guile/libguile/numbers.c

/* Copyright (C) 1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc.
 *
 * Portions Copyright 1990, 1991, 1992, 1993 by AT&T Bell Laboratories
 * and Bellcore.  See scm_divide.
 *
 *
 * This 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.
 *
 * This 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 this library; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
 */


/* General assumptions:
 * All objects satisfying SCM_COMPLEXP() have a non-zero complex component.
 * All objects satisfying SCM_BIGP() are too large to fit in a fixnum.
 * If an object satisfies integer?, it's either an inum, a bignum, or a real.
 * If floor (r) == r, r is an int, and mpz_set_d will DTRT.
 * All objects satisfying SCM_FRACTIONP are never an integer.
 */

/* TODO:
   
   - see if special casing bignums and reals in integer-exponent when
     possible (to use mpz_pow and mpf_pow_ui) is faster.

   - look in to better short-circuiting of common cases in
     integer-expt and elsewhere.

   - see if direct mpz operations can help in ash and elsewhere.

 */

#ifdef HAVE_CONFIG_H
#  include <config.h>
#endif

#include <math.h>
#include <ctype.h>
#include <string.h>

#if HAVE_COMPLEX_H
#include <complex.h>
#endif

#include "libguile/_scm.h"
#include "libguile/feature.h"
#include "libguile/ports.h"
#include "libguile/root.h"
#include "libguile/smob.h"
#include "libguile/strings.h"

#include "libguile/validate.h"
#include "libguile/numbers.h"
#include "libguile/deprecation.h"

#include "libguile/eq.h"

#include "libguile/discouraged.h"

/* values per glibc, if not already defined */
#ifndef M_LOG10E
#define M_LOG10E   0.43429448190325182765
#endif
#ifndef M_PI
#define M_PI       3.14159265358979323846
#endif



/*
  Wonder if this might be faster for some of our code?  A switch on
  the numtag would jump directly to the right case, and the
  SCM_I_NUMTAG code might be faster than repeated SCM_FOOP tests...

  #define SCM_I_NUMTAG_NOTNUM 0
  #define SCM_I_NUMTAG_INUM 1
  #define SCM_I_NUMTAG_BIG scm_tc16_big
  #define SCM_I_NUMTAG_REAL scm_tc16_real
  #define SCM_I_NUMTAG_COMPLEX scm_tc16_complex
  #define SCM_I_NUMTAG(x) \
    (SCM_I_INUMP(x) ? SCM_I_NUMTAG_INUM \
       : (SCM_IMP(x) ? SCM_I_NUMTAG_NOTNUM \
         : (((0xfcff & SCM_CELL_TYPE (x)) == scm_tc7_number) ? SCM_TYP16(x) \
           : SCM_I_NUMTAG_NOTNUM)))
*/
/* the macro above will not work as is with fractions */


#define SCM_SWAP(x, y) do { SCM __t = x; x = y; y = __t; } while (0)

/* FLOBUFLEN is the maximum number of characters neccessary for the
 * printed or scm_string representation of an inexact number.
 */
#define FLOBUFLEN (40+2*(sizeof(double)/sizeof(char)*SCM_CHAR_BIT*3+9)/10)

#if defined (SCO)
#if ! defined (HAVE_ISNAN)
#define HAVE_ISNAN
static int
isnan (double x)
{
  return (IsNANorINF (x) && NaN (x) && ! IsINF (x)) ? 1 : 0;
}
#endif
#if ! defined (HAVE_ISINF)
#define HAVE_ISINF
static int
isinf (double x)
{
  return (IsNANorINF (x) && IsINF (x)) ? 1 : 0;
}

#endif
#endif


/* mpz_cmp_d in gmp 4.1.3 doesn't recognise infinities, so xmpz_cmp_d uses
   an explicit check.  In some future gmp (don't know what version number),
   mpz_cmp_d is supposed to do this itself.  */
#if 1
#define xmpz_cmp_d(z, d)                                \
  (xisinf (d) ? (d < 0.0 ? 1 : -1) : mpz_cmp_d (z, d))
#else
#define xmpz_cmp_d(z, d)  mpz_cmp_d (z, d)
#endif

/* For reference, sparc solaris 7 has infinities (IEEE) but doesn't have
   isinf.  It does have finite and isnan though, hence the use of those.
   fpclass would be a possibility on that system too.  */
static int
xisinf (double x)
{
#if defined (HAVE_ISINF)
  return isinf (x);
#elif defined (HAVE_FINITE) && defined (HAVE_ISNAN)
  return (! (finite (x) || isnan (x)));
#else
  return 0;
#endif
}

static int
xisnan (double x)
{
#if defined (HAVE_ISNAN)
  return isnan (x);
#else
  return 0;
#endif
}

#if defined (GUILE_I)
#if HAVE_COMPLEX_DOUBLE

/* For an SCM object Z which is a complex number (ie. satisfies
   SCM_COMPLEXP), return its value as a C level "complex double". */
#define SCM_COMPLEX_VALUE(z)                                    \
  (SCM_COMPLEX_REAL (z) + GUILE_I * SCM_COMPLEX_IMAG (z))

static inline SCM scm_from_complex_double (complex double z) SCM_UNUSED;

/* Convert a C "complex double" to an SCM value. */
static inline SCM
scm_from_complex_double (complex double z)
{
  return scm_c_make_rectangular (creal (z), cimag (z));
}

#endif /* HAVE_COMPLEX_DOUBLE */
#endif /* GUILE_I */



static mpz_t z_negative_one;



SCM
scm_i_mkbig ()
{
  /* Return a newly created bignum. */
  SCM z = scm_double_cell (scm_tc16_big, 0, 0, 0);
  mpz_init (SCM_I_BIG_MPZ (z));
  return z;
}

SCM
scm_i_long2big (long x)
{
  /* Return a newly created bignum initialized to X. */
  SCM z = scm_double_cell (scm_tc16_big, 0, 0, 0);
  mpz_init_set_si (SCM_I_BIG_MPZ (z), x);
  return z;
}

SCM
scm_i_ulong2big (unsigned long x)
{
  /* Return a newly created bignum initialized to X. */
  SCM z = scm_double_cell (scm_tc16_big, 0, 0, 0);
  mpz_init_set_ui (SCM_I_BIG_MPZ (z), x);
  return z;
}

SCM
scm_i_clonebig (SCM src_big, int same_sign_p)
{
  /* Copy src_big's value, negate it if same_sign_p is false, and return. */
  SCM z = scm_double_cell (scm_tc16_big, 0, 0, 0);
  mpz_init_set (SCM_I_BIG_MPZ (z), SCM_I_BIG_MPZ (src_big));
  if (!same_sign_p)
    mpz_neg (SCM_I_BIG_MPZ (z), SCM_I_BIG_MPZ (z));
  return z;
}

int
scm_i_bigcmp (SCM x, SCM y)
{
  /* Return neg if x < y, pos if x > y, and 0 if x == y */
  /* presume we already know x and y are bignums */
  int result = mpz_cmp (SCM_I_BIG_MPZ (x), SCM_I_BIG_MPZ (y));
  scm_remember_upto_here_2 (x, y);
  return result;
}

SCM
scm_i_dbl2big (double d)
{
  /* results are only defined if d is an integer */
  SCM z = scm_double_cell (scm_tc16_big, 0, 0, 0);
  mpz_init_set_d (SCM_I_BIG_MPZ (z), d);
  return z;
}

/* Convert a integer in double representation to a SCM number. */

SCM
scm_i_dbl2num (double u)
{
  /* SCM_MOST_POSITIVE_FIXNUM+1 and SCM_MOST_NEGATIVE_FIXNUM are both
     powers of 2, so there's no rounding when making "double" values
     from them.  If plain SCM_MOST_POSITIVE_FIXNUM was used it could
     get rounded on a 64-bit machine, hence the "+1".

     The use of floor() to force to an integer value ensures we get a
     "numerically closest" value without depending on how a
     double->long cast or how mpz_set_d will round.  For reference,
     double->long probably follows the hardware rounding mode,
     mpz_set_d truncates towards zero.  */

  /* XXX - what happens when SCM_MOST_POSITIVE_FIXNUM etc is not
     representable as a double? */

  if (u < (double) (SCM_MOST_POSITIVE_FIXNUM+1)
      && u >= (double) SCM_MOST_NEGATIVE_FIXNUM)
    return SCM_I_MAKINUM ((long) u);
  else
    return scm_i_dbl2big (u);
}

/* scm_i_big2dbl() rounds to the closest representable double, in accordance
   with R5RS exact->inexact.

   The approach is to use mpz_get_d to pick out the high DBL_MANT_DIG bits
   (ie. truncate towards zero), then adjust to get the closest double by
   examining the next lower bit and adding 1 (to the absolute value) if
   necessary.

   Bignums exactly half way between representable doubles are rounded to the
   next higher absolute value (ie. away from zero).  This seems like an
   adequate interpretation of R5RS "numerically closest", and it's easier
   and faster than a full "nearest-even" style.

   The bit test must be done on the absolute value of the mpz_t, which means
   we need to use mpz_getlimbn.  mpz_tstbit is not right, it treats
   negatives as twos complement.

   In current gmp 4.1.3, mpz_get_d rounding is unspecified.  It ends up
   following the hardware rounding mode, but applied to the absolute value
   of the mpz_t operand.  This is not what we want so we put the high
   DBL_MANT_DIG bits into a temporary.  In some future gmp, don't know when,
   mpz_get_d is supposed to always truncate towards zero.

   ENHANCE-ME: The temporary init+clear to force the rounding in gmp 4.1.3
   is a slowdown.  It'd be faster to pick out the relevant high bits with
   mpz_getlimbn if we could be bothered coding that, and if the new
   truncating gmp doesn't come out.  */

double
scm_i_big2dbl (SCM b)
{
  double result;
  size_t bits;

  bits = mpz_sizeinbase (SCM_I_BIG_MPZ (b), 2);

#if 1
  {
    /* Current GMP, eg. 4.1.3, force truncation towards zero */
    mpz_t  tmp;
    if (bits > DBL_MANT_DIG)
      {
        size_t  shift = bits - DBL_MANT_DIG;
        mpz_init2 (tmp, DBL_MANT_DIG);
        mpz_tdiv_q_2exp (tmp, SCM_I_BIG_MPZ (b), shift);
        result = ldexp (mpz_get_d (tmp), shift);
        mpz_clear (tmp);
      }
    else
      {
        result = mpz_get_d (SCM_I_BIG_MPZ (b));
      }
  }
#else
  /* Future GMP */
  result = mpz_get_d (SCM_I_BIG_MPZ (b));
#endif

  if (bits > DBL_MANT_DIG)
    {
      unsigned long  pos = bits - DBL_MANT_DIG - 1;
      /* test bit number "pos" in absolute value */
      if (mpz_getlimbn (SCM_I_BIG_MPZ (b), pos / GMP_NUMB_BITS)
          & ((mp_limb_t) 1 << (pos % GMP_NUMB_BITS)))
        {
          result += ldexp ((double) mpz_sgn (SCM_I_BIG_MPZ (b)), pos + 1);
        }
    }

  scm_remember_upto_here_1 (b);
  return result;
}

SCM
scm_i_normbig (SCM b)
{
  /* convert a big back to a fixnum if it'll fit */
  /* presume b is a bignum */
  if (mpz_fits_slong_p (SCM_I_BIG_MPZ (b)))
    {
      long val = mpz_get_si (SCM_I_BIG_MPZ (b));
      if (SCM_FIXABLE (val))
        b = SCM_I_MAKINUM (val);
    }
  return b;
}

static SCM_C_INLINE_KEYWORD SCM
scm_i_mpz2num (mpz_t b)
{
  /* convert a mpz number to a SCM number. */
  if (mpz_fits_slong_p (b))
    {
      long val = mpz_get_si (b);
      if (SCM_FIXABLE (val))
        return SCM_I_MAKINUM (val);
    }

  {
    SCM z = scm_double_cell (scm_tc16_big, 0, 0, 0);
    mpz_init_set (SCM_I_BIG_MPZ (z), b);
    return z;
  }
}

/* this is needed when we want scm_divide to make a float, not a ratio, even if passed two ints */
static SCM scm_divide2real (SCM x, SCM y);

static SCM
scm_i_make_ratio (SCM numerator, SCM denominator)
#define FUNC_NAME "make-ratio"
{
  /* First make sure the arguments are proper.
   */
  if (SCM_I_INUMP (denominator))
    {
      if (scm_is_eq (denominator, SCM_INUM0))
	scm_num_overflow ("make-ratio");
      if (scm_is_eq (denominator, SCM_I_MAKINUM(1)))
	return numerator;
    }
  else 
    {
      if (!(SCM_BIGP(denominator)))
	SCM_WRONG_TYPE_ARG (2, denominator);
    }
  if (!SCM_I_INUMP (numerator) && !SCM_BIGP (numerator))
    SCM_WRONG_TYPE_ARG (1, numerator);

  /* Then flip signs so that the denominator is positive.
   */
  if (scm_is_true (scm_negative_p (denominator)))
    {
      numerator = scm_difference (numerator, SCM_UNDEFINED);
      denominator = scm_difference (denominator, SCM_UNDEFINED);
    }

  /* Now consider for each of the four fixnum/bignum combinations
     whether the rational number is really an integer.
  */
  if (SCM_I_INUMP (numerator))
    {
      long  x = SCM_I_INUM (numerator);
      if (scm_is_eq (numerator, SCM_INUM0))
	return SCM_INUM0;
      if (SCM_I_INUMP (denominator))
	{
	  long y;
	  y = SCM_I_INUM (denominator);
	  if (x == y)
	    return SCM_I_MAKINUM(1);
	  if ((x % y) == 0)
	    return SCM_I_MAKINUM (x / y);
	}
      else
        {
          /* When x == SCM_MOST_NEGATIVE_FIXNUM we could have the negative
             of that value for the denominator, as a bignum.  Apart from
             that case, abs(bignum) > abs(inum) so inum/bignum is not an
             integer.  */
          if (x == SCM_MOST_NEGATIVE_FIXNUM
              && mpz_cmp_ui (SCM_I_BIG_MPZ (denominator),
                             - SCM_MOST_NEGATIVE_FIXNUM) == 0)
	    return SCM_I_MAKINUM(-1);
        }
    }
  else if (SCM_BIGP (numerator))
    {
      if (SCM_I_INUMP (denominator))
	{
	  long yy = SCM_I_INUM (denominator);
	  if (mpz_divisible_ui_p (SCM_I_BIG_MPZ (numerator), yy))
	    return scm_divide (numerator, denominator);
	}
      else
	{
	  if (scm_is_eq (numerator, denominator))
	    return SCM_I_MAKINUM(1);
	  if (mpz_divisible_p (SCM_I_BIG_MPZ (numerator),
			       SCM_I_BIG_MPZ (denominator)))
	    return scm_divide(numerator, denominator);
	}
    }

  /* No, it's a proper fraction.
   */
  {
    SCM divisor = scm_gcd (numerator, denominator);
    if (!(scm_is_eq (divisor, SCM_I_MAKINUM(1))))
      {
	numerator = scm_divide (numerator, divisor);
	denominator = scm_divide (denominator, divisor);
      }
      
    return scm_double_cell (scm_tc16_fraction,
			    SCM_UNPACK (numerator),
			    SCM_UNPACK (denominator), 0);
  }
}
#undef FUNC_NAME

double
scm_i_fraction2double (SCM z)
{
  return scm_to_double (scm_divide2real (SCM_FRACTION_NUMERATOR (z), 
					 SCM_FRACTION_DENOMINATOR (z)));
}

SCM_DEFINE (scm_exact_p, "exact?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is an exact number, @code{#f}\n"
	    "otherwise.")
#define FUNC_NAME s_scm_exact_p
{
  if (SCM_I_INUMP (x))
    return SCM_BOOL_T;
  if (SCM_BIGP (x))
    return SCM_BOOL_T;
  if (SCM_FRACTIONP (x))
    return SCM_BOOL_T;
  if (SCM_NUMBERP (x))
    return SCM_BOOL_F;
  SCM_WRONG_TYPE_ARG (1, x);
}
#undef FUNC_NAME


SCM_DEFINE (scm_odd_p, "odd?", 1, 0, 0, 
            (SCM n),
	    "Return @code{#t} if @var{n} is an odd number, @code{#f}\n"
	    "otherwise.")
#define FUNC_NAME s_scm_odd_p
{
  if (SCM_I_INUMP (n))
    {
      long val = SCM_I_INUM (n);
      return scm_from_bool ((val & 1L) != 0);
    }
  else if (SCM_BIGP (n))
    {
      int odd_p = mpz_odd_p (SCM_I_BIG_MPZ (n));
      scm_remember_upto_here_1 (n);
      return scm_from_bool (odd_p);
    }
  else if (scm_is_true (scm_inf_p (n)))
    return SCM_BOOL_T;
  else if (SCM_REALP (n))
    {
      double rem = fabs (fmod (SCM_REAL_VALUE(n), 2.0));
      if (rem == 1.0)
	return SCM_BOOL_T;
      else if (rem == 0.0)
	return SCM_BOOL_F;
      else
	SCM_WRONG_TYPE_ARG (1, n);
    }
  else
    SCM_WRONG_TYPE_ARG (1, n);
}
#undef FUNC_NAME


SCM_DEFINE (scm_even_p, "even?", 1, 0, 0, 
            (SCM n),
	    "Return @code{#t} if @var{n} is an even number, @code{#f}\n"
	    "otherwise.")
#define FUNC_NAME s_scm_even_p
{
  if (SCM_I_INUMP (n))
    {
      long val = SCM_I_INUM (n);
      return scm_from_bool ((val & 1L) == 0);
    }
  else if (SCM_BIGP (n))
    {
      int even_p = mpz_even_p (SCM_I_BIG_MPZ (n));
      scm_remember_upto_here_1 (n);
      return scm_from_bool (even_p);
    }
  else if (scm_is_true (scm_inf_p (n)))
    return SCM_BOOL_T;
  else if (SCM_REALP (n))
    {
      double rem = fabs (fmod (SCM_REAL_VALUE(n), 2.0));
      if (rem == 1.0)
	return SCM_BOOL_F;
      else if (rem == 0.0)
	return SCM_BOOL_T;
      else
	SCM_WRONG_TYPE_ARG (1, n);
    }
  else
    SCM_WRONG_TYPE_ARG (1, n);
}
#undef FUNC_NAME

SCM_DEFINE (scm_inf_p, "inf?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is either @samp{+inf.0}\n"
	    "or @samp{-inf.0}, @code{#f} otherwise.")
#define FUNC_NAME s_scm_inf_p
{
  if (SCM_REALP (x))
    return scm_from_bool (xisinf (SCM_REAL_VALUE (x)));
  else if (SCM_COMPLEXP (x))
    return scm_from_bool (xisinf (SCM_COMPLEX_REAL (x))
			  || xisinf (SCM_COMPLEX_IMAG (x)));
  else
    return SCM_BOOL_F;
}
#undef FUNC_NAME

SCM_DEFINE (scm_nan_p, "nan?", 1, 0, 0, 
            (SCM n),
	    "Return @code{#t} if @var{n} is a NaN, @code{#f}\n"
	    "otherwise.")
#define FUNC_NAME s_scm_nan_p
{
  if (SCM_REALP (n))
    return scm_from_bool (xisnan (SCM_REAL_VALUE (n)));
  else if (SCM_COMPLEXP (n))
    return scm_from_bool (xisnan (SCM_COMPLEX_REAL (n))
		     || xisnan (SCM_COMPLEX_IMAG (n)));
  else
    return SCM_BOOL_F;
}
#undef FUNC_NAME

/* Guile's idea of infinity.  */
static double guile_Inf;

/* Guile's idea of not a number.  */
static double guile_NaN;

static void
guile_ieee_init (void)
{
#if defined (HAVE_ISINF) || defined (HAVE_FINITE)

/* Some version of gcc on some old version of Linux used to crash when
   trying to make Inf and NaN.  */

#ifdef INFINITY
  /* C99 INFINITY, when available.
     FIXME: The standard allows for INFINITY to be something that overflows
     at compile time.  We ought to have a configure test to check for that
     before trying to use it.  (But in practice we believe this is not a
     problem on any system guile is likely to target.)  */
  guile_Inf = INFINITY;
#elif HAVE_DINFINITY
  /* OSF */
  extern unsigned int DINFINITY[2];
  union
  {
    double d;
    int i[2];
  } alias;
  alias.i[0] = DINFINITY[0];
  alias.i[1] = DINFINITY[1];
  guile_Inf = alias.d;
#else
  double tmp = 1e+10;
  guile_Inf = tmp;
  for (;;)
    {
      guile_Inf *= 1e+10;
      if (guile_Inf == tmp)
	break;
      tmp = guile_Inf;
    }
#endif

#endif

#if defined (HAVE_ISNAN)

#if defined __GNUC__ && defined __alpha__ && !defined _IEEE_FP
  /* On Alpha GCC must be passed `-mieee' to provide proper NaN handling.
     See http://lists.gnu.org/archive/html/bug-gnulib/2009-05/msg00010.html
     for more details.  */
# error NaN handling will not work when compiling without -mieee
#endif

#ifdef NAN
  /* C99 NAN, when available */
  guile_NaN = NAN;
#elif HAVE_DQNAN
  {
    /* OSF */
    extern unsigned int DQNAN[2];
    union
    {
      double d;
      int i[2];
    } alias;
    alias.i[0] = DQNAN[0];
    alias.i[1] = DQNAN[1];
    guile_NaN = alias.d;
  }
#else
  guile_NaN = guile_Inf / guile_Inf;
#endif

#endif
}

SCM_DEFINE (scm_inf, "inf", 0, 0, 0, 
            (void),
	    "Return Inf.")
#define FUNC_NAME s_scm_inf
{
  static int initialized = 0;
  if (! initialized)
    {
      guile_ieee_init ();
      initialized = 1;
    }
  return scm_from_double (guile_Inf);
}
#undef FUNC_NAME

SCM_DEFINE (scm_nan, "nan", 0, 0, 0, 
            (void),
	    "Return NaN.")
#define FUNC_NAME s_scm_nan
{
  static int initialized = 0;
  if (!initialized)
    {
      guile_ieee_init ();
      initialized = 1;
    }
  return scm_from_double (guile_NaN);
}
#undef FUNC_NAME


SCM_PRIMITIVE_GENERIC (scm_abs, "abs", 1, 0, 0,
		       (SCM x),
		       "Return the absolute value of @var{x}.")
#define FUNC_NAME
{
  if (SCM_I_INUMP (x))
    {
      long int xx = SCM_I_INUM (x);
      if (xx >= 0)
	return x;
      else if (SCM_POSFIXABLE (-xx))
	return SCM_I_MAKINUM (-xx);
      else
	return scm_i_long2big (-xx);
    }
  else if (SCM_BIGP (x))
    {
      const int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
      if (sgn < 0)
	return scm_i_clonebig (x, 0);
      else
	return x;
    }
  else if (SCM_REALP (x))
    {
      /* note that if x is a NaN then xx<0 is false so we return x unchanged */
      double xx = SCM_REAL_VALUE (x);
      if (xx < 0.0)
        return scm_from_double (-xx);
      else
        return x;
    }
  else if (SCM_FRACTIONP (x))
    {
      if (scm_is_false (scm_negative_p (SCM_FRACTION_NUMERATOR (x))))
	return x;
      return scm_i_make_ratio (scm_difference (SCM_FRACTION_NUMERATOR (x), SCM_UNDEFINED),
			     SCM_FRACTION_DENOMINATOR (x));
    }
  else
    SCM_WTA_DISPATCH_1 (g_scm_abs, x, 1, s_scm_abs);
}
#undef FUNC_NAME


SCM_GPROC (s_quotient, "quotient", 2, 0, 0, scm_quotient, g_quotient);
/* "Return the quotient of the numbers @var{x} and @var{y}."
 */
SCM
scm_quotient (SCM x, SCM y)
{
  if (SCM_I_INUMP (x))
    {
      long xx = SCM_I_INUM (x);
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  if (yy == 0)
	    scm_num_overflow (s_quotient);
	  else
	    {
	      long z = xx / yy;
	      if (SCM_FIXABLE (z))
		return SCM_I_MAKINUM (z);
	      else
		return scm_i_long2big (z);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  if ((SCM_I_INUM (x) == SCM_MOST_NEGATIVE_FIXNUM)
	      && (mpz_cmp_ui (SCM_I_BIG_MPZ (y),
                              - SCM_MOST_NEGATIVE_FIXNUM) == 0))
            {
              /* Special case:  x == fixnum-min && y == abs (fixnum-min) */
	      scm_remember_upto_here_1 (y);
              return SCM_I_MAKINUM (-1);
            }
	  else
	    return SCM_I_MAKINUM (0);
	}
      else
	SCM_WTA_DISPATCH_2 (g_quotient, x, y, SCM_ARG2, s_quotient);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  if (yy == 0)
	    scm_num_overflow (s_quotient);
	  else if (yy == 1)
	    return x;
	  else
	    {
	      SCM result = scm_i_mkbig ();
	      if (yy < 0)
		{
		  mpz_tdiv_q_ui (SCM_I_BIG_MPZ (result),
				 SCM_I_BIG_MPZ (x),
				 - yy);
		  mpz_neg (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (result));
		}
	      else
		mpz_tdiv_q_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (x), yy);
	      scm_remember_upto_here_1 (x);
	      return scm_i_normbig (result);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  SCM result = scm_i_mkbig ();
	  mpz_tdiv_q (SCM_I_BIG_MPZ (result),
		      SCM_I_BIG_MPZ (x),
		      SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  return scm_i_normbig (result);
	}
      else
	SCM_WTA_DISPATCH_2 (g_quotient, x, y, SCM_ARG2, s_quotient);
    }
  else
    SCM_WTA_DISPATCH_2 (g_quotient, x, y, SCM_ARG1, s_quotient);
}

SCM_GPROC (s_remainder, "remainder", 2, 0, 0, scm_remainder, g_remainder);
/* "Return the remainder of the numbers @var{x} and @var{y}.\n"
 * "@lisp\n"
 * "(remainder 13 4) @result{} 1\n"
 * "(remainder -13 4) @result{} -1\n"
 * "@end lisp"
 */
SCM
scm_remainder (SCM x, SCM y)
{
  if (SCM_I_INUMP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  if (yy == 0)
	    scm_num_overflow (s_remainder);
	  else
	    {
	      long z = SCM_I_INUM (x) % yy;
	      return SCM_I_MAKINUM (z);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  if ((SCM_I_INUM (x) == SCM_MOST_NEGATIVE_FIXNUM)
	      && (mpz_cmp_ui (SCM_I_BIG_MPZ (y),
                              - SCM_MOST_NEGATIVE_FIXNUM) == 0))
            {
              /* Special case:  x == fixnum-min && y == abs (fixnum-min) */
	      scm_remember_upto_here_1 (y);
              return SCM_I_MAKINUM (0);
            }
	  else
	    return x;
	}
      else
	SCM_WTA_DISPATCH_2 (g_remainder, x, y, SCM_ARG2, s_remainder);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  if (yy == 0)
	    scm_num_overflow (s_remainder);
	  else
	    {
	      SCM result = scm_i_mkbig ();
	      if (yy < 0)
		yy = - yy;
	      mpz_tdiv_r_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ(x), yy);
	      scm_remember_upto_here_1 (x);
	      return scm_i_normbig (result);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  SCM result = scm_i_mkbig ();
	  mpz_tdiv_r (SCM_I_BIG_MPZ (result),
		      SCM_I_BIG_MPZ (x),
		      SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  return scm_i_normbig (result);
	}
      else
	SCM_WTA_DISPATCH_2 (g_remainder, x, y, SCM_ARG2, s_remainder);
    }
  else
    SCM_WTA_DISPATCH_2 (g_remainder, x, y, SCM_ARG1, s_remainder);
}


SCM_GPROC (s_modulo, "modulo", 2, 0, 0, scm_modulo, g_modulo);
/* "Return the modulo of the numbers @var{x} and @var{y}.\n"
 * "@lisp\n"
 * "(modulo 13 4) @result{} 1\n"
 * "(modulo -13 4) @result{} 3\n"
 * "@end lisp"
 */
SCM
scm_modulo (SCM x, SCM y)
{
  if (SCM_I_INUMP (x))
    {
      long xx = SCM_I_INUM (x);
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  if (yy == 0)
	    scm_num_overflow (s_modulo);
	  else
	    {
	      /* C99 specifies that "%" is the remainder corresponding to a
                 quotient rounded towards zero, and that's also traditional
                 for machine division, so z here should be well defined.  */
	      long z = xx % yy;
	      long result;

	      if (yy < 0)
		{
		  if (z > 0)
		    result = z + yy;
		  else
		    result = z;
		}
	      else
		{
		  if (z < 0)
		    result = z + yy;
		  else
		    result = z;
		}
	      return SCM_I_MAKINUM (result);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  int sgn_y = mpz_sgn (SCM_I_BIG_MPZ (y));
	    {
	      mpz_t z_x;
	      SCM result;

	      if (sgn_y < 0)
		{
		  SCM pos_y = scm_i_clonebig (y, 0);
		  /* do this after the last scm_op */
		  mpz_init_set_si (z_x, xx);
		  result = pos_y; /* re-use this bignum */
		  mpz_mod (SCM_I_BIG_MPZ (result),
			   z_x,
			   SCM_I_BIG_MPZ (pos_y));        
		  scm_remember_upto_here_1 (pos_y);
		}
	      else
		{
		  result = scm_i_mkbig ();
		  /* do this after the last scm_op */
		  mpz_init_set_si (z_x, xx);
		  mpz_mod (SCM_I_BIG_MPZ (result),
			   z_x,
			   SCM_I_BIG_MPZ (y));        
		  scm_remember_upto_here_1 (y);
		}
        
	      if ((sgn_y < 0) && mpz_sgn (SCM_I_BIG_MPZ (result)) != 0)
		mpz_add (SCM_I_BIG_MPZ (result),
			 SCM_I_BIG_MPZ (y),
			 SCM_I_BIG_MPZ (result));
	      scm_remember_upto_here_1 (y);
	      /* and do this before the next one */
	      mpz_clear (z_x);
	      return scm_i_normbig (result);
	    }
	}
      else
	SCM_WTA_DISPATCH_2 (g_modulo, x, y, SCM_ARG2, s_modulo);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  if (yy == 0)
	    scm_num_overflow (s_modulo);
	  else
	    {
	      SCM result = scm_i_mkbig ();
	      mpz_mod_ui (SCM_I_BIG_MPZ (result),
			  SCM_I_BIG_MPZ (x),
			  (yy < 0) ? - yy : yy);
	      scm_remember_upto_here_1 (x);
	      if ((yy < 0) && (mpz_sgn (SCM_I_BIG_MPZ (result)) != 0))
		mpz_sub_ui (SCM_I_BIG_MPZ (result),
			    SCM_I_BIG_MPZ (result),
			    - yy);
	      return scm_i_normbig (result);
	    }
	}
      else if (SCM_BIGP (y))
	{
	    {
	      SCM result = scm_i_mkbig ();
	      int y_sgn = mpz_sgn (SCM_I_BIG_MPZ (y));
	      SCM pos_y = scm_i_clonebig (y, y_sgn >= 0);
	      mpz_mod (SCM_I_BIG_MPZ (result),
		       SCM_I_BIG_MPZ (x),
		       SCM_I_BIG_MPZ (pos_y));
        
	      scm_remember_upto_here_1 (x);
	      if ((y_sgn < 0) && (mpz_sgn (SCM_I_BIG_MPZ (result)) != 0))
		mpz_add (SCM_I_BIG_MPZ (result),
			 SCM_I_BIG_MPZ (y),
			 SCM_I_BIG_MPZ (result));
	      scm_remember_upto_here_2 (y, pos_y);
	      return scm_i_normbig (result);
	    }
	}
      else
	SCM_WTA_DISPATCH_2 (g_modulo, x, y, SCM_ARG2, s_modulo);
    }
  else
    SCM_WTA_DISPATCH_2 (g_modulo, x, y, SCM_ARG1, s_modulo);
}

SCM_GPROC1 (s_gcd, "gcd", scm_tc7_asubr, scm_gcd, g_gcd);
/* "Return the greatest common divisor of all arguments.\n"
 * "If called without arguments, 0 is returned."
 */
SCM
scm_gcd (SCM x, SCM y)
{
  if (SCM_UNBNDP (y))
    return SCM_UNBNDP (x) ? SCM_INUM0 : scm_abs (x);
  
  if (SCM_I_INUMP (x))
    {
      if (SCM_I_INUMP (y))
        {
          long xx = SCM_I_INUM (x);
          long yy = SCM_I_INUM (y);
          long u = xx < 0 ? -xx : xx;
          long v = yy < 0 ? -yy : yy;
          long result;
          if (xx == 0)
	    result = v;
	  else if (yy == 0)
	    result = u;
	  else
	    {
	      long k = 1;
	      long t;
	      /* Determine a common factor 2^k */
	      while (!(1 & (u | v)))
		{
		  k <<= 1;
		  u >>= 1;
		  v >>= 1;
		}
	      /* Now, any factor 2^n can be eliminated */
	      if (u & 1)
		t = -v;
	      else
		{
		  t = u;
		b3:
		  t = SCM_SRS (t, 1);
		}
	      if (!(1 & t))
		goto b3;
	      if (t > 0)
		u = t;
	      else
		v = -t;
	      t = u - v;
	      if (t != 0)
		goto b3;
	      result = u * k;
	    }
          return (SCM_POSFIXABLE (result)
		  ? SCM_I_MAKINUM (result)
		  : scm_i_long2big (result));
        }
      else if (SCM_BIGP (y))
        {
          SCM_SWAP (x, y);
          goto big_inum;
        }
      else
        SCM_WTA_DISPATCH_2 (g_gcd, x, y, SCM_ARG2, s_gcd);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
        {
          unsigned long result;
          long yy;
        big_inum:
          yy = SCM_I_INUM (y);
          if (yy == 0)
            return scm_abs (x);
          if (yy < 0)
	    yy = -yy;
          result = mpz_gcd_ui (NULL, SCM_I_BIG_MPZ (x), yy);
          scm_remember_upto_here_1 (x);
          return (SCM_POSFIXABLE (result) 
		  ? SCM_I_MAKINUM (result)
		  : scm_from_ulong (result));
        }
      else if (SCM_BIGP (y))
        {
          SCM result = scm_i_mkbig ();
          mpz_gcd (SCM_I_BIG_MPZ (result),
		   SCM_I_BIG_MPZ (x),
		   SCM_I_BIG_MPZ (y));
          scm_remember_upto_here_2 (x, y);
          return scm_i_normbig (result);
        }
      else
        SCM_WTA_DISPATCH_2 (g_gcd, x, y, SCM_ARG2, s_gcd);
    }
  else
    SCM_WTA_DISPATCH_2 (g_gcd, x, y, SCM_ARG1, s_gcd);
}

SCM_GPROC1 (s_lcm, "lcm", scm_tc7_asubr, scm_lcm, g_lcm);
/* "Return the least common multiple of the arguments.\n"
 * "If called without arguments, 1 is returned."
 */
SCM
scm_lcm (SCM n1, SCM n2)
{
  if (SCM_UNBNDP (n2))
    {
      if (SCM_UNBNDP (n1))
        return SCM_I_MAKINUM (1L);
      n2 = SCM_I_MAKINUM (1L);
    }

  SCM_GASSERT2 (SCM_I_INUMP (n1) || SCM_BIGP (n1),
                g_lcm, n1, n2, SCM_ARG1, s_lcm);
  SCM_GASSERT2 (SCM_I_INUMP (n2) || SCM_BIGP (n2),
                g_lcm, n1, n2, SCM_ARGn, s_lcm);

  if (SCM_I_INUMP (n1))
    {
      if (SCM_I_INUMP (n2))
        {
          SCM d = scm_gcd (n1, n2);
          if (scm_is_eq (d, SCM_INUM0))
            return d;
          else
            return scm_abs (scm_product (n1, scm_quotient (n2, d)));
        }
      else
        {
          /* inum n1, big n2 */
        inumbig:
          {
            SCM result = scm_i_mkbig ();
            long nn1 = SCM_I_INUM (n1);
            if (nn1 == 0) return SCM_INUM0;
            if (nn1 < 0) nn1 = - nn1;
            mpz_lcm_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (n2), nn1);
            scm_remember_upto_here_1 (n2);
            return result;
          }
        }
    }
  else
    {
      /* big n1 */
      if (SCM_I_INUMP (n2))
        {
          SCM_SWAP (n1, n2);
          goto inumbig;
        }
      else
        {
          SCM result = scm_i_mkbig ();
          mpz_lcm(SCM_I_BIG_MPZ (result),
                  SCM_I_BIG_MPZ (n1),
                  SCM_I_BIG_MPZ (n2));
          scm_remember_upto_here_2(n1, n2);
          /* shouldn't need to normalize b/c lcm of 2 bigs should be big */
          return result;
        }
    }
}

/* Emulating 2's complement bignums with sign magnitude arithmetic:

   Logand:
   X	Y	Result	Method:
		 (len)
   +	+	+ x	(map digit:logand X Y)
   +	-	+ x	(map digit:logand X (lognot (+ -1 Y)))
   -	+	+ y	(map digit:logand (lognot (+ -1 X)) Y)
   -	-	-	(+ 1 (map digit:logior (+ -1 X) (+ -1 Y)))

   Logior:
   X	Y	Result	Method:

   +	+	+	(map digit:logior X Y)
   +	-	- y	(+ 1 (map digit:logand (lognot X) (+ -1 Y)))
   -	+	- x	(+ 1 (map digit:logand (+ -1 X) (lognot Y)))
   -	-	- x	(+ 1 (map digit:logand (+ -1 X) (+ -1 Y)))

   Logxor:
   X	Y	Result	Method:

   +	+	+	(map digit:logxor X Y)
   +	-	-	(+ 1 (map digit:logxor X (+ -1 Y)))
   -	+	-	(+ 1 (map digit:logxor (+ -1 X) Y))
   -	-	+	(map digit:logxor (+ -1 X) (+ -1 Y))

   Logtest:
   X	Y	Result

   +	+	(any digit:logand X Y)
   +	-	(any digit:logand X (lognot (+ -1 Y)))
   -	+	(any digit:logand (lognot (+ -1 X)) Y)
   -	-	#t

*/

SCM_DEFINE1 (scm_logand, "logand", scm_tc7_asubr,
             (SCM n1, SCM n2),
	     "Return the bitwise AND of the integer arguments.\n\n"
	     "@lisp\n"
	     "(logand) @result{} -1\n"
	     "(logand 7) @result{} 7\n"
	     "(logand #b111 #b011 #b001) @result{} 1\n"
	     "@end lisp")
#define FUNC_NAME s_scm_logand
{
  long int nn1;

  if (SCM_UNBNDP (n2))
    {
      if (SCM_UNBNDP (n1))
	return SCM_I_MAKINUM (-1);
      else if (!SCM_NUMBERP (n1))
	SCM_WRONG_TYPE_ARG (SCM_ARG1, n1);
      else if (SCM_NUMBERP (n1))
	return n1;
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG1, n1);
    }

  if (SCM_I_INUMP (n1))
    {
      nn1 = SCM_I_INUM (n1);
      if (SCM_I_INUMP (n2))
	{
	  long nn2 = SCM_I_INUM (n2);
	  return SCM_I_MAKINUM (nn1 & nn2);
	}
      else if SCM_BIGP (n2)
	{
	intbig: 
	  if (n1 == 0)
	    return SCM_INUM0;
	  {
	    SCM result_z = scm_i_mkbig ();
	    mpz_t nn1_z;
	    mpz_init_set_si (nn1_z, nn1);
	    mpz_and (SCM_I_BIG_MPZ (result_z), nn1_z, SCM_I_BIG_MPZ (n2));
	    scm_remember_upto_here_1 (n2);
	    mpz_clear (nn1_z);
	    return scm_i_normbig (result_z);
	  }
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, n2);
    }
  else if (SCM_BIGP (n1))
    {
      if (SCM_I_INUMP (n2))
	{
	  SCM_SWAP (n1, n2);
	  nn1 = SCM_I_INUM (n1);
	  goto intbig;
	}
      else if (SCM_BIGP (n2))
	{
	  SCM result_z = scm_i_mkbig ();
	  mpz_and (SCM_I_BIG_MPZ (result_z),
		   SCM_I_BIG_MPZ (n1),
		   SCM_I_BIG_MPZ (n2));
	  scm_remember_upto_here_2 (n1, n2);
	  return scm_i_normbig (result_z);
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, n2);
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG1, n1);
}
#undef FUNC_NAME


SCM_DEFINE1 (scm_logior, "logior", scm_tc7_asubr,
             (SCM n1, SCM n2),
	     "Return the bitwise OR of the integer arguments.\n\n"
	     "@lisp\n"
	     "(logior) @result{} 0\n"
	     "(logior 7) @result{} 7\n"
	     "(logior #b000 #b001 #b011) @result{} 3\n"
	    "@end lisp")
#define FUNC_NAME s_scm_logior
{
  long int nn1;

  if (SCM_UNBNDP (n2))
    {
      if (SCM_UNBNDP (n1))
	return SCM_INUM0;
      else if (SCM_NUMBERP (n1))
	return n1;
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG1, n1);
    }

  if (SCM_I_INUMP (n1))
    {
      nn1 = SCM_I_INUM (n1);
      if (SCM_I_INUMP (n2))
	{
	  long nn2 = SCM_I_INUM (n2);
	  return SCM_I_MAKINUM (nn1 | nn2);
	}
      else if (SCM_BIGP (n2))
	{
	intbig:
	  if (nn1 == 0)
	    return n2;
	  {
	    SCM result_z = scm_i_mkbig ();
	    mpz_t nn1_z;
	    mpz_init_set_si (nn1_z, nn1);
	    mpz_ior (SCM_I_BIG_MPZ (result_z), nn1_z, SCM_I_BIG_MPZ (n2));
	    scm_remember_upto_here_1 (n2);
	    mpz_clear (nn1_z);
	    return scm_i_normbig (result_z);
	  }
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, n2);
    }
  else if (SCM_BIGP (n1))
    {
      if (SCM_I_INUMP (n2))
	{
	  SCM_SWAP (n1, n2); 
	  nn1 = SCM_I_INUM (n1);
	  goto intbig;
	}
      else if (SCM_BIGP (n2))
	{
	  SCM result_z = scm_i_mkbig ();
	  mpz_ior (SCM_I_BIG_MPZ (result_z),
		   SCM_I_BIG_MPZ (n1),
		   SCM_I_BIG_MPZ (n2));
	  scm_remember_upto_here_2 (n1, n2);
	  return scm_i_normbig (result_z);
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, n2);
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG1, n1);
}
#undef FUNC_NAME


SCM_DEFINE1 (scm_logxor, "logxor", scm_tc7_asubr,
             (SCM n1, SCM n2),
	     "Return the bitwise XOR of the integer arguments.  A bit is\n"
	     "set in the result if it is set in an odd number of arguments.\n"
	     "@lisp\n"
	     "(logxor) @result{} 0\n"
	     "(logxor 7) @result{} 7\n"
	     "(logxor #b000 #b001 #b011) @result{} 2\n"
	     "(logxor #b000 #b001 #b011 #b011) @result{} 1\n"
	    "@end lisp")
#define FUNC_NAME s_scm_logxor
{
  long int nn1;

  if (SCM_UNBNDP (n2))
    {
      if (SCM_UNBNDP (n1))
	return SCM_INUM0;
      else if (SCM_NUMBERP (n1))
	return n1;
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG1, n1);
    }

  if (SCM_I_INUMP (n1))
    {
      nn1 = SCM_I_INUM (n1);
      if (SCM_I_INUMP (n2))
	{
	  long nn2 = SCM_I_INUM (n2);
	  return SCM_I_MAKINUM (nn1 ^ nn2);
	}
      else if (SCM_BIGP (n2))
	{
	intbig:
	  {
	    SCM result_z = scm_i_mkbig ();
	    mpz_t nn1_z;
	    mpz_init_set_si (nn1_z, nn1);
	    mpz_xor (SCM_I_BIG_MPZ (result_z), nn1_z, SCM_I_BIG_MPZ (n2));
	    scm_remember_upto_here_1 (n2);
	    mpz_clear (nn1_z);
	    return scm_i_normbig (result_z);
	  }
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, n2);
    }
  else if (SCM_BIGP (n1))
    {
      if (SCM_I_INUMP (n2))
	{
	  SCM_SWAP (n1, n2);
	  nn1 = SCM_I_INUM (n1);
	  goto intbig;
	}
      else if (SCM_BIGP (n2))
	{
	  SCM result_z = scm_i_mkbig ();
	  mpz_xor (SCM_I_BIG_MPZ (result_z),
		   SCM_I_BIG_MPZ (n1),
		   SCM_I_BIG_MPZ (n2));
	  scm_remember_upto_here_2 (n1, n2);
	  return scm_i_normbig (result_z);
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, n2);
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG1, n1);
}
#undef FUNC_NAME


SCM_DEFINE (scm_logtest, "logtest", 2, 0, 0,
            (SCM j, SCM k),
	    "Test whether @var{j} and @var{k} have any 1 bits in common.\n"
	    "This is equivalent to @code{(not (zero? (logand j k)))}, but\n"
	    "without actually calculating the @code{logand}, just testing\n"
	    "for non-zero.\n"
	    "\n"
	    "@lisp\n"
	    "(logtest #b0100 #b1011) @result{} #f\n"
	    "(logtest #b0100 #b0111) @result{} #t\n"
	    "@end lisp")
#define FUNC_NAME s_scm_logtest
{
  long int nj;

  if (SCM_I_INUMP (j))
    {
      nj = SCM_I_INUM (j);
      if (SCM_I_INUMP (k))
	{
	  long nk = SCM_I_INUM (k);
	  return scm_from_bool (nj & nk);
	}
      else if (SCM_BIGP (k))
	{
	intbig: 
	  if (nj == 0)
	    return SCM_BOOL_F;
	  {
	    SCM result;
	    mpz_t nj_z;
	    mpz_init_set_si (nj_z, nj);
	    mpz_and (nj_z, nj_z, SCM_I_BIG_MPZ (k));
	    scm_remember_upto_here_1 (k);
	    result = scm_from_bool (mpz_sgn (nj_z) != 0);
	    mpz_clear (nj_z);
	    return result;
	  }
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, k);
    }
  else if (SCM_BIGP (j))
    {
      if (SCM_I_INUMP (k))
	{
	  SCM_SWAP (j, k);
	  nj = SCM_I_INUM (j);
	  goto intbig;
	}
      else if (SCM_BIGP (k))
	{
	  SCM result;
	  mpz_t result_z;
	  mpz_init (result_z);
	  mpz_and (result_z,
		   SCM_I_BIG_MPZ (j),
		   SCM_I_BIG_MPZ (k));
	  scm_remember_upto_here_2 (j, k);
	  result = scm_from_bool (mpz_sgn (result_z) != 0);
	  mpz_clear (result_z);
	  return result;
	}
      else
	SCM_WRONG_TYPE_ARG (SCM_ARG2, k);
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG1, j);
}
#undef FUNC_NAME


SCM_DEFINE (scm_logbit_p, "logbit?", 2, 0, 0,
            (SCM index, SCM j),
	    "Test whether bit number @var{index} in @var{j} is set.\n"
	    "@var{index} starts from 0 for the least significant bit.\n"
	    "\n"
	    "@lisp\n"
	    "(logbit? 0 #b1101) @result{} #t\n"
	    "(logbit? 1 #b1101) @result{} #f\n"
	    "(logbit? 2 #b1101) @result{} #t\n"
	    "(logbit? 3 #b1101) @result{} #t\n"
	    "(logbit? 4 #b1101) @result{} #f\n"
	    "@end lisp")
#define FUNC_NAME s_scm_logbit_p
{
  unsigned long int iindex;
  iindex = scm_to_ulong (index);

  if (SCM_I_INUMP (j))
    {
      /* bits above what's in an inum follow the sign bit */
      iindex = min (iindex, SCM_LONG_BIT - 1);
      return scm_from_bool ((1L << iindex) & SCM_I_INUM (j));
    }
  else if (SCM_BIGP (j))
    {
      int val = mpz_tstbit (SCM_I_BIG_MPZ (j), iindex);
      scm_remember_upto_here_1 (j);
      return scm_from_bool (val);
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG2, j);
}
#undef FUNC_NAME


SCM_DEFINE (scm_lognot, "lognot", 1, 0, 0, 
            (SCM n),
	    "Return the integer which is the ones-complement of the integer\n"
	    "argument.\n"
	    "\n"
	    "@lisp\n"
	    "(number->string (lognot #b10000000) 2)\n"
	    "   @result{} \"-10000001\"\n"
	    "(number->string (lognot #b0) 2)\n"
	    "   @result{} \"-1\"\n"
	    "@end lisp")
#define FUNC_NAME s_scm_lognot
{
  if (SCM_I_INUMP (n)) {
    /* No overflow here, just need to toggle all the bits making up the inum.
       Enhancement: No need to strip the tag and add it back, could just xor
       a block of 1 bits, if that worked with the various debug versions of
       the SCM typedef.  */
    return SCM_I_MAKINUM (~ SCM_I_INUM (n));

  } else if (SCM_BIGP (n)) {
    SCM result = scm_i_mkbig ();
    mpz_com (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (n));
    scm_remember_upto_here_1 (n);
    return result;

  } else {
    SCM_WRONG_TYPE_ARG (SCM_ARG1, n);
  }
}
#undef FUNC_NAME

/* returns 0 if IN is not an integer.  OUT must already be
   initialized. */
static int
coerce_to_big (SCM in, mpz_t out)
{
  if (SCM_BIGP (in))
    mpz_set (out, SCM_I_BIG_MPZ (in));
  else if (SCM_I_INUMP (in))
    mpz_set_si (out, SCM_I_INUM (in));
  else
    return 0;

  return 1;
}

SCM_DEFINE (scm_modulo_expt, "modulo-expt", 3, 0, 0,
            (SCM n, SCM k, SCM m),
            "Return @var{n} raised to the integer exponent\n"
	    "@var{k}, modulo @var{m}.\n"
	    "\n"
	    "@lisp\n"
	    "(modulo-expt 2 3 5)\n"
	    "   @result{} 3\n"
	    "@end lisp")
#define FUNC_NAME s_scm_modulo_expt
{
  mpz_t n_tmp; 
  mpz_t k_tmp; 
  mpz_t m_tmp; 
    
  /* There are two classes of error we might encounter --
     1) Math errors, which we'll report by calling scm_num_overflow,
     and
     2) wrong-type errors, which of course we'll report by calling
     SCM_WRONG_TYPE_ARG.
     We don't report those errors immediately, however; instead we do
     some cleanup first.  These variables tell us which error (if
     any) we should report after cleaning up.  
  */
  int report_overflow = 0;

  int position_of_wrong_type = 0;
  SCM value_of_wrong_type = SCM_INUM0;

  SCM result = SCM_UNDEFINED;

  mpz_init (n_tmp);
  mpz_init (k_tmp);
  mpz_init (m_tmp);
    
  if (scm_is_eq (m, SCM_INUM0))
    {
      report_overflow = 1;
      goto cleanup;
    }
  
  if (!coerce_to_big (n, n_tmp))
    {
      value_of_wrong_type = n;
      position_of_wrong_type = 1;
      goto cleanup;
    }

  if (!coerce_to_big (k, k_tmp))
    {
      value_of_wrong_type = k;
      position_of_wrong_type = 2;
      goto cleanup;
    }

  if (!coerce_to_big (m, m_tmp))
    {
      value_of_wrong_type = m;
      position_of_wrong_type = 3;
      goto cleanup;
    }

  /* if the exponent K is negative, and we simply call mpz_powm, we
     will get a divide-by-zero exception when an inverse 1/n mod m
     doesn't exist (or is not unique).  Since exceptions are hard to
     handle, we'll attempt the inversion "by hand" -- that way, we get
     a simple failure code, which is easy to handle. */
  
  if (-1 == mpz_sgn (k_tmp))
    {
      if (!mpz_invert (n_tmp, n_tmp, m_tmp))
        {
          report_overflow = 1;
          goto cleanup;
        }
      mpz_neg (k_tmp, k_tmp);
    }

  result = scm_i_mkbig ();
  mpz_powm (SCM_I_BIG_MPZ (result),
            n_tmp,
            k_tmp,
            m_tmp);

  if (mpz_sgn (m_tmp) < 0 && mpz_sgn (SCM_I_BIG_MPZ (result)) != 0)
    mpz_add (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (result), m_tmp);

 cleanup:
  mpz_clear (m_tmp);
  mpz_clear (k_tmp);
  mpz_clear (n_tmp);

  if (report_overflow)
    scm_num_overflow (FUNC_NAME);

  if (position_of_wrong_type)
    SCM_WRONG_TYPE_ARG (position_of_wrong_type,
                        value_of_wrong_type);
  
  return scm_i_normbig (result);
}
#undef FUNC_NAME

SCM_DEFINE (scm_integer_expt, "integer-expt", 2, 0, 0,
            (SCM n, SCM k),
	    "Return @var{n} raised to the power @var{k}.  @var{k} must be an\n"
	    "exact integer, @var{n} can be any number.\n"
	    "\n"
	    "Negative @var{k} is supported, and results in @math{1/n^abs(k)}\n"
	    "in the usual way.  @math{@var{n}^0} is 1, as usual, and that\n"
	    "includes @math{0^0} is 1.\n"
	    "\n"
	    "@lisp\n"
	    "(integer-expt 2 5)   @result{} 32\n"
	    "(integer-expt -3 3)  @result{} -27\n"
	    "(integer-expt 5 -3)  @result{} 1/125\n"
	    "(integer-expt 0 0)   @result{} 1\n"
	    "@end lisp")
#define FUNC_NAME s_scm_integer_expt
{
  long i2 = 0;
  SCM z_i2 = SCM_BOOL_F;
  int i2_is_big = 0;
  SCM acc = SCM_I_MAKINUM (1L);

  /* 0^0 == 1 according to R5RS */
  if (scm_is_eq (n, SCM_INUM0) || scm_is_eq (n, acc))
    return scm_is_false (scm_zero_p(k)) ? n : acc;
  else if (scm_is_eq (n, SCM_I_MAKINUM (-1L)))
    return scm_is_false (scm_even_p (k)) ? n : acc;

  if (SCM_I_INUMP (k))
    i2 = SCM_I_INUM (k);
  else if (SCM_BIGP (k))
    {
      z_i2 = scm_i_clonebig (k, 1);
      scm_remember_upto_here_1 (k);
      i2_is_big = 1;
    }
  else
    SCM_WRONG_TYPE_ARG (2, k);
  
  if (i2_is_big)
    {
      if (mpz_sgn(SCM_I_BIG_MPZ (z_i2)) == -1)
        {
          mpz_neg (SCM_I_BIG_MPZ (z_i2), SCM_I_BIG_MPZ (z_i2));
          n = scm_divide (n, SCM_UNDEFINED);
        }
      while (1)
        {
          if (mpz_sgn(SCM_I_BIG_MPZ (z_i2)) == 0)
            {
              return acc;
            }
          if (mpz_cmp_ui(SCM_I_BIG_MPZ (z_i2), 1) == 0)
            {
              return scm_product (acc, n);
            }
          if (mpz_tstbit(SCM_I_BIG_MPZ (z_i2), 0))
            acc = scm_product (acc, n);
          n = scm_product (n, n);
          mpz_fdiv_q_2exp (SCM_I_BIG_MPZ (z_i2), SCM_I_BIG_MPZ (z_i2), 1);
        }
    }
  else
    {
      if (i2 < 0)
        {
          i2 = -i2;
          n = scm_divide (n, SCM_UNDEFINED);
        }
      while (1)
        {
          if (0 == i2)
            return acc;
          if (1 == i2)
            return scm_product (acc, n);
          if (i2 & 1)
            acc = scm_product (acc, n);
          n = scm_product (n, n);
          i2 >>= 1;
        }
    }
}
#undef FUNC_NAME

SCM_DEFINE (scm_ash, "ash", 2, 0, 0,
            (SCM n, SCM cnt),
	    "Return @var{n} shifted left by @var{cnt} bits, or shifted right\n"
	    "if @var{cnt} is negative.  This is an ``arithmetic'' shift.\n"
	    "\n"
	    "This is effectively a multiplication by 2^@var{cnt}, and when\n"
	    "@var{cnt} is negative it's a division, rounded towards negative\n"
	    "infinity.  (Note that this is not the same rounding as\n"
	    "@code{quotient} does.)\n"
	    "\n"
	    "With @var{n} viewed as an infinite precision twos complement,\n"
	    "@code{ash} means a left shift introducing zero bits, or a right\n"
	    "shift dropping bits.\n"
	    "\n"
	    "@lisp\n"
	    "(number->string (ash #b1 3) 2)     @result{} \"1000\"\n"
	    "(number->string (ash #b1010 -1) 2) @result{} \"101\"\n"
	    "\n"
	    ";; -23 is bits ...11101001, -6 is bits ...111010\n"
	    "(ash -23 -2) @result{} -6\n"
	    "@end lisp")
#define FUNC_NAME s_scm_ash
{
  long bits_to_shift;
  bits_to_shift = scm_to_long (cnt);

  if (SCM_I_INUMP (n))
    {
      long nn = SCM_I_INUM (n);

      if (bits_to_shift > 0)
        {
          /* Left shift of bits_to_shift >= SCM_I_FIXNUM_BIT-1 will always
             overflow a non-zero fixnum.  For smaller shifts we check the
             bits going into positions above SCM_I_FIXNUM_BIT-1.  If they're
             all 0s for nn>=0, or all 1s for nn<0 then there's no overflow.
             Those bits are "nn >> (SCM_I_FIXNUM_BIT-1 -
             bits_to_shift)".  */

          if (nn == 0)
            return n;

          if (bits_to_shift < SCM_I_FIXNUM_BIT-1
              && ((unsigned long)
                  (SCM_SRS (nn, (SCM_I_FIXNUM_BIT-1 - bits_to_shift)) + 1)
                  <= 1))
            {
              return SCM_I_MAKINUM (nn << bits_to_shift);
            }
          else
            {
              SCM result = scm_i_long2big (nn);
              mpz_mul_2exp (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (result),
                            bits_to_shift);
              return result;
            }
        }
      else
        {
          bits_to_shift = -bits_to_shift;
          if (bits_to_shift >= SCM_LONG_BIT)
            return (nn >= 0 ? SCM_I_MAKINUM (0) : SCM_I_MAKINUM(-1));
          else
            return SCM_I_MAKINUM (SCM_SRS (nn, bits_to_shift));
        }

    }
  else if (SCM_BIGP (n))
    {
      SCM result;

      if (bits_to_shift == 0)
        return n;

      result = scm_i_mkbig ();
      if (bits_to_shift >= 0)
        {
          mpz_mul_2exp (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (n),
                        bits_to_shift);
          return result;
        }
      else
        {
          /* GMP doesn't have an fdiv_q_2exp variant returning just a long, so
             we have to allocate a bignum even if the result is going to be a
             fixnum.  */
          mpz_fdiv_q_2exp (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (n),
                           -bits_to_shift);
          return scm_i_normbig (result);
        }

    }
  else
    {
      SCM_WRONG_TYPE_ARG (SCM_ARG1, n);
    }
}
#undef FUNC_NAME


SCM_DEFINE (scm_bit_extract, "bit-extract", 3, 0, 0,
            (SCM n, SCM start, SCM end),
	    "Return the integer composed of the @var{start} (inclusive)\n"
	    "through @var{end} (exclusive) bits of @var{n}.  The\n"
	    "@var{start}th bit becomes the 0-th bit in the result.\n"
	    "\n"
	    "@lisp\n"
	    "(number->string (bit-extract #b1101101010 0 4) 2)\n"
	    "   @result{} \"1010\"\n"
	    "(number->string (bit-extract #b1101101010 4 9) 2)\n"
	    "   @result{} \"10110\"\n"
	    "@end lisp")
#define FUNC_NAME s_scm_bit_extract
{
  unsigned long int istart, iend, bits;
  istart = scm_to_ulong (start);
  iend = scm_to_ulong (end);
  SCM_ASSERT_RANGE (3, end, (iend >= istart));

  /* how many bits to keep */
  bits = iend - istart;

  if (SCM_I_INUMP (n))
    {
      long int in = SCM_I_INUM (n);

      /* When istart>=SCM_I_FIXNUM_BIT we can just limit the shift to
         SCM_I_FIXNUM_BIT-1 to get either 0 or -1 per the sign of "in". */
      in = SCM_SRS (in, min (istart, SCM_I_FIXNUM_BIT-1));

      if (in < 0 && bits >= SCM_I_FIXNUM_BIT)
	{
	  /* Since we emulate two's complement encoded numbers, this
	   * special case requires us to produce a result that has
	   * more bits than can be stored in a fixnum.
	   */
          SCM result = scm_i_long2big (in);
          mpz_fdiv_r_2exp (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (result),
                           bits);
          return result;
	}

      /* mask down to requisite bits */
      bits = min (bits, SCM_I_FIXNUM_BIT);
      return SCM_I_MAKINUM (in & ((1L << bits) - 1));
    }
  else if (SCM_BIGP (n))
    {
      SCM result;
      if (bits == 1)
        {
          result = SCM_I_MAKINUM (mpz_tstbit (SCM_I_BIG_MPZ (n), istart));
        }
      else
        {
          /* ENHANCE-ME: It'd be nice not to allocate a new bignum when
             bits<SCM_I_FIXNUM_BIT.  Would want some help from GMP to get
             such bits into a ulong.  */
          result = scm_i_mkbig ();
          mpz_fdiv_q_2exp (SCM_I_BIG_MPZ(result), SCM_I_BIG_MPZ(n), istart);
          mpz_fdiv_r_2exp (SCM_I_BIG_MPZ(result), SCM_I_BIG_MPZ(result), bits);
          result = scm_i_normbig (result);
        }
      scm_remember_upto_here_1 (n);
      return result;
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG1, n);
}
#undef FUNC_NAME


static const char scm_logtab[] = {
  0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4
};

SCM_DEFINE (scm_logcount, "logcount", 1, 0, 0,
            (SCM n),
	    "Return the number of bits in integer @var{n}.  If integer is\n"
	    "positive, the 1-bits in its binary representation are counted.\n"
	    "If negative, the 0-bits in its two's-complement binary\n"
	    "representation are counted.  If 0, 0 is returned.\n"
	    "\n"
	    "@lisp\n"
	    "(logcount #b10101010)\n"
	    "   @result{} 4\n"
	    "(logcount 0)\n"
	    "   @result{} 0\n"
	    "(logcount -2)\n"
	    "   @result{} 1\n"
	    "@end lisp")
#define FUNC_NAME s_scm_logcount
{
  if (SCM_I_INUMP (n))
    {
      unsigned long int c = 0;
      long int nn = SCM_I_INUM (n);
      if (nn < 0)
        nn = -1 - nn;
      while (nn)
        {
          c += scm_logtab[15 & nn];
          nn >>= 4;
        }
      return SCM_I_MAKINUM (c);
    }
  else if (SCM_BIGP (n))
    {
      unsigned long count;
      if (mpz_sgn (SCM_I_BIG_MPZ (n)) >= 0)
        count = mpz_popcount (SCM_I_BIG_MPZ (n));
      else
        count = mpz_hamdist (SCM_I_BIG_MPZ (n), z_negative_one);
      scm_remember_upto_here_1 (n);
      return SCM_I_MAKINUM (count);
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG1, n);
}
#undef FUNC_NAME


static const char scm_ilentab[] = {
  0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
};


SCM_DEFINE (scm_integer_length, "integer-length", 1, 0, 0,
            (SCM n),
	    "Return the number of bits necessary to represent @var{n}.\n"
	    "\n"
	    "@lisp\n"
	    "(integer-length #b10101010)\n"
	    "   @result{} 8\n"
	    "(integer-length 0)\n"
	    "   @result{} 0\n"
	    "(integer-length #b1111)\n"
	    "   @result{} 4\n"
	    "@end lisp")
#define FUNC_NAME s_scm_integer_length
{
  if (SCM_I_INUMP (n))
    {
      unsigned long int c = 0;
      unsigned int l = 4;
      long int nn = SCM_I_INUM (n);
      if (nn < 0)
	nn = -1 - nn;
      while (nn)
	{
	  c += 4;
	  l = scm_ilentab [15 & nn];
	  nn >>= 4;
	}
      return SCM_I_MAKINUM (c - 4 + l);
    }
  else if (SCM_BIGP (n))
    {
      /* mpz_sizeinbase looks at the absolute value of negatives, whereas we
	 want a ones-complement.  If n is ...111100..00 then mpz_sizeinbase is
	 1 too big, so check for that and adjust.  */
      size_t size = mpz_sizeinbase (SCM_I_BIG_MPZ (n), 2);
      if (mpz_sgn (SCM_I_BIG_MPZ (n)) < 0
	  && mpz_scan0 (SCM_I_BIG_MPZ (n),  /* no 0 bits above the lowest 1 */
			mpz_scan1 (SCM_I_BIG_MPZ (n), 0)) == ULONG_MAX)
	size--;
      scm_remember_upto_here_1 (n);
      return SCM_I_MAKINUM (size);
    }
  else
    SCM_WRONG_TYPE_ARG (SCM_ARG1, n);
}
#undef FUNC_NAME

/*** NUMBERS -> STRINGS ***/
#define SCM_MAX_DBL_PREC  60
#define SCM_MAX_DBL_RADIX 36

/* this is an array starting with radix 2, and ending with radix SCM_MAX_DBL_RADIX */
static int scm_dblprec[SCM_MAX_DBL_RADIX - 1];
static double fx_per_radix[SCM_MAX_DBL_RADIX - 1][SCM_MAX_DBL_PREC];

static
void init_dblprec(int *prec, int radix) {
   /* determine floating point precision by adding successively
      smaller increments to 1.0 until it is considered == 1.0 */
   double f = ((double)1.0)/radix;
   double fsum = 1.0 + f;

   *prec = 0;
   while (fsum != 1.0)
   {
      if (++(*prec) > SCM_MAX_DBL_PREC)
         fsum = 1.0;
      else
      {
         f /= radix;
         fsum = f + 1.0;
      }
   }
   (*prec) -= 1;
}

static
void init_fx_radix(double *fx_list, int radix)
{
  /* initialize a per-radix list of tolerances.  When added
     to a number < 1.0, we can determine if we should raund
     up and quit converting a number to a string. */
   int i;
   fx_list[0] = 0.0;
   fx_list[1] = 0.5;
   for( i = 2 ; i < SCM_MAX_DBL_PREC; ++i ) 
     fx_list[i] = (fx_list[i-1] / radix);
}

/* use this array as a way to generate a single digit */
static const char*number_chars="0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ";

static size_t
idbl2str (double f, char *a, int radix)
{
   int efmt, dpt, d, i, wp;
   double *fx;
#ifdef DBL_MIN_10_EXP
   double f_cpy;
   int exp_cpy;
#endif /* DBL_MIN_10_EXP */
   size_t ch = 0;
   int exp = 0;

   if(radix < 2 || 
      radix > SCM_MAX_DBL_RADIX)
   {
      /* revert to existing behavior */
      radix = 10;
   }

   wp = scm_dblprec[radix-2];
   fx = fx_per_radix[radix-2];

  if (f == 0.0)
    {
#ifdef HAVE_COPYSIGN
      double sgn = copysign (1.0, f);

      if (sgn < 0.0)
	a[ch++] = '-';
#endif
      goto zero;	/*{a[0]='0'; a[1]='.'; a[2]='0'; return 3;} */
    }

  if (xisinf (f))
    {
      if (f < 0)
	strcpy (a, "-inf.0");
      else
	strcpy (a, "+inf.0");
      return ch+6;
    }
  else if (xisnan (f))
    {
      strcpy (a, "+nan.0");
      return ch+6;
    }

  if (f < 0.0)
    {
      f = -f;
      a[ch++] = '-';
    }

#ifdef DBL_MIN_10_EXP  /* Prevent unnormalized values, as from 
			  make-uniform-vector, from causing infinite loops. */
  /* just do the checking...if it passes, we do the conversion for our
     radix again below */
  f_cpy = f;
  exp_cpy = exp;

  while (f_cpy < 1.0)
    {
      f_cpy *= 10.0;
      if (exp_cpy-- < DBL_MIN_10_EXP)
	{
	  a[ch++] = '#';
	  a[ch++] = '.';
	  a[ch++] = '#';
	  return ch;
	}
    }
  while (f_cpy > 10.0)
    {
      f_cpy *= 0.10;
      if (exp_cpy++ > DBL_MAX_10_EXP)
	{
	  a[ch++] = '#';
	  a[ch++] = '.';
	  a[ch++] = '#';
	  return ch;
	}
    }
#endif

  while (f < 1.0)
    {
      f *= radix;
      exp--;
    }
  while (f > radix)
    {
      f /= radix;
      exp++;
    }

  if (f + fx[wp] >= radix)
    {
      f = 1.0;
      exp++;
    }
 zero:
#ifdef ENGNOT 
  /* adding 9999 makes this equivalent to abs(x) % 3 */
  dpt = (exp + 9999) % 3;
  exp -= dpt++;
  efmt = 1;
#else
  efmt = (exp < -3) || (exp > wp + 2);
  if (!efmt)
    {
      if (exp < 0)
	{
	  a[ch++] = '0';
	  a[ch++] = '.';
	  dpt = exp;
	  while (++dpt)
	    a[ch++] = '0';
	}
      else
	dpt = exp + 1;
    }
  else
    dpt = 1;
#endif

  do
    {
      d = f;
      f -= d;
      a[ch++] = number_chars[d];
      if (f < fx[wp])
	break;
      if (f + fx[wp] >= 1.0)
	{
          a[ch - 1] = number_chars[d+1]; 
	  break;
	}
      f *= radix;
      if (!(--dpt))
	a[ch++] = '.';
    }
  while (wp--);

  if (dpt > 0)
    {
#ifndef ENGNOT
      if ((dpt > 4) && (exp > 6))
	{
	  d = (a[0] == '-' ? 2 : 1);
	  for (i = ch++; i > d; i--)
	    a[i] = a[i - 1];
	  a[d] = '.';
	  efmt = 1;
	}
      else
#endif
	{
	  while (--dpt)
	    a[ch++] = '0';
	  a[ch++] = '.';
	}
    }
  if (a[ch - 1] == '.')
    a[ch++] = '0';		/* trailing zero */
  if (efmt && exp)
    {
      a[ch++] = 'e';
      if (exp < 0)
	{
	  exp = -exp;
	  a[ch++] = '-';
	}
      for (i = radix; i <= exp; i *= radix);
      for (i /= radix; i; i /= radix)
	{
          a[ch++] = number_chars[exp / i];
	  exp %= i;
	}
    }
  return ch;
}


static size_t
icmplx2str (double real, double imag, char *str, int radix)
{
  size_t i;
  
  i = idbl2str (real, str, radix);
  if (imag != 0.0)
    {
      /* Don't output a '+' for negative numbers or for Inf and
	 NaN.  They will provide their own sign. */
      if (0 <= imag && !xisinf (imag) && !xisnan (imag))
	str[i++] = '+';
      i += idbl2str (imag, &str[i], radix);
      str[i++] = 'i';
    }
  return i;
}

static size_t
iflo2str (SCM flt, char *str, int radix)
{
  size_t i;
  if (SCM_REALP (flt))
    i = idbl2str (SCM_REAL_VALUE (flt), str, radix);
  else
    i = icmplx2str (SCM_COMPLEX_REAL (flt), SCM_COMPLEX_IMAG (flt),
		    str, radix);
  return i;
}

/* convert a scm_t_intmax to a string (unterminated).  returns the number of
   characters in the result. 
   rad is output base
   p is destination: worst case (base 2) is SCM_INTBUFLEN  */
size_t
scm_iint2str (scm_t_intmax num, int rad, char *p)
{
  if (num < 0)
    {
      *p++ = '-';
      return scm_iuint2str (-num, rad, p) + 1;
    }
  else
    return scm_iuint2str (num, rad, p);
}

/* convert a scm_t_intmax to a string (unterminated).  returns the number of
   characters in the result. 
   rad is output base
   p is destination: worst case (base 2) is SCM_INTBUFLEN  */
size_t
scm_iuint2str (scm_t_uintmax num, int rad, char *p)
{
  size_t j = 1;
  size_t i;
  scm_t_uintmax n = num;

  for (n /= rad; n > 0; n /= rad)
    j++;

  i = j;
  n = num;
  while (i--)
    {
      int d = n % rad;

      n /= rad;
      p[i] = d + ((d < 10) ? '0' : 'a' - 10);
    }
  return j;
}

SCM_DEFINE (scm_number_to_string, "number->string", 1, 1, 0,
            (SCM n, SCM radix),
	    "Return a string holding the external representation of the\n"
	    "number @var{n} in the given @var{radix}.  If @var{n} is\n"
	    "inexact, a radix of 10 will be used.")
#define FUNC_NAME s_scm_number_to_string
{
  int base;

  if (SCM_UNBNDP (radix))
    base = 10;
  else
    base = scm_to_signed_integer (radix, 2, 36);

  if (SCM_I_INUMP (n))
    {
      char num_buf [SCM_INTBUFLEN];
      size_t length = scm_iint2str (SCM_I_INUM (n), base, num_buf);
      return scm_from_locale_stringn (num_buf, length);
    }
  else if (SCM_BIGP (n))
    {
      char *str = mpz_get_str (NULL, base, SCM_I_BIG_MPZ (n));
      scm_remember_upto_here_1 (n);
      return scm_take_locale_string (str);
    }
  else if (SCM_FRACTIONP (n))
    {
      return scm_string_append (scm_list_3 (scm_number_to_string (SCM_FRACTION_NUMERATOR (n), radix),
					    scm_from_locale_string ("/"), 
					    scm_number_to_string (SCM_FRACTION_DENOMINATOR (n), radix)));
    }
  else if (SCM_INEXACTP (n))
    {
      char num_buf [FLOBUFLEN];
      return scm_from_locale_stringn (num_buf, iflo2str (n, num_buf, base));
    }
  else
    SCM_WRONG_TYPE_ARG (1, n);
}
#undef FUNC_NAME


/* These print routines used to be stubbed here so that scm_repl.c
   wouldn't need SCM_BIGDIG conditionals (pre GMP) */

int
scm_print_real (SCM sexp, SCM port, scm_print_state *pstate SCM_UNUSED)
{
  char num_buf[FLOBUFLEN];
  scm_lfwrite (num_buf, iflo2str (sexp, num_buf, 10), port);
  return !0;
}

void
scm_i_print_double (double val, SCM port)
{
  char num_buf[FLOBUFLEN];
  scm_lfwrite (num_buf, idbl2str (val, num_buf, 10), port);
}

int
scm_print_complex (SCM sexp, SCM port, scm_print_state *pstate SCM_UNUSED)

{
  char num_buf[FLOBUFLEN];
  scm_lfwrite (num_buf, iflo2str (sexp, num_buf, 10), port);
  return !0;
}

void
scm_i_print_complex (double real, double imag, SCM port)
{
  char num_buf[FLOBUFLEN];
  scm_lfwrite (num_buf, icmplx2str (real, imag, num_buf, 10), port);
}

int
scm_i_print_fraction (SCM sexp, SCM port, scm_print_state *pstate SCM_UNUSED)
{
  SCM str;
  str = scm_number_to_string (sexp, SCM_UNDEFINED);
  scm_lfwrite (scm_i_string_chars (str), scm_i_string_length (str), port);
  scm_remember_upto_here_1 (str);
  return !0;
}

int
scm_bigprint (SCM exp, SCM port, scm_print_state *pstate SCM_UNUSED)
{
  char *str = mpz_get_str (NULL, 10, SCM_I_BIG_MPZ (exp));
  scm_remember_upto_here_1 (exp);
  scm_lfwrite (str, (size_t) strlen (str), port);
  free (str);
  return !0;
}
/*** END nums->strs ***/


/*** STRINGS -> NUMBERS ***/

/* The following functions implement the conversion from strings to numbers.
 * The implementation somehow follows the grammar for numbers as it is given
 * in R5RS.  Thus, the functions resemble syntactic units (<ureal R>,
 * <uinteger R>, ...) that are used to build up numbers in the grammar.  Some
 * points should be noted about the implementation:
 * * Each function keeps a local index variable 'idx' that points at the
 * current position within the parsed string.  The global index is only
 * updated if the function could parse the corresponding syntactic unit
 * successfully.
 * * Similarly, the functions keep track of indicators of inexactness ('#',
 * '.' or exponents) using local variables ('hash_seen', 'x').  Again, the
 * global exactness information is only updated after each part has been
 * successfully parsed.
 * * Sequences of digits are parsed into temporary variables holding fixnums.
 * Only if these fixnums would overflow, the result variables are updated
 * using the standard functions scm_add, scm_product, scm_divide etc.  Then,
 * the temporary variables holding the fixnums are cleared, and the process
 * starts over again.  If for example fixnums were able to store five decimal
 * digits, a number 1234567890 would be parsed in two parts 12345 and 67890,
 * and the result was computed as 12345 * 100000 + 67890.  In other words,
 * only every five digits two bignum operations were performed.
 */

enum t_exactness {NO_EXACTNESS, INEXACT, EXACT};

/* R5RS, section 7.1.1, lexical structure of numbers: <uinteger R>. */

/* In non ASCII-style encodings the following macro might not work. */
#define XDIGIT2UINT(d)                                  \
  (isdigit ((int) (unsigned char) d)                    \
   ? (d) - '0'                                          \
   : tolower ((int) (unsigned char) d) - 'a' + 10)

static SCM
mem2uinteger (const char* mem, size_t len, unsigned int *p_idx,
	      unsigned int radix, enum t_exactness *p_exactness)
{
  unsigned int idx = *p_idx;
  unsigned int hash_seen = 0;
  scm_t_bits shift = 1;
  scm_t_bits add = 0;
  unsigned int digit_value;
  SCM result;
  char c;

  if (idx == len)
    return SCM_BOOL_F;

  c = mem[idx];
  if (!isxdigit ((int) (unsigned char) c))
    return SCM_BOOL_F;
  digit_value = XDIGIT2UINT (c);
  if (digit_value >= radix)
    return SCM_BOOL_F;

  idx++;
  result = SCM_I_MAKINUM (digit_value);
  while (idx != len)
    {
      char c = mem[idx];
      if (isxdigit ((int) (unsigned char) c))
	{
	  if (hash_seen)
	    break;
	  digit_value = XDIGIT2UINT (c);
	  if (digit_value >= radix)
	    break;
	}
      else if (c == '#')
	{
	  hash_seen = 1;
	  digit_value = 0;
	}
      else
	break;

      idx++;
      if (SCM_MOST_POSITIVE_FIXNUM / radix < shift)
	{
	  result = scm_product (result, SCM_I_MAKINUM (shift));
	  if (add > 0)
	    result = scm_sum (result, SCM_I_MAKINUM (add));

	  shift = radix;
	  add = digit_value;
	}
      else
	{
	  shift = shift * radix;
	  add = add * radix + digit_value;
	}
    };

  if (shift > 1)
    result = scm_product (result, SCM_I_MAKINUM (shift));
  if (add > 0)
    result = scm_sum (result, SCM_I_MAKINUM (add));

  *p_idx = idx;
  if (hash_seen)
    *p_exactness = INEXACT;

  return result;
}


/* R5RS, section 7.1.1, lexical structure of numbers: <decimal 10>.  Only
 * covers the parts of the rules that start at a potential point.  The value
 * of the digits up to the point have been parsed by the caller and are given
 * in variable result.  The content of *p_exactness indicates, whether a hash
 * has already been seen in the digits before the point.
 */

/* In non ASCII-style encodings the following macro might not work. */
#define DIGIT2UINT(d) ((d) - '0')

static SCM
mem2decimal_from_point (SCM result, const char* mem, size_t len, 
			unsigned int *p_idx, enum t_exactness *p_exactness)
{
  unsigned int idx = *p_idx;
  enum t_exactness x = *p_exactness;

  if (idx == len)
    return result;

  if (mem[idx] == '.')
    {
      scm_t_bits shift = 1;
      scm_t_bits add = 0;
      unsigned int digit_value;
      SCM big_shift = SCM_I_MAKINUM (1);

      idx++;
      while (idx != len)
	{
	  char c = mem[idx];
	  if (isdigit ((int) (unsigned char) c))
	    {
	      if (x == INEXACT)
		return SCM_BOOL_F;
	      else
		digit_value = DIGIT2UINT (c);
	    }
	  else if (c == '#')
	    {
	      x = INEXACT;
	      digit_value = 0;
	    }
	  else
	    break;

	  idx++;
	  if (SCM_MOST_POSITIVE_FIXNUM / 10 < shift)
	    {
	      big_shift = scm_product (big_shift, SCM_I_MAKINUM (shift));
	      result = scm_product (result, SCM_I_MAKINUM (shift));
	      if (add > 0)
		result = scm_sum (result, SCM_I_MAKINUM (add));
	      
	      shift = 10;
	      add = digit_value;
	    }
	  else
	    {
	      shift = shift * 10;
	      add = add * 10 + digit_value;
	    }
	};

      if (add > 0)
	{
	  big_shift = scm_product (big_shift, SCM_I_MAKINUM (shift));
	  result = scm_product (result, SCM_I_MAKINUM (shift));
	  result = scm_sum (result, SCM_I_MAKINUM (add));
	}

      result = scm_divide (result, big_shift);

      /* We've seen a decimal point, thus the value is implicitly inexact. */
      x = INEXACT;
    }

  if (idx != len)
    {
      int sign = 1;
      unsigned int start;
      char c;
      int exponent;
      SCM e;

      /* R5RS, section 7.1.1, lexical structure of numbers: <suffix> */

      switch (mem[idx])
	{
	case 'd': case 'D':
	case 'e': case 'E':
	case 'f': case 'F':
	case 'l': case 'L':
	case 's': case 'S':
	  idx++;
          if (idx == len)
            return SCM_BOOL_F;

	  start = idx;
	  c = mem[idx];
	  if (c == '-')
	    {
	      idx++;
              if (idx == len)
                return SCM_BOOL_F;

	      sign = -1;
	      c = mem[idx];
	    }
	  else if (c == '+')
	    {
	      idx++;
              if (idx == len)
                return SCM_BOOL_F;

	      sign = 1;
	      c = mem[idx];
	    }
	  else
	    sign = 1;

	  if (!isdigit ((int) (unsigned char) c))
	    return SCM_BOOL_F;

	  idx++;
	  exponent = DIGIT2UINT (c);
	  while (idx != len)
	    {
	      char c = mem[idx];
	      if (isdigit ((int) (unsigned char) c))
		{
		  idx++;
		  if (exponent <= SCM_MAXEXP)
		    exponent = exponent * 10 + DIGIT2UINT (c);
		}
	      else
		break;
	    }

	  if (exponent > SCM_MAXEXP)
	    {
	      size_t exp_len = idx - start;
	      SCM exp_string = scm_from_locale_stringn (&mem[start], exp_len);
	      SCM exp_num = scm_string_to_number (exp_string, SCM_UNDEFINED);
	      scm_out_of_range ("string->number", exp_num);
	    }

	  e = scm_integer_expt (SCM_I_MAKINUM (10), SCM_I_MAKINUM (exponent));
	  if (sign == 1)
	    result = scm_product (result, e);
	  else
	    result = scm_divide2real (result, e);

	  /* We've seen an exponent, thus the value is implicitly inexact. */
	  x = INEXACT;

	  break;

	default:
	  break;
	}
    }

  *p_idx = idx;
  if (x == INEXACT)
    *p_exactness = x;

  return result;
}


/* R5RS, section 7.1.1, lexical structure of numbers: <ureal R> */

static SCM
mem2ureal (const char* mem, size_t len, unsigned int *p_idx,
	   unsigned int radix, enum t_exactness *p_exactness)
{
  unsigned int idx = *p_idx;
  SCM result;

  /* Start off believing that the number will be exact.  This changes
     to INEXACT if we see a decimal point or a hash. */
  enum t_exactness x = EXACT;

  if (idx == len)
    return SCM_BOOL_F;

  if (idx+5 <= len && !strncmp (mem+idx, "inf.0", 5))
    {
      *p_idx = idx+5;
      return scm_inf ();
    }

  if (idx+4 < len && !strncmp (mem+idx, "nan.", 4))
    {
      /* Cobble up the fractional part.  We might want to set the
	 NaN's mantissa from it. */
      idx += 4;
      mem2uinteger (mem, len, &idx, 10, &x);
      *p_idx = idx;
      return scm_nan ();
    }

  if (mem[idx] == '.')
    {
      if (radix != 10)
	return SCM_BOOL_F;
      else if (idx + 1 == len)
	return SCM_BOOL_F;
      else if (!isdigit ((int) (unsigned char) mem[idx + 1]))
	return SCM_BOOL_F;
      else
	result = mem2decimal_from_point (SCM_I_MAKINUM (0), mem, len,
					 p_idx, &x);
    }
  else
    {
      SCM uinteger;

      uinteger = mem2uinteger (mem, len, &idx, radix, &x);
      if (scm_is_false (uinteger))
	return SCM_BOOL_F;

      if (idx == len)
	result = uinteger;
      else if (mem[idx] == '/')
	{
	  SCM divisor;

	  idx++;
          if (idx == len)
            return SCM_BOOL_F;

          divisor = mem2uinteger (mem, len, &idx, radix, &x);
	  if (scm_is_false (divisor))
	    return SCM_BOOL_F;

	  /* both are int/big here, I assume */
	  result = scm_i_make_ratio (uinteger, divisor);
	}
      else if (radix == 10)
	{
	  result = mem2decimal_from_point (uinteger, mem, len, &idx, &x);
	  if (scm_is_false (result))
	    return SCM_BOOL_F;
	}
      else
	result = uinteger;

      *p_idx = idx;
    }

  /* Update *p_exactness if the number just read was inexact.  This is
     important for complex numbers, so that a complex number is
     treated as inexact overall if either its real or imaginary part
     is inexact.
  */
  if (x == INEXACT)
    *p_exactness = x;

  /* When returning an inexact zero, make sure it is represented as a
     floating point value so that we can change its sign. 
  */
  if (scm_is_eq (result, SCM_I_MAKINUM(0)) && *p_exactness == INEXACT)
    result = scm_from_double (0.0);

  return result;
}


/* R5RS, section 7.1.1, lexical structure of numbers: <complex R> */

static SCM
mem2complex (const char* mem, size_t len, unsigned int idx,
	     unsigned int radix, enum t_exactness *p_exactness)
{
  char c;
  int sign = 0;
  SCM ureal;

  if (idx == len)
    return SCM_BOOL_F;

  c = mem[idx];
  if (c == '+')
    {
      idx++;
      sign = 1;
    }
  else if (c == '-')
    {
      idx++;
      sign = -1;
    }

  if (idx == len)
    return SCM_BOOL_F;

  ureal = mem2ureal (mem, len, &idx, radix, p_exactness);
  if (scm_is_false (ureal))
    {
      /* input must be either +i or -i */

      if (sign == 0)
	return SCM_BOOL_F;

      if (mem[idx] == 'i' || mem[idx] == 'I')
	{
	  idx++;
	  if (idx != len)
	    return SCM_BOOL_F;
	  
	  return scm_make_rectangular (SCM_I_MAKINUM (0), SCM_I_MAKINUM (sign));
	}
      else
	return SCM_BOOL_F;
    }
  else
    {
      if (sign == -1 && scm_is_false (scm_nan_p (ureal)))
	ureal = scm_difference (ureal, SCM_UNDEFINED);

      if (idx == len)
	return ureal;

      c = mem[idx];
      switch (c)
	{
	case 'i': case 'I':
	  /* either +<ureal>i or -<ureal>i */

	  idx++;
	  if (sign == 0)
	    return SCM_BOOL_F;
	  if (idx != len)
	    return SCM_BOOL_F;
	  return scm_make_rectangular (SCM_I_MAKINUM (0), ureal);

	case '@':
	  /* polar input: <real>@<real>. */

	  idx++;
	  if (idx == len)
	    return SCM_BOOL_F;
	  else
	    {
	      int sign;
	      SCM angle;
	      SCM result;

	      c = mem[idx];
	      if (c == '+')
		{
		  idx++;
                  if (idx == len)
                    return SCM_BOOL_F;
		  sign = 1;
		}
	      else if (c == '-')
		{
		  idx++;
                  if (idx == len)
                    return SCM_BOOL_F;
		  sign = -1;
		}
	      else
		sign = 1;

	      angle = mem2ureal (mem, len, &idx, radix, p_exactness);
	      if (scm_is_false (angle))
		return SCM_BOOL_F;
	      if (idx != len)
		return SCM_BOOL_F;

	      if (sign == -1 && scm_is_false (scm_nan_p (ureal)))
		angle = scm_difference (angle, SCM_UNDEFINED);

	      result = scm_make_polar (ureal, angle);
	      return result;
	    }
	case '+':
	case '-':
	  /* expecting input matching <real>[+-]<ureal>?i */

	  idx++;
	  if (idx == len)
	    return SCM_BOOL_F;
	  else
	    {
	      int sign = (c == '+') ? 1 : -1;
	      SCM imag = mem2ureal (mem, len, &idx, radix, p_exactness);

	      if (scm_is_false (imag))
		imag = SCM_I_MAKINUM (sign);
	      else if (sign == -1 && scm_is_false (scm_nan_p (ureal)))
		imag = scm_difference (imag, SCM_UNDEFINED);

	      if (idx == len)
		return SCM_BOOL_F;
	      if (mem[idx] != 'i' && mem[idx] != 'I')
		return SCM_BOOL_F;

	      idx++;
	      if (idx != len)
		return SCM_BOOL_F;

	      return scm_make_rectangular (ureal, imag);
	    }
	default:
	  return SCM_BOOL_F;
	}
    }
}


/* R5RS, section 7.1.1, lexical structure of numbers: <number> */

enum t_radix {NO_RADIX=0, DUAL=2, OCT=8, DEC=10, HEX=16};

SCM
scm_c_locale_stringn_to_number (const char* mem, size_t len,
				unsigned int default_radix)
{
  unsigned int idx = 0;
  unsigned int radix = NO_RADIX;
  enum t_exactness forced_x = NO_EXACTNESS;
  enum t_exactness implicit_x = EXACT;
  SCM result;

  /* R5RS, section 7.1.1, lexical structure of numbers: <prefix R> */
  while (idx + 2 < len && mem[idx] == '#')
    {
      switch (mem[idx + 1])
	{
	case 'b': case 'B':
	  if (radix != NO_RADIX)
	    return SCM_BOOL_F;
	  radix = DUAL;
	  break;
	case 'd': case 'D':
	  if (radix != NO_RADIX)
	    return SCM_BOOL_F;
	  radix = DEC;
	  break;
	case 'i': case 'I':
	  if (forced_x != NO_EXACTNESS)
	    return SCM_BOOL_F;
	  forced_x = INEXACT;
	  break;
	case 'e': case 'E':
	  if (forced_x != NO_EXACTNESS)
	    return SCM_BOOL_F;
	  forced_x = EXACT;
	  break;
	case 'o': case 'O':
	  if (radix != NO_RADIX)
	    return SCM_BOOL_F;
	  radix = OCT;
	  break;
	case 'x': case 'X':
	  if (radix != NO_RADIX)
	    return SCM_BOOL_F;
	  radix = HEX;
	  break;
	default:
	  return SCM_BOOL_F;
	}
      idx += 2;
    }

  /* R5RS, section 7.1.1, lexical structure of numbers: <complex R> */
  if (radix == NO_RADIX)
    result = mem2complex (mem, len, idx, default_radix, &implicit_x);
  else
    result = mem2complex (mem, len, idx, (unsigned int) radix, &implicit_x);

  if (scm_is_false (result))
    return SCM_BOOL_F;

  switch (forced_x)
    {
    case EXACT:
      if (SCM_INEXACTP (result))
	return scm_inexact_to_exact (result);
      else
	return result;
    case INEXACT:
      if (SCM_INEXACTP (result))
	return result;
      else
	return scm_exact_to_inexact (result);
    case NO_EXACTNESS:
    default:
      if (implicit_x == INEXACT)
	{
	  if (SCM_INEXACTP (result))
	    return result;
	  else
	    return scm_exact_to_inexact (result);
	}
      else
	return result;
    }
}


SCM_DEFINE (scm_string_to_number, "string->number", 1, 1, 0,
            (SCM string, SCM radix),
	    "Return a number of the maximally precise representation\n"
	    "expressed by the given @var{string}. @var{radix} must be an\n"
	    "exact integer, either 2, 8, 10, or 16. If supplied, @var{radix}\n"
	    "is a default radix that may be overridden by an explicit radix\n"
	    "prefix in @var{string} (e.g. \"#o177\"). If @var{radix} is not\n"
	    "supplied, then the default radix is 10. If string is not a\n"
	    "syntactically valid notation for a number, then\n"
	    "@code{string->number} returns @code{#f}.") 
#define FUNC_NAME s_scm_string_to_number
{
  SCM answer;
  unsigned int base;
  SCM_VALIDATE_STRING (1, string);

  if (SCM_UNBNDP (radix))
    base = 10;
  else
    base = scm_to_unsigned_integer (radix, 2, INT_MAX);

  answer = scm_c_locale_stringn_to_number (scm_i_string_chars (string),
					   scm_i_string_length (string),
					   base);
  scm_remember_upto_here_1 (string);
  return answer;
}
#undef FUNC_NAME


/*** END strs->nums ***/


SCM
scm_bigequal (SCM x, SCM y)
{
  int result = mpz_cmp (SCM_I_BIG_MPZ (x), SCM_I_BIG_MPZ (y));
  scm_remember_upto_here_2 (x, y);
  return scm_from_bool (0 == result);
}

SCM
scm_real_equalp (SCM x, SCM y)
{
  return scm_from_bool (SCM_REAL_VALUE (x) == SCM_REAL_VALUE (y));
}

SCM
scm_complex_equalp (SCM x, SCM y)
{
  return scm_from_bool (SCM_COMPLEX_REAL (x) == SCM_COMPLEX_REAL (y)
		   && SCM_COMPLEX_IMAG (x) == SCM_COMPLEX_IMAG (y));
}

SCM
scm_i_fraction_equalp (SCM x, SCM y)
{
  if (scm_is_false (scm_equal_p (SCM_FRACTION_NUMERATOR (x),
			       SCM_FRACTION_NUMERATOR (y)))
      || scm_is_false (scm_equal_p (SCM_FRACTION_DENOMINATOR (x),
				  SCM_FRACTION_DENOMINATOR (y))))
    return SCM_BOOL_F;
  else
    return SCM_BOOL_T;
}


SCM_DEFINE (scm_number_p, "number?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is a number, @code{#f}\n"
	    "otherwise.")
#define FUNC_NAME s_scm_number_p
{
  return scm_from_bool (SCM_NUMBERP (x));
}
#undef FUNC_NAME

SCM_DEFINE (scm_complex_p, "complex?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is a complex number, @code{#f}\n"
	    "otherwise.  Note that the sets of real, rational and integer\n"
	    "values form subsets of the set of complex numbers, i. e. the\n"
	    "predicate will also be fulfilled if @var{x} is a real,\n"
	    "rational or integer number.")
#define FUNC_NAME s_scm_complex_p
{
  /* all numbers are complex. */
  return scm_number_p (x);
}
#undef FUNC_NAME

SCM_DEFINE (scm_real_p, "real?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is a real number, @code{#f}\n"
	    "otherwise.  Note that the set of integer values forms a subset of\n"
	    "the set of real numbers, i. e. the predicate will also be\n"
	    "fulfilled if @var{x} is an integer number.")
#define FUNC_NAME s_scm_real_p
{
  /* we can't represent irrational numbers. */
  return scm_rational_p (x);
}
#undef FUNC_NAME

SCM_DEFINE (scm_rational_p, "rational?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is a rational number, @code{#f}\n"
	    "otherwise.  Note that the set of integer values forms a subset of\n"
	    "the set of rational numbers, i. e. the predicate will also be\n"
	    "fulfilled if @var{x} is an integer number.")
#define FUNC_NAME s_scm_rational_p
{
  if (SCM_I_INUMP (x))
    return SCM_BOOL_T;
  else if (SCM_IMP (x))
    return SCM_BOOL_F;
  else if (SCM_BIGP (x))
    return SCM_BOOL_T;
  else if (SCM_FRACTIONP (x))
    return SCM_BOOL_T;
  else if (SCM_REALP (x))
    /* due to their limited precision, all floating point numbers are
       rational as well. */
    return SCM_BOOL_T;
  else
    return SCM_BOOL_F;
}
#undef FUNC_NAME

SCM_DEFINE (scm_integer_p, "integer?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is an integer number, @code{#f}\n"
	    "else.")
#define FUNC_NAME s_scm_integer_p
{
  double r;
  if (SCM_I_INUMP (x))
    return SCM_BOOL_T;
  if (SCM_IMP (x))
    return SCM_BOOL_F;
  if (SCM_BIGP (x))
    return SCM_BOOL_T;
  if (!SCM_INEXACTP (x))
    return SCM_BOOL_F;
  if (SCM_COMPLEXP (x))
    return SCM_BOOL_F;
  r = SCM_REAL_VALUE (x);
  /* +/-inf passes r==floor(r), making those #t */
  if (r == floor (r))
    return SCM_BOOL_T;
  return SCM_BOOL_F;
}
#undef FUNC_NAME


SCM_DEFINE (scm_inexact_p, "inexact?", 1, 0, 0, 
            (SCM x),
	    "Return @code{#t} if @var{x} is an inexact number, @code{#f}\n"
	    "else.")
#define FUNC_NAME s_scm_inexact_p
{
  if (SCM_INEXACTP (x))
    return SCM_BOOL_T;
  if (SCM_NUMBERP (x))
    return SCM_BOOL_F;
  SCM_WRONG_TYPE_ARG (1, x);
}
#undef FUNC_NAME


SCM_GPROC1 (s_eq_p, "=", scm_tc7_rpsubr, scm_num_eq_p, g_eq_p);
/* "Return @code{#t} if all parameters are numerically equal."  */
SCM
scm_num_eq_p (SCM x, SCM y)
{
 again:
  if (SCM_I_INUMP (x))
    {
      long xx = SCM_I_INUM (x);
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  return scm_from_bool (xx == yy);
	}
      else if (SCM_BIGP (y))
	return SCM_BOOL_F;
      else if (SCM_REALP (y))
        {
          /* On a 32-bit system an inum fits a double, we can cast the inum
             to a double and compare.

             But on a 64-bit system an inum is bigger than a double and
             casting it to a double (call that dxx) will round.  dxx is at
             worst 1 bigger or smaller than xx, so if dxx==yy we know yy is
             an integer and fits a long.  So we cast yy to a long and
             compare with plain xx.

             An alternative (for any size system actually) would be to check
             yy is an integer (with floor) and is in range of an inum
             (compare against appropriate powers of 2) then test
             xx==(long)yy.  It's just a matter of which casts/comparisons
             might be fastest or easiest for the cpu.  */

          double yy = SCM_REAL_VALUE (y);
          return scm_from_bool ((double) xx == yy
				&& (DBL_MANT_DIG >= SCM_I_FIXNUM_BIT-1
				    || xx == (long) yy));
        }
      else if (SCM_COMPLEXP (y))
	return scm_from_bool (((double) xx == SCM_COMPLEX_REAL (y))
			 && (0.0 == SCM_COMPLEX_IMAG (y)));
      else if (SCM_FRACTIONP (y))
	return SCM_BOOL_F;
      else
	SCM_WTA_DISPATCH_2 (g_eq_p, x, y, SCM_ARGn, s_eq_p);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	return SCM_BOOL_F;
      else if (SCM_BIGP (y))
	{
	  int cmp = mpz_cmp (SCM_I_BIG_MPZ (x), SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  return scm_from_bool (0 == cmp);
	}
      else if (SCM_REALP (y))
	{
	  int cmp;
	  if (xisnan (SCM_REAL_VALUE (y)))
	    return SCM_BOOL_F;
	  cmp = xmpz_cmp_d (SCM_I_BIG_MPZ (x), SCM_REAL_VALUE (y));
	  scm_remember_upto_here_1 (x);
	  return scm_from_bool (0 == cmp);
	}
      else if (SCM_COMPLEXP (y))
	{
	  int cmp;
	  if (0.0 != SCM_COMPLEX_IMAG (y))
	    return SCM_BOOL_F;
	  if (xisnan (SCM_COMPLEX_REAL (y)))
	    return SCM_BOOL_F;
	  cmp = xmpz_cmp_d (SCM_I_BIG_MPZ (x), SCM_COMPLEX_REAL (y));
	  scm_remember_upto_here_1 (x);
	  return scm_from_bool (0 == cmp);
	}
      else if (SCM_FRACTIONP (y))
	return SCM_BOOL_F;
      else
	SCM_WTA_DISPATCH_2 (g_eq_p, x, y, SCM_ARGn, s_eq_p);
    }
  else if (SCM_REALP (x))
    {
      double xx = SCM_REAL_VALUE (x);
      if (SCM_I_INUMP (y))
        {
          /* see comments with inum/real above */
          long yy = SCM_I_INUM (y);
          return scm_from_bool (xx == (double) yy
				&& (DBL_MANT_DIG >= SCM_I_FIXNUM_BIT-1
				    || (long) xx == yy));
        }
      else if (SCM_BIGP (y))
	{
	  int cmp;
	  if (xisnan (SCM_REAL_VALUE (x)))
	    return SCM_BOOL_F;
	  cmp = xmpz_cmp_d (SCM_I_BIG_MPZ (y), SCM_REAL_VALUE (x));
	  scm_remember_upto_here_1 (y);
	  return scm_from_bool (0 == cmp);
	}
      else if (SCM_REALP (y))
	return scm_from_bool (SCM_REAL_VALUE (x) == SCM_REAL_VALUE (y));
      else if (SCM_COMPLEXP (y))
	return scm_from_bool ((SCM_REAL_VALUE (x) == SCM_COMPLEX_REAL (y))
			 && (0.0 == SCM_COMPLEX_IMAG (y)));
      else if (SCM_FRACTIONP (y))
        {
          double  xx = SCM_REAL_VALUE (x);
          if (xisnan (xx))
            return SCM_BOOL_F;
          if (xisinf (xx))
            return scm_from_bool (xx < 0.0);
          x = scm_inexact_to_exact (x);  /* with x as frac or int */
          goto again;
        }
      else
	SCM_WTA_DISPATCH_2 (g_eq_p, x, y, SCM_ARGn, s_eq_p);
    }
  else if (SCM_COMPLEXP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_from_bool ((SCM_COMPLEX_REAL (x) == (double) SCM_I_INUM (y))
			 && (SCM_COMPLEX_IMAG (x) == 0.0));
      else if (SCM_BIGP (y))
	{
	  int cmp;
	  if (0.0 != SCM_COMPLEX_IMAG (x))
	    return SCM_BOOL_F;
	  if (xisnan (SCM_COMPLEX_REAL (x)))
	    return SCM_BOOL_F;
	  cmp = xmpz_cmp_d (SCM_I_BIG_MPZ (y), SCM_COMPLEX_REAL (x));
	  scm_remember_upto_here_1 (y);
	  return scm_from_bool (0 == cmp);
	}
      else if (SCM_REALP (y))
	return scm_from_bool ((SCM_COMPLEX_REAL (x) == SCM_REAL_VALUE (y))
			 && (SCM_COMPLEX_IMAG (x) == 0.0));
      else if (SCM_COMPLEXP (y))
	return scm_from_bool ((SCM_COMPLEX_REAL (x) == SCM_COMPLEX_REAL (y))
			 && (SCM_COMPLEX_IMAG (x) == SCM_COMPLEX_IMAG (y)));
      else if (SCM_FRACTIONP (y))
        {
          double  xx;
          if (SCM_COMPLEX_IMAG (x) != 0.0)
            return SCM_BOOL_F;
          xx = SCM_COMPLEX_REAL (x);
          if (xisnan (xx))
            return SCM_BOOL_F;
          if (xisinf (xx))
            return scm_from_bool (xx < 0.0);
          x = scm_inexact_to_exact (x);  /* with x as frac or int */
          goto again;
        }
      else
	SCM_WTA_DISPATCH_2 (g_eq_p, x, y, SCM_ARGn, s_eq_p);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y))
	return SCM_BOOL_F;
      else if (SCM_BIGP (y))
	return SCM_BOOL_F;
      else if (SCM_REALP (y))
        {
          double yy = SCM_REAL_VALUE (y);
          if (xisnan (yy))
            return SCM_BOOL_F;
          if (xisinf (yy))
            return scm_from_bool (0.0 < yy);
          y = scm_inexact_to_exact (y);  /* with y as frac or int */
          goto again;
        }
      else if (SCM_COMPLEXP (y))
        {
          double yy;
          if (SCM_COMPLEX_IMAG (y) != 0.0)
            return SCM_BOOL_F;
          yy = SCM_COMPLEX_REAL (y);
          if (xisnan (yy))
            return SCM_BOOL_F;
          if (xisinf (yy))
            return scm_from_bool (0.0 < yy);
          y = scm_inexact_to_exact (y);  /* with y as frac or int */
          goto again;
        }
      else if (SCM_FRACTIONP (y))
	return scm_i_fraction_equalp (x, y);
      else
	SCM_WTA_DISPATCH_2 (g_eq_p, x, y, SCM_ARGn, s_eq_p);
    }
  else
    SCM_WTA_DISPATCH_2 (g_eq_p, x, y, SCM_ARG1, s_eq_p);
}


/* OPTIMIZE-ME: For int/frac and frac/frac compares, the multiplications
   done are good for inums, but for bignums an answer can almost always be
   had by just examining a few high bits of the operands, as done by GMP in
   mpq_cmp.  flonum/frac compares likewise, but with the slight complication
   of the float exponent to take into account.  */

SCM_GPROC1 (s_less_p, "<", scm_tc7_rpsubr, scm_less_p, g_less_p);
/* "Return @code{#t} if the list of parameters is monotonically\n"
 * "increasing."
 */
SCM
scm_less_p (SCM x, SCM y)
{
 again:
  if (SCM_I_INUMP (x))
    {
      long xx = SCM_I_INUM (x);
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  return scm_from_bool (xx < yy);
	}
      else if (SCM_BIGP (y))
	{
	  int sgn = mpz_sgn (SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_1 (y);
	  return scm_from_bool (sgn > 0);
	}
      else if (SCM_REALP (y))
	return scm_from_bool ((double) xx < SCM_REAL_VALUE (y));
      else if (SCM_FRACTIONP (y))
        {
          /* "x < a/b" becomes "x*b < a" */
        int_frac:
          x = scm_product (x, SCM_FRACTION_DENOMINATOR (y));
          y = SCM_FRACTION_NUMERATOR (y);
          goto again;
        }
      else
	SCM_WTA_DISPATCH_2 (g_less_p, x, y, SCM_ARGn, s_less_p);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
	  scm_remember_upto_here_1 (x);
	  return scm_from_bool (sgn < 0);
	}
      else if (SCM_BIGP (y))
	{
	  int cmp = mpz_cmp (SCM_I_BIG_MPZ (x), SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  return scm_from_bool (cmp < 0);
	}
      else if (SCM_REALP (y))
	{
	  int cmp;
	  if (xisnan (SCM_REAL_VALUE (y)))
	    return SCM_BOOL_F;
	  cmp = xmpz_cmp_d (SCM_I_BIG_MPZ (x), SCM_REAL_VALUE (y));
	  scm_remember_upto_here_1 (x);
	  return scm_from_bool (cmp < 0);
	}
      else if (SCM_FRACTIONP (y))
        goto int_frac;
      else
	SCM_WTA_DISPATCH_2 (g_less_p, x, y, SCM_ARGn, s_less_p);
    }
  else if (SCM_REALP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_from_bool (SCM_REAL_VALUE (x) < (double) SCM_I_INUM (y));
      else if (SCM_BIGP (y))
	{
	  int cmp;
	  if (xisnan (SCM_REAL_VALUE (x)))
	    return SCM_BOOL_F;
	  cmp = xmpz_cmp_d (SCM_I_BIG_MPZ (y), SCM_REAL_VALUE (x));
	  scm_remember_upto_here_1 (y);
	  return scm_from_bool (cmp > 0);
	}
      else if (SCM_REALP (y))
	return scm_from_bool (SCM_REAL_VALUE (x) < SCM_REAL_VALUE (y));
      else if (SCM_FRACTIONP (y))
        {
          double  xx = SCM_REAL_VALUE (x);
	  if (xisnan (xx))
	    return SCM_BOOL_F;
          if (xisinf (xx))
            return scm_from_bool (xx < 0.0);
          x = scm_inexact_to_exact (x);  /* with x as frac or int */
          goto again;
        }
      else
	SCM_WTA_DISPATCH_2 (g_less_p, x, y, SCM_ARGn, s_less_p);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y) || SCM_BIGP (y))
        {
          /* "a/b < y" becomes "a < y*b" */
          y = scm_product (y, SCM_FRACTION_DENOMINATOR (x));
          x = SCM_FRACTION_NUMERATOR (x);
          goto again;
        }
      else if (SCM_REALP (y))
        {
          double yy = SCM_REAL_VALUE (y);
          if (xisnan (yy))
            return SCM_BOOL_F;
          if (xisinf (yy))
            return scm_from_bool (0.0 < yy);
          y = scm_inexact_to_exact (y);  /* with y as frac or int */
          goto again;
        }
      else if (SCM_FRACTIONP (y))
        {
          /* "a/b < c/d" becomes "a*d < c*b" */
          SCM new_x = scm_product (SCM_FRACTION_NUMERATOR (x),
                                   SCM_FRACTION_DENOMINATOR (y));
          SCM new_y = scm_product (SCM_FRACTION_NUMERATOR (y),
                                   SCM_FRACTION_DENOMINATOR (x));
          x = new_x;
          y = new_y;
          goto again;
        }
      else
	SCM_WTA_DISPATCH_2 (g_less_p, x, y, SCM_ARGn, s_less_p);
    }
  else
    SCM_WTA_DISPATCH_2 (g_less_p, x, y, SCM_ARG1, s_less_p);
}


SCM_GPROC1 (s_scm_gr_p, ">", scm_tc7_rpsubr, scm_gr_p, g_gr_p);
/* "Return @code{#t} if the list of parameters is monotonically\n"
 * "decreasing."
 */
#define FUNC_NAME s_scm_gr_p
SCM
scm_gr_p (SCM x, SCM y)
{
  if (!SCM_NUMBERP (x))
    SCM_WTA_DISPATCH_2 (g_gr_p, x, y, SCM_ARG1, FUNC_NAME);
  else if (!SCM_NUMBERP (y))
    SCM_WTA_DISPATCH_2 (g_gr_p, x, y, SCM_ARG2, FUNC_NAME);
  else
    return scm_less_p (y, x);
}
#undef FUNC_NAME


SCM_GPROC1 (s_scm_leq_p, "<=", scm_tc7_rpsubr, scm_leq_p, g_leq_p);
/* "Return @code{#t} if the list of parameters is monotonically\n"
 * "non-decreasing."
 */
#define FUNC_NAME s_scm_leq_p
SCM
scm_leq_p (SCM x, SCM y)
{
  if (!SCM_NUMBERP (x))
    SCM_WTA_DISPATCH_2 (g_leq_p, x, y, SCM_ARG1, FUNC_NAME);
  else if (!SCM_NUMBERP (y))
    SCM_WTA_DISPATCH_2 (g_leq_p, x, y, SCM_ARG2, FUNC_NAME);
  else if (scm_is_true (scm_nan_p (x)) || scm_is_true (scm_nan_p (y)))
    return SCM_BOOL_F;
  else
    return scm_not (scm_less_p (y, x));
}
#undef FUNC_NAME


SCM_GPROC1 (s_scm_geq_p, ">=", scm_tc7_rpsubr, scm_geq_p, g_geq_p);
/* "Return @code{#t} if the list of parameters is monotonically\n"
 * "non-increasing."
 */
#define FUNC_NAME s_scm_geq_p
SCM
scm_geq_p (SCM x, SCM y)
{
  if (!SCM_NUMBERP (x))
    SCM_WTA_DISPATCH_2 (g_geq_p, x, y, SCM_ARG1, FUNC_NAME);
  else if (!SCM_NUMBERP (y))
    SCM_WTA_DISPATCH_2 (g_geq_p, x, y, SCM_ARG2, FUNC_NAME);
  else if (scm_is_true (scm_nan_p (x)) || scm_is_true (scm_nan_p (y)))
    return SCM_BOOL_F;
  else
    return scm_not (scm_less_p (x, y));
}
#undef FUNC_NAME


SCM_GPROC (s_zero_p, "zero?", 1, 0, 0, scm_zero_p, g_zero_p);
/* "Return @code{#t} if @var{z} is an exact or inexact number equal to\n"
 * "zero."
 */
SCM
scm_zero_p (SCM z)
{
  if (SCM_I_INUMP (z))
    return scm_from_bool (scm_is_eq (z, SCM_INUM0));
  else if (SCM_BIGP (z))
    return SCM_BOOL_F;
  else if (SCM_REALP (z))
    return scm_from_bool (SCM_REAL_VALUE (z) == 0.0);
  else if (SCM_COMPLEXP (z))
    return scm_from_bool (SCM_COMPLEX_REAL (z) == 0.0
		     && SCM_COMPLEX_IMAG (z) == 0.0);
  else if (SCM_FRACTIONP (z))
    return SCM_BOOL_F;
  else
    SCM_WTA_DISPATCH_1 (g_zero_p, z, SCM_ARG1, s_zero_p);
}


SCM_GPROC (s_positive_p, "positive?", 1, 0, 0, scm_positive_p, g_positive_p);
/* "Return @code{#t} if @var{x} is an exact or inexact number greater than\n"
 * "zero."
 */
SCM
scm_positive_p (SCM x)
{
  if (SCM_I_INUMP (x))
    return scm_from_bool (SCM_I_INUM (x) > 0);
  else if (SCM_BIGP (x))
    {
      int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
      scm_remember_upto_here_1 (x);
      return scm_from_bool (sgn > 0);
    }
  else if (SCM_REALP (x))
    return scm_from_bool(SCM_REAL_VALUE (x) > 0.0);
  else if (SCM_FRACTIONP (x))
    return scm_positive_p (SCM_FRACTION_NUMERATOR (x));
  else
    SCM_WTA_DISPATCH_1 (g_positive_p, x, SCM_ARG1, s_positive_p);
}


SCM_GPROC (s_negative_p, "negative?", 1, 0, 0, scm_negative_p, g_negative_p);
/* "Return @code{#t} if @var{x} is an exact or inexact number less than\n"
 * "zero."
 */
SCM
scm_negative_p (SCM x)
{
  if (SCM_I_INUMP (x))
    return scm_from_bool (SCM_I_INUM (x) < 0);
  else if (SCM_BIGP (x))
    {
      int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
      scm_remember_upto_here_1 (x);
      return scm_from_bool (sgn < 0);
    }
  else if (SCM_REALP (x))
    return scm_from_bool(SCM_REAL_VALUE (x) < 0.0);
  else if (SCM_FRACTIONP (x))
    return scm_negative_p (SCM_FRACTION_NUMERATOR (x));
  else
    SCM_WTA_DISPATCH_1 (g_negative_p, x, SCM_ARG1, s_negative_p);
}


/* scm_min and scm_max return an inexact when either argument is inexact, as
   required by r5rs.  On that basis, for exact/inexact combinations the
   exact is converted to inexact to compare and possibly return.  This is
   unlike scm_less_p above which takes some trouble to preserve all bits in
   its test, such trouble is not required for min and max.  */

SCM_GPROC1 (s_max, "max", scm_tc7_asubr, scm_max, g_max);
/* "Return the maximum of all parameter values."
 */
SCM
scm_max (SCM x, SCM y)
{
  if (SCM_UNBNDP (y))
    {
      if (SCM_UNBNDP (x))
	SCM_WTA_DISPATCH_0 (g_max, s_max);
      else if (SCM_I_INUMP(x) || SCM_BIGP(x) || SCM_REALP(x) || SCM_FRACTIONP(x))
	return x;
      else
	SCM_WTA_DISPATCH_1 (g_max, x, SCM_ARG1, s_max);
    }
  
  if (SCM_I_INUMP (x))
    {
      long xx = SCM_I_INUM (x);
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  return (xx < yy) ? y : x;
	}
      else if (SCM_BIGP (y))
	{
	  int sgn = mpz_sgn (SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_1 (y);
	  return (sgn < 0) ? x : y;
	}
      else if (SCM_REALP (y))
	{
	  double z = xx;
	  /* if y==NaN then ">" is false and we return NaN */
	  return (z > SCM_REAL_VALUE (y)) ? scm_from_double (z) : y;
	}
      else if (SCM_FRACTIONP (y))
	{
        use_less:
          return (scm_is_false (scm_less_p (x, y)) ? x : y);
	}
      else
	SCM_WTA_DISPATCH_2 (g_max, x, y, SCM_ARGn, s_max);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
	  scm_remember_upto_here_1 (x);
	  return (sgn < 0) ? y : x;
	}
      else if (SCM_BIGP (y))
	{
	  int cmp = mpz_cmp (SCM_I_BIG_MPZ (x), SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  return (cmp > 0) ? x : y;
	}
      else if (SCM_REALP (y))
	{
          /* if y==NaN then xx>yy is false, so we return the NaN y */
          double xx, yy;
        big_real:
          xx = scm_i_big2dbl (x);
          yy = SCM_REAL_VALUE (y);
	  return (xx > yy ? scm_from_double (xx) : y);
	}
      else if (SCM_FRACTIONP (y))
	{
          goto use_less;
	}
      else
	SCM_WTA_DISPATCH_2 (g_max, x, y, SCM_ARGn, s_max);
    }
  else if (SCM_REALP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  double z = SCM_I_INUM (y);
	  /* if x==NaN then "<" is false and we return NaN */
	  return (SCM_REAL_VALUE (x) < z) ? scm_from_double (z) : x;
	}
      else if (SCM_BIGP (y))
	{
          SCM_SWAP (x, y);
          goto big_real;
	}
      else if (SCM_REALP (y))
	{
	  /* if x==NaN then our explicit check means we return NaN
	     if y==NaN then ">" is false and we return NaN
	     calling isnan is unavoidable, since it's the only way to know
	     which of x or y causes any compares to be false */
	  double xx = SCM_REAL_VALUE (x);
	  return (xisnan (xx) || xx > SCM_REAL_VALUE (y)) ? x : y;
	}
      else if (SCM_FRACTIONP (y))
	{
	  double yy = scm_i_fraction2double (y);
	  double xx = SCM_REAL_VALUE (x);
	  return (xx < yy) ? scm_from_double (yy) : x;
	}
      else
	SCM_WTA_DISPATCH_2 (g_max, x, y, SCM_ARGn, s_max);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y))
	{
          goto use_less;
	}
      else if (SCM_BIGP (y))
	{
          goto use_less;
	}
      else if (SCM_REALP (y))
	{
	  double xx = scm_i_fraction2double (x);
	  return (xx < SCM_REAL_VALUE (y)) ? y : scm_from_double (xx);
	}
      else if (SCM_FRACTIONP (y))
	{
          goto use_less;
	}
      else
	SCM_WTA_DISPATCH_2 (g_max, x, y, SCM_ARGn, s_max);
    }
  else
    SCM_WTA_DISPATCH_2 (g_max, x, y, SCM_ARG1, s_max);
}


SCM_GPROC1 (s_min, "min", scm_tc7_asubr, scm_min, g_min);
/* "Return the minium of all parameter values."
 */
SCM
scm_min (SCM x, SCM y)
{
  if (SCM_UNBNDP (y))
    {
      if (SCM_UNBNDP (x))
	SCM_WTA_DISPATCH_0 (g_min, s_min);
      else if (SCM_I_INUMP(x) || SCM_BIGP(x) || SCM_REALP(x) || SCM_FRACTIONP(x))
	return x;
      else
	SCM_WTA_DISPATCH_1 (g_min, x, SCM_ARG1, s_min);
    }
  
  if (SCM_I_INUMP (x))
    {
      long xx = SCM_I_INUM (x);
      if (SCM_I_INUMP (y))
	{
	  long yy = SCM_I_INUM (y);
	  return (xx < yy) ? x : y;
	}
      else if (SCM_BIGP (y))
	{
	  int sgn = mpz_sgn (SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_1 (y);
	  return (sgn < 0) ? y : x;
	}
      else if (SCM_REALP (y))
	{
	  double z = xx;
	  /* if y==NaN then "<" is false and we return NaN */
	  return (z < SCM_REAL_VALUE (y)) ? scm_from_double (z) : y;
	}
      else if (SCM_FRACTIONP (y))
	{
        use_less:
          return (scm_is_false (scm_less_p (x, y)) ? y : x);
	}
      else
	SCM_WTA_DISPATCH_2 (g_min, x, y, SCM_ARGn, s_min);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
	  scm_remember_upto_here_1 (x);
	  return (sgn < 0) ? x : y;
	}
      else if (SCM_BIGP (y))
	{
	  int cmp = mpz_cmp (SCM_I_BIG_MPZ (x), SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  return (cmp > 0) ? y : x;
	}
      else if (SCM_REALP (y))
	{
          /* if y==NaN then xx<yy is false, so we return the NaN y */
          double xx, yy;
        big_real:
          xx = scm_i_big2dbl (x);
          yy = SCM_REAL_VALUE (y);
	  return (xx < yy ? scm_from_double (xx) : y);
	}
      else if (SCM_FRACTIONP (y))
	{
          goto use_less;
	}
      else
	SCM_WTA_DISPATCH_2 (g_min, x, y, SCM_ARGn, s_min);
    }
  else if (SCM_REALP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  double z = SCM_I_INUM (y);
	  /* if x==NaN then "<" is false and we return NaN */
	  return (z < SCM_REAL_VALUE (x)) ? scm_from_double (z) : x;
	}
      else if (SCM_BIGP (y))
	{
          SCM_SWAP (x, y);
          goto big_real;
	}
      else if (SCM_REALP (y))
	{
	  /* if x==NaN then our explicit check means we return NaN
	     if y==NaN then "<" is false and we return NaN
	     calling isnan is unavoidable, since it's the only way to know
	     which of x or y causes any compares to be false */
	  double xx = SCM_REAL_VALUE (x);
	  return (xisnan (xx) || xx < SCM_REAL_VALUE (y)) ? x : y;
	}
      else if (SCM_FRACTIONP (y))
	{
	  double yy = scm_i_fraction2double (y);
	  double xx = SCM_REAL_VALUE (x);
	  return (yy < xx) ? scm_from_double (yy) : x;
	}
      else
	SCM_WTA_DISPATCH_2 (g_min, x, y, SCM_ARGn, s_min);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y))
	{
          goto use_less;
	}
      else if (SCM_BIGP (y))
	{
          goto use_less;
	}
      else if (SCM_REALP (y))
	{
	  double xx = scm_i_fraction2double (x);
	  return (SCM_REAL_VALUE (y) < xx) ? y : scm_from_double (xx);
	}
      else if (SCM_FRACTIONP (y))
	{
          goto use_less;
	}
      else
	SCM_WTA_DISPATCH_2 (g_max, x, y, SCM_ARGn, s_max);
    }
  else
    SCM_WTA_DISPATCH_2 (g_min, x, y, SCM_ARG1, s_min);
}


SCM_GPROC1 (s_sum, "+", scm_tc7_asubr, scm_sum, g_sum);
/* "Return the sum of all parameter values.  Return 0 if called without\n"
 * "any parameters." 
 */
SCM
scm_sum (SCM x, SCM y)
{
  if (SCM_UNLIKELY (SCM_UNBNDP (y)))
    {
      if (SCM_NUMBERP (x)) return x;
      if (SCM_UNBNDP (x)) return SCM_INUM0;
      SCM_WTA_DISPATCH_1 (g_sum, x, SCM_ARG1, s_sum);
    }

  if (SCM_LIKELY (SCM_I_INUMP (x)))
    {
      if (SCM_LIKELY (SCM_I_INUMP (y)))
        {
          long xx = SCM_I_INUM (x);
          long yy = SCM_I_INUM (y);
          long int z = xx + yy;
          return SCM_FIXABLE (z) ? SCM_I_MAKINUM (z) : scm_i_long2big (z);
        }
      else if (SCM_BIGP (y))
        {
          SCM_SWAP (x, y);
          goto add_big_inum;
        }
      else if (SCM_REALP (y))
        {
          long int xx = SCM_I_INUM (x);
          return scm_from_double (xx + SCM_REAL_VALUE (y));
        }
      else if (SCM_COMPLEXP (y))
        {
          long int xx = SCM_I_INUM (x);
          return scm_c_make_rectangular (xx + SCM_COMPLEX_REAL (y),
                                   SCM_COMPLEX_IMAG (y));
        }
      else if (SCM_FRACTIONP (y))
	return scm_i_make_ratio (scm_sum (SCM_FRACTION_NUMERATOR (y), 
					scm_product (x, SCM_FRACTION_DENOMINATOR (y))),
			       SCM_FRACTION_DENOMINATOR (y));
      else
        SCM_WTA_DISPATCH_2 (g_sum, x, y, SCM_ARGn, s_sum);
    } else if (SCM_BIGP (x))
      {
	if (SCM_I_INUMP (y))
	  {
	    long int inum;
	    int bigsgn;
	  add_big_inum:
	    inum = SCM_I_INUM (y);      
	    if (inum == 0)
	      return x;
	    bigsgn = mpz_sgn (SCM_I_BIG_MPZ (x));
	    if (inum < 0)
	      {
		SCM result = scm_i_mkbig ();
		mpz_sub_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (x), - inum);
		scm_remember_upto_here_1 (x);
		/* we know the result will have to be a bignum */
		if (bigsgn == -1)
		  return result;
		return scm_i_normbig (result);
	      }
	    else
	      {
		SCM result = scm_i_mkbig ();
		mpz_add_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (x), inum);
		scm_remember_upto_here_1 (x);
		/* we know the result will have to be a bignum */
		if (bigsgn == 1)
		  return result;
		return scm_i_normbig (result);        
	      }
	  }
	else if (SCM_BIGP (y))
	  {
	    SCM result = scm_i_mkbig ();
	    int sgn_x = mpz_sgn (SCM_I_BIG_MPZ (x)); 
	    int sgn_y = mpz_sgn (SCM_I_BIG_MPZ (y)); 
	    mpz_add (SCM_I_BIG_MPZ (result),
		     SCM_I_BIG_MPZ (x),
		     SCM_I_BIG_MPZ (y));
	    scm_remember_upto_here_2 (x, y);
	    /* we know the result will have to be a bignum */
	    if (sgn_x == sgn_y)
	      return result;
	    return scm_i_normbig (result);
	  }
	else if (SCM_REALP (y))
	  {
	    double result = mpz_get_d (SCM_I_BIG_MPZ (x)) + SCM_REAL_VALUE (y);
	    scm_remember_upto_here_1 (x);
	    return scm_from_double (result);
	  }
	else if (SCM_COMPLEXP (y))
	  {
	    double real_part = (mpz_get_d (SCM_I_BIG_MPZ (x))
				+ SCM_COMPLEX_REAL (y));
	    scm_remember_upto_here_1 (x);
	    return scm_c_make_rectangular (real_part, SCM_COMPLEX_IMAG (y));
	  }
	else if (SCM_FRACTIONP (y))
	  return scm_i_make_ratio (scm_sum (SCM_FRACTION_NUMERATOR (y), 
					  scm_product (x, SCM_FRACTION_DENOMINATOR (y))),
				 SCM_FRACTION_DENOMINATOR (y));
	else
	  SCM_WTA_DISPATCH_2 (g_sum, x, y, SCM_ARGn, s_sum);
      }
  else if (SCM_REALP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_from_double (SCM_REAL_VALUE (x) + SCM_I_INUM (y));
      else if (SCM_BIGP (y))
	{
	  double result = mpz_get_d (SCM_I_BIG_MPZ (y)) + SCM_REAL_VALUE (x);
	  scm_remember_upto_here_1 (y);
	  return scm_from_double (result);
	}
      else if (SCM_REALP (y))
	return scm_from_double (SCM_REAL_VALUE (x) + SCM_REAL_VALUE (y));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (SCM_REAL_VALUE (x) + SCM_COMPLEX_REAL (y),
				 SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	return scm_from_double (SCM_REAL_VALUE (x) + scm_i_fraction2double (y));
      else
	SCM_WTA_DISPATCH_2 (g_sum, x, y, SCM_ARGn, s_sum);
    }
  else if (SCM_COMPLEXP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) + SCM_I_INUM (y),
				 SCM_COMPLEX_IMAG (x));
      else if (SCM_BIGP (y))
	{
	  double real_part = (mpz_get_d (SCM_I_BIG_MPZ (y))
			      + SCM_COMPLEX_REAL (x));
	  scm_remember_upto_here_1 (y);
	  return scm_c_make_rectangular (real_part, SCM_COMPLEX_IMAG (x));
	}
      else if (SCM_REALP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) + SCM_REAL_VALUE (y),
				 SCM_COMPLEX_IMAG (x));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) + SCM_COMPLEX_REAL (y),
				 SCM_COMPLEX_IMAG (x) + SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) + scm_i_fraction2double (y),
				 SCM_COMPLEX_IMAG (x));
      else
	SCM_WTA_DISPATCH_2 (g_sum, x, y, SCM_ARGn, s_sum);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_i_make_ratio (scm_sum (SCM_FRACTION_NUMERATOR (x), 
					scm_product (y, SCM_FRACTION_DENOMINATOR (x))),
			       SCM_FRACTION_DENOMINATOR (x));
      else if (SCM_BIGP (y))
	return scm_i_make_ratio (scm_sum (SCM_FRACTION_NUMERATOR (x), 
					scm_product (y, SCM_FRACTION_DENOMINATOR (x))),
			       SCM_FRACTION_DENOMINATOR (x));
      else if (SCM_REALP (y))
	return scm_from_double (SCM_REAL_VALUE (y) + scm_i_fraction2double (x));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (y) + scm_i_fraction2double (x),
				 SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	/* a/b + c/d = (ad + bc) / bd */
	return scm_i_make_ratio (scm_sum (scm_product (SCM_FRACTION_NUMERATOR (x), SCM_FRACTION_DENOMINATOR (y)),
					scm_product (SCM_FRACTION_NUMERATOR (y), SCM_FRACTION_DENOMINATOR (x))),
			       scm_product (SCM_FRACTION_DENOMINATOR (x), SCM_FRACTION_DENOMINATOR (y)));
      else
	SCM_WTA_DISPATCH_2 (g_sum, x, y, SCM_ARGn, s_sum);
    }
  else
    SCM_WTA_DISPATCH_2 (g_sum, x, y, SCM_ARG1, s_sum);
}


SCM_DEFINE (scm_oneplus, "1+", 1, 0, 0, 
            (SCM x),
	    "Return @math{@var{x}+1}.")
#define FUNC_NAME s_scm_oneplus
{
  return scm_sum (x, SCM_I_MAKINUM (1));
}
#undef FUNC_NAME


SCM_GPROC1 (s_difference, "-", scm_tc7_asubr, scm_difference, g_difference);
/* If called with one argument @var{z1}, -@var{z1} returned. Otherwise
 * the sum of all but the first argument are subtracted from the first
 * argument.  */
#define FUNC_NAME s_difference
SCM
scm_difference (SCM x, SCM y)
{
  if (SCM_UNLIKELY (SCM_UNBNDP (y)))
    {
      if (SCM_UNBNDP (x))
        SCM_WTA_DISPATCH_0 (g_difference, s_difference);
      else 
        if (SCM_I_INUMP (x))
          {
            long xx = -SCM_I_INUM (x);
            if (SCM_FIXABLE (xx))
              return SCM_I_MAKINUM (xx);
            else
              return scm_i_long2big (xx);
          }
        else if (SCM_BIGP (x))
          /* Must scm_i_normbig here because -SCM_MOST_NEGATIVE_FIXNUM is a
             bignum, but negating that gives a fixnum.  */
          return scm_i_normbig (scm_i_clonebig (x, 0));
        else if (SCM_REALP (x))
          return scm_from_double (-SCM_REAL_VALUE (x));
        else if (SCM_COMPLEXP (x))
          return scm_c_make_rectangular (-SCM_COMPLEX_REAL (x),
                                   -SCM_COMPLEX_IMAG (x));
	else if (SCM_FRACTIONP (x))
	  return scm_i_make_ratio (scm_difference (SCM_FRACTION_NUMERATOR (x), SCM_UNDEFINED),
				 SCM_FRACTION_DENOMINATOR (x));
        else
          SCM_WTA_DISPATCH_1 (g_difference, x, SCM_ARG1, s_difference);
    }
  
  if (SCM_LIKELY (SCM_I_INUMP (x)))
    {
      if (SCM_LIKELY (SCM_I_INUMP (y)))
	{
	  long int xx = SCM_I_INUM (x);
	  long int yy = SCM_I_INUM (y);
	  long int z = xx - yy;
	  if (SCM_FIXABLE (z))
	    return SCM_I_MAKINUM (z);
	  else
	    return scm_i_long2big (z);
	}
      else if (SCM_BIGP (y))
	{
	  /* inum-x - big-y */
	  long xx = SCM_I_INUM (x);

	  if (xx == 0)
	    return scm_i_clonebig (y, 0);
	  else
	    {
	      int sgn_y = mpz_sgn (SCM_I_BIG_MPZ (y));
	      SCM result = scm_i_mkbig ();

	      if (xx >= 0)
		mpz_ui_sub (SCM_I_BIG_MPZ (result), xx, SCM_I_BIG_MPZ (y));
	      else
		{
		  /* x - y == -(y + -x) */
		  mpz_add_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (y), -xx);
		  mpz_neg (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (result));
		}
	      scm_remember_upto_here_1 (y);

	      if ((xx < 0 && (sgn_y > 0)) || ((xx > 0) && sgn_y < 0))
		/* we know the result will have to be a bignum */
		return result;
	      else
		return scm_i_normbig (result);
	    }
	}
      else if (SCM_REALP (y))
	{
	  long int xx = SCM_I_INUM (x);
	  return scm_from_double (xx - SCM_REAL_VALUE (y));
	}
      else if (SCM_COMPLEXP (y))
	{
	  long int xx = SCM_I_INUM (x);
	  return scm_c_make_rectangular (xx - SCM_COMPLEX_REAL (y),
				   - SCM_COMPLEX_IMAG (y));
	}
      else if (SCM_FRACTIONP (y))
	/* a - b/c = (ac - b) / c */
	return scm_i_make_ratio (scm_difference (scm_product (x, SCM_FRACTION_DENOMINATOR (y)),
					       SCM_FRACTION_NUMERATOR (y)),
			       SCM_FRACTION_DENOMINATOR (y));
      else
	SCM_WTA_DISPATCH_2 (g_difference, x, y, SCM_ARGn, s_difference);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  /* big-x - inum-y */
	  long yy = SCM_I_INUM (y);
	  int sgn_x = mpz_sgn (SCM_I_BIG_MPZ (x));

	  scm_remember_upto_here_1 (x);
	  if (sgn_x == 0)
	    return (SCM_FIXABLE (-yy) ?
		    SCM_I_MAKINUM (-yy) : scm_from_long (-yy));
	  else
	    {
	      SCM result = scm_i_mkbig ();

	      if (yy >= 0)
		mpz_sub_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (x), yy);
	      else
		mpz_add_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (x), -yy);
	      scm_remember_upto_here_1 (x);

	      if ((sgn_x < 0 && (yy > 0)) || ((sgn_x > 0) && yy < 0))
		/* we know the result will have to be a bignum */
		return result;
	      else
		return scm_i_normbig (result);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  int sgn_x = mpz_sgn (SCM_I_BIG_MPZ (x)); 
	  int sgn_y = mpz_sgn (SCM_I_BIG_MPZ (y)); 
	  SCM result = scm_i_mkbig ();
	  mpz_sub (SCM_I_BIG_MPZ (result),
		   SCM_I_BIG_MPZ (x),
		   SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  /* we know the result will have to be a bignum */
	  if ((sgn_x == 1) && (sgn_y == -1))
	    return result;
	  if ((sgn_x == -1) && (sgn_y == 1))
	    return result;
	  return scm_i_normbig (result);
	}
      else if (SCM_REALP (y))
	{
	  double result = mpz_get_d (SCM_I_BIG_MPZ (x)) - SCM_REAL_VALUE (y);
	  scm_remember_upto_here_1 (x);
	  return scm_from_double (result);
	}
      else if (SCM_COMPLEXP (y))
	{
	  double real_part = (mpz_get_d (SCM_I_BIG_MPZ (x))
			      - SCM_COMPLEX_REAL (y));
	  scm_remember_upto_here_1 (x);
	  return scm_c_make_rectangular (real_part, - SCM_COMPLEX_IMAG (y));      
	}
      else if (SCM_FRACTIONP (y))
	return scm_i_make_ratio (scm_difference (scm_product (x, SCM_FRACTION_DENOMINATOR (y)),
					       SCM_FRACTION_NUMERATOR (y)),
			       SCM_FRACTION_DENOMINATOR (y));
      else SCM_WTA_DISPATCH_2 (g_difference, x, y, SCM_ARGn, s_difference);
    }
  else if (SCM_REALP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_from_double (SCM_REAL_VALUE (x) - SCM_I_INUM (y));
      else if (SCM_BIGP (y))
	{
	  double result = SCM_REAL_VALUE (x) - mpz_get_d (SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_1 (x);
	  return scm_from_double (result);      
	}
      else if (SCM_REALP (y))
	return scm_from_double (SCM_REAL_VALUE (x) - SCM_REAL_VALUE (y));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (SCM_REAL_VALUE (x) - SCM_COMPLEX_REAL (y),
				 -SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	return scm_from_double (SCM_REAL_VALUE (x) - scm_i_fraction2double (y));
      else
	SCM_WTA_DISPATCH_2 (g_difference, x, y, SCM_ARGn, s_difference);
    }
  else if (SCM_COMPLEXP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) - SCM_I_INUM (y),
				 SCM_COMPLEX_IMAG (x));
      else if (SCM_BIGP (y))
	{
	  double real_part = (SCM_COMPLEX_REAL (x)
			      - mpz_get_d (SCM_I_BIG_MPZ (y)));
	  scm_remember_upto_here_1 (x);
	  return scm_c_make_rectangular (real_part, SCM_COMPLEX_IMAG (y));      
	}
      else if (SCM_REALP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) - SCM_REAL_VALUE (y),
				 SCM_COMPLEX_IMAG (x));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) - SCM_COMPLEX_REAL (y),
				 SCM_COMPLEX_IMAG (x) - SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) - scm_i_fraction2double (y),
				 SCM_COMPLEX_IMAG (x));
      else
	SCM_WTA_DISPATCH_2 (g_difference, x, y, SCM_ARGn, s_difference);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y))
	/* a/b - c = (a - cb) / b */
	return scm_i_make_ratio (scm_difference (SCM_FRACTION_NUMERATOR (x), 
					       scm_product(y, SCM_FRACTION_DENOMINATOR (x))),
			       SCM_FRACTION_DENOMINATOR (x));
      else if (SCM_BIGP (y))
	return scm_i_make_ratio (scm_difference (SCM_FRACTION_NUMERATOR (x), 
					       scm_product(y, SCM_FRACTION_DENOMINATOR (x))),
			       SCM_FRACTION_DENOMINATOR (x));
      else if (SCM_REALP (y))
	return scm_from_double (scm_i_fraction2double (x) - SCM_REAL_VALUE (y));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (scm_i_fraction2double (x) - SCM_COMPLEX_REAL (y),
				 -SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	/* a/b - c/d = (ad - bc) / bd */
	return scm_i_make_ratio (scm_difference (scm_product (SCM_FRACTION_NUMERATOR (x), SCM_FRACTION_DENOMINATOR (y)),
					       scm_product (SCM_FRACTION_NUMERATOR (y), SCM_FRACTION_DENOMINATOR (x))),
			       scm_product (SCM_FRACTION_DENOMINATOR (x), SCM_FRACTION_DENOMINATOR (y)));
      else
	SCM_WTA_DISPATCH_2 (g_difference, x, y, SCM_ARGn, s_difference);
    }
  else
    SCM_WTA_DISPATCH_2 (g_difference, x, y, SCM_ARG1, s_difference);
}
#undef FUNC_NAME


SCM_DEFINE (scm_oneminus, "1-", 1, 0, 0, 
            (SCM x),
	    "Return @math{@var{x}-1}.")
#define FUNC_NAME s_scm_oneminus
{
  return scm_difference (x, SCM_I_MAKINUM (1));
}
#undef FUNC_NAME


SCM_GPROC1 (s_product, "*", scm_tc7_asubr, scm_product, g_product);
/* "Return the product of all arguments.  If called without arguments,\n"
 * "1 is returned."
 */
SCM
scm_product (SCM x, SCM y)
{
  if (SCM_UNLIKELY (SCM_UNBNDP (y)))
    {
      if (SCM_UNBNDP (x))
	return SCM_I_MAKINUM (1L);
      else if (SCM_NUMBERP (x))
	return x;
      else
	SCM_WTA_DISPATCH_1 (g_product, x, SCM_ARG1, s_product);
    }
  
  if (SCM_LIKELY (SCM_I_INUMP (x)))
    {
      long xx;

    intbig:
      xx = SCM_I_INUM (x);

      switch (xx)
	{
        case 0: return x; break;
        case 1: return y; break;
	}

      if (SCM_LIKELY (SCM_I_INUMP (y)))
	{
	  long yy = SCM_I_INUM (y);
	  long kk = xx * yy;
	  SCM k = SCM_I_MAKINUM (kk);
	  if ((kk == SCM_I_INUM (k)) && (kk / xx == yy))
	    return k;
	  else
	    {
	      SCM result = scm_i_long2big (xx);
	      mpz_mul_si (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (result), yy);
	      return scm_i_normbig (result);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  SCM result = scm_i_mkbig ();
	  mpz_mul_si (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (y), xx);
	  scm_remember_upto_here_1 (y);
	  return result;
	}
      else if (SCM_REALP (y))
	return scm_from_double (xx * SCM_REAL_VALUE (y));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (xx * SCM_COMPLEX_REAL (y),
				 xx * SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	return scm_i_make_ratio (scm_product (x, SCM_FRACTION_NUMERATOR (y)),
			       SCM_FRACTION_DENOMINATOR (y));
      else
	SCM_WTA_DISPATCH_2 (g_product, x, y, SCM_ARGn, s_product);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  SCM_SWAP (x, y);
	  goto intbig;
	}
      else if (SCM_BIGP (y))
	{
	  SCM result = scm_i_mkbig ();
	  mpz_mul (SCM_I_BIG_MPZ (result),
		   SCM_I_BIG_MPZ (x),
		   SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_2 (x, y);
	  return result;
	}
      else if (SCM_REALP (y))
	{
	  double result = mpz_get_d (SCM_I_BIG_MPZ (x)) * SCM_REAL_VALUE (y);
	  scm_remember_upto_here_1 (x);
	  return scm_from_double (result);
	}
      else if (SCM_COMPLEXP (y))
	{
	  double z = mpz_get_d (SCM_I_BIG_MPZ (x));
	  scm_remember_upto_here_1 (x);
	  return scm_c_make_rectangular (z * SCM_COMPLEX_REAL (y),
				   z * SCM_COMPLEX_IMAG (y));
	}
      else if (SCM_FRACTIONP (y))
	return scm_i_make_ratio (scm_product (x, SCM_FRACTION_NUMERATOR (y)),
			       SCM_FRACTION_DENOMINATOR (y));
      else
	SCM_WTA_DISPATCH_2 (g_product, x, y, SCM_ARGn, s_product);
    }
  else if (SCM_REALP (x))
    {
      if (SCM_I_INUMP (y))
        {
          /* inexact*exact0 => exact 0, per R5RS "Exactness" section */
          if (scm_is_eq (y, SCM_INUM0))
            return y;
          return scm_from_double (SCM_I_INUM (y) * SCM_REAL_VALUE (x));
        }
      else if (SCM_BIGP (y))
	{
	  double result = mpz_get_d (SCM_I_BIG_MPZ (y)) * SCM_REAL_VALUE (x);
	  scm_remember_upto_here_1 (y);
	  return scm_from_double (result);
	}
      else if (SCM_REALP (y))
	return scm_from_double (SCM_REAL_VALUE (x) * SCM_REAL_VALUE (y));
      else if (SCM_COMPLEXP (y))
	return scm_c_make_rectangular (SCM_REAL_VALUE (x) * SCM_COMPLEX_REAL (y),
				 SCM_REAL_VALUE (x) * SCM_COMPLEX_IMAG (y));
      else if (SCM_FRACTIONP (y))
	return scm_from_double (SCM_REAL_VALUE (x) * scm_i_fraction2double (y));
      else
	SCM_WTA_DISPATCH_2 (g_product, x, y, SCM_ARGn, s_product);
    }
  else if (SCM_COMPLEXP (x))
    {
      if (SCM_I_INUMP (y))
        {
          /* inexact*exact0 => exact 0, per R5RS "Exactness" section */
          if (scm_is_eq (y, SCM_INUM0))
            return y;
          return scm_c_make_rectangular (SCM_I_INUM (y) * SCM_COMPLEX_REAL (x),
                                         SCM_I_INUM (y) * SCM_COMPLEX_IMAG (x));
        }
      else if (SCM_BIGP (y))
	{
	  double z = mpz_get_d (SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_1 (y);
	  return scm_c_make_rectangular (z * SCM_COMPLEX_REAL (x),
				   z * SCM_COMPLEX_IMAG (x));
	}
      else if (SCM_REALP (y))
	return scm_c_make_rectangular (SCM_REAL_VALUE (y) * SCM_COMPLEX_REAL (x),
				 SCM_REAL_VALUE (y) * SCM_COMPLEX_IMAG (x));
      else if (SCM_COMPLEXP (y))
	{
	  return scm_c_make_rectangular (SCM_COMPLEX_REAL (x) * SCM_COMPLEX_REAL (y)
				   - SCM_COMPLEX_IMAG (x) * SCM_COMPLEX_IMAG (y),
				   SCM_COMPLEX_REAL (x) * SCM_COMPLEX_IMAG (y)
				   + SCM_COMPLEX_IMAG (x) * SCM_COMPLEX_REAL (y));
	}
      else if (SCM_FRACTIONP (y))
	{
	  double yy = scm_i_fraction2double (y);
	  return scm_c_make_rectangular (yy * SCM_COMPLEX_REAL (x),
				   yy * SCM_COMPLEX_IMAG (x));
	}
      else
	SCM_WTA_DISPATCH_2 (g_product, x, y, SCM_ARGn, s_product);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y))
	return scm_i_make_ratio (scm_product (y, SCM_FRACTION_NUMERATOR (x)),
			       SCM_FRACTION_DENOMINATOR (x));
      else if (SCM_BIGP (y))
	return scm_i_make_ratio (scm_product (y, SCM_FRACTION_NUMERATOR (x)),
			       SCM_FRACTION_DENOMINATOR (x));
      else if (SCM_REALP (y))
	return scm_from_double (scm_i_fraction2double (x) * SCM_REAL_VALUE (y));
      else if (SCM_COMPLEXP (y))
	{
	  double xx = scm_i_fraction2double (x);
	  return scm_c_make_rectangular (xx * SCM_COMPLEX_REAL (y),
				   xx * SCM_COMPLEX_IMAG (y));
	}
      else if (SCM_FRACTIONP (y))
	/* a/b * c/d = ac / bd */
	return scm_i_make_ratio (scm_product (SCM_FRACTION_NUMERATOR (x),
					    SCM_FRACTION_NUMERATOR (y)),
			       scm_product (SCM_FRACTION_DENOMINATOR (x),
					    SCM_FRACTION_DENOMINATOR (y)));
      else
	SCM_WTA_DISPATCH_2 (g_product, x, y, SCM_ARGn, s_product);
    }
  else
    SCM_WTA_DISPATCH_2 (g_product, x, y, SCM_ARG1, s_product);
}

#if ((defined (HAVE_ISINF) && defined (HAVE_ISNAN)) \
     || (defined (HAVE_FINITE) && defined (HAVE_ISNAN)))
#define ALLOW_DIVIDE_BY_ZERO
/* #define ALLOW_DIVIDE_BY_EXACT_ZERO */
#endif

/* The code below for complex division is adapted from the GNU
   libstdc++, which adapted it from f2c's libF77, and is subject to
   this copyright:  */

/****************************************************************
Copyright 1990, 1991, 1992, 1993 by AT&T Bell Laboratories and Bellcore.

Permission to use, copy, modify, and distribute this software
and its documentation for any purpose and without fee is hereby
granted, provided that the above copyright notice appear in all
copies and that both that the copyright notice and this
permission notice and warranty disclaimer appear in supporting
documentation, and that the names of AT&T Bell Laboratories or
Bellcore or any of their entities not be used in advertising or
publicity pertaining to distribution of the software without
specific, written prior permission.

AT&T and Bellcore disclaim all warranties with regard to this
software, including all implied warranties of merchantability
and fitness.  In no event shall AT&T or Bellcore be liable for
any special, indirect or consequential damages or any damages
whatsoever resulting from loss of use, data or profits, whether
in an action of contract, negligence or other tortious action,
arising out of or in connection with the use or performance of
this software.
****************************************************************/

SCM_GPROC1 (s_divide, "/", scm_tc7_asubr, scm_divide, g_divide);
/* Divide the first argument by the product of the remaining
   arguments.  If called with one argument @var{z1}, 1/@var{z1} is
   returned.  */
#define FUNC_NAME s_divide
static SCM
scm_i_divide (SCM x, SCM y, int inexact)
{
  double a;

  if (SCM_UNLIKELY (SCM_UNBNDP (y)))
    {
      if (SCM_UNBNDP (x))
	SCM_WTA_DISPATCH_0 (g_divide, s_divide);
      else if (SCM_I_INUMP (x))
	{
	  long xx = SCM_I_INUM (x);
	  if (xx == 1 || xx == -1)
	    return x;
#ifndef ALLOW_DIVIDE_BY_EXACT_ZERO
	  else if (xx == 0)
	    scm_num_overflow (s_divide);
#endif
	  else
	    {
	      if (inexact)
		return scm_from_double (1.0 / (double) xx);
	      else return scm_i_make_ratio (SCM_I_MAKINUM(1), x);
	    }
	}
      else if (SCM_BIGP (x))
	{
	  if (inexact)
	    return scm_from_double (1.0 / scm_i_big2dbl (x));
	  else return scm_i_make_ratio (SCM_I_MAKINUM(1), x);
	}
      else if (SCM_REALP (x))
	{
	  double xx = SCM_REAL_VALUE (x);
#ifndef ALLOW_DIVIDE_BY_ZERO
	  if (xx == 0.0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_from_double (1.0 / xx);
	}
      else if (SCM_COMPLEXP (x))
	{
	  double r = SCM_COMPLEX_REAL (x);
	  double i = SCM_COMPLEX_IMAG (x);
	  if (fabs(r) <= fabs(i))
	    {
	      double t = r / i;
	      double d = i * (1.0 + t * t);
	      return scm_c_make_rectangular (t / d, -1.0 / d);
	    }
	  else
	    {
	      double t = i / r;
	      double d = r * (1.0 + t * t);
	      return scm_c_make_rectangular (1.0 / d, -t / d);
	    }
	}
      else if (SCM_FRACTIONP (x))
	return scm_i_make_ratio (SCM_FRACTION_DENOMINATOR (x),
			       SCM_FRACTION_NUMERATOR (x));
      else
	SCM_WTA_DISPATCH_1 (g_divide, x, SCM_ARG1, s_divide);
    }

  if (SCM_LIKELY (SCM_I_INUMP (x)))
    {
      long xx = SCM_I_INUM (x);
      if (SCM_LIKELY (SCM_I_INUMP (y)))
	{
	  long yy = SCM_I_INUM (y);
	  if (yy == 0)
	    {
#ifndef ALLOW_DIVIDE_BY_EXACT_ZERO
	      scm_num_overflow (s_divide);
#else
	      return scm_from_double ((double) xx / (double) yy);
#endif
	    }
	  else if (xx % yy != 0)
	    {
	      if (inexact)
		return scm_from_double ((double) xx / (double) yy);
	      else return scm_i_make_ratio (x, y);
	    }
	  else
	    {
	      long z = xx / yy;
	      if (SCM_FIXABLE (z))
		return SCM_I_MAKINUM (z);
	      else
		return scm_i_long2big (z);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  if (inexact)
	    return scm_from_double ((double) xx / scm_i_big2dbl (y));
	  else return scm_i_make_ratio (x, y);
	}
      else if (SCM_REALP (y))
	{
	  double yy = SCM_REAL_VALUE (y);
#ifndef ALLOW_DIVIDE_BY_ZERO
	  if (yy == 0.0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_from_double ((double) xx / yy);
	}
      else if (SCM_COMPLEXP (y))
	{
	  a = xx;
	complex_div: /* y _must_ be a complex number */
	  {
	    double r = SCM_COMPLEX_REAL (y);
	    double i = SCM_COMPLEX_IMAG (y);
	    if (fabs(r) <= fabs(i))
	      {
		double t = r / i;
		double d = i * (1.0 + t * t);
		return scm_c_make_rectangular ((a * t) / d,  -a / d);
	      }
	    else
	      {
		double t = i / r;
		double d = r * (1.0 + t * t);
		return scm_c_make_rectangular (a / d,  -(a * t) / d);
	      }
	  }
	}
      else if (SCM_FRACTIONP (y))
	/* a / b/c = ac / b */
	return scm_i_make_ratio (scm_product (x, SCM_FRACTION_DENOMINATOR (y)),
			       SCM_FRACTION_NUMERATOR (y));
      else
	SCM_WTA_DISPATCH_2 (g_divide, x, y, SCM_ARGn, s_divide);
    }
  else if (SCM_BIGP (x))
    {
      if (SCM_I_INUMP (y))
	{
	  long int yy = SCM_I_INUM (y);
	  if (yy == 0)
	    {
#ifndef ALLOW_DIVIDE_BY_EXACT_ZERO
	      scm_num_overflow (s_divide);
#else
	      int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
	      scm_remember_upto_here_1 (x);
	      return (sgn == 0) ? scm_nan () : scm_inf ();
#endif
	    }
	  else if (yy == 1)
	    return x;
	  else
	    {
	      /* FIXME: HMM, what are the relative performance issues here?
		 We need to test.  Is it faster on average to test
		 divisible_p, then perform whichever operation, or is it
		 faster to perform the integer div opportunistically and
		 switch to real if there's a remainder?  For now we take the
		 middle ground: test, then if divisible, use the faster div
		 func. */

	      long abs_yy = yy < 0 ? -yy : yy;
	      int divisible_p = mpz_divisible_ui_p (SCM_I_BIG_MPZ (x), abs_yy);

	      if (divisible_p)
		{
		  SCM result = scm_i_mkbig ();
		  mpz_divexact_ui (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (x), abs_yy);
		  scm_remember_upto_here_1 (x);
		  if (yy < 0)
		    mpz_neg (SCM_I_BIG_MPZ (result), SCM_I_BIG_MPZ (result));
		  return scm_i_normbig (result);
		}
	      else
		{
		  if (inexact)
		    return scm_from_double (scm_i_big2dbl (x) / (double) yy);
		  else return scm_i_make_ratio (x, y);
		}
	    }
	}
      else if (SCM_BIGP (y))
	{
	  int y_is_zero = (mpz_sgn (SCM_I_BIG_MPZ (y)) == 0);
	  if (y_is_zero)
	    {
#ifndef ALLOW_DIVIDE_BY_EXACT_ZERO
	      scm_num_overflow (s_divide);
#else
	      int sgn = mpz_sgn (SCM_I_BIG_MPZ (x));
	      scm_remember_upto_here_1 (x);
	      return (sgn == 0) ? scm_nan () : scm_inf ();
#endif
	    }
	  else
	    {
	      /* big_x / big_y */
              if (inexact)
                {
                  /* It's easily possible for the ratio x/y to fit a double
                     but one or both x and y be too big to fit a double,
                     hence the use of mpq_get_d rather than converting and
                     dividing.  */
                  mpq_t q;
                  *mpq_numref(q) = *SCM_I_BIG_MPZ (x);
                  *mpq_denref(q) = *SCM_I_BIG_MPZ (y);
                  return scm_from_double (mpq_get_d (q));
                }
              else
                {
                  int divisible_p = mpz_divisible_p (SCM_I_BIG_MPZ (x),
                                                     SCM_I_BIG_MPZ (y));
                  if (divisible_p)
                    {
                      SCM result = scm_i_mkbig ();
                      mpz_divexact (SCM_I_BIG_MPZ (result),
                                    SCM_I_BIG_MPZ (x),
                                    SCM_I_BIG_MPZ (y));
                      scm_remember_upto_here_2 (x, y);
                      return scm_i_normbig (result);
                    }
                  else
                    return scm_i_make_ratio (x, y);
                }
	    }
	}
      else if (SCM_REALP (y))
	{
	  double yy = SCM_REAL_VALUE (y);
#ifndef ALLOW_DIVIDE_BY_ZERO
	  if (yy == 0.0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_from_double (scm_i_big2dbl (x) / yy);
	}
      else if (SCM_COMPLEXP (y))
	{
	  a = scm_i_big2dbl (x);
	  goto complex_div;
	}
      else if (SCM_FRACTIONP (y))
	return scm_i_make_ratio (scm_product (x, SCM_FRACTION_DENOMINATOR (y)),
			       SCM_FRACTION_NUMERATOR (y));
      else
	SCM_WTA_DISPATCH_2 (g_divide, x, y, SCM_ARGn, s_divide);
    }
  else if (SCM_REALP (x))
    {
      double rx = SCM_REAL_VALUE (x);
      if (SCM_I_INUMP (y))
	{
	  long int yy = SCM_I_INUM (y);
#ifndef ALLOW_DIVIDE_BY_EXACT_ZERO
	  if (yy == 0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_from_double (rx / (double) yy);
	}
      else if (SCM_BIGP (y))
	{
	  double dby = mpz_get_d (SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_1 (y);
	  return scm_from_double (rx / dby);
	}
      else if (SCM_REALP (y))
	{
	  double yy = SCM_REAL_VALUE (y);
#ifndef ALLOW_DIVIDE_BY_ZERO
	  if (yy == 0.0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_from_double (rx / yy);
	}
      else if (SCM_COMPLEXP (y))
	{
	  a = rx;
	  goto complex_div;
	}
      else if (SCM_FRACTIONP (y))
	return scm_from_double (rx / scm_i_fraction2double (y));
      else
	SCM_WTA_DISPATCH_2 (g_divide, x, y, SCM_ARGn, s_divide);
    }
  else if (SCM_COMPLEXP (x))
    {
      double rx = SCM_COMPLEX_REAL (x);
      double ix = SCM_COMPLEX_IMAG (x);
      if (SCM_I_INUMP (y))
	{
	  long int yy = SCM_I_INUM (y);
#ifndef ALLOW_DIVIDE_BY_EXACT_ZERO
	  if (yy == 0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    {
	      double d = yy;
	      return scm_c_make_rectangular (rx / d, ix / d);
	    }
	}
      else if (SCM_BIGP (y))
	{
	  double dby = mpz_get_d (SCM_I_BIG_MPZ (y));
	  scm_remember_upto_here_1 (y);
	  return scm_c_make_rectangular (rx / dby, ix / dby);
	}
      else if (SCM_REALP (y))
	{
	  double yy = SCM_REAL_VALUE (y);
#ifndef ALLOW_DIVIDE_BY_ZERO
	  if (yy == 0.0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_c_make_rectangular (rx / yy, ix / yy);
	}
      else if (SCM_COMPLEXP (y))
	{
	  double ry = SCM_COMPLEX_REAL (y);
	  double iy = SCM_COMPLEX_IMAG (y);
	  if (fabs(ry) <= fabs(iy))
	    {
	      double t = ry / iy;
	      double d = iy * (1.0 + t * t);
	      return scm_c_make_rectangular ((rx * t + ix) / d, (ix * t - rx) / d);
	    }
	  else
	    {
	      double t = iy / ry;
	      double d = ry * (1.0 + t * t);
	      return scm_c_make_rectangular ((rx + ix * t) / d, (ix - rx * t) / d);
	    }
	}
      else if (SCM_FRACTIONP (y))
	{
	  double yy = scm_i_fraction2double (y);
	  return scm_c_make_rectangular (rx / yy, ix / yy);
	}
      else
	SCM_WTA_DISPATCH_2 (g_divide, x, y, SCM_ARGn, s_divide);
    }
  else if (SCM_FRACTIONP (x))
    {
      if (SCM_I_INUMP (y)) 
	{
	  long int yy = SCM_I_INUM (y);
#ifndef ALLOW_DIVIDE_BY_EXACT_ZERO
	  if (yy == 0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_i_make_ratio (SCM_FRACTION_NUMERATOR (x),
				   scm_product (SCM_FRACTION_DENOMINATOR (x), y));
	} 
      else if (SCM_BIGP (y)) 
	{
	  return scm_i_make_ratio (SCM_FRACTION_NUMERATOR (x),
				 scm_product (SCM_FRACTION_DENOMINATOR (x), y));
	} 
      else if (SCM_REALP (y)) 
	{
	  double yy = SCM_REAL_VALUE (y);
#ifndef ALLOW_DIVIDE_BY_ZERO
	  if (yy == 0.0)
	    scm_num_overflow (s_divide);
	  else
#endif
	    return scm_from_double (scm_i_fraction2double (x) / yy);
	}
      else if (SCM_COMPLEXP (y)) 
	{
	  a = scm_i_fraction2double (x);
	  goto complex_div;
	} 
      else if (SCM_FRACTIONP (y))
	return scm_i_make_ratio (scm_product (SCM_FRACTION_NUMERATOR (x), SCM_FRACTION_DENOMINATOR (y)),
			       scm_product (SCM_FRACTION_NUMERATOR (y), SCM_FRACTION_DENOMINATOR (x)));
      else 
	SCM_WTA_DISPATCH_2 (g_divide, x, y, SCM_ARGn, s_divide);
    }
  else
    SCM_WTA_DISPATCH_2 (g_divide, x, y, SCM_ARG1, s_divide);
}

SCM
scm_divide (SCM x, SCM y)
{
  return scm_i_divide (x, y, 0);
}

static SCM scm_divide2real (SCM x, SCM y)
{
  return scm_i_divide (x, y, 1);
}
#undef FUNC_NAME


double
scm_asinh (double x)
{
#if HAVE_ASINH
  return asinh (x);
#else
#define asinh scm_asinh
  return log (x + sqrt (x * x + 1));
#endif
}
SCM_GPROC1 (s_asinh, "$asinh", scm_tc7_dsubr, (SCM (*)()) asinh, g_asinh);
/* "Return the inverse hyperbolic sine of @var{x}."
 */


double
scm_acosh (double x)
{
#if HAVE_ACOSH
  return acosh (x);
#else
#define acosh scm_acosh
  return log (x + sqrt (x * x - 1));
#endif
}
SCM_GPROC1 (s_acosh, "$acosh", scm_tc7_dsubr, (SCM (*)()) acosh, g_acosh);
/* "Return the inverse hyperbolic cosine of @var{x}."
 */


double
scm_atanh (double x)
{
#if HAVE_ATANH
  return atanh (x);
#else
#define atanh scm_atanh
  return 0.5 * log ((1 + x) / (1 - x));
#endif
}
SCM_GPROC1 (s_atanh, "$atanh", scm_tc7_dsubr, (SCM (*)()) atanh, g_atanh);
/* "Return the inverse hyperbolic tangent of @var{x}."
 */


double
scm_c_truncate (double x)
{
#if HAVE_TRUNC
  return trunc (x);
#else
  if (x < 0.0)
    return -floor (-x);
  return floor (x);
#endif
}

/* scm_c_round is done using floor(x+0.5) to round to nearest and with
   half-way case (ie. when x is an integer plus 0.5) going upwards.
   Then half-way cases are identified and adjusted down if the
   round-upwards didn't give the desired even integer.

   "plus_half == result" identifies a half-way case.  If plus_half, which is
   x + 0.5, is an integer then x must be an integer plus 0.5.

   An odd "result" value is identified with result/2 != floor(result/2).
   This is done with plus_half, since that value is ready for use sooner in
   a pipelined cpu, and we're already requiring plus_half == result.

   Note however that we need to be careful when x is big and already an
   integer.  In that case "x+0.5" may round to an adjacent integer, causing
   us to return such a value, incorrectly.  For instance if the hardware is
   in the usual default nearest-even rounding, then for x = 0x1FFFFFFFFFFFFF
   (ie. 53 one bits) we will have x+0.5 = 0x20000000000000 and that value
   returned.  Or if the hardware is in round-upwards mode, then other bigger
   values like say x == 2^128 will see x+0.5 rounding up to the next higher
   representable value, 2^128+2^76 (or whatever), again incorrect.

   These bad roundings of x+0.5 are avoided by testing at the start whether
   x is already an integer.  If it is then clearly that's the desired result
   already.  And if it's not then the exponent must be small enough to allow
   an 0.5 to be represented, and hence added without a bad rounding.  */

double
scm_c_round (double x)
{
  double plus_half, result;

  if (x == floor (x))
    return x;

  plus_half = x + 0.5;
  result = floor (plus_half);
  /* Adjust so that the rounding is towards even.  */
  return ((plus_half == result && plus_half / 2 != floor (plus_half / 2))
	  ? result - 1
	  : result);
}

SCM_DEFINE (scm_truncate_number, "truncate", 1, 0, 0,
	    (SCM x),
	    "Round the number @var{x} towards zero.")
#define FUNC_NAME s_scm_truncate_number
{
  if (scm_is_false (scm_negative_p (x)))
    return scm_floor (x);
  else
    return scm_ceiling (x);
}
#undef FUNC_NAME

static SCM exactly_one_half;

SCM_DEFINE (scm_round_number, "round", 1, 0, 0,
	    (SCM x),
	    "Round the number @var{x} towards the nearest integer. "
	    "When it is exactly halfway between two integers, "
	    "round towards the even one.")
#define FUNC_NAME s_scm_round_number
{
  if (SCM_I_INUMP (x) || SCM_BIGP (x))
    return x;
  else if (SCM_REALP (x))
    return scm_from_double (scm_c_round (SCM_REAL_VALUE (x)));
  else
    {
      /* OPTIMIZE-ME: Fraction case could be done more efficiently by a
         single quotient+remainder division then examining to see which way
         the rounding should go.  */
      SCM plus_half = scm_sum (x, exactly_one_half);
      SCM result = scm_floor (plus_half);
      /* Adjust so that the rounding is towards even.  */
      if (scm_is_true (scm_num_eq_p (plus_half, result))
          && scm_is_true (scm_odd_p (result)))
        return scm_difference (result, SCM_I_MAKINUM (1));
      else
        return result;
    }
}
#undef FUNC_NAME

SCM_PRIMITIVE_GENERIC (scm_floor, "floor", 1, 0, 0,
		       (SCM x),
		       "Round the number @var{x} towards minus infinity.")
#define FUNC_NAME s_scm_floor
{
  if (SCM_I_INUMP (x) || SCM_BIGP (x))
    return x;
  else if (SCM_REALP (x))
    return scm_from_double (floor (SCM_REAL_VALUE (x)));
  else if (SCM_FRACTIONP (x))
    {
      SCM q = scm_quotient (SCM_FRACTION_NUMERATOR (x),
			    SCM_FRACTION_DENOMINATOR (x));
      if (scm_is_false (scm_negative_p (x)))
	{
	  /* For positive x, rounding towards zero is correct. */
	  return q;
	}
      else
	{
	  /* For negative x, we need to return q-1 unless x is an
	     integer.  But fractions are never integer, per our
	     assumptions. */
	  return scm_difference (q, SCM_I_MAKINUM (1));
	}
    }
  else
    SCM_WTA_DISPATCH_1 (g_scm_floor, x, 1, s_scm_floor);
}  
#undef FUNC_NAME

SCM_PRIMITIVE_GENERIC (scm_ceiling, "ceiling", 1, 0, 0,
		       (SCM x),
		       "Round the number @var{x} towards infinity.")
#define FUNC_NAME s_scm_ceiling
{
  if (SCM_I_INUMP (x) || SCM_BIGP (x))
    return x;
  else if (SCM_REALP (x))
    return scm_from_double (ceil (SCM_REAL_VALUE (x)));
  else if (SCM_FRACTIONP (x))
    {
      SCM q = scm_quotient (SCM_FRACTION_NUMERATOR (x),
			    SCM_FRACTION_DENOMINATOR (x));
      if (scm_is_false (scm_positive_p (x)))
	{
	  /* For negative x, rounding towards zero is correct. */
	  return q;
	}
      else
	{
	  /* For positive x, we need to return q+1 unless x is an
	     integer.  But fractions are never integer, per our
	     assumptions. */
	  return scm_sum (q, SCM_I_MAKINUM (1));
	}
    }
  else
    SCM_WTA_DISPATCH_1 (g_scm_ceiling, x, 1, s_scm_ceiling);
}
#undef FUNC_NAME

SCM_GPROC1 (s_i_sqrt, "$sqrt", scm_tc7_dsubr, (SCM (*)()) sqrt, g_i_sqrt);
/* "Return the square root of the real number @var{x}."
 */
SCM_GPROC1 (s_i_abs, "$abs", scm_tc7_dsubr, (SCM (*)()) fabs, g_i_abs);
/* "Return the absolute value of the real number @var{x}."
 */
SCM_GPROC1 (s_i_exp, "$exp", scm_tc7_dsubr, (SCM (*)()) exp, g_i_exp);
/* "Return the @var{x}th power of e."
 */
SCM_GPROC1 (s_i_log, "$log", scm_tc7_dsubr, (SCM (*)()) log, g_i_log);
/* "Return the natural logarithm of the real number @var{x}."
 */
SCM_GPROC1 (s_i_sin, "$sin", scm_tc7_dsubr, (SCM (*)()) sin, g_i_sin);
/* "Return the sine of the real number @var{x}."
 */
SCM_GPROC1 (s_i_cos, "$cos", scm_tc7_dsubr, (SCM (*)()) cos, g_i_cos);
/* "Return the cosine of the real number @var{x}."
 */
SCM_GPROC1 (s_i_tan, "$tan", scm_tc7_dsubr, (SCM (*)()) tan, g_i_tan);
/* "Return the tangent of the real number @var{x}."
 */
SCM_GPROC1 (s_i_asin, "$asin", scm_tc7_dsubr, (SCM (*)()) asin, g_i_asin);
/* "Return the arc sine of the real number @var{x}."
 */
SCM_GPROC1 (s_i_acos, "$acos", scm_tc7_dsubr, (SCM (*)()) acos, g_i_acos);
/* "Return the arc cosine of the real number @var{x}."
 */
SCM_GPROC1 (s_i_atan, "$atan", scm_tc7_dsubr, (SCM (*)()) atan, g_i_atan);
/* "Return the arc tangent of the real number @var{x}."
 */
SCM_GPROC1 (s_i_sinh, "$sinh", scm_tc7_dsubr, (SCM (*)()) sinh, g_i_sinh);
/* "Return the hyperbolic sine of the real number @var{x}."
 */
SCM_GPROC1 (s_i_cosh, "$cosh", scm_tc7_dsubr, (SCM (*)()) cosh, g_i_cosh);
/* "Return the hyperbolic cosine of the real number @var{x}."
 */
SCM_GPROC1 (s_i_tanh, "$tanh", scm_tc7_dsubr, (SCM (*)()) tanh, g_i_tanh);
/* "Return the hyperbolic tangent of the real number @var{x}."
 */

struct dpair
{
  double x, y;
};

static void scm_two_doubles (SCM x,
			     SCM y,
			     const char *sstring,
			     struct dpair * xy);

static void
scm_two_doubles (SCM x, SCM y, const char *sstring, struct dpair *xy)
{
  if (SCM_I_INUMP (x))
    xy->x = SCM_I_INUM (x);
  else if (SCM_BIGP (x))
    xy->x = scm_i_big2dbl (x);
  else if (SCM_REALP (x))
    xy->x = SCM_REAL_VALUE (x);
  else if (SCM_FRACTIONP (x))
    xy->x = scm_i_fraction2double (x);
  else
    scm_wrong_type_arg (sstring, SCM_ARG1, x);

  if (SCM_I_INUMP (y))
    xy->y = SCM_I_INUM (y);
  else if (SCM_BIGP (y))
    xy->y = scm_i_big2dbl (y);
  else if (SCM_REALP (y))
    xy->y = SCM_REAL_VALUE (y);
  else if (SCM_FRACTIONP (y))
    xy->y = scm_i_fraction2double (y);
  else
    scm_wrong_type_arg (sstring, SCM_ARG2, y);
}


SCM_DEFINE (scm_sys_expt, "$expt", 2, 0, 0,
            (SCM x, SCM y),
	    "Return @var{x} raised to the power of @var{y}. This\n"
	    "procedure does not accept complex arguments.") 
#define FUNC_NAME s_scm_sys_expt
{
  struct dpair xy;
  scm_two_doubles (x, y, FUNC_NAME, &xy);
  return scm_from_double (pow (xy.x, xy.y));
}
#undef FUNC_NAME


SCM_DEFINE (scm_sys_atan2, "$atan2", 2, 0, 0,
            (SCM x, SCM y),
	    "Return the arc tangent of the two arguments @var{x} and\n"
	    "@var{y}. This is similar to calculating the arc tangent of\n"
	    "@var{x} / @var{y}, except that the signs of both arguments\n"
	    "are used to determine the quadrant of the result. This\n"
	    "procedure does not accept complex arguments.")
#define FUNC_NAME s_scm_sys_atan2
{
  struct dpair xy;
  scm_two_doubles (x, y, FUNC_NAME, &xy);
  return scm_from_double (atan2 (xy.x, xy.y));
}
#undef FUNC_NAME

SCM
scm_c_make_rectangular (double re, double im)
{
  if (im == 0.0)
    return scm_from_double (re);
  else
    {
      SCM z;
      SCM_NEWSMOB (z, scm_tc16_complex, scm_gc_malloc (sizeof (scm_t_complex),
						       "complex"));
      SCM_COMPLEX_REAL (z) = re;
      SCM_COMPLEX_IMAG (z) = im;
      return z;
    }
}

SCM_DEFINE (scm_make_rectangular, "make-rectangular", 2, 0, 0,
            (SCM real_part, SCM imaginary_part),
	    "Return a complex number constructed of the given @var{real-part} "
	    "and @var{imaginary-part} parts.")
#define FUNC_NAME s_scm_make_rectangular
{
  struct dpair xy;
  scm_two_doubles (real_part, imaginary_part, FUNC_NAME, &xy);
  return scm_c_make_rectangular (xy.x, xy.y);
}
#undef FUNC_NAME

SCM
scm_c_make_polar (double mag, double ang)
{
  double s, c;

  /* The sincos(3) function is undocumented an broken on Tru64.  Thus we only
     use it on Glibc-based systems that have it (it's a GNU extension).  See
     http://lists.gnu.org/archive/html/guile-user/2009-04/msg00033.html for
     details.  */
#if (defined HAVE_SINCOS) && (defined __GLIBC__) && (defined _GNU_SOURCE)
  sincos (ang, &s, &c);
#else
  s = sin (ang);
  c = cos (ang);
#endif
  return scm_c_make_rectangular (mag * c, mag * s);
}

SCM_DEFINE (scm_make_polar, "make-polar", 2, 0, 0,
            (SCM x, SCM y),
	    "Return the complex number @var{x} * e^(i * @var{y}).")
#define FUNC_NAME s_scm_make_polar
{
  struct dpair xy;
  scm_two_doubles (x, y, FUNC_NAME, &xy);
  return scm_c_make_polar (xy.x, xy.y);
}
#undef FUNC_NAME


SCM_GPROC (s_real_part, "real-part", 1, 0, 0, scm_real_part, g_real_part);
/* "Return the real part of the number @var{z}."
 */
SCM
scm_real_part (SCM z)
{
  if (SCM_I_INUMP (z))
    return z;
  else if (SCM_BIGP (z))
    return z;
  else if (SCM_REALP (z))
    return z;
  else if (SCM_COMPLEXP (z))
    return scm_from_double (SCM_COMPLEX_REAL (z));
  else if (SCM_FRACTIONP (z))
    return z;
  else
    SCM_WTA_DISPATCH_1 (g_real_part, z, SCM_ARG1, s_real_part);
}


SCM_GPROC (s_imag_part, "imag-part", 1, 0, 0, scm_imag_part, g_imag_part);
/* "Return the imaginary part of the number @var{z}."
 */
SCM
scm_imag_part (SCM z)
{
  if (SCM_I_INUMP (z))
    return SCM_INUM0;
  else if (SCM_BIGP (z))
    return SCM_INUM0;
  else if (SCM_REALP (z))
    return scm_flo0;
  else if (SCM_COMPLEXP (z))
    return scm_from_double (SCM_COMPLEX_IMAG (z));
  else if (SCM_FRACTIONP (z))
    return SCM_INUM0;
  else
    SCM_WTA_DISPATCH_1 (g_imag_part, z, SCM_ARG1, s_imag_part);
}

SCM_GPROC (s_numerator, "numerator", 1, 0, 0, scm_numerator, g_numerator);
/* "Return the numerator of the number @var{z}."
 */
SCM
scm_numerator (SCM z)
{
  if (SCM_I_INUMP (z))
    return z;
  else if (SCM_BIGP (z))
    return z;
  else if (SCM_FRACTIONP (z))
    return SCM_FRACTION_NUMERATOR (z);
  else if (SCM_REALP (z))
    return scm_exact_to_inexact (scm_numerator (scm_inexact_to_exact (z)));
  else
    SCM_WTA_DISPATCH_1 (g_numerator, z, SCM_ARG1, s_numerator);
}


SCM_GPROC (s_denominator, "denominator", 1, 0, 0, scm_denominator, g_denominator);
/* "Return the denominator of the number @var{z}."
 */
SCM
scm_denominator (SCM z)
{
  if (SCM_I_INUMP (z))
    return SCM_I_MAKINUM (1);
  else if (SCM_BIGP (z)) 
    return SCM_I_MAKINUM (1);
  else if (SCM_FRACTIONP (z))
    return SCM_FRACTION_DENOMINATOR (z);
  else if (SCM_REALP (z))
    return scm_exact_to_inexact (scm_denominator (scm_inexact_to_exact (z)));
  else
    SCM_WTA_DISPATCH_1 (g_denominator, z, SCM_ARG1, s_denominator);
}

SCM_GPROC (s_magnitude, "magnitude", 1, 0, 0, scm_magnitude, g_magnitude);
/* "Return the magnitude of the number @var{z}. This is the same as\n"
 * "@code{abs} for real arguments, but also allows complex numbers."
 */
SCM
scm_magnitude (SCM z)
{
  if (SCM_I_INUMP (z))
    {
      long int zz = SCM_I_INUM (z);
      if (zz >= 0)
	return z;
      else if (SCM_POSFIXABLE (-zz))
	return SCM_I_MAKINUM (-zz);
      else
	return scm_i_long2big (-zz);
    }
  else if (SCM_BIGP (z))
    {
      int sgn = mpz_sgn (SCM_I_BIG_MPZ (z));
      scm_remember_upto_here_1 (z);
      if (sgn < 0)
	return scm_i_clonebig (z, 0);
      else
	return z;
    }
  else if (SCM_REALP (z))
    return scm_from_double (fabs (SCM_REAL_VALUE (z)));
  else if (SCM_COMPLEXP (z))
    return scm_from_double (hypot (SCM_COMPLEX_REAL (z), SCM_COMPLEX_IMAG (z)));
  else if (SCM_FRACTIONP (z))
    {
      if (scm_is_false (scm_negative_p (SCM_FRACTION_NUMERATOR (z))))
	return z;
      return scm_i_make_ratio (scm_difference (SCM_FRACTION_NUMERATOR (z), SCM_UNDEFINED),
			     SCM_FRACTION_DENOMINATOR (z));
    }
  else
    SCM_WTA_DISPATCH_1 (g_magnitude, z, SCM_ARG1, s_magnitude);
}


SCM_GPROC (s_angle, "angle", 1, 0, 0, scm_angle, g_angle);
/* "Return the angle of the complex number @var{z}."
 */
SCM
scm_angle (SCM z)
{
  /* atan(0,-1) is pi and it'd be possible to have that as a constant like
     scm_flo0 to save allocating a new flonum with scm_from_double each time.
     But if atan2 follows the floating point rounding mode, then the value
     is not a constant.  Maybe it'd be close enough though.  */
  if (SCM_I_INUMP (z))
    {
      if (SCM_I_INUM (z) >= 0)
        return scm_flo0;
      else
	return scm_from_double (atan2 (0.0, -1.0));
    }
  else if (SCM_BIGP (z))
    {
      int sgn = mpz_sgn (SCM_I_BIG_MPZ (z));
      scm_remember_upto_here_1 (z);
      if (sgn < 0)
	return scm_from_double (atan2 (0.0, -1.0));
      else
        return scm_flo0;
    }
  else if (SCM_REALP (z))
    {
      if (SCM_REAL_VALUE (z) >= 0)
        return scm_flo0;
      else
        return scm_from_double (atan2 (0.0, -1.0));
    }
  else if (SCM_COMPLEXP (z))
    return scm_from_double (atan2 (SCM_COMPLEX_IMAG (z), SCM_COMPLEX_REAL (z)));
  else if (SCM_FRACTIONP (z))
    {
      if (scm_is_false (scm_negative_p (SCM_FRACTION_NUMERATOR (z))))
	return scm_flo0;
      else return scm_from_double (atan2 (0.0, -1.0));
    }
  else
    SCM_WTA_DISPATCH_1 (g_angle, z, SCM_ARG1, s_angle);
}


SCM_GPROC (s_exact_to_inexact, "exact->inexact", 1, 0, 0, scm_exact_to_inexact, g_exact_to_inexact);
/* Convert the number @var{x} to its inexact representation.\n" 
 */
SCM
scm_exact_to_inexact (SCM z)
{
  if (SCM_I_INUMP (z))
    return scm_from_double ((double) SCM_I_INUM (z));
  else if (SCM_BIGP (z))
    return scm_from_double (scm_i_big2dbl (z));
  else if (SCM_FRACTIONP (z))
    return scm_from_double (scm_i_fraction2double (z));
  else if (SCM_INEXACTP (z))
    return z;
  else
    SCM_WTA_DISPATCH_1 (g_exact_to_inexact, z, 1, s_exact_to_inexact);
}


SCM_DEFINE (scm_inexact_to_exact, "inexact->exact", 1, 0, 0, 
            (SCM z),
	    "Return an exact number that is numerically closest to @var{z}.")
#define FUNC_NAME s_scm_inexact_to_exact
{
  if (SCM_I_INUMP (z))
    return z;
  else if (SCM_BIGP (z))
    return z;
  else if (SCM_REALP (z))
    {
      if (xisinf (SCM_REAL_VALUE (z)) || xisnan (SCM_REAL_VALUE (z)))
	SCM_OUT_OF_RANGE (1, z);
      else
	{
	  mpq_t frac;
	  SCM q;
	  
	  mpq_init (frac);
	  mpq_set_d (frac, SCM_REAL_VALUE (z));
	  q = scm_i_make_ratio (scm_i_mpz2num (mpq_numref (frac)),
			      scm_i_mpz2num (mpq_denref (frac)));

	  /* When scm_i_make_ratio throws, we leak the memory allocated
	     for frac...
	   */
	  mpq_clear (frac);
	  return q;
	}
    }
  else if (SCM_FRACTIONP (z))
    return z;
  else
    SCM_WRONG_TYPE_ARG (1, z);
}
#undef FUNC_NAME

SCM_DEFINE (scm_rationalize, "rationalize", 2, 0, 0, 
            (SCM x, SCM eps),
	    "Returns the @emph{simplest} rational number differing\n"
	    "from @var{x} by no more than @var{eps}.\n"
	    "\n"
	    "As required by @acronym{R5RS}, @code{rationalize} only returns an\n"
	    "exact result when both its arguments are exact.  Thus, you might need\n"
	    "to use @code{inexact->exact} on the arguments.\n"
	    "\n"
	    "@lisp\n"
	    "(rationalize (inexact->exact 1.2) 1/100)\n"
	    "@result{} 6/5\n"
	    "@end lisp")
#define FUNC_NAME s_scm_rationalize
{
  if (SCM_I_INUMP (x))
    return x;
  else if (SCM_BIGP (x))
    return x;
  else if ((SCM_REALP (x)) || SCM_FRACTIONP (x)) 
    {
      /* Use continued fractions to find closest ratio.  All
	 arithmetic is done with exact numbers.
      */

      SCM ex = scm_inexact_to_exact (x);
      SCM int_part = scm_floor (ex);
      SCM tt = SCM_I_MAKINUM (1);
      SCM a1 = SCM_I_MAKINUM (0), a2 = SCM_I_MAKINUM (1), a = SCM_I_MAKINUM (0);
      SCM b1 = SCM_I_MAKINUM (1), b2 = SCM_I_MAKINUM (0), b = SCM_I_MAKINUM (0);
      SCM rx;
      int i = 0;

      if (scm_is_true (scm_num_eq_p (ex, int_part)))
	return ex;
      
      ex = scm_difference (ex, int_part);            /* x = x-int_part */
      rx = scm_divide (ex, SCM_UNDEFINED); 	       /* rx = 1/x */

      /* We stop after a million iterations just to be absolutely sure
	 that we don't go into an infinite loop.  The process normally
	 converges after less than a dozen iterations.
      */

      eps = scm_abs (eps);
      while (++i < 1000000)
	{
	  a = scm_sum (scm_product (a1, tt), a2);    /* a = a1*tt + a2 */
	  b = scm_sum (scm_product (b1, tt), b2);    /* b = b1*tt + b2 */
	  if (scm_is_false (scm_zero_p (b)) &&         /* b != 0 */
	      scm_is_false 
	      (scm_gr_p (scm_abs (scm_difference (ex, scm_divide (a, b))),
			 eps)))                      /* abs(x-a/b) <= eps */
	    {
	      SCM res = scm_sum (int_part, scm_divide (a, b));
	      if (scm_is_false (scm_exact_p (x))
		  || scm_is_false (scm_exact_p (eps)))
		return scm_exact_to_inexact (res);
	      else
		return res;
	    }
	  rx = scm_divide (scm_difference (rx, tt),  /* rx = 1/(rx - tt) */
			   SCM_UNDEFINED);
	  tt = scm_floor (rx);                       /* tt = floor (rx) */
	  a2 = a1;
	  b2 = b1;
	  a1 = a;
	  b1 = b;
	}
      scm_num_overflow (s_scm_rationalize);
    }
  else
    SCM_WRONG_TYPE_ARG (1, x);
}
#undef FUNC_NAME

/* conversion functions */

int
scm_is_integer (SCM val)
{
  return scm_is_true (scm_integer_p (val));
}

int
scm_is_signed_integer (SCM val, scm_t_intmax min, scm_t_intmax max)
{
  if (SCM_I_INUMP (val))
    {
      scm_t_signed_bits n = SCM_I_INUM (val);
      return n >= min && n <= max;
    }
  else if (SCM_BIGP (val))
    {
      if (min >= SCM_MOST_NEGATIVE_FIXNUM && max <= SCM_MOST_POSITIVE_FIXNUM)
	return 0;
      else if (min >= LONG_MIN && max <= LONG_MAX)
	{
	  if (mpz_fits_slong_p (SCM_I_BIG_MPZ (val)))
	    {
	      long n = mpz_get_si (SCM_I_BIG_MPZ (val));
	      return n >= min && n <= max;
	    }
	  else
	    return 0;
	}
      else
	{
	  scm_t_intmax n;
	  size_t count;

	  if (mpz_sizeinbase (SCM_I_BIG_MPZ (val), 2) 
	      > CHAR_BIT*sizeof (scm_t_uintmax))
	    return 0;
	  
	  mpz_export (&n, &count, 1, sizeof (scm_t_uintmax), 0, 0,
		      SCM_I_BIG_MPZ (val));

	  if (mpz_sgn (SCM_I_BIG_MPZ (val)) >= 0)
	    {
	      if (n < 0)
		return 0;
	    }
	  else
	    {
	      n = -n;
	      if (n >= 0)
		return 0;
	    }

	  return n >= min && n <= max;
	}
    }
  else
    return 0;
}

int
scm_is_unsigned_integer (SCM val, scm_t_uintmax min, scm_t_uintmax max)
{
  if (SCM_I_INUMP (val))
    {
      scm_t_signed_bits n = SCM_I_INUM (val);
      return n >= 0 && ((scm_t_uintmax)n) >= min && ((scm_t_uintmax)n) <= max;
    }
  else if (SCM_BIGP (val))
    {
      if (max <= SCM_MOST_POSITIVE_FIXNUM)
	return 0;
      else if (max <= ULONG_MAX)
	{
	  if (mpz_fits_ulong_p (SCM_I_BIG_MPZ (val)))
	    {
	      unsigned long n = mpz_get_ui (SCM_I_BIG_MPZ (val));
	      return n >= min && n <= max;
	    }
	  else
	    return 0;
	}
      else
	{
	  scm_t_uintmax n;
	  size_t count;

	  if (mpz_sgn (SCM_I_BIG_MPZ (val)) < 0)
	    return 0;

	  if (mpz_sizeinbase (SCM_I_BIG_MPZ (val), 2)
	      > CHAR_BIT*sizeof (scm_t_uintmax))
	    return 0;
	  
	  mpz_export (&n, &count, 1, sizeof (scm_t_uintmax), 0, 0,
		      SCM_I_BIG_MPZ (val));

	  return n >= min && n <= max;
	}
    }
  else
    return 0;
}

static void
scm_i_range_error (SCM bad_val, SCM min, SCM max)
{
  scm_error (scm_out_of_range_key,
	     NULL,
	     "Value out of range ~S to ~S: ~S",
             scm_list_3 (min, max, bad_val),
             scm_list_1 (bad_val));
}

#define TYPE                     scm_t_intmax
#define TYPE_MIN                 min
#define TYPE_MAX                 max
#define SIZEOF_TYPE              0
#define SCM_TO_TYPE_PROTO(arg)   scm_to_signed_integer (arg, scm_t_intmax min, scm_t_intmax max)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_signed_integer (arg)
#include "libguile/conv-integer.i.c"

#define TYPE                     scm_t_uintmax
#define TYPE_MIN                 min
#define TYPE_MAX                 max
#define SIZEOF_TYPE              0
#define SCM_TO_TYPE_PROTO(arg)   scm_to_unsigned_integer (arg, scm_t_uintmax min, scm_t_uintmax max)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_unsigned_integer (arg)
#include "libguile/conv-uinteger.i.c"

#define TYPE                     scm_t_int8
#define TYPE_MIN                 SCM_T_INT8_MIN
#define TYPE_MAX                 SCM_T_INT8_MAX
#define SIZEOF_TYPE              1
#define SCM_TO_TYPE_PROTO(arg)   scm_to_int8 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_int8 (arg)
#include "libguile/conv-integer.i.c"

#define TYPE                     scm_t_uint8
#define TYPE_MIN                 0
#define TYPE_MAX                 SCM_T_UINT8_MAX
#define SIZEOF_TYPE              1
#define SCM_TO_TYPE_PROTO(arg)   scm_to_uint8 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_uint8 (arg)
#include "libguile/conv-uinteger.i.c"

#define TYPE                     scm_t_int16
#define TYPE_MIN                 SCM_T_INT16_MIN
#define TYPE_MAX                 SCM_T_INT16_MAX
#define SIZEOF_TYPE              2
#define SCM_TO_TYPE_PROTO(arg)   scm_to_int16 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_int16 (arg)
#include "libguile/conv-integer.i.c"

#define TYPE                     scm_t_uint16
#define TYPE_MIN                 0
#define TYPE_MAX                 SCM_T_UINT16_MAX
#define SIZEOF_TYPE              2
#define SCM_TO_TYPE_PROTO(arg)   scm_to_uint16 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_uint16 (arg)
#include "libguile/conv-uinteger.i.c"

#define TYPE                     scm_t_int32
#define TYPE_MIN                 SCM_T_INT32_MIN
#define TYPE_MAX                 SCM_T_INT32_MAX
#define SIZEOF_TYPE              4
#define SCM_TO_TYPE_PROTO(arg)   scm_to_int32 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_int32 (arg)
#include "libguile/conv-integer.i.c"

#define TYPE                     scm_t_uint32
#define TYPE_MIN                 0
#define TYPE_MAX                 SCM_T_UINT32_MAX
#define SIZEOF_TYPE              4
#define SCM_TO_TYPE_PROTO(arg)   scm_to_uint32 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_uint32 (arg)
#include "libguile/conv-uinteger.i.c"

#define TYPE                     scm_t_int64
#define TYPE_MIN                 SCM_T_INT64_MIN
#define TYPE_MAX                 SCM_T_INT64_MAX
#define SIZEOF_TYPE              8
#define SCM_TO_TYPE_PROTO(arg)   scm_to_int64 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_int64 (arg)
#include "libguile/conv-integer.i.c"

#define TYPE                     scm_t_uint64
#define TYPE_MIN                 0
#define TYPE_MAX                 SCM_T_UINT64_MAX
#define SIZEOF_TYPE              8
#define SCM_TO_TYPE_PROTO(arg)   scm_to_uint64 (arg)
#define SCM_FROM_TYPE_PROTO(arg) scm_from_uint64 (arg)
#include "libguile/conv-uinteger.i.c"

void
scm_to_mpz (SCM val, mpz_t rop)
{
  if (SCM_I_INUMP (val))
    mpz_set_si (rop, SCM_I_INUM (val));
  else if (SCM_BIGP (val))
    mpz_set (rop, SCM_I_BIG_MPZ (val));
  else
    scm_wrong_type_arg_msg (NULL, 0, val, "exact integer");
}

SCM
scm_from_mpz (mpz_t val)
{
  return scm_i_mpz2num (val);
}

int
scm_is_real (SCM val)
{
  return scm_is_true (scm_real_p (val));
}

int
scm_is_rational (SCM val)
{
  return scm_is_true (scm_rational_p (val));
}

double
scm_to_double (SCM val)
{
  if (SCM_I_INUMP (val))
    return SCM_I_INUM (val);
  else if (SCM_BIGP (val))
    return scm_i_big2dbl (val);
  else if (SCM_FRACTIONP (val))
    return scm_i_fraction2double (val);
  else if (SCM_REALP (val))
    return SCM_REAL_VALUE (val);
  else
    scm_wrong_type_arg_msg (NULL, 0, val, "real number");
}

SCM
scm_from_double (double val)
{
  SCM z = scm_double_cell (scm_tc16_real, 0, 0, 0);
  SCM_REAL_VALUE (z) = val;
  return z;
}

#if SCM_ENABLE_DISCOURAGED == 1

float
scm_num2float (SCM num, unsigned long int pos, const char *s_caller)
{
  if (SCM_BIGP (num))
    {
      float res = mpz_get_d (SCM_I_BIG_MPZ (num));
      if (!xisinf (res))
	return res;
      else
	scm_out_of_range (NULL, num);
    }
  else
    return scm_to_double (num);
}

double
scm_num2double (SCM num, unsigned long int pos, const char *s_caller)
{
  if (SCM_BIGP (num))
    {
      double res = mpz_get_d (SCM_I_BIG_MPZ (num));
      if (!xisinf (res))
	return res;
      else
	scm_out_of_range (NULL, num);
    }
  else
    return scm_to_double (num);
}

#endif

int
scm_is_complex (SCM val)
{
  return scm_is_true (scm_complex_p (val));
}

double
scm_c_real_part (SCM z)
{
  if (SCM_COMPLEXP (z))
    return SCM_COMPLEX_REAL (z);
  else
    {
      /* Use the scm_real_part to get proper error checking and
	 dispatching.
      */
      return scm_to_double (scm_real_part (z)); 
    }
}

double
scm_c_imag_part (SCM z)
{
  if (SCM_COMPLEXP (z))
    return SCM_COMPLEX_IMAG (z);
  else
    {
      /* Use the scm_imag_part to get proper error checking and
	 dispatching.  The result will almost always be 0.0, but not
	 always.
      */
      return scm_to_double (scm_imag_part (z));
    }
}

double
scm_c_magnitude (SCM z)
{
  return scm_to_double (scm_magnitude (z));
}

double
scm_c_angle (SCM z)
{
  return scm_to_double (scm_angle (z));
}

int
scm_is_number (SCM z)
{
  return scm_is_true (scm_number_p (z));
}


/* In the following functions we dispatch to the real-arg funcs like log()
   when we know the arg is real, instead of just handing everything to
   clog() for instance.  This is in case clog() doesn't optimize for a
   real-only case, and because we have to test SCM_COMPLEXP anyway so may as
   well use it to go straight to the applicable C func.  */

SCM_DEFINE (scm_log, "log", 1, 0, 0,
            (SCM z),
	    "Return the natural logarithm of @var{z}.")
#define FUNC_NAME s_scm_log
{
  if (SCM_COMPLEXP (z))
    {
#if HAVE_COMPLEX_DOUBLE && HAVE_CLOG && defined (SCM_COMPLEX_VALUE)
      return scm_from_complex_double (clog (SCM_COMPLEX_VALUE (z)));
#else
      double re = SCM_COMPLEX_REAL (z);
      double im = SCM_COMPLEX_IMAG (z);
      return scm_c_make_rectangular (log (hypot (re, im)),
                                     atan2 (im, re));
#endif
    }
  else
    {
      /* ENHANCE-ME: When z is a bignum the logarithm will fit a double
         although the value itself overflows.  */
      double re = scm_to_double (z);
      double l = log (fabs (re));
      if (re >= 0.0)
        return scm_from_double (l);
      else
        return scm_c_make_rectangular (l, M_PI);
    }
}
#undef FUNC_NAME


SCM_DEFINE (scm_log10, "log10", 1, 0, 0,
            (SCM z),
	    "Return the base 10 logarithm of @var{z}.")
#define FUNC_NAME s_scm_log10
{
  if (SCM_COMPLEXP (z))
    {
      /* Mingw has clog() but not clog10().  (Maybe it'd be worth using
         clog() and a multiply by M_LOG10E, rather than the fallback
         log10+hypot+atan2.)  */
#if HAVE_COMPLEX_DOUBLE && HAVE_CLOG10 && defined (SCM_COMPLEX_VALUE)
      return scm_from_complex_double (clog10 (SCM_COMPLEX_VALUE (z)));
#else
      double re = SCM_COMPLEX_REAL (z);
      double im = SCM_COMPLEX_IMAG (z);
      return scm_c_make_rectangular (log10 (hypot (re, im)),
                                     M_LOG10E * atan2 (im, re));
#endif
    }
  else
    {
      /* ENHANCE-ME: When z is a bignum the logarithm will fit a double
         although the value itself overflows.  */
      double re = scm_to_double (z);
      double l = log10 (fabs (re));
      if (re >= 0.0)
        return scm_from_double (l);
      else
        return scm_c_make_rectangular (l, M_LOG10E * M_PI);
    }
}
#undef FUNC_NAME


SCM_DEFINE (scm_exp, "exp", 1, 0, 0,
            (SCM z),
	    "Return @math{e} to the power of @var{z}, where @math{e} is the\n"
	    "base of natural logarithms (2.71828@dots{}).")
#define FUNC_NAME s_scm_exp
{
  if (SCM_COMPLEXP (z))
    {
#if HAVE_COMPLEX_DOUBLE && HAVE_CEXP && defined (SCM_COMPLEX_VALUE)
      return scm_from_complex_double (cexp (SCM_COMPLEX_VALUE (z)));
#else
      return scm_c_make_polar (exp (SCM_COMPLEX_REAL (z)),
                               SCM_COMPLEX_IMAG (z));
#endif
    }
  else
    {
      /* When z is a negative bignum the conversion to double overflows,
         giving -infinity, but that's ok, the exp is still 0.0.  */
      return scm_from_double (exp (scm_to_double (z)));
    }
}
#undef FUNC_NAME


SCM_DEFINE (scm_sqrt, "sqrt", 1, 0, 0,
            (SCM x),
	    "Return the square root of @var{z}.  Of the two possible roots\n"
	    "(positive and negative), the one with the a positive real part\n"
	    "is returned, or if that's zero then a positive imaginary part.\n"
	    "Thus,\n"
	    "\n"
	    "@example\n"
	    "(sqrt 9.0)       @result{} 3.0\n"
	    "(sqrt -9.0)      @result{} 0.0+3.0i\n"
	    "(sqrt 1.0+1.0i)  @result{} 1.09868411346781+0.455089860562227i\n"
	    "(sqrt -1.0-1.0i) @result{} 0.455089860562227-1.09868411346781i\n"
	    "@end example")
#define FUNC_NAME s_scm_sqrt
{
  if (SCM_COMPLEXP (x))
    {
#if HAVE_COMPLEX_DOUBLE && HAVE_USABLE_CSQRT && defined (SCM_COMPLEX_VALUE)
      return scm_from_complex_double (csqrt (SCM_COMPLEX_VALUE (x)));
#else
      double re = SCM_COMPLEX_REAL (x);
      double im = SCM_COMPLEX_IMAG (x);
      return scm_c_make_polar (sqrt (hypot (re, im)),
                               0.5 * atan2 (im, re));
#endif
    }
  else
    {
      double xx = scm_to_double (x);
      if (xx < 0)
        return scm_c_make_rectangular (0.0, sqrt (-xx));
      else
        return scm_from_double (sqrt (xx));
    }
}
#undef FUNC_NAME



void
scm_init_numbers ()
{
  int i;

  mpz_init_set_si (z_negative_one, -1);

  /* It may be possible to tune the performance of some algorithms by using
   * the following constants to avoid the creation of bignums.  Please, before
   * using these values, remember the two rules of program optimization:
   * 1st Rule:  Don't do it.  2nd Rule (experts only):  Don't do it yet. */
  scm_c_define ("most-positive-fixnum",
		SCM_I_MAKINUM (SCM_MOST_POSITIVE_FIXNUM));
  scm_c_define ("most-negative-fixnum",
		SCM_I_MAKINUM (SCM_MOST_NEGATIVE_FIXNUM));

  scm_add_feature ("complex");
  scm_add_feature ("inexact");
  scm_flo0 = scm_from_double (0.0);

  /* determine floating point precision */
  for (i=2; i <= SCM_MAX_DBL_RADIX; ++i)
    {
      init_dblprec(&scm_dblprec[i-2],i);
      init_fx_radix(fx_per_radix[i-2],i);
    }
#ifdef DBL_DIG
  /* hard code precision for base 10 if the preprocessor tells us to... */
      scm_dblprec[10-2] = (DBL_DIG > 20) ? 20 : DBL_DIG;
#endif

  exactly_one_half = scm_permanent_object (scm_divide (SCM_I_MAKINUM (1),
						       SCM_I_MAKINUM (2)));
#include "libguile/numbers.x"
}

/*
  Local Variables:
  c-file-style: "gnu"
  End:
*/