 0, already handled. */
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (x)) /* x = +Inf */
        {
          MPFR_ASSERTD (MPFR_IS_POS (x));
          if (n < 0) /* (1+Inf)^n = +0 for n < 0 */
            MPFR_SET_ZERO (y);
          else /* n > 0: (1+Inf)^n = +Inf */
            MPFR_SET_INF (y);
          MPFR_SET_POS (y);
          MPFR_RET (0); /* exact 0 or infinity */
        }
    }

  /* (1+x)^n = NaN for x < -1 */
  compared = mpfr_cmp_si (x, -1);
  if (compared < 0)
    {
      MPFR_SET_NAN (y);
      MPFR_RET_NAN;
    }

  /* compound(x,0) gives 1 for x >= -1 */
  if (n == 0)
    return mpfr_set_ui (y, 1, rnd_mode);

  if (compared == 0)
    {
      if (n < 0)
        {
          /* compound(-1,n) = +Inf with divide-by-zero exception */
          MPFR_SET_INF (y);
          MPFR_SET_DIVBY0 ();
        }
      else
        {
          /* compound(-1,n) = +0 */
          MPFR_SET_ZERO (y);
        }
      MPFR_SET_POS (y);
      MPFR_RET (0);
    }

  if (n == 1)
    return mpfr_add_ui (y, x, 1, rnd_mode);

  MPFR_SAVE_EXPO_MARK (expo);

  py = MPFR_GET_PREC (y);
  prec = py + MPFR_INT_CEIL_LOG2 (py) + 6;

  mpfr_init2 (t, prec);
  mpfr_init2 (u, prec);

  k = MPFR_INT_CEIL_LOG2(SAFE_ABS (unsigned long, n));  /* thus |n| <= 2^k */

  /* We compute u=log2p1(x) with prec+extra bits, since we lose some bits
     in 2^u. */
  extra = 0;

  /* Now, x > -1 and it cannot be zero. And n < 0 or n > 1. */
  MPFR_ZIV_INIT (loop, prec);
  for (nloop = 0; ; nloop++)
    {
      unsigned int inex;
      mpfr_exp_t e, e2, ex;
      mpfr_prec_t precu = MPFR_ADD_PREC (prec, extra);
      mpfr_rnd_t rnd2;

      /* Exact cases like compound(0.5,2) = 9/4 must be detected, since
         except for 1+x power of 2, the log2p1 below will be inexact,
         so that in the Ziv test, inexact != 0 and MPFR_CAN_ROUND will
         fail (even for RNDN, as the ternary value cannot be determined),
         yielding an infinite loop.
         For an exact case in precision prec(y), 1+x will necessarily
         be exact in precision prec(y), thus also in prec(t), since
         prec(t) >= prec(y); and we can use mpfr_pow_si under this
         condition.
         Let's do this detection first as it is fast (just an addition),
         and in case of success, we get the result immediately. Moreover,
         this will succeed on many inexact cases (as soon as 1+x fits in
         the working precision).
         Note: we could also do the detection without doing the addition,
         but the gain (in case of failure) should not be noticeable due to
         the much slower mpfr_log2p1 just below. */
      if (mpfr_add_ui (t, x, 1, MPFR_RNDZ) == 0)
        {
          MPFR_CLEAR_FLAGS ();
          inexact = mpfr_pow_si (y, t, n, rnd_mode);
          MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
          goto end;
        }

      /* We compute (1+x)^n as 2^(n*log2p1(x)),
         and we round toward 1, thus we round n*log2p1(x) toward 0,
         which implies we round log2p1(x) toward 0.
         Around 0, log2p1(x)/x ~= 1/log(2) ~= 1.44, so that log2p1(x)
         will not underflow, and overflow is not possible either. */
      inex = mpfr_log2p1 (u, x, MPFR_RNDZ) != 0;  /* denoted u1 */
      e = MPFR_GET_EXP (u);
      /* |u1 - log2(1+x)| <= ulp(u1) = 2^(e-precu) */
      /* The following multiplication cannot underflow. And it cannot
         overflow either since u comes from a log and n cannot be a
         huge integer in practice. */
      inex |= mpfr_mul_si (u, u, n, MPFR_RNDZ) != 0;  /* denoted u2 */
      e2 = MPFR_GET_EXP (u);
      /* |u2 - n*log2(1+x)| <= 2^(e2-precu) + |n|*2^(e-precu)
                            <= 2^(e2-precu) + 2^(e+k-precu)
         where |n| <= 2^k, and e2 is the exponent of u2.
         Since MPFR_RNDZ was used for u2, |u1| < 2^e and |n| <= 2^k,
         we have |u2| < 2^(e+k), thus e2 <= e+k, and
         |u2 - n*log2(1+x)| <= 2^(e+k+1-precu). */
      MPFR_ASSERTD (e2 <= e + k);
      e += k + 1;
      /* With this new e, |u2 - n*log2(1+x)| <= 2^(e-precu). */

      /* Detect overflow: since we rounded n*log2p1(x) toward 0,
         if n*log2p1(x) >= __gmpfr_emax, we are sure there is overflow. */
      if (mpfr_cmp_si (u, __gmpfr_emax) >= 0)
        {
          MPFR_ZIV_FREE (loop);
          mpfr_clear (t);
          mpfr_clear (u);
          MPFR_SAVE_EXPO_FREE (expo);
          return mpfr_overflow (y, rnd_mode, 1);
        }

      /* Detect underflow: similarly, since we rounded n*log2p1(x) toward 0,
         if n*log2p1(x) < __gmpfr_emin-1, we are sure there is underflow. */
      if (mpfr_cmp_si (u, __gmpfr_emin - 1) < 0)
        {
          MPFR_ZIV_FREE (loop);
          mpfr_clear (t);
          mpfr_clear (u);
          MPFR_SAVE_EXPO_FREE (expo);
          return mpfr_underflow (y,
                            rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode, 1);
        }

      /* Detect cases where result is 1 or 1+ulp(1) or 1-1/2*ulp(1):
         |2^u - 1| = |exp(u*log(2)) - 1| <= |u|*log(2) < |u| */
      if (nloop == 0 && MPFR_GET_EXP(u) < - py)
        {
          /* Since ulp(1) = 2^(1-py), we have |u| < 1/4*ulp(1). */
          /* mpfr_compound_near_one must be called in the extended
             exponent range, so that 1 is representable. */
          inexact = mpfr_compound_near_one (y, MPFR_SIGN (u), rnd_mode);
          goto end;
        }

      /* round 2^u toward 1 */
      rnd2 = MPFR_IS_POS (u) ? MPFR_RNDD : MPFR_RNDU;
      inex |= mpfr_exp2 (t, u, rnd2) != 0;
      /* We had |u2 - n*log2(1+x)| < 2^(e-precu)
         thus u2 = n*log2(1+x) + delta with |delta| < 2^(e-precu)
         then 2^u2 = (1+x)^n * 2^delta with |delta| < 2^(e-precu).
         When e-precu <= -1, we have |delta| < 0.5,
         |2^delta - 1| <= |delta|, thus
         |t - (1+x)^n| <= ulp(t) + |t|*2^(e-precu)
                       < 2^(EXP(t)-prec) + 2^(EXP(t)+e-precu).
         If e-precu >= 0, then the rounding error on u2 is too large,
         and we have to loop again (except in some particular cases
         below, for which the result is computed in another way). */
      if (e < precu)
        {
          mpfr_exp_t e3 = (precu - prec >= e) ? 1 : e + 1 - (precu - prec);
          /* e3-prec = 1 + max(-prec,e-precu), thus
             |t - (1+x)^n| < 2^(EXP(t)+e3-prec) = 2^e3 ulp(t) */
          if (MPFR_LIKELY (!inex ||
                           MPFR_CAN_ROUND (t, prec - e3, py, rnd_mode)))
            break;

          /* If t fits in the target precision (or with 1 more bit), then
             we can round, assuming the working precision is large enough,
             but the above MPFR_CAN_ROUND() will fail because we cannot
             determine the ternary value.
             However, since we rounded t toward 1, we can determine it.
             Since the error in the approximation t is at most 2^e3 ulp(t),
             this error should be less than 1/2 ulp(y), thus we should have
             prec - py >= e3 + 1. */
          if (mpfr_min_prec (t) <= py + 1 && prec - py >= e3 + 1)
            {
              /* we add/subtract one ulp to get the correct rounding */
              if (rnd2 == MPFR_RNDD) /* t was rounded downwards */
                mpfr_nextabove (t);
              else
                mpfr_nextbelow (t);
              break;
            }
        }

      /* Detect particular cases where Ziv's strategy may take too much
         memory and be too long, i.e. when x^n fits in the target precision
         (+ 1 additional bit for rounding to nearest) and the exact result
         (1+x)^n is very close to x^n.
         Necessarily, x is a large even integer and n > 0 (thus n > 1).
         Since this does not depend on the working precision, we only
         check this at the first iteration (nloop == 0).
         Hence the first "if" below and the kx < ex test of the second "if"
         (x is an even integer iff its least bit 1 has exponent >= 1).
         The second test of the second "if" corresponds to another simple
         condition that implies that x^n fits in the target precision.
         Here are the details:
         Let k be the minimum length of the significand of x, and x' the odd
         (integer) significand of x. This means  that 2^(k-1) <= x' < 2^k.
         Thus 2^(n*(k-1)) <= (x')^n < 2^(k*n), and x^n has between n*(k-1)+1
         and k*n bits. So x^n can fit into p bits only if p >= n*(k-1)+1,
         i.e. n*(k-1) <= p-1.
         Note that x >= 2^k, so that x^n >= 2^(k*n). Since raw overflow
         has already been detected, k*n cannot overflow if computed with
         the mpfr_exp_t type. Hence the second test of the second "if",
         which cannot overflow. */
      MPFR_ASSERTD (n < 0 || n > 1);
      if (nloop == 0 && n > 1 && (ex = MPFR_GET_EXP (x)) >= 17)
        {
          mpfr_prec_t kx = mpfr_min_prec (x);
          mpfr_prec_t p = py + (rnd_mode == MPFR_RNDN);

          MPFR_LOG_MSG (("Check if x^n fits... n=%ld kx=%Pd p=%Pd\n",
                         n, kx, p));
          if (kx < ex && n * (mpfr_exp_t) (kx - 1) <= p - 1)
            {
              mpfr_t v;

              /* Check whether x^n really fits into p bits. */
              mpfr_init2 (v, p);
              inexact = mpfr_pow_ui (v, x, n, MPFR_RNDZ);
              if (inexact == 0)
                {
                  MPFR_LOG_MSG (("x^n fits into p bits\n", 0));
                  /* (x+1)^n = x^n * (1 + 1/x)^n
                     For directed rounding, we can round when (1 + 1/x)^n
                     < 1 + 2^-p, and then the result is x^n,
                     except for rounding up. Indeed, if (1 + 1/x)^n < 1 + 2^-p,
                     1 <= (x+1)^n < x^n * (1 + 2^-p) = x^n + x^n/2^p
                     < x^n + ulp(x^n).
                     For rounding to nearest, we can round when (1 + 1/x)^n
                     < 1 + 2^-p, and then the result is x^n when x^n fits
                     into p-1 bits, and nextabove(x^n) otherwise. */
                  mpfr_ui_div (t, 1, x, MPFR_RNDU);
                  mpfr_add_ui (t, t, 1, MPFR_RNDU);
                  mpfr_pow_ui (t, t, n, MPFR_RNDU);
                  mpfr_sub_ui (t, t, 1, MPFR_RNDU);
                  /* t cannot be zero */
                  if (MPFR_GET_EXP(t) < - py)
                    {
                      mpfr_set (y, v, MPFR_RNDZ);
                      if ((rnd_mode == MPFR_RNDN && mpfr_min_prec (v) == p)
                          || rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA)
                        {
                          /* round up */
                          mpfr_nextabove (y);
                          inexact = 1;
                        }
                      else
                        inexact = -1;
                      mpfr_clear (v);
                      goto end;
                    }
                }
              mpfr_clear (v);
            }
        }

      MPFR_ZIV_NEXT (loop, prec);
      mpfr_set_prec (t, prec);
      MPFR_ASSERTN (e2 <= MPFR_PREC_MAX);
      extra = e2 > 0 ? e2 : 0;
      mpfr_set_prec (u, MPFR_ADD_PREC (prec, extra));
    }

  inexact = mpfr_set (y, t, rnd_mode);

 end:
  MPFR_ZIV_FREE (loop);
  mpfr_clear (t);
  mpfr_clear (u);

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

int
mpfr_compound (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y, mpfr_rnd_t rnd_mode)
{
  int inexact, compared, nloop;
  mpfr_exp_t k;
  mpfr_t t, u;
  mpfr_prec_t pz, prec, extra;
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);

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

  /* Special cases (copied from mpfr_compound_si) */
  if (MPFR_IS_SINGULAR (x))
    {
      if (MPFR_IS_INF (x) && MPFR_IS_NEG (x))
        {
          /* compound(-Inf,y) is NaN */
          MPFR_SET_NAN (z);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_ZERO (x) || MPFR_IS_ZERO (y))
        {
          /* compound(0,y) = 1 even when y is NaN;
             compound(x,0) = 1 for x >= -1 or NaN (the only special value
             of x that is not concerned is -Inf, already handled). */
          return mpfr_set_ui (z, 1, rnd_mode);
        }
      else if (MPFR_IS_NAN (x) || MPFR_IS_NAN (y))
        {
          /* compound(NaN,y) is NaN, except for y = 0, already handled.
             compound(x,NaN) is NaN, except for x = 0, already handled. */
          MPFR_SET_NAN (z);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (x)) /* x = +Inf */
        {
          MPFR_ASSERTD (MPFR_IS_POS (x) &&
                        !MPFR_IS_NAN (y) &&
                        MPFR_NOTZERO (y));
          if (MPFR_IS_NEG (y)) /* (1+Inf)^y = +0 for y < 0 */
            MPFR_SET_ZERO (z);
          else /* y > 0: (1+Inf)^y = +Inf */
            MPFR_SET_INF (z);
          MPFR_SET_POS (z);
          MPFR_RET (0); /* exact 0 or infinity */
        }
    }

  if (MPFR_IS_NAN (y))
    {
      /* compound(x,NaN) is NaN, except for x = 0, already handled. */
      MPFR_SET_NAN (z);
      MPFR_RET_NAN;
    }

  compared = mpfr_cmp_si (x, -1);
  /* (1+x)^y = NaN for x < -1 */
  if (compared < 0)
    {
      MPFR_SET_NAN (z);
      MPFR_RET_NAN;
    }

  /* compound(x,0) gives 1 for x >= -1 */
  if (MPFR_IS_ZERO (y))
    return mpfr_set_ui (z, 1, rnd_mode);

  if (compared == 0)
    {
      if (MPFR_IS_NEG (y))
        {
          /* compound(-1,y) = +Inf, and if y is finite,
             raise the divide-by-zero flag.
             Warning! Testing y must be done before changing z,
             in case y and z are the same object. */
          if (! MPFR_IS_INF (y))
            MPFR_SET_DIVBY0 ();
          MPFR_SET_INF (z);
        }
      else
        {
          /* compound(-1,y) = +0 */
          MPFR_SET_ZERO (z);
        }
      MPFR_SET_POS (z);
      MPFR_RET (0);
    }

  /* now x > -1 and it cannot be zero */
  if (MPFR_IS_INF (y))
    {
      /* (1+x)^+Inf = +0 for -1 < x < 0
       * (1+x)^-Inf = +0 for x > 0
       * (1+x)^+Inf = +Inf for x > 0
       * (1+x)^-Inf = +Inf for -1 < x < 0
       */
      if (MPFR_IS_POS (x) ^ MPFR_IS_POS (y))
        MPFR_SET_ZERO (z);
      else
        MPFR_SET_INF (z);
      MPFR_SET_POS (z);
      MPFR_RET (0);
    }

  MPFR_ASSERTD (MPFR_IS_PURE_FP (y));

  MPFR_SAVE_EXPO_MARK (expo);
  pz = MPFR_GET_PREC (z);
  prec = pz + MPFR_INT_CEIL_LOG2 (pz) + 6;

  mpfr_init2 (t, prec);
  mpfr_init2 (u, prec);

  k = MPFR_GET_EXP (y);
  if (k <= 0)
    k = 0;

  /* We compute u=log2p1(x) with prec+extra bits, since we lose some bits
     in 2^u. */
  extra = 0;

  MPFR_ZIV_INIT (loop, prec);
  for (nloop = 0; ; nloop++)
    {
      unsigned int inex;
      mpfr_exp_t e, e2, ex;
      mpfr_prec_t precu = MPFR_ADD_PREC (prec, extra);
      mpfr_prec_t new_extra;
      mpfr_rnd_t rnd2;

      /* Exact cases like compound(0.5,2) = 9/4 must be detected, since
         except for 1+x power of 2, the log2p1 below will be inexact,
         so that in the Ziv test, inexact != 0 and MPFR_CAN_ROUND will
         fail (even for RNDN, as the ternary value cannot be determined),
         yielding an infinite loop.
         For an exact case in precision prec(z), 1+x will necessarily
         be exact in precision prec(z), thus also in prec(t), since
         prec(t) >= prec(z); and we can use mpfr_pow_si under this
         condition.
         Let's do this detection first as it is fast (just an addition),
         and in case of success, we get the result immediately. Moreover,
         this will succeed on many inexact cases (as soon as 1+x fits in
         the working precision).
         Note: we could also do the detection without doing the addition,
         but the gain (in case of failure) should not be noticeable due to
         the much slower mpfr_log2p1 just below. */
      if (mpfr_add_ui (t, x, 1, MPFR_RNDZ) == 0)
        {
          MPFR_CLEAR_FLAGS ();
          inexact = mpfr_pow (z, t, y, rnd_mode);
          MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
          goto end2;
        }

      /* We compute (1+x)^y as 2^(y*log2p1(x)),
         and we round toward 1, thus we round y*log2p1(x) toward 0. */
      inex = mpfr_log2p1 (u, x, MPFR_RNDZ) != 0;
      e = MPFR_GET_EXP (u);
      /* |u - log2(1+x)| <= ulp(u) = 2^(e-precu) */
      inex |= mpfr_mul (u, u, y, MPFR_RNDZ) != 0;
      /* If u=0, it means |u| < 2^(__gmpfr_emin-1) modulo rounding errors,
         thus surely the "near_one" condition is satisfied. */
      if (MPFR_IS_ZERO (u))
        goto near_one;
      e2 = MPFR_GET_EXP (u);
      /* |u - y*log2(1+x)| <= 2^(e2-precu) + |y|*2^(e-precu)
                           <= 2^(e2-precu) + 2^(e+k-precu).
         Since |u_old| < 2^e and |y| < 2^k, we have |u| < 2^(e+k)
         thus e2 <= e + k and 2^(e2-precu) + 2^(e+k-precu) <= 2^(e+k+1-precu).
      */
      MPFR_ASSERTD (e2 <= e + k);
      e += k + 1;
      MPFR_ASSERTN (e2 <= MPFR_PREC_MAX);
      new_extra = e2 > 0 ? e2 : 0;
      /* |u - y*log2(1+x)| <= 2^(e-precu) */
      /* detect overflow: since we rounded y*log2p1(x) toward 0,
         if y*log2p1(x) >= __gmpfr_emax, we are sure there is overflow. */
      if (mpfr_cmp_si (u, __gmpfr_emax) >= 0)
        {
          MPFR_ZIV_FREE (loop);
          mpfr_clear (t);
          mpfr_clear (u);
          MPFR_SAVE_EXPO_FREE (expo);
          return mpfr_overflow (z, rnd_mode, 1);
        }
      /* detect underflow: similarly, since we rounded y*log2p1(x) toward 0,
         if y*log2p1(x) < __gmpfr_emin-1, we are sure there is underflow. */
      if (mpfr_cmp_si (u, __gmpfr_emin - 1) < 0)
        {
          MPFR_ZIV_FREE (loop);
          mpfr_clear (t);
          mpfr_clear (u);
          MPFR_SAVE_EXPO_FREE (expo);
          return mpfr_underflow (z,
                            rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode, 1);
        }
      /* Detect cases where result is 1 or 1+ulp(1) or 1-1/2*ulp(1):
         |2^u - 1| = |exp(u*log(2)) - 1| <= |u|*log(2) < |u| */
      if (nloop == 0 && e2 < - pz) /* e2 = EXP (u) */
        {
        near_one:
          /* since ulp(1) = 2^(1-pz), we have |u| < 1/4*ulp(1) */
          /* mpfr_compound_near_one must be called in the extended
             exponent range, so that 1 is representable. */
          inexact = mpfr_compound_near_one (z, MPFR_SIGN (u), rnd_mode);
          goto end2;
        }
      /* round 2^u toward 1 */
      rnd2 = MPFR_IS_POS (u) ? MPFR_RNDD : MPFR_RNDU;
      inex |= mpfr_exp2 (t, u, rnd2) != 0;
      /* we had |u - y*log2(1+x)| < 2^(e-precu)
         thus u = y*log2(1+x) + delta with |delta| < 2^(e-precu)
         then 2^u = (1+x)^y * 2^delta with |delta| < 2^(e-precu).
         When e-precu <= -1, |delta| < 0.5, |2^delta - 1| <= |delta| thus
         |t - (1+x)^y| <= ulp(t) + |t|*2^(e-precu)
                       < 2^(EXP(t)-prec) + 2^(EXP(t)+e-precu) */
      if (e < precu)
        {
          e = (precu - prec >= e) ? 1 : e + 1 - (precu - prec);
          /* now |t - (1+x)^y| < 2^(EXP(t)+e-prec) */

          if (MPFR_LIKELY (!inex || MPFR_CAN_ROUND (t, prec - e, pz, rnd_mode)))
            break;

          /* If t fits in the target precision (or with 1 more bit), then we
             can round, assuming the working precision is large enough, but
             the above MPFR_CAN_ROUND() will fail because we cannot determine
             the ternary value. However, since we rounded t toward 1, we can
             determine it. Since the error in the approximation t is at most
             2^e ulp(t), this error should be less than 1/2 ulp(y), thus we
             should have prec - pz >= e + 1. */
          if (mpfr_min_prec (t) <= pz + 1 && prec - pz >= e + 1)
            {
              /* we add/subtract one ulp to get the correct rounding */
              if (rnd2 == MPFR_RNDD) /* t was rounded downwards */
                mpfr_nextabove (t);
              else
                mpfr_nextbelow (t);
              break;
            }
        }

      /* Detect particular cases where Ziv's strategy may take too much
         memory and be too long, i.e. when x^y fits in the target precision
         (+ 1 additional bit for rounding to nearest) and the exact result
         (1+x)^y is very close to x^y.
         Necessarily, x is a large even integer and y > 0 (thus y > 1).
         Since this does not depend on the working precision, we only
         check this at the first iteration (nloop == 0).
         Hence the first "if" below and the kx < ex test of the second "if"
         (x is an even integer iff its least bit 1 has exponent >= 1).
         The second test of the second "if" corresponds to another simple
         condition that implies that x^y fits in the target precision.
         Here are the details:
         Let k be the minimum length of the significand of x, and x' the odd
         (integer) significand of x. This means  that 2^(k-1) <= x' < 2^k.
         Thus 2^(y*(k-1)) <= (x')^y < 2^(k*y), and x^y has between y*(k-1)+1
         and k*y bits. So x^y can fit into p bits only if p >= y*(k-1)+1,
         i.e. y*(k-1) <= p-1.
         Note that x >= 2^k, so that x^y >= 2^(k*y). Since raw overflow
         has already been detected, k*y cannot overflow if computed with
         the mpfr_exp_t type. Hence the second test of the second "if",
         which cannot overflow. */
      if (nloop == 0 && mpfr_cmp_ui (y, 1) > 1 &&
          (ex = MPFR_GET_EXP (x)) >= 17)
        {
          mpfr_prec_t p = pz + (rnd_mode == MPFR_RNDN);
          mpfr_t v;

          MPFR_LOG_MSG (("Check if x^y fits... p=%Pd\n", p));
          /* Check whether x^y really fits into p bits. */
          mpfr_init2 (v, p);
          inexact = mpfr_pow (v, x, y, MPFR_RNDZ);
          if (inexact == 0)
            {
              MPFR_LOG_MSG (("x^y fits into p bits\n", 0));
              /* (x+1)^y = x^y * (1 + 1/x)^y
                 For directed rounding, we can round when
                 (1 + 1/x)^y < 1 + 2^-p, and then the result is x^y,
                 except for rounding up. Indeed, if (1 + 1/x)^y < 1 + 2^-p,
                 1 <= (x+1)^y < x^y * (1 + 2^-p) = x^y + x^y/2^p
                   < x^y + ulp(x^y).
                 For rounding to nearest, we can round when
                 (1 + 1/x)^y < 1 + 2^-p, and then the result is x^y when
                 x^y fits into p-1 bits, and nextabove(x^y) otherwise. */
              mpfr_ui_div (t, 1, x, MPFR_RNDU);
              mpfr_add_ui (t, t, 1, MPFR_RNDU);
              mpfr_pow (t, t, y, MPFR_RNDU);
              mpfr_sub_ui (t, t, 1, MPFR_RNDU);
              /* t cannot be zero */
              if (MPFR_GET_EXP(t) < - pz)
                {
                  mpfr_set (z, v, MPFR_RNDZ);
                  if ((rnd_mode == MPFR_RNDN && mpfr_min_prec (v) == p)
                      || rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA)
                    {
                      /* round up */
                      mpfr_nextabove (z);
                      inexact = 1;
                    }
                  else
                    inexact = -1;
                  mpfr_clear (v);
                  goto end2;
                }
            }
          mpfr_clear (v);
        }

      MPFR_ZIV_NEXT (loop, prec);
      mpfr_set_prec (t, prec);
      extra = new_extra;
      mpfr_set_prec (u, MPFR_ADD_PREC (prec, extra));
    }

  inexact = mpfr_set (z, t, rnd_mode);

 end2:
  MPFR_ZIV_FREE (loop);
  mpfr_clear (t);
  mpfr_clear (u);

  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (z, inexact, rnd_mode);
}