xref: /src/external/lgpl3/mpfr/dist/tests/texp.c

/* Test file for mpfr_exp.

Copyright 1999, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free Software Foundation, Inc.
Contributed by the Arenaire and Cacao 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. */

#include <stdio.h>
#include <stdlib.h>
#include <limits.h>

#include "mpfr-test.h"

#ifdef CHECK_EXTERNAL
static int
test_exp (mpfr_ptr a, mpfr_srcptr b, mpfr_rnd_t rnd_mode)
{
  int res;
  int ok = rnd_mode == MPFR_RNDN && mpfr_number_p (b) && mpfr_get_prec (a)>=53;
  if (ok)
    {
      mpfr_print_raw (b);
    }
  res = mpfr_exp (a, b, rnd_mode);
  if (ok)
    {
      printf (" ");
      mpfr_print_raw (a);
      printf ("\n");
    }
  return res;
}
#else
#define test_exp mpfr_exp
#endif

/* returns the number of ulp of error */
static void
check3 (const char *op, mpfr_rnd_t rnd, const char *res)
{
  mpfr_t x, y;

  mpfr_inits2 (53, x, y, (mpfr_ptr) 0);
  /* y negative. If we forget to set the sign in mpfr_exp, we'll see it. */
  mpfr_set_si (y, -1, MPFR_RNDN);
  mpfr_set_str1 (x, op);
  test_exp (y, x, rnd);
  if (mpfr_cmp_str1 (y, res) )
    {
      printf ("mpfr_exp failed for x=%s, rnd=%s\n",
              op, mpfr_print_rnd_mode (rnd));
      printf ("expected result is %s, got ", res);
      mpfr_out_str (stdout, 10, 0, y, MPFR_RNDN);
      putchar('\n');
      exit (1);
    }
  mpfr_clears (x, y, (mpfr_ptr) 0);
}

/* expx is the value of exp(X) rounded toward -infinity */
static void
check_worst_case (const char *Xs, const char *expxs)
{
  mpfr_t x, y;

  mpfr_inits2 (53, x, y, (mpfr_ptr) 0);
  mpfr_set_str1(x, Xs);
  test_exp(y, x, MPFR_RNDD);
  if (mpfr_cmp_str1 (y, expxs))
    {
      printf ("exp(x) rounded toward -infinity is wrong\n");
      exit(1);
    }
  mpfr_set_str1(x, Xs);
  test_exp(x, x, MPFR_RNDU);
  mpfr_nexttoinf (y);
  if (mpfr_cmp(x,y))
    {
      printf ("exp(x) rounded toward +infinity is wrong\n");
      exit(1);
    }
  mpfr_clears (x, y, (mpfr_ptr) 0);
}

/* worst cases communicated by Jean-Michel Muller and Vincent Lefevre */
static int
check_worst_cases (void)
{
  mpfr_t x; mpfr_t y;

  mpfr_init(x);
  mpfr_set_prec (x, 53);

  check_worst_case("4.44089209850062517562e-16", "1.00000000000000022204");
  check_worst_case("6.39488462184069720009e-14", "1.00000000000006372680");
  check_worst_case("1.84741111297455401935e-12", "1.00000000000184718907");
  check_worst_case("1.76177628026265550074e-10", "1.00000000017617751702");
  check3("1.76177628026265550074e-10", MPFR_RNDN, "1.00000000017617773906");
  check_worst_case("7.54175277499595900852e-10", "1.00000000075417516676");
  check3("7.54175277499595900852e-10", MPFR_RNDN, "1.00000000075417538881");
  /* bug found by Vincent Lefe`vre on December 8, 1999 */
  check3("-5.42410311287441459172e+02", MPFR_RNDN, "2.7176584868845723e-236");
  /* further cases communicated by Vincent Lefe`vre on January 27, 2000 */
  check3("-1.32920285897904911589e-10", MPFR_RNDN, "0.999999999867079769622");
  check3("-1.44037948245738330735e-10", MPFR_RNDN, "0.9999999998559621072757");
  check3("-1.66795910430705305937e-10", MPFR_RNDZ, "0.9999999998332040895832");
  check3("-1.64310953745426656203e-10", MPFR_RNDN, "0.9999999998356891017792");
  check3("-1.38323574826034659172e-10", MPFR_RNDZ, "0.9999999998616764251835");
  check3("-1.23621668465115401498e-10", MPFR_RNDZ, "0.9999999998763783315425");

  mpfr_set_prec (x, 601);
  mpfr_set_str (x, "0.88b6ba510e10450edc258748bc9dfdd466f21b47ed264cdf24aa8f64af1f3fad9ec2301d43c0743f534b5aa20091ff6d352df458ef1ba519811ef6f5b11853534fd8fa32764a0a6d2d0dd20@0", 16, MPFR_RNDZ);
  mpfr_init2 (y, 601);
  mpfr_exp_2 (y, x, MPFR_RNDD);
  mpfr_exp_3 (x, x, MPFR_RNDD);
  if (mpfr_cmp (x, y))
    {
      printf ("mpfr_exp_2 and mpfr_exp_3 differ for prec=601\n");
      printf ("mpfr_exp_2 gives ");
      mpfr_out_str (stdout, 2, 0, y, MPFR_RNDN);
      printf ("\nmpfr_exp_3 gives ");
      mpfr_out_str (stdout, 2, 0, x, MPFR_RNDN);
      printf ("\n");
      exit (1);
    }

  mpfr_set_prec (x, 13001);
  mpfr_set_prec (y, 13001);
  mpfr_urandomb (x, RANDS);
  mpfr_exp_3 (y, x, MPFR_RNDN);
  mpfr_exp_2 (x, x, MPFR_RNDN);
  if (mpfr_cmp (x, y))
    {
      printf ("mpfr_exp_2 and mpfr_exp_3 differ for prec=13001\n");
      exit (1);
    }

  mpfr_clear (x);
  mpfr_clear (y);
  return 0;
}

static void
compare_exp2_exp3 (mpfr_prec_t p0, mpfr_prec_t p1)
{
  mpfr_t x, y, z;
  mpfr_prec_t prec;
  mpfr_rnd_t rnd;

  mpfr_init (x);
  mpfr_init (y);
  mpfr_init (z);
  for (prec = p0; prec <= p1; prec ++)
    {
      mpfr_set_prec (x, prec);
      mpfr_set_prec (y, prec);
      mpfr_set_prec (z, prec);
      do
        mpfr_urandomb (x, RANDS);
      while (MPFR_IS_ZERO (x));  /* 0 is handled by mpfr_exp only */
      rnd = RND_RAND ();
      mpfr_exp_2 (y, x, rnd);
      mpfr_exp_3 (z, x, rnd);
      if (mpfr_cmp (y,z))
        {
          printf ("mpfr_exp_2 and mpfr_exp_3 disagree for rnd=%s and\nx=",
                  mpfr_print_rnd_mode (rnd));
          mpfr_print_binary (x);
          puts ("");
          printf ("mpfr_exp_2 gives ");
          mpfr_print_binary (y);
          puts ("");
          printf ("mpfr_exp_3 gives ");
          mpfr_print_binary (z);
          puts ("");
          exit (1);
        }
  }

  mpfr_clear (x);
  mpfr_clear (y);
  mpfr_clear (z);
}

static void
check_large (void)
{
  mpfr_t x, z;
  mpfr_prec_t prec;

  /* bug found by Patrick Pe'lissier on 7 Jun 2004 */
  prec = 203780;
  mpfr_init2 (x, prec);
  mpfr_init2 (z, prec);
  mpfr_set_ui (x, 3, MPFR_RNDN);
  mpfr_sqrt (x, x, MPFR_RNDN);
  mpfr_sub_ui (x, x, 1, MPFR_RNDN);
  mpfr_exp_3 (z, x, MPFR_RNDN);
  mpfr_clear (x);
  mpfr_clear (z);
}

#define TEST_FUNCTION test_exp
#define TEST_RANDOM_EMIN -36
#define TEST_RANDOM_EMAX 36
#include "tgeneric.c"

static void
check_special (void)
{
  mpfr_t x, y, z;
  mpfr_exp_t emin, emax;

  emin = mpfr_get_emin ();
  emax = mpfr_get_emax ();

  mpfr_init (x);
  mpfr_init (y);
  mpfr_init (z);

  /* check exp(NaN) = NaN */
  mpfr_set_nan (x);
  test_exp (y, x, MPFR_RNDN);
  if (!mpfr_nan_p (y))
    {
      printf ("Error for exp(NaN)\n");
      exit (1);
    }

  /* check exp(+inf) = +inf */
  mpfr_set_inf (x, 1);
  test_exp (y, x, MPFR_RNDN);
  if (!mpfr_inf_p (y) || mpfr_sgn (y) < 0)
    {
      printf ("Error for exp(+inf)\n");
      exit (1);
    }

  /* check exp(-inf) = +0 */
  mpfr_set_inf (x, -1);
  test_exp (y, x, MPFR_RNDN);
  if (mpfr_cmp_ui (y, 0) || mpfr_sgn (y) < 0)
    {
      printf ("Error for exp(-inf)\n");
      exit (1);
    }

  /* Check overflow. Corner case of mpfr_exp_2 */
  mpfr_set_prec (x, 64);
  mpfr_set_emax (MPFR_EMAX_DEFAULT);
  mpfr_set_emin (MPFR_EMIN_DEFAULT);
  mpfr_set_str (x,
    "0.1011000101110010000101111111010100001100000001110001100111001101E30",
                2, MPFR_RNDN);
  mpfr_exp (x, x, MPFR_RNDD);
  if (mpfr_cmp_str (x,
".1111111111111111111111111111111111111111111111111111111111111111E1073741823",
                    2, MPFR_RNDN) != 0)
    {
      printf ("Wrong overflow detection in mpfr_exp\n");
      mpfr_dump (x);
      exit (1);
    }
  /* Check underflow. Corner case of mpfr_exp_2 */
  mpfr_set_str (x,
"-0.1011000101110010000101111111011111010001110011110111100110101100E30",
                2, MPFR_RNDN);
  mpfr_exp (x, x, MPFR_RNDN);
  if (mpfr_cmp_str (x, "0.1E-1073741823", 2, MPFR_RNDN) != 0)
    {
      printf ("Wrong underflow (1) detection in mpfr_exp\n");
      mpfr_dump (x);
      exit (1);
    }
  mpfr_set_str (x,
"-0.1011001101110010000101111111011111010001110011110111100110111101E30",
                2, MPFR_RNDN);
  mpfr_exp (x, x, MPFR_RNDN);
  if (mpfr_cmp_ui (x, 0) != 0)
    {
      printf ("Wrong underflow (2) detection in mpfr_exp\n");
      mpfr_dump (x);
      exit (1);
    }
  /* Check overflow. Corner case of mpfr_exp_3 */
  if (MPFR_PREC_MAX >= MPFR_EXP_THRESHOLD + 10 && MPFR_PREC_MAX >= 64)
    {
      /* this ensures that for small MPFR_EXP_THRESHOLD, the following
         mpfr_set_str conversion is exact */
      mpfr_set_prec (x, (MPFR_EXP_THRESHOLD + 10 > 64)
                       ? MPFR_EXP_THRESHOLD + 10 : 64);
      mpfr_set_str (x,
       "0.1011000101110010000101111111010100001100000001110001100111001101E30",
                    2, MPFR_RNDN);
      mpfr_clear_overflow ();
      mpfr_exp (x, x, MPFR_RNDD);
      if (!mpfr_overflow_p ())
        {
          printf ("Wrong overflow detection in mpfr_exp_3\n");
          mpfr_dump (x);
          exit (1);
        }
      /* Check underflow. Corner case of mpfr_exp_3 */
      mpfr_set_str (x,
      "-0.1011000101110010000101111111011111010001110011110111100110101100E30",
                    2, MPFR_RNDN);
      mpfr_clear_underflow ();
      mpfr_exp (x, x, MPFR_RNDN);
      if (!mpfr_underflow_p ())
        {
          printf ("Wrong underflow detection in mpfr_exp_3\n");
          mpfr_dump (x);
          exit (1);
        }
      mpfr_set_prec (x, 53);
    }

  /* check overflow */
  set_emax (10);
  mpfr_set_ui (x, 7, MPFR_RNDN);
  test_exp (y, x, MPFR_RNDN);
  if (!mpfr_inf_p (y) || mpfr_sgn (y) < 0)
    {
      printf ("Error for exp(7) for emax=10\n");
      exit (1);
    }
  set_emax (emax);

  /* check underflow */
  set_emin (-10);
  mpfr_set_si (x, -9, MPFR_RNDN);
  test_exp (y, x, MPFR_RNDN);
  if (mpfr_cmp_ui (y, 0) || mpfr_sgn (y) < 0)
    {
      printf ("Error for exp(-9) for emin=-10\n");
      printf ("Expected +0\n");
      printf ("Got      "); mpfr_print_binary (y); puts ("");
      exit (1);
    }
  set_emin (emin);

  /* check case EXP(x) < -precy */
  mpfr_set_prec (y, 2);
  mpfr_set_str_binary (x, "-0.1E-3");
  test_exp (y, x, MPFR_RNDD);
  if (mpfr_cmp_ui_2exp (y, 3, -2))
    {
      printf ("Error for exp(-1/16), prec=2, RNDD\n");
      printf ("expected 0.11, got ");
      mpfr_dump (y);
      exit (1);
    }
  test_exp (y, x, MPFR_RNDZ);
  if (mpfr_cmp_ui_2exp (y, 3, -2))
    {
      printf ("Error for exp(-1/16), prec=2, RNDZ\n");
      printf ("expected 0.11, got ");
      mpfr_dump (y);
      exit (1);
    }
  mpfr_set_str_binary (x, "0.1E-3");
  test_exp (y, x, MPFR_RNDN);
  if (mpfr_cmp_ui (y, 1))
    {
      printf ("Error for exp(1/16), prec=2, RNDN\n");
      exit (1);
    }
  test_exp (y, x, MPFR_RNDU);
  if (mpfr_cmp_ui_2exp (y, 3, -1))
    {
      printf ("Error for exp(1/16), prec=2, RNDU\n");
      exit (1);
    }

  /* bug reported by Franky Backeljauw, 28 Mar 2003 */
  mpfr_set_prec (x, 53);
  mpfr_set_prec (y, 53);
  mpfr_set_str_binary (x, "1.1101011000111101011110000111010010101001101001110111e28");
  test_exp (y, x, MPFR_RNDN);

  mpfr_set_prec (x, 153);
  mpfr_set_prec (z, 153);
  mpfr_set_str_binary (x, "1.1101011000111101011110000111010010101001101001110111e28");
  test_exp (z, x, MPFR_RNDN);
  mpfr_prec_round (z, 53, MPFR_RNDN);

  if (mpfr_cmp (y, z))
    {
      printf ("Error in mpfr_exp for large argument\n");
      exit (1);
    }

  /* corner cases in mpfr_exp_3 */
  mpfr_set_prec (x, 2);
  mpfr_set_ui (x, 1, MPFR_RNDN);
  mpfr_set_prec (y, 2);
  mpfr_exp_3 (y, x, MPFR_RNDN);

  /* Check some little things about overflow detection */
  set_emin (-125);
  set_emax (128);
  mpfr_set_prec (x, 107);
  mpfr_set_prec (y, 107);
  mpfr_set_str_binary (x, "0.11110000000000000000000000000000000000000000000"
                       "0000000000000000000000000000000000000000000000000000"
                       "00000000E4");
  test_exp (y, x, MPFR_RNDN);
  if (mpfr_cmp_str (y, "0.11000111100001100110010101111101011010010101010000"
                    "1101110111100010111001011111111000110111001011001101010"
                    "01E22", 2, MPFR_RNDN))
    {
      printf ("Special overflow error (1)\n");
      mpfr_dump (y);
      exit (1);
    }

  set_emin (emin);
  set_emax (emax);

  /* Check for overflow producing a segfault with HUGE exponent */
  mpfr_set_ui  (x, 3, MPFR_RNDN);
  mpfr_mul_2ui (x, x, 32, MPFR_RNDN);
  test_exp (y, x, MPFR_RNDN); /* Can't test return value: May overflow or not*/

  /* Bug due to wrong approximation of (x)/log2 */
  mpfr_set_prec (x, 163);

  mpfr_set_str (x, "-4.28ac8fceeadcda06bb56359017b1c81b85b392e7", 16,
                MPFR_RNDN);
  mpfr_exp (x, x, MPFR_RNDN);
  if (mpfr_cmp_str (x, "3.fffffffffffffffffffffffffffffffffffffffe8@-2",
                    16, MPFR_RNDN))
    {
      printf ("Error for x= -4.28ac8fceeadcda06bb56359017b1c81b85b392e7");
      printf ("expected  3.fffffffffffffffffffffffffffffffffffffffe8@-2");
      printf ("Got       ");
      mpfr_out_str (stdout, 16, 0, x, MPFR_RNDN); putchar ('\n');
    }

  /* bug found by Guillaume Melquiond, 13 Sep 2005 */
  mpfr_set_prec (x, 53);
  mpfr_set_str_binary (x, "-1E-400");
  mpfr_exp (x, x, MPFR_RNDZ);
  if (mpfr_cmp_ui (x, 1) == 0)
    {
      printf ("Error for exp(-2^(-400))\n");
      exit (1);
    }

  mpfr_clear (x);
  mpfr_clear (y);
  mpfr_clear (z);
}

/* check sign of inexact flag */
static void
check_inexact (void)
{
  mpfr_t x, y;
  int inexact;

  mpfr_init2 (x, 53);
  mpfr_init2 (y, 53);

  mpfr_set_str_binary (x,
        "1.0000000000001001000110100100101000001101101011100101e2");
  inexact = test_exp (y, x, MPFR_RNDN);
  if (inexact <= 0)
    {
      printf ("Wrong inexact flag (Got %d instead of 1)\n", inexact);
      exit (1);
    }

  mpfr_clear (x);
  mpfr_clear (y);
}

static void
check_exp10(void)
{
  mpfr_t x;
  int inexact;

  mpfr_init2 (x, 200);
  mpfr_set_ui(x, 4, MPFR_RNDN);

  inexact = mpfr_exp10 (x, x, MPFR_RNDN);
  if (mpfr_cmp_ui(x, 10*10*10*10))
    {
      printf ("exp10: Wrong returned value\n");
      exit (1);
    }
  if (inexact != 0)
    {
      printf ("exp10: Wrong inexact flag\n");
      exit (1);
    }

  mpfr_clear (x);
}

static void
overflowed_exp0 (void)
{
  mpfr_t x, y;
  int emax, i, inex, rnd, err = 0;
  mpfr_exp_t old_emax;

  old_emax = mpfr_get_emax ();

  mpfr_init2 (x, 8);
  mpfr_init2 (y, 8);

  for (emax = -1; emax <= 0; emax++)
    {
      mpfr_set_ui_2exp (y, 1, emax, MPFR_RNDN);
      mpfr_nextbelow (y);
      set_emax (emax);  /* 1 is not representable. */
      /* and if emax < 0, 1 - eps is not representable either. */
      for (i = -1; i <= 1; i++)
        RND_LOOP (rnd)
        {
          mpfr_set_si_2exp (x, i, -512 * ABS (i), MPFR_RNDN);
          mpfr_clear_flags ();
          inex = mpfr_exp (x, x, (mpfr_rnd_t) rnd);
          if ((i >= 0 || emax < 0 || rnd == MPFR_RNDN || rnd == MPFR_RNDU) &&
              ! mpfr_overflow_p ())
            {
              printf ("Error in overflowed_exp0 (i = %d, rnd = %s):\n"
                      "  The overflow flag is not set.\n",
                      i, mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
              err = 1;
            }
          if (rnd == MPFR_RNDZ || rnd == MPFR_RNDD)
            {
              if (inex >= 0)
                {
                  printf ("Error in overflowed_exp0 (i = %d, rnd = %s):\n"
                          "  The inexact value must be negative.\n",
                          i, mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                  err = 1;
                }
              if (! mpfr_equal_p (x, y))
                {
                  printf ("Error in overflowed_exp0 (i = %d, rnd = %s):\n"
                          "  Got ", i, mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                  mpfr_print_binary (x);
                  printf (" instead of 0.11111111E%d.\n", emax);
                  err = 1;
                }
            }
          else
            {
              if (inex <= 0)
                {
                  printf ("Error in overflowed_exp0 (i = %d, rnd = %s):\n"
                          "  The inexact value must be positive.\n",
                          i, mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                  err = 1;
                }
              if (! (mpfr_inf_p (x) && MPFR_SIGN (x) > 0))
                {
                  printf ("Error in overflowed_exp0 (i = %d, rnd = %s):\n"
                          "  Got ", i, mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                  mpfr_print_binary (x);
                  printf (" instead of +Inf.\n");
                  err = 1;
                }
            }
        }
      set_emax (old_emax);
    }

  if (err)
    exit (1);
  mpfr_clear (x);
  mpfr_clear (y);
}

/* This bug occurs in mpfr_exp_2 on a Linux-64 machine, r5475. */
static void
bug20080731 (void)
{
  mpfr_exp_t emin;
  mpfr_t x, y1, y2;
  mpfr_prec_t prec = 64;

  emin = mpfr_get_emin ();
  set_emin (MPFR_EMIN_MIN);

  mpfr_init2 (x, 200);
  mpfr_set_str (x, "-2.c5c85fdf473de6af278ece700fcbdabd03cd0cb9ca62d8b62c@7",
                16, MPFR_RNDN);

  mpfr_init2 (y1, prec);
  mpfr_exp (y1, x, MPFR_RNDU);

  /* Compute the result with a higher internal precision. */
  mpfr_init2 (y2, 300);
  mpfr_exp (y2, x, MPFR_RNDU);
  mpfr_prec_round (y2, prec, MPFR_RNDU);

  if (mpfr_cmp0 (y1, y2) != 0)
    {
      printf ("Error in bug20080731\nExpected ");
      mpfr_out_str (stdout, 16, 0, y2, MPFR_RNDN);
      printf ("\nGot      ");
      mpfr_out_str (stdout, 16, 0, y1, MPFR_RNDN);
      printf ("\n");
      exit (1);
    }

  mpfr_clears (x, y1, y2, (mpfr_ptr) 0);
  set_emin (emin);
}

/* Emulate mpfr_exp with mpfr_exp_3 in the general case. */
static int
exp_3 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  int inexact;

  inexact = mpfr_exp_3 (y, x, rnd_mode);
  return mpfr_check_range (y, inexact, rnd_mode);
}

static void
underflow_up (int extended_emin)
{
  mpfr_t minpos, x, y, t, t2;
  int precx, precy;
  int inex;
  int rnd;
  int e3;
  int i, j;

  mpfr_init2 (minpos, 2);
  mpfr_set_ui (minpos, 0, MPFR_RNDN);
  mpfr_nextabove (minpos);

  /* Let's test values near the underflow boundary.
   *
   * Minimum representable positive number: minpos = 2^(emin - 1).
   * Let's choose an MPFR number x = log(minpos) + eps, with |eps| small
   * (note: eps cannot be 0, and cannot be a rational number either).
   * Then exp(x) = minpos * exp(eps) ~= minpos * (1 + eps + eps^2).
   * We will compute y = rnd(exp(x)) in some rounding mode, precision p.
   *   1. If eps > 0, then in any rounding mode:
   *        rnd(exp(x)) >= minpos and no underflow.
   *      So, let's take x1 = rndu(log(minpos)) in some precision.
   *   2. If eps < 0, then exp(x) < minpos and the result will be either 0
   *      or minpos. An underflow always occurs in MPFR_RNDZ and MPFR_RNDD,
   *      but not necessarily in MPFR_RNDN and MPFR_RNDU (this is underflow
   *      after rounding in an unbounded exponent range). If -a < eps < -b,
   *        minpos * (1 - a) < exp(x) < minpos * (1 - b + b^2).
   *      - If eps > -2^(-p), no underflow in MPFR_RNDU.
   *      - If eps > -2^(-p-1), no underflow in MPFR_RNDN.
   *      - If eps < - (2^(-p-1) + 2^(-2p-1)), underflow in MPFR_RNDN.
   *      - If eps < - (2^(-p) + 2^(-2p+1)), underflow in MPFR_RNDU.
   *      - In MPFR_RNDN, result is minpos iff exp(eps) > 1/2, i.e.
   *        - log(2) < eps < ...
   *
   * Moreover, since precy < MPFR_EXP_THRESHOLD (to avoid tests that take
   * too much time), mpfr_exp() always selects mpfr_exp_2(); so, we need
   * to test mpfr_exp_3() too. This will be done via the e3 variable:
   *   e3 = 0: mpfr_exp(), thus mpfr_exp_2().
   *   e3 = 1: mpfr_exp_3(), via the exp_3() wrapper.
   * i.e.: inex = e3 ? exp_3 (y, x, rnd) : mpfr_exp (y, x, rnd);
   */

  /* Case eps > 0. In revision 5461 (trunk) on a 64-bit Linux machine:
   *   Incorrect flags in underflow_up, eps > 0, MPFR_RNDN and extended emin
   *   for precx = 96, precy = 16, mpfr_exp_3
   *   Got 9 instead of 8.
   * Note: testing this case in several precisions for x and y introduces
   * some useful random. Indeed, the bug is not always triggered.
   * Fixed in r5469.
   */
  for (precx = 16; precx <= 128; precx += 16)
    {
      mpfr_init2 (x, precx);
      mpfr_log (x, minpos, MPFR_RNDU);
      for (precy = 16; precy <= 128; precy += 16)
        {
          mpfr_init2 (y, precy);

          for (e3 = 0; e3 <= 1; e3++)
            {
              RND_LOOP (rnd)
                {
                  int err = 0;

                  mpfr_clear_flags ();
                  inex = e3 ? exp_3 (y, x, (mpfr_rnd_t) rnd)
                    : mpfr_exp (y, x, (mpfr_rnd_t) rnd);
                  if (__gmpfr_flags != MPFR_FLAGS_INEXACT)
                    {
                      printf ("Incorrect flags in underflow_up, eps > 0, %s",
                              mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                      if (extended_emin)
                        printf (" and extended emin");
                      printf ("\nfor precx = %d, precy = %d, %s\n",
                              precx, precy, e3 ? "mpfr_exp_3" : "mpfr_exp");
                      printf ("Got %u instead of %u.\n", __gmpfr_flags,
                              (unsigned int) MPFR_FLAGS_INEXACT);
                      err = 1;
                    }
                  if (mpfr_cmp0 (y, minpos) < 0)
                    {
                      printf ("Incorrect result in underflow_up, eps > 0, %s",
                              mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                      if (extended_emin)
                        printf (" and extended emin");
                      printf ("\nfor precx = %d, precy = %d, %s\n",
                              precx, precy, e3 ? "mpfr_exp_3" : "mpfr_exp");
                      mpfr_dump (y);
                      err = 1;
                    }
                  MPFR_ASSERTN (inex != 0);
                  if (rnd == MPFR_RNDD || rnd == MPFR_RNDZ)
                    MPFR_ASSERTN (inex < 0);
                  if (rnd == MPFR_RNDU)
                    MPFR_ASSERTN (inex > 0);
                  if (err)
                    exit (1);
                }
            }

          mpfr_clear (y);
        }
      mpfr_clear (x);
    }

  /* Case - log(2) < eps < 0 in MPFR_RNDN, starting with small-precision x;
   * only check the result and the ternary value.
   * Previous to r5453 (trunk), on 32-bit and 64-bit machines, this fails
   * for precx = 65 and precy = 16, e.g.:
   *   exp_2.c:264:  assertion failed: ...
   * because mpfr_sub (r, x, r, MPFR_RNDU); yields a null value. This is
   * fixed in r5453 by going to next Ziv's iteration.
   */
  for (precx = sizeof(mpfr_exp_t) * CHAR_BIT + 1; precx <= 81; precx += 8)
    {
      mpfr_init2 (x, precx);
      mpfr_log (x, minpos, MPFR_RNDD);  /* |ulp| <= 1/2 */
      for (precy = 16; precy <= 128; precy += 16)
        {
          mpfr_init2 (y, precy);
          inex = mpfr_exp (y, x, MPFR_RNDN);
          if (inex <= 0 || mpfr_cmp0 (y, minpos) != 0)
            {
              printf ("Error in underflow_up, - log(2) < eps < 0");
              if (extended_emin)
                printf (" and extended emin");
              printf (" for prec = %d\nExpected ", precy);
              mpfr_out_str (stdout, 16, 0, minpos, MPFR_RNDN);
              printf (" (minimum positive MPFR number) and inex > 0\nGot ");
              mpfr_out_str (stdout, 16, 0, y, MPFR_RNDN);
              printf ("\nwith inex = %d\n", inex);
              exit (1);
            }
          mpfr_clear (y);
        }
      mpfr_clear (x);
    }

  /* Cases eps ~ -2^(-p) and eps ~ -2^(-p-1). More precisely,
   *   _ for j = 0, eps > -2^(-(p+i)),
   *   _ for j = 1, eps < - (2^(-(p+i)) + 2^(1-2(p+i))),
   * where i = 0 or 1.
   */
  mpfr_inits2 (2, t, t2, (mpfr_ptr) 0);
  for (precy = 16; precy <= 128; precy += 16)
    {
      mpfr_set_ui_2exp (t, 1, - precy, MPFR_RNDN);         /* 2^(-p) */
      mpfr_set_ui_2exp (t2, 1, 1 - 2 * precy, MPFR_RNDN);  /* 2^(-2p+1) */
      precx = sizeof(mpfr_exp_t) * CHAR_BIT + 2 * precy + 8;
      mpfr_init2 (x, precx);
      mpfr_init2 (y, precy);
      for (i = 0; i <= 1; i++)
        {
          for (j = 0; j <= 1; j++)
            {
              if (j == 0)
                {
                  /* Case eps > -2^(-(p+i)). */
                  mpfr_log (x, minpos, MPFR_RNDU);
                }
              else  /* j == 1 */
                {
                  /* Case eps < - (2^(-(p+i)) + 2^(1-2(p+i))). */
                  mpfr_log (x, minpos, MPFR_RNDD);
                  inex = mpfr_sub (x, x, t2, MPFR_RNDN);
                  MPFR_ASSERTN (inex == 0);
                }
              inex = mpfr_sub (x, x, t, MPFR_RNDN);
              MPFR_ASSERTN (inex == 0);

              RND_LOOP (rnd)
                for (e3 = 0; e3 <= 1; e3++)
                  {
                    int err = 0;
                    unsigned int flags;

                    flags = MPFR_FLAGS_INEXACT |
                      (((rnd == MPFR_RNDU || rnd == MPFR_RNDA)
                             && (i == 1 || j == 0)) ||
                       (rnd == MPFR_RNDN && (i == 1 && j == 0)) ?
                       0 : MPFR_FLAGS_UNDERFLOW);
                    mpfr_clear_flags ();
                    inex = e3 ? exp_3 (y, x, (mpfr_rnd_t) rnd)
                      : mpfr_exp (y, x, (mpfr_rnd_t) rnd);
                    if (__gmpfr_flags != flags)
                      {
                        printf ("Incorrect flags in underflow_up, %s",
                                mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                        if (extended_emin)
                          printf (" and extended emin");
                        printf ("\nfor precx = %d, precy = %d, ",
                                precx, precy);
                        if (j == 0)
                          printf ("eps >~ -2^(-%d)", precy + i);
                        else
                          printf ("eps <~ - (2^(-%d) + 2^(%d))",
                                  precy + i, 1 - 2 * (precy + i));
                        printf (", %s\n", e3 ? "mpfr_exp_3" : "mpfr_exp");
                        printf ("Got %u instead of %u.\n",
                                __gmpfr_flags, flags);
                        err = 1;
                      }
                    if (rnd == MPFR_RNDU || rnd == MPFR_RNDA || rnd == MPFR_RNDN ?
                        mpfr_cmp0 (y, minpos) != 0 : MPFR_NOTZERO (y))
                      {
                        printf ("Incorrect result in underflow_up, %s",
                                mpfr_print_rnd_mode ((mpfr_rnd_t) rnd));
                        if (extended_emin)
                          printf (" and extended emin");
                        printf ("\nfor precx = %d, precy = %d, ",
                                precx, precy);
                        if (j == 0)
                          printf ("eps >~ -2^(-%d)", precy + i);
                        else
                          printf ("eps <~ - (2^(-%d) + 2^(%d))",
                                  precy + i, 1 - 2 * (precy + i));
                        printf (", %s\n", e3 ? "mpfr_exp_3" : "mpfr_exp");
                        mpfr_dump (y);
                        err = 1;
                      }
                    if (err)
                      exit (1);
                  }  /* for (e3 ...) */
            }  /* for (j ...) */
          mpfr_div_2si (t, t, 1, MPFR_RNDN);
          mpfr_div_2si (t2, t2, 2, MPFR_RNDN);
        }  /* for (i ...) */
      mpfr_clears (x, y, (mpfr_ptr) 0);
    }  /* for (precy ...) */
  mpfr_clears (t, t2, (mpfr_ptr) 0);

  /* Case exp(eps) ~= 1/2, i.e. eps ~= - log(2).
   * We choose x0 and x1 with high enough precision such that:
   *   x0 = rndd(rndd(log(minpos)) - rndu(log(2)))
   *   x1 = rndu(rndu(log(minpos)) - rndd(log(2)))
   * In revision 5507 (trunk) on a 64-bit Linux machine, this fails:
   *   Error in underflow_up, eps >~ - log(2) and extended emin
   *   for precy = 16, mpfr_exp
   *   Expected 1.0@-1152921504606846976 (minimum positive MPFR number),
   *   inex > 0 and flags = 9
   *   Got 0
   *   with inex = -1 and flags = 9
   * due to a double-rounding problem in mpfr_mul_2si when rescaling
   * the result.
   */
  mpfr_inits2 (sizeof(mpfr_exp_t) * CHAR_BIT + 64, x, t, (mpfr_ptr) 0);
  for (i = 0; i <= 1; i++)
    {
      mpfr_log (x, minpos, i ? MPFR_RNDU : MPFR_RNDD);
      mpfr_const_log2 (t, i ? MPFR_RNDD : MPFR_RNDU);
      mpfr_sub (x, x, t, i ? MPFR_RNDU : MPFR_RNDD);
      for (precy = 16; precy <= 128; precy += 16)
        {
          mpfr_init2 (y, precy);
          for (e3 = 0; e3 <= 1; e3++)
            {
              unsigned int flags, uflags =
                MPFR_FLAGS_INEXACT | MPFR_FLAGS_UNDERFLOW;

              mpfr_clear_flags ();
              inex = e3 ? exp_3 (y, x, MPFR_RNDN) : mpfr_exp (y, x, MPFR_RNDN);
              flags = __gmpfr_flags;
              if (flags != uflags ||
                  (i ? (inex <= 0 || mpfr_cmp0 (y, minpos) != 0)
                     : (inex >= 0 || MPFR_NOTZERO (y))))
                {
                  printf ("Error in underflow_up, eps %c~ - log(2)",
                          i ? '>' : '<');
                  if (extended_emin)
                    printf (" and extended emin");
                  printf ("\nfor precy = %d, %s\nExpected ", precy,
                          e3 ? "mpfr_exp_3" : "mpfr_exp");
                  if (i)
                    {
                      mpfr_out_str (stdout, 16, 0, minpos, MPFR_RNDN);
                      printf (" (minimum positive MPFR number),\ninex >");
                    }
                  else
                    {
                      printf ("+0, inex <");
                    }
                  printf (" 0 and flags = %u\nGot ", uflags);
                  mpfr_out_str (stdout, 16, 0, y, MPFR_RNDN);
                  printf ("\nwith inex = %d and flags = %u\n", inex, flags);
                  exit (1);
                }
            }
          mpfr_clear (y);
        }
    }
  mpfr_clears (x, t, (mpfr_ptr) 0);

  mpfr_clear (minpos);
}

static void
underflow (void)
{
  mpfr_exp_t emin;

  underflow_up (0);

  emin = mpfr_get_emin ();
  set_emin (MPFR_EMIN_MIN);
  if (mpfr_get_emin () != emin)
    {
      underflow_up (1);
      set_emin (emin);
    }
}

int
main (int argc, char *argv[])
{
  tests_start_mpfr ();

  if (argc > 1)
    check_large ();

  check_inexact ();
  check_special ();

  test_generic (2, 100, 100);

  compare_exp2_exp3 (20, 1000);
  check_worst_cases();
  check3("0.0", MPFR_RNDU, "1.0");
  check3("-1e-170", MPFR_RNDU, "1.0");
  check3("-1e-170", MPFR_RNDN, "1.0");
  check3("-8.88024741073346941839e-17", MPFR_RNDU, "1.0");
  check3("8.70772839244701057915e-01", MPFR_RNDN, "2.38875626491680437269");
  check3("1.0", MPFR_RNDN, "2.71828182845904509080");
  check3("-3.42135637628104173534e-07", MPFR_RNDZ, "0.999999657864420798958");
  /* worst case for argument reduction, very near from 5*log(2),
     thanks to Jean-Michel Muller  */
  check3("3.4657359027997265421", MPFR_RNDN, "32.0");
  check3("3.4657359027997265421", MPFR_RNDU, "32.0");
  check3("3.4657359027997265421", MPFR_RNDD, "31.999999999999996447");
  check3("2.26523754332090625496e+01", MPFR_RNDD, "6.8833785261699581146e9");
  check3("1.31478962104089092122e+01", MPFR_RNDZ, "5.12930793917860137299e+05");
  check3("4.25637507920002378103e-01", MPFR_RNDU, "1.53056585656161181497e+00");
  check3("6.26551618962329307459e-16", MPFR_RNDU, "1.00000000000000066613e+00");
  check3("-3.35589513871216568383e-03",MPFR_RNDD, "9.96649729583626853291e-01");
  check3("1.95151388850007272424e+01", MPFR_RNDU, "2.98756340674767792225e+08");
  check3("2.45045953503350730784e+01", MPFR_RNDN, "4.38743344916128387451e+10");
  check3("2.58165606081678085104e+01", MPFR_RNDD, "1.62925781879432281494e+11");
  check3("-2.36539020084338638128e+01",MPFR_RNDZ, "5.33630792749924762447e-11");
  check3("2.39211946135858077866e+01", MPFR_RNDU, "2.44817704330214385986e+10");
  check3("-2.78190533055889162029e+01",MPFR_RNDZ, "8.2858803483596879512e-13");
  check3("2.64028186174889789584e+01", MPFR_RNDD, "2.9281844652878973388e11");
  check3("2.92086338843268329413e+01", MPFR_RNDZ, "4.8433797301907177734e12");
  check3("-2.46355324071459982349e+01",MPFR_RNDZ, "1.9995129297760994791e-11");
  check3("-2.23509444608605427618e+01",MPFR_RNDZ, "1.9638492867489702307e-10");
  check3("-2.41175390197331687148e+01",MPFR_RNDD, "3.3564940885530624592e-11");
  check3("2.46363885231578088053e+01", MPFR_RNDU, "5.0055014282693267822e10");
  check3("111.1263531080090984914932050742208957672119140625", MPFR_RNDN, "1.8262572323517295459e48");
  check3("-3.56196340354684821250e+02",MPFR_RNDN, "2.0225297096141478156e-155");
  check3("6.59678273772710895173e+02", MPFR_RNDU, "3.1234469273830195529e286");
  check3("5.13772529701934331570e+02", MPFR_RNDD, "1.3445427121297197752e223");
  check3("3.57430211008718345056e+02", MPFR_RNDD, "1.6981197246857298443e155");
  check3("3.82001814471465536371e+02", MPFR_RNDU, "7.9667300591087367805e165");
  check3("5.92396038219384422518e+02", MPFR_RNDD, "1.880747529554661989e257");
  check3("-5.02678550462488090034e+02",MPFR_RNDU, "4.8919201895446217839e-219");
  check3("5.30015757134837031117e+02", MPFR_RNDD, "1.5237672861171573939e230");
  check3("5.16239362447650933063e+02", MPFR_RNDZ, "1.5845518406744492105e224");
  check3("6.00812634798592370977e-01", MPFR_RNDN, "1.823600119339019443");
  check_exp10 ();

  bug20080731 ();

  overflowed_exp0 ();
  underflow ();

  data_check ("data/exp", mpfr_exp, "mpfr_exp");
  bad_cases (mpfr_exp, mpfr_log, "mpfr_exp", 0, -256, 255, 4, 128, 800, 50);

  tests_end_mpfr ();
  return 0;
}