text: /**********************************************************************
  
    numeric.c -
  
    $Author: nagachika $
    created at: Fri Aug 13 18:33:09 JST 1993
  
    Copyright (C) 1993-2007 Yukihiro Matsumoto
  
  **********************************************************************/
  
  #include "ruby/ruby.h"
  #include "ruby/encoding.h"
  #include "ruby/util.h"
  #include "internal.h"
  #include "id.h"
  #include <ctype.h>
  #include <math.h>
  #include <stdio.h>
  
  #if defined(__FreeBSD__) && __FreeBSD__ < 4
  #include <floatingpoint.h>
  #endif
  
  #ifdef HAVE_FLOAT_H
  #include <float.h>
  #endif
  
  #ifdef HAVE_IEEEFP_H
  #include <ieeefp.h>
  #endif
  
  #if !defined HAVE_ISFINITE && !defined isfinite
  #if defined HAVE_FINITE && !defined finite && !defined _WIN32
  extern int finite(double);
  # define HAVE_ISFINITE 1
  # define isfinite(x) finite(x)
  #endif
  #endif
  
  /* use IEEE 64bit values if not defined */
  #ifndef FLT_RADIX
  #define FLT_RADIX 2
  #endif
  #ifndef FLT_ROUNDS
  #define FLT_ROUNDS 1
  #endif
  #ifndef DBL_MIN
  #define DBL_MIN 2.2250738585072014e-308
  #endif
  #ifndef DBL_MAX
  #define DBL_MAX 1.7976931348623157e+308
  #endif
  #ifndef DBL_MIN_EXP
  #define DBL_MIN_EXP (-1021)
  #endif
  #ifndef DBL_MAX_EXP
  #define DBL_MAX_EXP 1024
  #endif
  #ifndef DBL_MIN_10_EXP
  #define DBL_MIN_10_EXP (-307)
  #endif
  #ifndef DBL_MAX_10_EXP
  #define DBL_MAX_10_EXP 308
  #endif
  #ifndef DBL_DIG
  #define DBL_DIG 15
  #endif
  #ifndef DBL_MANT_DIG
  #define DBL_MANT_DIG 53
  #endif
  #ifndef DBL_EPSILON
  #define DBL_EPSILON 2.2204460492503131e-16
  #endif
  
  #ifdef HAVE_INFINITY
  #elif !defined(WORDS_BIGENDIAN) /* BYTE_ORDER == LITTLE_ENDIAN */
  const union bytesequence4_or_float rb_infinity = {{0x00, 0x00, 0x80, 0x7f}};
  #else
  const union bytesequence4_or_float rb_infinity = {{0x7f, 0x80, 0x00, 0x00}};
  #endif
  
  #ifdef HAVE_NAN
  #elif !defined(WORDS_BIGENDIAN) /* BYTE_ORDER == LITTLE_ENDIAN */
  const union bytesequence4_or_float rb_nan = {{0x00, 0x00, 0xc0, 0x7f}};
  #else
  const union bytesequence4_or_float rb_nan = {{0x7f, 0xc0, 0x00, 0x00}};
  #endif
  
  #ifndef HAVE_ROUND
  double
  round(double x)
  {
      double f;
  
      if (x > 0.0) {
  	f = floor(x);
  	x = f + (x - f >= 0.5);
      }
      else if (x < 0.0) {
  	f = ceil(x);
  	x = f - (f - x >= 0.5);
      }
      return x;
  }
  #endif
  
  static VALUE fix_uminus(VALUE num);
  static VALUE fix_mul(VALUE x, VALUE y);
  static VALUE int_pow(long x, unsigned long y);
  
  static ID id_coerce, id_to_i, id_eq, id_div, id_cmp;
  
  VALUE rb_cNumeric;
  VALUE rb_cFloat;
  VALUE rb_cInteger;
  VALUE rb_cFixnum;
  
  VALUE rb_eZeroDivError;
  VALUE rb_eFloatDomainError;
  
  static ID id_to, id_by;
  
  void
  rb_num_zerodiv(void)
  {
      rb_raise(rb_eZeroDivError, "divided by 0");
  }
  
  /* experimental API */
  int
  rb_num_to_uint(VALUE val, unsigned int *ret)
  {
  #define NUMERR_TYPE     1
  #define NUMERR_NEGATIVE 2
  #define NUMERR_TOOLARGE 3
      if (FIXNUM_P(val)) {
  	long v = FIX2LONG(val);
  #if SIZEOF_INT < SIZEOF_LONG
  	if (v > (long)UINT_MAX) return NUMERR_TOOLARGE;
  #endif
  	if (v < 0) return NUMERR_NEGATIVE;
  	*ret = (unsigned int)v;
  	return 0;
      }
  
      if (RB_TYPE_P(val, T_BIGNUM)) {
  	if (RBIGNUM_NEGATIVE_P(val)) return NUMERR_NEGATIVE;
  #if SIZEOF_INT < SIZEOF_LONG
  	/* long is 64bit */
  	return NUMERR_TOOLARGE;
  #else
  	/* long is 32bit */
  	if (rb_absint_size(val, NULL) > sizeof(int)) return NUMERR_TOOLARGE;
  	*ret = (unsigned int)rb_big2ulong((VALUE)val);
  	return 0;
  #endif
      }
      return NUMERR_TYPE;
  }
  
  #define method_basic_p(klass) rb_method_basic_definition_p(klass, mid)
  
  static inline int
  positive_int_p(VALUE num)
  {
      const ID mid = '>';
  
      if (FIXNUM_P(num)) {
  	if (method_basic_p(rb_cFixnum))
  	    return (SIGNED_VALUE)num > 0;
      }
      else if (RB_TYPE_P(num, T_BIGNUM)) {
  	if (method_basic_p(rb_cBignum))
  	    return RBIGNUM_POSITIVE_P(num);
      }
      return RTEST(rb_funcall(num, mid, 1, INT2FIX(0)));
  }
  
  static inline int
  negative_int_p(VALUE num)
  {
      const ID mid = '<';
  
      if (FIXNUM_P(num)) {
  	if (method_basic_p(rb_cFixnum))
  	    return (SIGNED_VALUE)num < 0;
      }
      else if (RB_TYPE_P(num, T_BIGNUM)) {
  	if (method_basic_p(rb_cBignum))
  	    return RBIGNUM_NEGATIVE_P(num);
      }
      return RTEST(rb_funcall(num, mid, 1, INT2FIX(0)));
  }
  
  int
  rb_num_negative_p(VALUE num)
  {
      return negative_int_p(num);
  }
  
  /*
   *  call-seq:
   *     num.coerce(numeric)  ->  array
   *
   *  If a +numeric is the same type as +num+, returns an array containing
   *  +numeric+ and +num+. Otherwise, returns an array with both a +numeric+ and
   *  +num+ represented as Float objects.
   *
   *  This coercion mechanism is used by Ruby to handle mixed-type numeric
   *  operations: it is intended to find a compatible common type between the two
   *  operands of the operator.
   *
   *     1.coerce(2.5)   #=> [2.5, 1.0]
   *     1.2.coerce(3)   #=> [3.0, 1.2]
   *     1.coerce(2)     #=> [2, 1]
   */
  
  static VALUE
  num_coerce(VALUE x, VALUE y)
  {
      if (CLASS_OF(x) == CLASS_OF(y))
  	return rb_assoc_new(y, x);
      x = rb_Float(x);
      y = rb_Float(y);
      return rb_assoc_new(y, x);
  }
  
  static VALUE
  coerce_body(VALUE *x)
  {
      return rb_funcall(x[1], id_coerce, 1, x[0]);
  }
  
  NORETURN(static void coerce_failed(VALUE x, VALUE y));
  static void
  coerce_failed(VALUE x, VALUE y)
  {
      rb_raise(rb_eTypeError, "%"PRIsVALUE" can't be coerced into %"PRIsVALUE,
  	     (rb_special_const_p(y)? rb_inspect(y) : rb_obj_class(y)),
  	     rb_obj_class(x));
  }
  
  static VALUE
  coerce_rescue(VALUE *x)
  {
      coerce_failed(x[0], x[1]);
      return Qnil;		/* dummy */
  }
  
  static int
  do_coerce(VALUE *x, VALUE *y, int err)
  {
      VALUE ary;
      VALUE a[2];
  
      a[0] = *x; a[1] = *y;
  
      if (!rb_respond_to(*y, id_coerce)) {
  	if (err) {
  	    coerce_rescue(a);
  	}
  	return FALSE;
      }
  
      ary = rb_rescue(coerce_body, (VALUE)a, err ? coerce_rescue : 0, (VALUE)a);
      if (!RB_TYPE_P(ary, T_ARRAY) || RARRAY_LEN(ary) != 2) {
  	if (err) {
  	    rb_raise(rb_eTypeError, "coerce must return [x, y]");
  	}
  	return FALSE;
      }
  
      *x = RARRAY_AREF(ary, 0);
      *y = RARRAY_AREF(ary, 1);
      return TRUE;
  }
  
  VALUE
  rb_num_coerce_bin(VALUE x, VALUE y, ID func)
  {
      do_coerce(&x, &y, TRUE);
      return rb_funcall(x, func, 1, y);
  }
  
  VALUE
  rb_num_coerce_cmp(VALUE x, VALUE y, ID func)
  {
      if (do_coerce(&x, &y, FALSE))
  	return rb_funcall(x, func, 1, y);
      return Qnil;
  }
  
  VALUE
  rb_num_coerce_relop(VALUE x, VALUE y, ID func)
  {
      VALUE c, x0 = x, y0 = y;
  
      if (!do_coerce(&x, &y, FALSE) ||
  	NIL_P(c = rb_funcall(x, func, 1, y))) {
  	rb_cmperr(x0, y0);
  	return Qnil;		/* not reached */
      }
      return c;
  }
  
  /*
   * Trap attempts to add methods to Numeric objects. Always raises a TypeError.
   *
   * Numerics should be values; singleton_methods should not be added to them.
   */
  
  static VALUE
  num_sadded(VALUE x, VALUE name)
  {
      ID mid = rb_to_id(name);
      /* ruby_frame = ruby_frame->prev; */ /* pop frame for "singleton_method_added" */
      rb_remove_method_id(rb_singleton_class(x), mid);
      rb_raise(rb_eTypeError,
  	     "can't define singleton method \"%s\" for %s",
  	     rb_id2name(mid),
  	     rb_obj_classname(x));
  
      UNREACHABLE;
  }
  
  /*
   * Numerics are immutable values, which should not be copied.
   *
   * Any attempt to use this method on a Numeric will raise a TypeError.
   */
  static VALUE
  num_init_copy(VALUE x, VALUE y)
  {
      rb_raise(rb_eTypeError, "can't copy %s", rb_obj_classname(x));
  
      UNREACHABLE;
  }
  
  /*
   *  call-seq:
   *     +num  ->  num
   *
   *  Unary Plus---Returns the receiver's value.
   */
  
  static VALUE
  num_uplus(VALUE num)
  {
      return num;
  }
  
  /*
   *  call-seq:
   *     num.i  ->  Complex(0,num)
   *
   *  Returns the corresponding imaginary number.
   *  Not available for complex numbers.
   */
  
  static VALUE
  num_imaginary(VALUE num)
  {
      return rb_complex_new(INT2FIX(0), num);
  }
  
  
  /*
   *  call-seq:
   *     -num  ->  numeric
   *
   *  Unary Minus---Returns the receiver's value, negated.
   */
  
  static VALUE
  num_uminus(VALUE num)
  {
      VALUE zero;
  
      zero = INT2FIX(0);
      do_coerce(&zero, &num, TRUE);
  
      return rb_funcall(zero, '-', 1, num);
  }
  
  /*
   *  call-seq:
   *     num.fdiv(numeric)  ->  float
   *
   *  Returns float division.
   */
  
  static VALUE
  num_fdiv(VALUE x, VALUE y)
  {
      return rb_funcall(rb_Float(x), '/', 1, y);
  }
  
  
  /*
   *  call-seq:
   *     num.div(numeric)  ->  integer
   *
   *  Uses +/+ to perform division, then converts the result to an integer.
   *  +numeric+ does not define the +/+ operator; this is left to subclasses.
   *
   *  Equivalent to <code>num.divmod(numeric)[0]</code>.
   *
   *  See Numeric#divmod.
   */
  
  static VALUE
  num_div(VALUE x, VALUE y)
  {
      if (rb_equal(INT2FIX(0), y)) rb_num_zerodiv();
      return rb_funcall(rb_funcall(x, '/', 1, y), rb_intern("floor"), 0);
  }
  
  
  /*
   *  call-seq:
   *     num.modulo(numeric)  ->  real
   *
   *     x.modulo(y) means x-y*(x/y).floor
   *
   *  Equivalent to <code>num.divmod(numeric)[1]</code>.
   *
   *  See Numeric#divmod.
   */
  
  static VALUE
  num_modulo(VALUE x, VALUE y)
  {
      return rb_funcall(x, '-', 1,
  		      rb_funcall(y, '*', 1,
  				 rb_funcall(x, rb_intern("div"), 1, y)));
  }
  
  /*
   *  call-seq:
   *     num.remainder(numeric)  ->  real
   *
   *     x.remainder(y) means x-y*(x/y).truncate
   *
   *  See Numeric#divmod.
   */
  
  static VALUE
  num_remainder(VALUE x, VALUE y)
  {
      VALUE z = rb_funcall(x, '%', 1, y);
  
      if ((!rb_equal(z, INT2FIX(0))) &&
  	((negative_int_p(x) &&
  	  positive_int_p(y)) ||
  	 (positive_int_p(x) &&
  	  negative_int_p(y)))) {
  	return rb_funcall(z, '-', 1, y);
      }
      return z;
  }
  
  /*
   *  call-seq:
   *     num.divmod(numeric)  ->  array
   *
   *  Returns an array containing the quotient and modulus obtained by dividing
   *  +num+ by +numeric+.
   *
   *  If <code>q, r = * x.divmod(y)</code>, then
   *
   *      q = floor(x/y)
   *      x = q*y+r
   *
   *  The quotient is rounded toward -infinity, as shown in the following table:
   *
   *     a    |  b  |  a.divmod(b)  |   a/b   | a.modulo(b) | a.remainder(b)
   *    ------+-----+---------------+---------+-------------+---------------
   *     13   |  4  |   3,    1     |   3     |    1        |     1
   *    ------+-----+---------------+---------+-------------+---------------
   *     13   | -4  |  -4,   -3     |  -4     |   -3        |     1
   *    ------+-----+---------------+---------+-------------+---------------
   *    -13   |  4  |  -4,    3     |  -4     |    3        |    -1
   *    ------+-----+---------------+---------+-------------+---------------
   *    -13   | -4  |   3,   -1     |   3     |   -1        |    -1
   *    ------+-----+---------------+---------+-------------+---------------
   *     11.5 |  4  |   2,    3.5   |   2.875 |    3.5      |     3.5
   *    ------+-----+---------------+---------+-------------+---------------
   *     11.5 | -4  |  -3,   -0.5   |  -2.875 |   -0.5      |     3.5
   *    ------+-----+---------------+---------+-------------+---------------
   *    -11.5 |  4  |  -3,    0.5   |  -2.875 |    0.5      |    -3.5
   *    ------+-----+---------------+---------+-------------+---------------
   *    -11.5 | -4  |   2,   -3.5   |   2.875 |   -3.5      |    -3.5
   *
   *
   *  Examples
   *
   *     11.divmod(3)         #=> [3, 2]
   *     11.divmod(-3)        #=> [-4, -1]
   *     11.divmod(3.5)       #=> [3, 0.5]
   *     (-11).divmod(3.5)    #=> [-4, 3.0]
   *     (11.5).divmod(3.5)   #=> [3, 1.0]
   */
  
  static VALUE
  num_divmod(VALUE x, VALUE y)
  {
      return rb_assoc_new(num_div(x, y), num_modulo(x, y));
  }
  
  /*
   *  call-seq:
   *     num.real?  ->  true or false
   *
   *  Returns +true+ if +num+ is a Real number. (i.e. not Complex).
   */
  
  static VALUE
  num_real_p(VALUE num)
  {
      return Qtrue;
  }
  
  /*
   *  call-seq:
   *     num.integer?  ->  true or false
   *
   *  Returns +true+ if +num+ is an Integer (including Fixnum and Bignum).
   *
   *      (1.0).integer? #=> false
   *      (1).integer?   #=> true
   */
  
  static VALUE
  num_int_p(VALUE num)
  {
      return Qfalse;
  }
  
  /*
   *  call-seq:
   *     num.abs        ->  numeric
   *     num.magnitude  ->  numeric
   *
   *  Returns the absolute value of +num+.
   *
   *     12.abs         #=> 12
   *     (-34.56).abs   #=> 34.56
   *     -34.56.abs     #=> 34.56
   *
   *  Numeric#magnitude is an alias of Numeric#abs.
   */
  
  static VALUE
  num_abs(VALUE num)
  {
      if (negative_int_p(num)) {
  	return rb_funcall(num, rb_intern("-@"), 0);
      }
      return num;
  }
  
  
  /*
   *  call-seq:
   *     num.zero?  ->  true or false
   *
   *  Returns +true+ if +num+ has a zero value.
   */
  
  static VALUE
  num_zero_p(VALUE num)
  {
      if (rb_equal(num, INT2FIX(0))) {
  	return Qtrue;
      }
      return Qfalse;
  }
  
  
  /*
   *  call-seq:
   *     num.nonzero?  ->  self or nil
   *
   *  Returns +self+ if +num+ is not zero, +nil+ otherwise.
   *
   *  This behavior is useful when chaining comparisons:
   *
   *     a = %w( z Bb bB bb BB a aA Aa AA A )
   *     b = a.sort {|a,b| (a.downcase <=> b.downcase).nonzero? || a <=> b }
   *     b   #=> ["A", "a", "AA", "Aa", "aA", "BB", "Bb", "bB", "bb", "z"]
   */
  
  static VALUE
  num_nonzero_p(VALUE num)
  {
      if (RTEST(rb_funcall(num, rb_intern("zero?"), 0, 0))) {
  	return Qnil;
      }
      return num;
  }
  
  /*
   *  call-seq:
   *     num.to_int  ->  integer
   *
   *  Invokes the child class's +to_i+ method to convert +num+ to an integer.
   *
   *      1.0.class => Float
   *      1.0.to_int.class => Fixnum
   *      1.0.to_i.class => Fixnum
   */
  
  static VALUE
  num_to_int(VALUE num)
  {
      return rb_funcall(num, id_to_i, 0, 0);
  }
  
  
  /********************************************************************
   *
   * Document-class: Float
   *
   *  Float objects represent inexact real numbers using the native
   *  architecture's double-precision floating point representation.
   *
   *  Floating point has a different arithmetic and is an inexact number.
   *  So you should know its esoteric system. see following:
   *
   *  - http://docs.sun.com/source/806-3568/ncg_goldberg.html
   *  - http://wiki.github.com/rdp/ruby_tutorials_core/ruby-talk-faq#wiki-floats_imprecise
   *  - http://en.wikipedia.org/wiki/Floating_point#Accuracy_problems
   */
  
  VALUE
  rb_float_new_in_heap(double d)
  {
      NEWOBJ_OF(flt, struct RFloat, rb_cFloat, T_FLOAT | (RGENGC_WB_PROTECTED_FLOAT ? FL_WB_PROTECTED : 0));
  
      flt->float_value = d;
      OBJ_FREEZE(flt);
      return (VALUE)flt;
  }
  
  /*
   *  call-seq:
   *     float.to_s  ->  string
   *
   *  Returns a string containing a representation of self. As well as a fixed or
   *  exponential form of the +float+, the call may return +NaN+, +Infinity+, and
   *  +-Infinity+.
   */
  
  static VALUE
  flo_to_s(VALUE flt)
  {
      char *ruby_dtoa(double d_, int mode, int ndigits, int *decpt, int *sign, char **rve);
      enum {decimal_mant = DBL_MANT_DIG-DBL_DIG};
      enum {float_dig = DBL_DIG+1};
      char buf[float_dig + (decimal_mant + CHAR_BIT - 1) / CHAR_BIT + 10];
      double value = RFLOAT_VALUE(flt);
      VALUE s;
      char *p, *e;
      int sign, decpt, digs;
  
      if (isinf(value))
  	return rb_usascii_str_new2(value < 0 ? "-Infinity" : "Infinity");
      else if (isnan(value))
  	return rb_usascii_str_new2("NaN");
  
      p = ruby_dtoa(value, 0, 0, &decpt, &sign, &e);
      s = sign ? rb_usascii_str_new_cstr("-") : rb_usascii_str_new(0, 0);
      if ((digs = (int)(e - p)) >= (int)sizeof(buf)) digs = (int)sizeof(buf) - 1;
      memcpy(buf, p, digs);
      xfree(p);
      if (decpt > 0) {
  	if (decpt < digs) {
  	    memmove(buf + decpt + 1, buf + decpt, digs - decpt);
  	    buf[decpt] = '.';
  	    rb_str_cat(s, buf, digs + 1);
  	}
  	else if (decpt <= DBL_DIG) {
  	    long len;
  	    char *ptr;
  	    rb_str_cat(s, buf, digs);
  	    rb_str_resize(s, (len = RSTRING_LEN(s)) + decpt - digs + 2);
  	    ptr = RSTRING_PTR(s) + len;
  	    if (decpt > digs) {
  		memset(ptr, '0', decpt - digs);
  		ptr += decpt - digs;
  	    }
  	    memcpy(ptr, ".0", 2);
  	}
  	else {
  	    goto exp;
  	}
      }
      else if (decpt > -4) {
  	long len;
  	char *ptr;
  	rb_str_cat(s, "0.", 2);
  	rb_str_resize(s, (len = RSTRING_LEN(s)) - decpt + digs);
  	ptr = RSTRING_PTR(s);
  	memset(ptr += len, '0', -decpt);
  	memcpy(ptr -= decpt, buf, digs);
      }
      else {
        exp:
  	if (digs > 1) {
  	    memmove(buf + 2, buf + 1, digs - 1);
  	}
  	else {
  	    buf[2] = '0';
  	    digs++;
  	}
  	buf[1] = '.';
  	rb_str_cat(s, buf, digs + 1);
  	rb_str_catf(s, "e%+03d", decpt - 1);
      }
      return s;
  }
  
  /*
   *  call-seq:
   *     float.coerce(numeric)  ->  array
   *
   *  Returns an array with both a +numeric+ and a +float+ represented as Float
   *  objects.
   *
   *  This is achieved by converting a +numeric+ to a Float.
   *
   *     1.2.coerce(3)       #=> [3.0, 1.2]
   *     2.5.coerce(1.1)     #=> [1.1, 2.5]
   */
  
  static VALUE
  flo_coerce(VALUE x, VALUE y)
  {
      return rb_assoc_new(rb_Float(y), x);
  }
  
  /*
   * call-seq:
   *    -float  ->  float
   *
   * Returns float, negated.
   */
  
  static VALUE
  flo_uminus(VALUE flt)
  {
      return DBL2NUM(-RFLOAT_VALUE(flt));
  }
  
  /*
   * call-seq:
   *   float + other  ->  float
   *
   * Returns a new float which is the sum of +float+ and +other+.
   */
  
  static VALUE
  flo_plus(VALUE x, VALUE y)
  {
      if (RB_TYPE_P(y, T_FIXNUM)) {
  	return DBL2NUM(RFLOAT_VALUE(x) + (double)FIX2LONG(y));
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return DBL2NUM(RFLOAT_VALUE(x) + rb_big2dbl(y));
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM(RFLOAT_VALUE(x) + RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, '+');
      }
  }
  
  /*
   * call-seq:
   *   float - other  ->  float
   *
   * Returns a new float which is the difference of +float+ and +other+.
   */
  
  static VALUE
  flo_minus(VALUE x, VALUE y)
  {
      if (RB_TYPE_P(y, T_FIXNUM)) {
  	return DBL2NUM(RFLOAT_VALUE(x) - (double)FIX2LONG(y));
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return DBL2NUM(RFLOAT_VALUE(x) - rb_big2dbl(y));
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM(RFLOAT_VALUE(x) - RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, '-');
      }
  }
  
  /*
   * call-seq:
   *   float * other  ->  float
   *
   * Returns a new float which is the product of +float+ and +other+.
   */
  
  static VALUE
  flo_mul(VALUE x, VALUE y)
  {
      if (RB_TYPE_P(y, T_FIXNUM)) {
  	return DBL2NUM(RFLOAT_VALUE(x) * (double)FIX2LONG(y));
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return DBL2NUM(RFLOAT_VALUE(x) * rb_big2dbl(y));
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM(RFLOAT_VALUE(x) * RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, '*');
      }
  }
  
  /*
   * call-seq:
   *   float / other  ->  float
   *
   * Returns a new float which is the result of dividing +float+ by +other+.
   */
  
  static VALUE
  flo_div(VALUE x, VALUE y)
  {
      long f_y;
      double d;
  
      if (RB_TYPE_P(y, T_FIXNUM)) {
  	f_y = FIX2LONG(y);
  	return DBL2NUM(RFLOAT_VALUE(x) / (double)f_y);
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	d = rb_big2dbl(y);
  	return DBL2NUM(RFLOAT_VALUE(x) / d);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM(RFLOAT_VALUE(x) / RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, '/');
      }
  }
  
  /*
   *  call-seq:
   *     float.fdiv(numeric)  ->  float
   *     float.quo(numeric)  ->  float
   *
   *  Returns <code>float / numeric</code>, same as Float#/.
   */
  
  static VALUE
  flo_quo(VALUE x, VALUE y)
  {
      return rb_funcall(x, '/', 1, y);
  }
  
  static void
  flodivmod(double x, double y, double *divp, double *modp)
  {
      double div, mod;
  
      if (y == 0.0) rb_num_zerodiv();
      if ((x == 0.0) || (isinf(y) && !isinf(x)))
          mod = x;
      else {
  #ifdef HAVE_FMOD
  	mod = fmod(x, y);
  #else
  	double z;
  
  	modf(x/y, &z);
  	mod = x - z * y;
  #endif
      }
      if (isinf(x) && !isinf(y) && !isnan(y))
  	div = x;
      else
  	div = (x - mod) / y;
      if (y*mod < 0) {
  	mod += y;
  	div -= 1.0;
      }
      if (modp) *modp = mod;
      if (divp) *divp = div;
  }
  
  /*
   * Returns the modulo of division of x by y.
   * An error will be raised if y == 0.
   */
  
  double
  ruby_float_mod(double x, double y)
  {
      double mod;
      flodivmod(x, y, 0, &mod);
      return mod;
  }
  
  
  /*
   *  call-seq:
   *     float % other        ->  float
   *     float.modulo(other)  ->  float
   *
   *  Return the modulo after division of +float+ by +other+.
   *
   *     6543.21.modulo(137)      #=> 104.21
   *     6543.21.modulo(137.24)   #=> 92.9299999999996
   */
  
  static VALUE
  flo_mod(VALUE x, VALUE y)
  {
      double fy;
  
      if (RB_TYPE_P(y, T_FIXNUM)) {
  	fy = (double)FIX2LONG(y);
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	fy = rb_big2dbl(y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	fy = RFLOAT_VALUE(y);
      }
      else {
  	return rb_num_coerce_bin(x, y, '%');
      }
      return DBL2NUM(ruby_float_mod(RFLOAT_VALUE(x), fy));
  }
  
  static VALUE
  dbl2ival(double d)
  {
      d = round(d);
      if (FIXABLE(d)) {
  	return LONG2FIX((long)d);
      }
      return rb_dbl2big(d);
  }
  
  /*
   *  call-seq:
   *     float.divmod(numeric)  ->  array
   *
   *  See Numeric#divmod.
   *
   *      42.0.divmod 6 #=> [7, 0.0]
   *      42.0.divmod 5 #=> [8, 2.0]
   */
  
  static VALUE
  flo_divmod(VALUE x, VALUE y)
  {
      double fy, div, mod;
      volatile VALUE a, b;
  
      if (RB_TYPE_P(y, T_FIXNUM)) {
  	fy = (double)FIX2LONG(y);
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	fy = rb_big2dbl(y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	fy = RFLOAT_VALUE(y);
      }
      else {
  	return rb_num_coerce_bin(x, y, rb_intern("divmod"));
      }
      flodivmod(RFLOAT_VALUE(x), fy, &div, &mod);
      a = dbl2ival(div);
      b = DBL2NUM(mod);
      return rb_assoc_new(a, b);
  }
  
  /*
   * call-seq:
   *
   *  float ** other  ->  float
   *
   * Raises +float+ to the power of +other+.
   *
   *    2.0**3      #=> 8.0
   */
  
  static VALUE
  flo_pow(VALUE x, VALUE y)
  {
      if (RB_TYPE_P(y, T_FIXNUM)) {
  	return DBL2NUM(pow(RFLOAT_VALUE(x), (double)FIX2LONG(y)));
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return DBL2NUM(pow(RFLOAT_VALUE(x), rb_big2dbl(y)));
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	{
  	    double dx = RFLOAT_VALUE(x);
  	    double dy = RFLOAT_VALUE(y);
  	    if (dx < 0 && dy != round(dy))
  		return rb_funcall(rb_complex_raw1(x), rb_intern("**"), 1, y);
  	    return DBL2NUM(pow(dx, dy));
  	}
      }
      else {
  	return rb_num_coerce_bin(x, y, rb_intern("**"));
      }
  }
  
  /*
   *  call-seq:
   *     num.eql?(numeric)  ->  true or false
   *
   *  Returns +true+ if +num+ and +numeric+ are the same type and have equal
   *  values.
   *
   *     1 == 1.0          #=> true
   *     1.eql?(1.0)       #=> false
   *     (1.0).eql?(1.0)   #=> true
   */
  
  static VALUE
  num_eql(VALUE x, VALUE y)
  {
      if (TYPE(x) != TYPE(y)) return Qfalse;
  
      return rb_equal(x, y);
  }
  
  /*
   *  call-seq:
   *     number <=> other  ->  0 or nil
   *
   *  Returns zero if +number+ equals +other+, otherwise +nil+ is returned if the
   *  two values are incomparable.
   */
  
  static VALUE
  num_cmp(VALUE x, VALUE y)
  {
      if (x == y) return INT2FIX(0);
      return Qnil;
  }
  
  static VALUE
  num_equal(VALUE x, VALUE y)
  {
      if (x == y) return Qtrue;
      return rb_funcall(y, id_eq, 1, x);
  }
  
  /*
   *  call-seq:
   *     float == obj  ->  true or false
   *
   *  Returns +true+ only if +obj+ has the same value as +float+. Contrast this
   *  with Float#eql?, which requires obj to be a Float.
   *
   *  The result of <code>NaN == NaN</code> is undefined, so the
   *  implementation-dependent value is returned.
   *
   *     1.0 == 1   #=> true
   *
   */
  
  static VALUE
  flo_eq(VALUE x, VALUE y)
  {
      volatile double a, b;
  
      if (RB_TYPE_P(y, T_FIXNUM) || RB_TYPE_P(y, T_BIGNUM)) {
          return rb_integer_float_eq(y, x);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	b = RFLOAT_VALUE(y);
  #if defined(_MSC_VER) && _MSC_VER < 1300
  	if (isnan(b)) return Qfalse;
  #endif
      }
      else {
  	return num_equal(x, y);
      }
      a = RFLOAT_VALUE(x);
  #if defined(_MSC_VER) && _MSC_VER < 1300
      if (isnan(a)) return Qfalse;
  #endif
      return (a == b)?Qtrue:Qfalse;
  }
  
  /*
   * call-seq:
   *   float.hash  ->  integer
   *
   * Returns a hash code for this float.
   */
  
  static VALUE
  flo_hash(VALUE num)
  {
      double d;
      st_index_t hash;
  
      d = RFLOAT_VALUE(num);
      /* normalize -0.0 to 0.0 */
      if (d == 0.0) d = 0.0;
      hash = rb_memhash(&d, sizeof(d));
      return LONG2FIX(hash);
  }
  
  VALUE
  rb_dbl_cmp(double a, double b)
  {
      if (isnan(a) || isnan(b)) return Qnil;
      if (a == b) return INT2FIX(0);
      if (a > b) return INT2FIX(1);
      if (a < b) return INT2FIX(-1);
      return Qnil;
  }
  
  /*
   *  call-seq:
   *     float <=> real  ->  -1, 0, +1 or nil
   *
   *  Returns -1, 0, +1 or nil depending on whether +float+ is less than, equal
   *  to, or greater than +real+. This is the basis for the tests in Comparable.
   *
   *  The result of <code>NaN <=> NaN</code> is undefined, so the
   *  implementation-dependent value is returned.
   *
   *  +nil+ is returned if the two values are incomparable.
   */
  
  static VALUE
  flo_cmp(VALUE x, VALUE y)
  {
      double a, b;
      VALUE i;
  
      a = RFLOAT_VALUE(x);
      if (isnan(a)) return Qnil;
      if (RB_TYPE_P(y, T_FIXNUM) || RB_TYPE_P(y, T_BIGNUM)) {
          VALUE rel = rb_integer_float_cmp(y, x);
          if (FIXNUM_P(rel))
              return INT2FIX(-FIX2INT(rel));
          return rel;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	b = RFLOAT_VALUE(y);
      }
      else {
  	if (isinf(a) && (i = rb_check_funcall(y, rb_intern("infinite?"), 0, 0)) != Qundef) {
  	    if (RTEST(i)) {
  		int j = rb_cmpint(i, x, y);
  		j = (a > 0.0) ? (j > 0 ? 0 : +1) : (j < 0 ? 0 : -1);
  		return INT2FIX(j);
  	    }
  	    if (a > 0.0) return INT2FIX(1);
  	    return INT2FIX(-1);
  	}
  	return rb_num_coerce_cmp(x, y, id_cmp);
      }
      return rb_dbl_cmp(a, b);
  }
  
  /*
   * call-seq:
   *   float > real  ->  true or false
   *
   * Returns +true+ if +float+ is greater than +real+.
   *
   * The result of <code>NaN > NaN</code> is undefined, so the
   * implementation-dependent value is returned.
   */
  
  static VALUE
  flo_gt(VALUE x, VALUE y)
  {
      double a, b;
  
      a = RFLOAT_VALUE(x);
      if (RB_TYPE_P(y, T_FIXNUM) || RB_TYPE_P(y, T_BIGNUM)) {
          VALUE rel = rb_integer_float_cmp(y, x);
          if (FIXNUM_P(rel))
              return -FIX2INT(rel) > 0 ? Qtrue : Qfalse;
          return Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	b = RFLOAT_VALUE(y);
  #if defined(_MSC_VER) && _MSC_VER < 1300
  	if (isnan(b)) return Qfalse;
  #endif
      }
      else {
  	return rb_num_coerce_relop(x, y, '>');
      }
  #if defined(_MSC_VER) && _MSC_VER < 1300
      if (isnan(a)) return Qfalse;
  #endif
      return (a > b)?Qtrue:Qfalse;
  }
  
  /*
   * call-seq:
   *   float >= real  ->  true or false
   *
   * Returns +true+ if +float+ is greater than or equal to +real+.
   *
   * The result of <code>NaN >= NaN</code> is undefined, so the
   * implementation-dependent value is returned.
   */
  
  static VALUE
  flo_ge(VALUE x, VALUE y)
  {
      double a, b;
  
      a = RFLOAT_VALUE(x);
      if (RB_TYPE_P(y, T_FIXNUM) || RB_TYPE_P(y, T_BIGNUM)) {
          VALUE rel = rb_integer_float_cmp(y, x);
          if (FIXNUM_P(rel))
              return -FIX2INT(rel) >= 0 ? Qtrue : Qfalse;
          return Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	b = RFLOAT_VALUE(y);
  #if defined(_MSC_VER) && _MSC_VER < 1300
  	if (isnan(b)) return Qfalse;
  #endif
      }
      else {
  	return rb_num_coerce_relop(x, y, rb_intern(">="));
      }
  #if defined(_MSC_VER) && _MSC_VER < 1300
      if (isnan(a)) return Qfalse;
  #endif
      return (a >= b)?Qtrue:Qfalse;
  }
  
  /*
   * call-seq:
   *   float < real  ->  true or false
   *
   * Returns +true+ if +float+ is less than +real+.
   *
   * The result of <code>NaN < NaN</code> is undefined, so the
   * implementation-dependent value is returned.
   */
  
  static VALUE
  flo_lt(VALUE x, VALUE y)
  {
      double a, b;
  
      a = RFLOAT_VALUE(x);
      if (RB_TYPE_P(y, T_FIXNUM) || RB_TYPE_P(y, T_BIGNUM)) {
          VALUE rel = rb_integer_float_cmp(y, x);
          if (FIXNUM_P(rel))
              return -FIX2INT(rel) < 0 ? Qtrue : Qfalse;
          return Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	b = RFLOAT_VALUE(y);
  #if defined(_MSC_VER) && _MSC_VER < 1300
  	if (isnan(b)) return Qfalse;
  #endif
      }
      else {
  	return rb_num_coerce_relop(x, y, '<');
      }
  #if defined(_MSC_VER) && _MSC_VER < 1300
      if (isnan(a)) return Qfalse;
  #endif
      return (a < b)?Qtrue:Qfalse;
  }
  
  /*
   * call-seq:
   *   float <= real  ->  true or false
   *
   * Returns +true+ if +float+ is less than or equal to +real+.
   *
   * The result of <code>NaN <= NaN</code> is undefined, so the
   * implementation-dependent value is returned.
   */
  
  static VALUE
  flo_le(VALUE x, VALUE y)
  {
      double a, b;
  
      a = RFLOAT_VALUE(x);
      if (RB_TYPE_P(y, T_FIXNUM) || RB_TYPE_P(y, T_BIGNUM)) {
          VALUE rel = rb_integer_float_cmp(y, x);
          if (FIXNUM_P(rel))
              return -FIX2INT(rel) <= 0 ? Qtrue : Qfalse;
          return Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	b = RFLOAT_VALUE(y);
  #if defined(_MSC_VER) && _MSC_VER < 1300
  	if (isnan(b)) return Qfalse;
  #endif
      }
      else {
  	return rb_num_coerce_relop(x, y, rb_intern("<="));
      }
  #if defined(_MSC_VER) && _MSC_VER < 1300
      if (isnan(a)) return Qfalse;
  #endif
      return (a <= b)?Qtrue:Qfalse;
  }
  
  /*
   *  call-seq:
   *     float.eql?(obj)  ->  true or false
   *
   *  Returns +true+ only if +obj+ is a Float with the same value as +float+.
   *  Contrast this with Float#==, which performs type conversions.
   *
   *  The result of <code>NaN.eql?(NaN)</code> is undefined, so the
   *  implementation-dependent value is returned.
   *
   *     1.0.eql?(1)   #=> false
   */
  
  static VALUE
  flo_eql(VALUE x, VALUE y)
  {
      if (RB_TYPE_P(y, T_FLOAT)) {
  	double a = RFLOAT_VALUE(x);
  	double b = RFLOAT_VALUE(y);
  #if defined(_MSC_VER) && _MSC_VER < 1300
  	if (isnan(a) || isnan(b)) return Qfalse;
  #endif
  	if (a == b)
  	    return Qtrue;
      }
      return Qfalse;
  }
  
  /*
   * call-seq:
   *   float.to_f  ->  self
   *
   * Since +float+ is already a float, returns +self+.
   */
  
  static VALUE
  flo_to_f(VALUE num)
  {
      return num;
  }
  
  /*
   *  call-seq:
   *     float.abs        ->  float
   *     float.magnitude  ->  float
   *
   *  Returns the absolute value of +float+.
   *
   *     (-34.56).abs   #=> 34.56
   *     -34.56.abs     #=> 34.56
   *
   */
  
  static VALUE
  flo_abs(VALUE flt)
  {
      double val = fabs(RFLOAT_VALUE(flt));
      return DBL2NUM(val);
  }
  
  /*
   *  call-seq:
   *     float.zero?  ->  true or false
   *
   *  Returns +true+ if +float+ is 0.0.
   *
   */
  
  static VALUE
  flo_zero_p(VALUE num)
  {
      if (RFLOAT_VALUE(num) == 0.0) {
  	return Qtrue;
      }
      return Qfalse;
  }
  
  /*
   *  call-seq:
   *     float.nan?  ->  true or false
   *
   *  Returns +true+ if +float+ is an invalid IEEE floating point number.
   *
   *     a = -1.0      #=> -1.0
   *     a.nan?        #=> false
   *     a = 0.0/0.0   #=> NaN
   *     a.nan?        #=> true
   */
  
  static VALUE
  flo_is_nan_p(VALUE num)
  {
      double value = RFLOAT_VALUE(num);
  
      return isnan(value) ? Qtrue : Qfalse;
  }
  
  /*
   *  call-seq:
   *     float.infinite?  ->  nil, -1, +1
   *
   *  Return values corresponding to the value of +float+:
   *
   *  +finite+::	    +nil+
   *  +-Infinity+::   +-1+
   *  ++Infinity+::   +1+
   *
   *  For example:
   *
   *     (0.0).infinite?        #=> nil
   *     (-1.0/0.0).infinite?   #=> -1
   *     (+1.0/0.0).infinite?   #=> 1
   */
  
  static VALUE
  flo_is_infinite_p(VALUE num)
  {
      double value = RFLOAT_VALUE(num);
  
      if (isinf(value)) {
  	return INT2FIX( value < 0 ? -1 : 1 );
      }
  
      return Qnil;
  }
  
  /*
   *  call-seq:
   *     float.finite?  ->  true or false
   *
   *  Returns +true+ if +float+ is a valid IEEE floating point number (it is not
   *  infinite, and Float#nan? is +false+).
   *
   */
  
  static VALUE
  flo_is_finite_p(VALUE num)
  {
      double value = RFLOAT_VALUE(num);
  
  #if HAVE_ISFINITE
      if (!isfinite(value))
  	return Qfalse;
  #else
      if (isinf(value) || isnan(value))
  	return Qfalse;
  #endif
  
      return Qtrue;
  }
  
  /*
   *  call-seq:
   *     float.floor  ->  integer
   *
   *  Returns the largest integer less than or equal to +float+.
   *
   *     1.2.floor      #=> 1
   *     2.0.floor      #=> 2
   *     (-1.2).floor   #=> -2
   *     (-2.0).floor   #=> -2
   */
  
  static VALUE
  flo_floor(VALUE num)
  {
      double f = floor(RFLOAT_VALUE(num));
      long val;
  
      if (!FIXABLE(f)) {
  	return rb_dbl2big(f);
      }
      val = (long)f;
      return LONG2FIX(val);
  }
  
  /*
   *  call-seq:
   *     float.ceil  ->  integer
   *
   *  Returns the smallest Integer greater than or equal to +float+.
   *
   *     1.2.ceil      #=> 2
   *     2.0.ceil      #=> 2
   *     (-1.2).ceil   #=> -1
   *     (-2.0).ceil   #=> -2
   */
  
  static VALUE
  flo_ceil(VALUE num)
  {
      double f = ceil(RFLOAT_VALUE(num));
      long val;
  
      if (!FIXABLE(f)) {
  	return rb_dbl2big(f);
      }
      val = (long)f;
      return LONG2FIX(val);
  }
  
  /*
   * Assumes num is an Integer, ndigits <= 0
   */
  static VALUE
  int_round_0(VALUE num, int ndigits)
  {
      VALUE n, f, h, r;
      long bytes;
      ID op;
      /* If 10**N / 2 > num, then return 0 */
      /* We have log_256(10) > 0.415241 and log_256(1/2) = -0.125, so */
      bytes = FIXNUM_P(num) ? sizeof(long) : rb_funcall(num, idSize, 0);
      if (-0.415241 * ndigits - 0.125 > bytes ) {
  	return INT2FIX(0);
      }
  
      f = int_pow(10, -ndigits);
      if (FIXNUM_P(num) && FIXNUM_P(f)) {
  	SIGNED_VALUE x = FIX2LONG(num), y = FIX2LONG(f);
  	int neg = x < 0;
  	if (neg) x = -x;
  	x = (x + y / 2) / y * y;
  	if (neg) x = -x;
  	return LONG2NUM(x);
      }
      if (RB_TYPE_P(f, T_FLOAT)) {
  	/* then int_pow overflow */
  	return INT2FIX(0);
      }
      h = rb_funcall(f, '/', 1, INT2FIX(2));
      r = rb_funcall(num, '%', 1, f);
      n = rb_funcall(num, '-', 1, r);
      op = negative_int_p(num) ? rb_intern("<=") : '<';
      if (!RTEST(rb_funcall(r, op, 1, h))) {
  	n = rb_funcall(n, '+', 1, f);
      }
      return n;
  }
  
  static VALUE
  flo_truncate(VALUE num);
  
  /*
   *  call-seq:
   *     float.round([ndigits])  ->  integer or float
   *
   *  Rounds +float+ to a given precision in decimal digits (default 0 digits).
   *
   *  Precision may be negative.  Returns a floating point number when +ndigits+
   *  is more than zero.
   *
   *     1.4.round      #=> 1
   *     1.5.round      #=> 2
   *     1.6.round      #=> 2
   *     (-1.5).round   #=> -2
   *
   *     1.234567.round(2)  #=> 1.23
   *     1.234567.round(3)  #=> 1.235
   *     1.234567.round(4)  #=> 1.2346
   *     1.234567.round(5)  #=> 1.23457
   *
   *     34567.89.round(-5) #=> 0
   *     34567.89.round(-4) #=> 30000
   *     34567.89.round(-3) #=> 35000
   *     34567.89.round(-2) #=> 34600
   *     34567.89.round(-1) #=> 34570
   *     34567.89.round(0)  #=> 34568
   *     34567.89.round(1)  #=> 34567.9
   *     34567.89.round(2)  #=> 34567.89
   *     34567.89.round(3)  #=> 34567.89
   *
   */
  
  static VALUE
  flo_round(int argc, VALUE *argv, VALUE num)
  {
      VALUE nd;
      double number, f;
      int ndigits = 0;
      int binexp;
      enum {float_dig = DBL_DIG+2};
  
      if (argc > 0 && rb_scan_args(argc, argv, "01", &nd) == 1) {
  	ndigits = NUM2INT(nd);
      }
      if (ndigits < 0) {
  	return int_round_0(flo_truncate(num), ndigits);
      }
      number  = RFLOAT_VALUE(num);
      if (ndigits == 0) {
  	return dbl2ival(number);
      }
      frexp(number, &binexp);
  
  /* Let `exp` be such that `number` is written as:"0.#{digits}e#{exp}",
     i.e. such that  10 ** (exp - 1) <= |number| < 10 ** exp
     Recall that up to float_dig digits can be needed to represent a double,
     so if ndigits + exp >= float_dig, the intermediate value (number * 10 ** ndigits)
     will be an integer and thus the result is the original number.
     If ndigits + exp <= 0, the result is 0 or "1e#{exp}", so
     if ndigits + exp < 0, the result is 0.
     We have:
  	2 ** (binexp-1) <= |number| < 2 ** binexp
  	10 ** ((binexp-1)/log_2(10)) <= |number| < 10 ** (binexp/log_2(10))
  	If binexp >= 0, and since log_2(10) = 3.322259:
  	   10 ** (binexp/4 - 1) < |number| < 10 ** (binexp/3)
  	   floor(binexp/4) <= exp <= ceil(binexp/3)
  	If binexp <= 0, swap the /4 and the /3
  	So if ndigits + floor(binexp/(4 or 3)) >= float_dig, the result is number
  	If ndigits + ceil(binexp/(3 or 4)) < 0 the result is 0
  */
      if (isinf(number) || isnan(number) ||
  	(ndigits >= float_dig - (binexp > 0 ? binexp / 4 : binexp / 3 - 1))) {
  	return num;
      }
      if (ndigits < - (binexp > 0 ? binexp / 3 + 1 : binexp / 4)) {
  	return DBL2NUM(0);
      }
      f = pow(10, ndigits);
      return DBL2NUM(round(number * f) / f);
  }
  
  /*
   *  call-seq:
   *     float.to_i      ->  integer
   *     float.to_int    ->  integer
   *     float.truncate  ->  integer
   *
   *  Returns the +float+ truncated to an Integer.
   *
   *  Synonyms are #to_i, #to_int, and #truncate.
   */
  
  static VALUE
  flo_truncate(VALUE num)
  {
      double f = RFLOAT_VALUE(num);
      long val;
  
      if (f > 0.0) f = floor(f);
      if (f < 0.0) f = ceil(f);
  
      if (!FIXABLE(f)) {
  	return rb_dbl2big(f);
      }
      val = (long)f;
      return LONG2FIX(val);
  }
  
  /*
   *  call-seq:
   *     num.floor  ->  integer
   *
   *  Returns the largest integer less than or equal to +num+.
   *
   *  Numeric implements this by converting an Integer to a Float and invoking
   *  Float#floor.
   *
   *     1.floor      #=> 1
   *     (-1).floor   #=> -1
   */
  
  static VALUE
  num_floor(VALUE num)
  {
      return flo_floor(rb_Float(num));
  }
  
  
  /*
   *  call-seq:
   *     num.ceil  ->  integer
   *
   *  Returns the smallest possible Integer that is greater than or equal to
   *  +num+.
   *
   *  Numeric achieves this by converting itself to a Float then invoking
   *  Float#ceil.
   *
   *     1.ceil        #=> 1
   *     1.2.ceil      #=> 2
   *     (-1.2).ceil   #=> -1
   *     (-1.0).ceil   #=> -1
   */
  
  static VALUE
  num_ceil(VALUE num)
  {
      return flo_ceil(rb_Float(num));
  }
  
  /*
   *  call-seq:
   *     num.round([ndigits])  ->  integer or float
   *
   *  Rounds +num+ to a given precision in decimal digits (default 0 digits).
   *
   *  Precision may be negative.  Returns a floating point number when +ndigits+
   *  is more than zero.
   *
   *  Numeric implements this by converting itself to a Float and invoking
   *  Float#round.
   */
  
  static VALUE
  num_round(int argc, VALUE* argv, VALUE num)
  {
      return flo_round(argc, argv, rb_Float(num));
  }
  
  /*
   *  call-seq:
   *     num.truncate  ->  integer
   *
   *  Returns +num+ truncated to an Integer.
   *
   *  Numeric implements this by converting its value to a Float and invoking
   *  Float#truncate.
   */
  
  static VALUE
  num_truncate(VALUE num)
  {
      return flo_truncate(rb_Float(num));
  }
  
  static double
  ruby_float_step_size(double beg, double end, double unit, int excl)
  {
      const double epsilon = DBL_EPSILON;
      double n = (end - beg)/unit;
      double err = (fabs(beg) + fabs(end) + fabs(end-beg)) / fabs(unit) * epsilon;
  
      if (isinf(unit)) {
  	return unit > 0 ? beg <= end : beg >= end;
      }
      if (unit == 0) {
  	return INFINITY;
      }
      if (err>0.5) err=0.5;
      if (excl) {
  	if (n<=0) return 0;
  	if (n<1)
  	    n = 0;
  	else
  	    n = floor(n - err);
      }
      else {
  	if (n<0) return 0;
  	n = floor(n + err);
      }
      return n+1;
  }
  
  int
  ruby_float_step(VALUE from, VALUE to, VALUE step, int excl)
  {
      if (RB_TYPE_P(from, T_FLOAT) || RB_TYPE_P(to, T_FLOAT) || RB_TYPE_P(step, T_FLOAT)) {
  	double beg = NUM2DBL(from);
  	double end = NUM2DBL(to);
  	double unit = NUM2DBL(step);
  	double n = ruby_float_step_size(beg, end, unit, excl);
  	long i;
  
  	if (isinf(unit)) {
  	    /* if unit is infinity, i*unit+beg is NaN */
  	    if (n) rb_yield(DBL2NUM(beg));
  	}
  	else if (unit == 0) {
  	    VALUE val = DBL2NUM(beg);
  	    for (;;)
  		rb_yield(val);
  	}
  	else {
  	    for (i=0; i<n; i++) {
  		double d = i*unit+beg;
  		if (unit >= 0 ? end < d : d < end) d = end;
  		rb_yield(DBL2NUM(d));
  	    }
  	}
  	return TRUE;
      }
      return FALSE;
  }
  
  VALUE
  ruby_num_interval_step_size(VALUE from, VALUE to, VALUE step, int excl)
  {
      if (FIXNUM_P(from) && FIXNUM_P(to) && FIXNUM_P(step)) {
  	long delta, diff;
  
  	diff = FIX2LONG(step);
  	if (diff == 0) {
  	    return DBL2NUM(INFINITY);
  	}
  	delta = FIX2LONG(to) - FIX2LONG(from);
  	if (diff < 0) {
  	    diff = -diff;
  	    delta = -delta;
  	}
  	if (excl) {
  	    delta--;
  	}
  	if (delta < 0) {
  	    return INT2FIX(0);
  	}
  	return ULONG2NUM(delta / diff + 1UL);
      }
      else if (RB_TYPE_P(from, T_FLOAT) || RB_TYPE_P(to, T_FLOAT) || RB_TYPE_P(step, T_FLOAT)) {
  	double n = ruby_float_step_size(NUM2DBL(from), NUM2DBL(to), NUM2DBL(step), excl);
  
  	if (isinf(n)) return DBL2NUM(n);
  	if (POSFIXABLE(n)) return LONG2FIX(n);
  	return rb_dbl2big(n);
      }
      else {
  	VALUE result;
  	ID cmp = '>';
  	switch (rb_cmpint(rb_num_coerce_cmp(step, INT2FIX(0), id_cmp), step, INT2FIX(0))) {
  	  case 0: return DBL2NUM(INFINITY);
  	  case -1: cmp = '<'; break;
  	}
  	if (RTEST(rb_funcall(from, cmp, 1, to))) return INT2FIX(0);
  	result = rb_funcall(rb_funcall(to, '-', 1, from), id_div, 1, step);
  	if (!excl || RTEST(rb_funcall(rb_funcall(from, '+', 1, rb_funcall(result, '*', 1, step)), cmp, 1, to))) {
  	    result = rb_funcall(result, '+', 1, INT2FIX(1));
  	}
  	return result;
      }
  }
  
  static int
  num_step_scan_args(int argc, const VALUE *argv, VALUE *to, VALUE *step)
  {
      VALUE hash;
      int desc;
  
      argc = rb_scan_args(argc, argv, "02:", to, step, &hash);
      if (!NIL_P(hash)) {
  	ID keys[2];
  	VALUE values[2];
  	keys[0] = id_to;
  	keys[1] = id_by;
  	rb_get_kwargs(hash, keys, 0, 2, values);
  	if (values[0] != Qundef) {
  	    if (argc > 0) rb_raise(rb_eArgError, "to is given twice");
  	    *to = values[0];
  	}
  	if (values[1] != Qundef) {
  	    if (argc > 1) rb_raise(rb_eArgError, "step is given twice");
  	    *step = values[1];
  	}
      }
      else {
  	/* compatibility */
  	if (argc > 1 && NIL_P(*step)) {
  	    rb_raise(rb_eTypeError, "step must be numeric");
  	}
  	if (rb_equal(*step, INT2FIX(0))) {
  	    rb_raise(rb_eArgError, "step can't be 0");
  	}
      }
      if (NIL_P(*step)) {
  	*step = INT2FIX(1);
      }
      desc = !positive_int_p(*step);
      if (NIL_P(*to)) {
  	*to = desc ? DBL2NUM(-INFINITY) : DBL2NUM(INFINITY);
      }
      return desc;
  }
  
  static VALUE
  num_step_size(VALUE from, VALUE args, VALUE eobj)
  {
      VALUE to, step;
      int argc = args ? RARRAY_LENINT(args) : 0;
      VALUE *argv = args ? RARRAY_PTR(args) : 0;
  
      num_step_scan_args(argc, argv, &to, &step);
  
      return ruby_num_interval_step_size(from, to, step, FALSE);
  }
  /*
   *  call-seq:
   *     num.step(by: step, to: limit]) {|i| block }  ->  self
   *     num.step(by: step, to: limit])               ->  an_enumerator
   *     num.step(limit=nil, step=1) {|i| block }     ->  self
   *     num.step(limit=nil, step=1)                  ->  an_enumerator
   *
   *  Invokes the given block with the sequence of numbers starting at +num+,
   *  incremented by +step+ (defaulted to +1+) on each call.
   *
   *  The loop finishes when the value to be passed to the block is greater than
   *  +limit+ (if +step+ is positive) or less than +limit+ (if +step+ is
   *  negative), where <i>limit</i> is defaulted to infinity.
   *
   *  In the recommended keyword argument style, either or both of
   *  +step+ and +limit+ (default infinity) can be omitted.  In the
   *  fixed position argument style, integer zero as a step
   *  (i.e. num.step(limit, 0)) is not allowed for historical
   *  compatibility reasons.
   *
   *  If all the arguments are integers, the loop operates using an integer
   *  counter.
   *
   *  If any of the arguments are floating point numbers, all are converted to floats, and the loop is executed the following expression:
   *
   *	floor(n + n*epsilon)+ 1
   *
   *  Where the +n+ is the following:
   *
   *	n = (limit - num)/step
   *
   *  Otherwise, the loop starts at +num+, uses either the less-than (<) or
   *  greater-than (>) operator to compare the counter against +limit+, and
   *  increments itself using the <code>+</code> operator.
   *
   *  If no block is given, an Enumerator is returned instead.
   *
   *  For example:
   *
   *     p 1.step.take(4)
   *     p 10.step(by: -1).take(4)
   *     3.step(to: 5) { |i| print i, " " }
   *     1.step(10, 2) { |i| print i, " " }
   *     Math::E.step(to: Math::PI, by: 0.2) { |f| print f, " " }
   *
   *  Will produce:
   *
   *     [1, 2, 3, 4]
   *     [10, 9, 8, 7]
   *     3 4 5
   *     1 3 5 7 9
   *     2.71828182845905 2.91828182845905 3.11828182845905
   */
  
  static VALUE
  num_step(int argc, VALUE *argv, VALUE from)
  {
      VALUE to, step;
      int desc, inf;
  
      RETURN_SIZED_ENUMERATOR(from, argc, argv, num_step_size);
  
      desc = num_step_scan_args(argc, argv, &to, &step);
      if (RTEST(rb_num_coerce_cmp(step, INT2FIX(0), id_eq))) {
  	inf = 1;
      }
      else if (RB_TYPE_P(to, T_FLOAT)) {
  	double f = RFLOAT_VALUE(to);
  	inf = isinf(f) && (signbit(f) ? desc : !desc);
      }
      else inf = 0;
  
      if (FIXNUM_P(from) && (inf || FIXNUM_P(to)) && FIXNUM_P(step)) {
  	long i = FIX2LONG(from);
  	long diff = FIX2LONG(step);
  
  	if (inf) {
  	    for (;; i += diff)
  		rb_yield(LONG2FIX(i));
  	}
  	else {
  	    long end = FIX2LONG(to);
  
  	    if (desc) {
  		for (; i >= end; i += diff)
  		    rb_yield(LONG2FIX(i));
  	    }
  	    else {
  		for (; i <= end; i += diff)
  		    rb_yield(LONG2FIX(i));
  	    }
  	}
      }
      else if (!ruby_float_step(from, to, step, FALSE)) {
  	VALUE i = from;
  
  	if (inf) {
  	    for (;; i = rb_funcall(i, '+', 1, step))
  		rb_yield(i);
  	}
  	else {
  	    ID cmp = desc ? '<' : '>';
  
  	    for (; !RTEST(rb_funcall(i, cmp, 1, to)); i = rb_funcall(i, '+', 1, step))
  		rb_yield(i);
  	}
      }
      return from;
  }
  
  #define LONG_MIN_MINUS_ONE ((double)LONG_MIN-1)
  #define LONG_MAX_PLUS_ONE (2*(double)(LONG_MAX/2+1))
  #define ULONG_MAX_PLUS_ONE (2*(double)(ULONG_MAX/2+1))
  #define LONG_MIN_MINUS_ONE_IS_LESS_THAN(n) \
    (LONG_MIN_MINUS_ONE == (double)LONG_MIN ? \
     LONG_MIN <= (n): \
     LONG_MIN_MINUS_ONE < (n))
  
  SIGNED_VALUE
  rb_num2long(VALUE val)
  {
    again:
      if (NIL_P(val)) {
  	rb_raise(rb_eTypeError, "no implicit conversion from nil to integer");
      }
  
      if (FIXNUM_P(val)) return FIX2LONG(val);
  
      else if (RB_TYPE_P(val, T_FLOAT)) {
  	if (RFLOAT_VALUE(val) < LONG_MAX_PLUS_ONE
  	    && LONG_MIN_MINUS_ONE_IS_LESS_THAN(RFLOAT_VALUE(val))) {
  	    return (long)RFLOAT_VALUE(val);
  	}
  	else {
  	    char buf[24];
  	    char *s;
  
  	    snprintf(buf, sizeof(buf), "%-.10g", RFLOAT_VALUE(val));
  	    if ((s = strchr(buf, ' ')) != 0) *s = '\0';
  	    rb_raise(rb_eRangeError, "float %s out of range of integer", buf);
  	}
      }
      else if (RB_TYPE_P(val, T_BIGNUM)) {
  	return rb_big2long(val);
      }
      else {
  	val = rb_to_int(val);
  	goto again;
      }
  }
  
  static unsigned long
  rb_num2ulong_internal(VALUE val, int *wrap_p)
  {
    again:
      if (NIL_P(val)) {
         rb_raise(rb_eTypeError, "no implicit conversion from nil to integer");
      }
  
      if (FIXNUM_P(val)) {
          long l = FIX2LONG(val); /* this is FIX2LONG, inteneded */
          if (wrap_p)
              *wrap_p = l < 0;
          return (unsigned long)l;
      }
      else if (RB_TYPE_P(val, T_FLOAT)) {
         if (RFLOAT_VALUE(val) < ULONG_MAX_PLUS_ONE
             && LONG_MIN_MINUS_ONE_IS_LESS_THAN(RFLOAT_VALUE(val))) {
             double d = RFLOAT_VALUE(val);
             if (wrap_p)
                 *wrap_p = d <= -1.0; /* NUM2ULONG(v) uses v.to_int conceptually.  */
             if (0 <= d)
                 return (unsigned long)d;
             return (unsigned long)(long)d;
         }
         else {
             char buf[24];
             char *s;
  
             snprintf(buf, sizeof(buf), "%-.10g", RFLOAT_VALUE(val));
             if ((s = strchr(buf, ' ')) != 0) *s = '\0';
             rb_raise(rb_eRangeError, "float %s out of range of integer", buf);
         }
      }
      else if (RB_TYPE_P(val, T_BIGNUM)) {
          {
              unsigned long ul = rb_big2ulong(val);
              if (wrap_p)
                  *wrap_p = RBIGNUM_NEGATIVE_P(val);
              return ul;
          }
      }
      else {
          val = rb_to_int(val);
          goto again;
      }
  }
  
  VALUE
  rb_num2ulong(VALUE val)
  {
      return rb_num2ulong_internal(val, NULL);
  }
  
  #if SIZEOF_INT < SIZEOF_LONG
  void
  rb_out_of_int(SIGNED_VALUE num)
  {
      rb_raise(rb_eRangeError, "integer %"PRIdVALUE " too %s to convert to `int'",
  	     num, num < 0 ? "small" : "big");
  }
  
  static void
  check_int(long num)
  {
      if ((long)(int)num != num) {
  	rb_out_of_int(num);
      }
  }
  
  static void
  check_uint(unsigned long num, int sign)
  {
      if (sign) {
  	/* minus */
  	if (num < (unsigned long)INT_MIN)
  	    rb_raise(rb_eRangeError, "integer %ld too small to convert to `unsigned int'", (long)num);
      }
      else {
  	/* plus */
  	if (UINT_MAX < num)
  	    rb_raise(rb_eRangeError, "integer %lu too big to convert to `unsigned int'", num);
      }
  }
  
  long
  rb_num2int(VALUE val)
  {
      long num = rb_num2long(val);
  
      check_int(num);
      return num;
  }
  
  long
  rb_fix2int(VALUE val)
  {
      long num = FIXNUM_P(val)?FIX2LONG(val):rb_num2long(val);
  
      check_int(num);
      return num;
  }
  
  unsigned long
  rb_num2uint(VALUE val)
  {
      int wrap;
      unsigned long num = rb_num2ulong_internal(val, &wrap);
  
      check_uint(num, wrap);
      return num;
  }
  
  unsigned long
  rb_fix2uint(VALUE val)
  {
      unsigned long num;
  
      if (!FIXNUM_P(val)) {
  	return rb_num2uint(val);
      }
      num = FIX2ULONG(val);
  
      check_uint(num, negative_int_p(val));
      return num;
  }
  #else
  long
  rb_num2int(VALUE val)
  {
      return rb_num2long(val);
  }
  
  long
  rb_fix2int(VALUE val)
  {
      return FIX2INT(val);
  }
  #endif
  
  void
  rb_out_of_short(SIGNED_VALUE num)
  {
      rb_raise(rb_eRangeError, "integer %"PRIdVALUE " too %s to convert to `short'",
  	     num, num < 0 ? "small" : "big");
  }
  
  static void
  check_short(long num)
  {
      if ((long)(short)num != num) {
  	rb_out_of_short(num);
      }
  }
  
  static void
  check_ushort(unsigned long num, int sign)
  {
      if (sign) {
  	/* minus */
  	if (num < (unsigned long)SHRT_MIN)
  	    rb_raise(rb_eRangeError, "integer %ld too small to convert to `unsigned short'", (long)num);
      }
      else {
  	/* plus */
  	if (USHRT_MAX < num)
  	    rb_raise(rb_eRangeError, "integer %lu too big to convert to `unsigned short'", num);
      }
  }
  
  short
  rb_num2short(VALUE val)
  {
      long num = rb_num2long(val);
  
      check_short(num);
      return num;
  }
  
  short
  rb_fix2short(VALUE val)
  {
      long num = FIXNUM_P(val)?FIX2LONG(val):rb_num2long(val);
  
      check_short(num);
      return num;
  }
  
  unsigned short
  rb_num2ushort(VALUE val)
  {
      int wrap;
      unsigned long num = rb_num2ulong_internal(val, &wrap);
  
      check_ushort(num, wrap);
      return num;
  }
  
  unsigned short
  rb_fix2ushort(VALUE val)
  {
      unsigned long num;
  
      if (!FIXNUM_P(val)) {
  	return rb_num2ushort(val);
      }
      num = FIX2ULONG(val);
  
      check_ushort(num, negative_int_p(val));
      return num;
  }
  
  VALUE
  rb_num2fix(VALUE val)
  {
      long v;
  
      if (FIXNUM_P(val)) return val;
  
      v = rb_num2long(val);
      if (!FIXABLE(v))
  	rb_raise(rb_eRangeError, "integer %ld out of range of fixnum", v);
      return LONG2FIX(v);
  }
  
  #if HAVE_LONG_LONG
  
  #define LLONG_MIN_MINUS_ONE ((double)LLONG_MIN-1)
  #define LLONG_MAX_PLUS_ONE (2*(double)(LLONG_MAX/2+1))
  #define ULLONG_MAX_PLUS_ONE (2*(double)(ULLONG_MAX/2+1))
  #ifndef ULLONG_MAX
  #define ULLONG_MAX ((unsigned LONG_LONG)LLONG_MAX*2+1)
  #endif
  #define LLONG_MIN_MINUS_ONE_IS_LESS_THAN(n) \
    (LLONG_MIN_MINUS_ONE == (double)LLONG_MIN ? \
     LLONG_MIN <= (n): \
     LLONG_MIN_MINUS_ONE < (n))
  
  LONG_LONG
  rb_num2ll(VALUE val)
  {
      if (NIL_P(val)) {
  	rb_raise(rb_eTypeError, "no implicit conversion from nil");
      }
  
      if (FIXNUM_P(val)) return (LONG_LONG)FIX2LONG(val);
  
      else if (RB_TYPE_P(val, T_FLOAT)) {
  	if (RFLOAT_VALUE(val) < LLONG_MAX_PLUS_ONE
              && (LLONG_MIN_MINUS_ONE_IS_LESS_THAN(RFLOAT_VALUE(val)))) {
  	    return (LONG_LONG)(RFLOAT_VALUE(val));
  	}
  	else {
  	    char buf[24];
  	    char *s;
  
  	    snprintf(buf, sizeof(buf), "%-.10g", RFLOAT_VALUE(val));
  	    if ((s = strchr(buf, ' ')) != 0) *s = '\0';
  	    rb_raise(rb_eRangeError, "float %s out of range of long long", buf);
  	}
      }
      else if (RB_TYPE_P(val, T_BIGNUM)) {
  	return rb_big2ll(val);
      }
      else if (RB_TYPE_P(val, T_STRING)) {
  	rb_raise(rb_eTypeError, "no implicit conversion from string");
      }
      else if (RB_TYPE_P(val, T_TRUE) || RB_TYPE_P(val, T_FALSE)) {
  	rb_raise(rb_eTypeError, "no implicit conversion from boolean");
      }
  
      val = rb_to_int(val);
      return NUM2LL(val);
  }
  
  unsigned LONG_LONG
  rb_num2ull(VALUE val)
  {
      if (RB_TYPE_P(val, T_NIL)) {
  	rb_raise(rb_eTypeError, "no implicit conversion from nil");
      }
      else if (RB_TYPE_P(val, T_FIXNUM)) {
  	return (LONG_LONG)FIX2LONG(val); /* this is FIX2LONG, inteneded */
      }
      else if (RB_TYPE_P(val, T_FLOAT)) {
  	if (RFLOAT_VALUE(val) < ULLONG_MAX_PLUS_ONE
              && LLONG_MIN_MINUS_ONE_IS_LESS_THAN(RFLOAT_VALUE(val))) {
              if (0 <= RFLOAT_VALUE(val))
                  return (unsigned LONG_LONG)(RFLOAT_VALUE(val));
  	    return (unsigned LONG_LONG)(LONG_LONG)(RFLOAT_VALUE(val));
  	}
  	else {
  	    char buf[24];
  	    char *s;
  
  	    snprintf(buf, sizeof(buf), "%-.10g", RFLOAT_VALUE(val));
  	    if ((s = strchr(buf, ' ')) != 0) *s = '\0';
  	    rb_raise(rb_eRangeError, "float %s out of range of unsgined long long", buf);
  	}
      }
      else if (RB_TYPE_P(val, T_BIGNUM)) {
  	return rb_big2ull(val);
      }
      else if (RB_TYPE_P(val, T_STRING)) {
  	rb_raise(rb_eTypeError, "no implicit conversion from string");
      }
      else if (RB_TYPE_P(val, T_TRUE) || RB_TYPE_P(val, T_FALSE)) {
  	rb_raise(rb_eTypeError, "no implicit conversion from boolean");
      }
  
      val = rb_to_int(val);
      return NUM2ULL(val);
  }
  
  #endif  /* HAVE_LONG_LONG */
  
  /*
   * Document-class: Integer
   *
   *  This class is the basis for the two concrete classes that hold whole
   *  numbers, Bignum and Fixnum.
   *
   */
  
  /*
   *  call-seq:
   *     int.to_i      ->  integer
   *
   *  As +int+ is already an Integer, all these methods simply return the receiver.
   *
   *  Synonyms are #to_int, #floor, #ceil, #truncate.
   */
  
  static VALUE
  int_to_i(VALUE num)
  {
      return num;
  }
  
  /*
   *  call-seq:
   *     int.integer?  ->  true
   *
   *  Since +int+ is already an Integer, this always returns +true+.
   */
  
  static VALUE
  int_int_p(VALUE num)
  {
      return Qtrue;
  }
  
  /*
   *  call-seq:
   *     int.odd?  ->  true or false
   *
   *  Returns +true+ if +int+ is an odd number.
   */
  
  static VALUE
  int_odd_p(VALUE num)
  {
      if (rb_funcall(num, '%', 1, INT2FIX(2)) != INT2FIX(0)) {
  	return Qtrue;
      }
      return Qfalse;
  }
  
  /*
   *  call-seq:
   *     int.even?  ->  true or false
   *
   *  Returns +true+ if +int+ is an even number.
   */
  
  static VALUE
  int_even_p(VALUE num)
  {
      if (rb_funcall(num, '%', 1, INT2FIX(2)) == INT2FIX(0)) {
  	return Qtrue;
      }
      return Qfalse;
  }
  
  /*
   *  call-seq:
   *     int.next  ->  integer
   *     int.succ  ->  integer
   *
   *  Returns the Integer equal to +int+ + 1.
   *
   *     1.next      #=> 2
   *     (-1).next   #=> 0
   */
  
  static VALUE
  fix_succ(VALUE num)
  {
      long i = FIX2LONG(num) + 1;
      return LONG2NUM(i);
  }
  
  /*
   *  call-seq:
   *     int.next  ->  integer
   *     int.succ  ->  integer
   *
   *  Returns the Integer equal to +int+ + 1, same as Fixnum#next.
   *
   *     1.next      #=> 2
   *     (-1).next   #=> 0
   */
  
  VALUE
  rb_int_succ(VALUE num)
  {
      if (FIXNUM_P(num)) {
  	long i = FIX2LONG(num) + 1;
  	return LONG2NUM(i);
      }
      if (RB_TYPE_P(num, T_BIGNUM)) {
  	return rb_big_plus(num, INT2FIX(1));
      }
      return rb_funcall(num, '+', 1, INT2FIX(1));
  }
  
  #define int_succ rb_int_succ
  
  /*
   *  call-seq:
   *     int.pred  ->  integer
   *
   *  Returns the Integer equal to +int+ - 1.
   *
   *     1.pred      #=> 0
   *     (-1).pred   #=> -2
   */
  
  VALUE
  rb_int_pred(VALUE num)
  {
      if (FIXNUM_P(num)) {
  	long i = FIX2LONG(num) - 1;
  	return LONG2NUM(i);
      }
      if (RB_TYPE_P(num, T_BIGNUM)) {
  	return rb_big_minus(num, INT2FIX(1));
      }
      return rb_funcall(num, '-', 1, INT2FIX(1));
  }
  
  #define int_pred rb_int_pred
  
  VALUE
  rb_enc_uint_chr(unsigned int code, rb_encoding *enc)
  {
      int n;
      VALUE str;
      switch (n = rb_enc_codelen(code, enc)) {
        case ONIGERR_INVALID_CODE_POINT_VALUE:
  	rb_raise(rb_eRangeError, "invalid codepoint 0x%X in %s", code, rb_enc_name(enc));
  	break;
        case ONIGERR_TOO_BIG_WIDE_CHAR_VALUE:
        case 0:
  	rb_raise(rb_eRangeError, "%u out of char range", code);
  	break;
      }
      str = rb_enc_str_new(0, n, enc);
      rb_enc_mbcput(code, RSTRING_PTR(str), enc);
      if (rb_enc_precise_mbclen(RSTRING_PTR(str), RSTRING_END(str), enc) != n) {
  	rb_raise(rb_eRangeError, "invalid codepoint 0x%X in %s", code, rb_enc_name(enc));
      }
      return str;
  }
  
  /*
   *  call-seq:
   *     int.chr([encoding])  ->  string
   *
   *  Returns a string containing the character represented by the +int+'s value
   *  according to +encoding+.
   *
   *     65.chr    #=> "A"
   *     230.chr   #=> "\346"
   *     255.chr(Encoding::UTF_8)   #=> "\303\277"
   */
  
  static VALUE
  int_chr(int argc, VALUE *argv, VALUE num)
  {
      char c;
      unsigned int i;
      rb_encoding *enc;
  
      if (rb_num_to_uint(num, &i) == 0) {
      }
      else if (FIXNUM_P(num)) {
  	rb_raise(rb_eRangeError, "%ld out of char range", FIX2LONG(num));
      }
      else {
  	rb_raise(rb_eRangeError, "bignum out of char range");
      }
  
      switch (argc) {
        case 0:
  	if (0xff < i) {
  	    enc = rb_default_internal_encoding();
  	    if (!enc) {
  		rb_raise(rb_eRangeError, "%d out of char range", i);
  	    }
  	    goto decode;
  	}
  	c = (char)i;
  	if (i < 0x80) {
  	    return rb_usascii_str_new(&c, 1);
  	}
  	else {
  	    return rb_str_new(&c, 1);
  	}
        case 1:
  	break;
        default:
  	rb_check_arity(argc, 0, 1);
  	break;
      }
      enc = rb_to_encoding(argv[0]);
      if (!enc) enc = rb_ascii8bit_encoding();
    decode:
      return rb_enc_uint_chr(i, enc);
  }
  
  /*
   *  call-seq:
   *     int.ord  ->  self
   *
   *  Returns the +int+ itself.
   *
   *     ?a.ord    #=> 97
   *
   *  This method is intended for compatibility to character constant in Ruby
   *  1.9.
   *
   *  For example, ?a.ord returns 97 both in 1.8 and 1.9.
   */
  
  static VALUE
  int_ord(VALUE num)
  {
      return num;
  }
  
  /********************************************************************
   *
   * Document-class: Fixnum
   *
   *  Holds Integer values that can be represented in a native machine word
   *  (minus 1 bit).  If any operation on a Fixnum exceeds this range, the value
   *  is automatically converted to a Bignum.
   *
   *  Fixnum objects have immediate value. This means that when they are assigned
   *  or passed as parameters, the actual object is passed, rather than a
   *  reference to that object.
   *
   *  Assignment does not alias Fixnum objects. There is effectively only one
   *  Fixnum object instance for any given integer value, so, for example, you
   *  cannot add a singleton method to a Fixnum. Any attempt to add a singleton
   *  method to a Fixnum object will raise a TypeError.
   */
  
  
  /*
   * call-seq:
   *   -fix  ->  integer
   *
   * Negates +fix+, which may return a Bignum.
   */
  
  static VALUE
  fix_uminus(VALUE num)
  {
      return LONG2NUM(-FIX2LONG(num));
  }
  
  VALUE
  rb_fix2str(VALUE x, int base)
  {
      extern const char ruby_digitmap[];
      char buf[SIZEOF_VALUE*CHAR_BIT + 2], *b = buf + sizeof buf;
      long val = FIX2LONG(x);
      int neg = 0;
  
      if (base < 2 || 36 < base) {
  	rb_raise(rb_eArgError, "invalid radix %d", base);
      }
      if (val == 0) {
  	return rb_usascii_str_new2("0");
      }
      if (val < 0) {
  	val = -val;
  	neg = 1;
      }
      *--b = '\0';
      do {
  	*--b = ruby_digitmap[(int)(val % base)];
      } while (val /= base);
      if (neg) {
  	*--b = '-';
      }
  
      return rb_usascii_str_new2(b);
  }
  
  /*
   *  call-seq:
   *     fix.to_s(base=10)  ->  string
   *
   *  Returns a string containing the representation of +fix+ radix +base+
   *  (between 2 and 36).
   *
   *     12345.to_s       #=> "12345"
   *     12345.to_s(2)    #=> "11000000111001"
   *     12345.to_s(8)    #=> "30071"
   *     12345.to_s(10)   #=> "12345"
   *     12345.to_s(16)   #=> "3039"
   *     12345.to_s(36)   #=> "9ix"
   *
   */
  static VALUE
  fix_to_s(int argc, VALUE *argv, VALUE x)
  {
      int base;
  
      if (argc == 0) base = 10;
      else {
  	VALUE b;
  
  	rb_scan_args(argc, argv, "01", &b);
  	base = NUM2INT(b);
      }
  
      return rb_fix2str(x, base);
  }
  
  /*
   * call-seq:
   *   fix + numeric  ->  numeric_result
   *
   * Performs addition: the class of the resulting object depends on the class of
   * +numeric+ and on the magnitude of the result. It may return a Bignum.
   */
  
  static VALUE
  fix_plus(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	long a, b, c;
  	VALUE r;
  
  	a = FIX2LONG(x);
  	b = FIX2LONG(y);
  	c = a + b;
  	r = LONG2NUM(c);
  
  	return r;
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_plus(y, x);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM((double)FIX2LONG(x) + RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, '+');
      }
  }
  
  /*
   * call-seq:
   *   fix - numeric  ->  numeric_result
   *
   * Performs subtraction: the class of the resulting object depends on the class
   * of +numeric+ and on the magnitude of the result. It may return a Bignum.
   */
  
  static VALUE
  fix_minus(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	long a, b, c;
  	VALUE r;
  
  	a = FIX2LONG(x);
  	b = FIX2LONG(y);
  	c = a - b;
  	r = LONG2NUM(c);
  
  	return r;
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	x = rb_int2big(FIX2LONG(x));
  	return rb_big_minus(x, y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM((double)FIX2LONG(x) - RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, '-');
      }
  }
  
  #define SQRT_LONG_MAX ((SIGNED_VALUE)1<<((SIZEOF_LONG*CHAR_BIT-1)/2))
  /*tests if N*N would overflow*/
  #define FIT_SQRT_LONG(n) (((n)<SQRT_LONG_MAX)&&((n)>=-SQRT_LONG_MAX))
  
  /*
   * call-seq:
   *   fix * numeric  ->  numeric_result
   *
   * Performs multiplication: the class of the resulting object depends on the
   * class of +numeric+ and on the magnitude of the result. It may return a
   * Bignum.
   */
  
  static VALUE
  fix_mul(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  #ifdef __HP_cc
  /* avoids an optimization bug of HP aC++/ANSI C B3910B A.06.05 [Jul 25 2005] */
  	volatile
  #endif
  	long a, b;
  #if SIZEOF_LONG * 2 <= SIZEOF_LONG_LONG
  	LONG_LONG d;
  #else
  	VALUE r;
  #endif
  
  	a = FIX2LONG(x);
  	b = FIX2LONG(y);
  
  #if SIZEOF_LONG * 2 <= SIZEOF_LONG_LONG
  	d = (LONG_LONG)a * b;
  	if (FIXABLE(d)) return LONG2FIX(d);
  	return rb_ll2inum(d);
  #else
  	if (a == 0) return x;
          if (MUL_OVERFLOW_FIXNUM_P(a, b))
  	    r = rb_big_mul(rb_int2big(a), rb_int2big(b));
          else
              r = LONG2FIX(a * b);
  	return r;
  #endif
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_mul(y, x);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM((double)FIX2LONG(x) * RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, '*');
      }
  }
  
  static void
  fixdivmod(long x, long y, long *divp, long *modp)
  {
      long div, mod;
  
      if (y == 0) rb_num_zerodiv();
      if (y < 0) {
  	if (x < 0)
  	    div = -x / -y;
  	else
  	    div = - (x / -y);
      }
      else {
  	if (x < 0)
  	    div = - (-x / y);
  	else
  	    div = x / y;
      }
      mod = x - div*y;
      if ((mod < 0 && y > 0) || (mod > 0 && y < 0)) {
  	mod += y;
  	div -= 1;
      }
      if (divp) *divp = div;
      if (modp) *modp = mod;
  }
  
  /*
   *  call-seq:
   *     fix.fdiv(numeric)  ->  float
   *
   *  Returns the floating point result of dividing +fix+ by +numeric+.
   *
   *     654321.fdiv(13731)      #=> 47.6528293642124
   *     654321.fdiv(13731.24)   #=> 47.6519964693647
   *
   */
  
  static VALUE
  fix_fdiv(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	return DBL2NUM((double)FIX2LONG(x) / (double)FIX2LONG(y));
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_fdiv(rb_int2big(FIX2LONG(x)), y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM((double)FIX2LONG(x) / RFLOAT_VALUE(y));
      }
      else {
  	return rb_num_coerce_bin(x, y, rb_intern("fdiv"));
      }
  }
  
  static VALUE
  fix_divide(VALUE x, VALUE y, ID op)
  {
      if (FIXNUM_P(y)) {
  	long div;
  
  	fixdivmod(FIX2LONG(x), FIX2LONG(y), &div, 0);
  	return LONG2NUM(div);
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	x = rb_int2big(FIX2LONG(x));
  	return rb_big_div(x, y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	{
  	    double div;
  
  	    if (op == '/') {
  		div = (double)FIX2LONG(x) / RFLOAT_VALUE(y);
  		return DBL2NUM(div);
  	    }
  	    else {
  		if (RFLOAT_VALUE(y) == 0) rb_num_zerodiv();
  		div = (double)FIX2LONG(x) / RFLOAT_VALUE(y);
  		return rb_dbl2big(floor(div));
  	    }
  	}
      }
      else {
  	if (RB_TYPE_P(y, T_RATIONAL) &&
  	    op == '/' && FIX2LONG(x) == 1)
  	    return rb_rational_reciprocal(y);
  	return rb_num_coerce_bin(x, y, op);
      }
  }
  
  /*
   * call-seq:
   *   fix / numeric  ->  numeric_result
   *
   * Performs division: the class of the resulting object depends on the class of
   * +numeric+ and on the magnitude of the result. It may return a Bignum.
   */
  
  static VALUE
  fix_div(VALUE x, VALUE y)
  {
      return fix_divide(x, y, '/');
  }
  
  /*
   * call-seq:
   *   fix.div(numeric)  ->  integer
   *
   * Performs integer division: returns integer result of dividing +fix+ by
   * +numeric+.
   */
  
  static VALUE
  fix_idiv(VALUE x, VALUE y)
  {
      return fix_divide(x, y, rb_intern("div"));
  }
  
  /*
   *  call-seq:
   *    fix % other        ->  real
   *    fix.modulo(other)  ->  real
   *
   *  Returns +fix+ modulo +other+.
   *
   *  See Numeric#divmod for more information.
   */
  
  static VALUE
  fix_mod(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	long mod;
  
  	fixdivmod(FIX2LONG(x), FIX2LONG(y), 0, &mod);
  	return LONG2NUM(mod);
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	x = rb_int2big(FIX2LONG(x));
  	return rb_big_modulo(x, y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	return DBL2NUM(ruby_float_mod((double)FIX2LONG(x), RFLOAT_VALUE(y)));
      }
      else {
  	return rb_num_coerce_bin(x, y, '%');
      }
  }
  
  /*
   *  call-seq:
   *     fix.divmod(numeric)  ->  array
   *
   *  See Numeric#divmod.
   */
  static VALUE
  fix_divmod(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	long div, mod;
  
  	fixdivmod(FIX2LONG(x), FIX2LONG(y), &div, &mod);
  
  	return rb_assoc_new(LONG2NUM(div), LONG2NUM(mod));
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	x = rb_int2big(FIX2LONG(x));
  	return rb_big_divmod(x, y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	{
  	    double div, mod;
  	    volatile VALUE a, b;
  
  	    flodivmod((double)FIX2LONG(x), RFLOAT_VALUE(y), &div, &mod);
  	    a = dbl2ival(div);
  	    b = DBL2NUM(mod);
  	    return rb_assoc_new(a, b);
  	}
      }
      else {
  	return rb_num_coerce_bin(x, y, rb_intern("divmod"));
      }
  }
  
  static VALUE
  int_pow(long x, unsigned long y)
  {
      int neg = x < 0;
      long z = 1;
  
      if (neg) x = -x;
      if (y & 1)
  	z = x;
      else
  	neg = 0;
      y &= ~1;
      do {
  	while (y % 2 == 0) {
  	    if (!FIT_SQRT_LONG(x)) {
  		VALUE v;
  	      bignum:
  		v = rb_big_pow(rb_int2big(x), LONG2NUM(y));
  		if (z != 1) v = rb_big_mul(rb_int2big(neg ? -z : z), v);
  		return v;
  	    }
  	    x = x * x;
  	    y >>= 1;
  	}
  	{
              if (MUL_OVERFLOW_FIXNUM_P(x, z)) {
  		goto bignum;
  	    }
  	    z = x * z;
  	}
      } while (--y);
      if (neg) z = -z;
      return LONG2NUM(z);
  }
  
  VALUE
  rb_int_positive_pow(long x, unsigned long y)
  {
      return int_pow(x, y);
  }
  
  /*
   *  call-seq:
   *    fix ** numeric  ->  numeric_result
   *
   *  Raises +fix+ to the power of +numeric+, which may be negative or
   *  fractional.
   *
   *    2 ** 3      #=> 8
   *    2 ** -1     #=> (1/2)
   *    2 ** 0.5    #=> 1.4142135623731
   */
  
  static VALUE
  fix_pow(VALUE x, VALUE y)
  {
      long a = FIX2LONG(x);
  
      if (FIXNUM_P(y)) {
  	long b = FIX2LONG(y);
  
  	if (a == 1) return INT2FIX(1);
  	if (a == -1) {
  	    if (b % 2 == 0)
  		return INT2FIX(1);
  	    else
  		return INT2FIX(-1);
  	}
  	if (b < 0)
  	    return rb_funcall(rb_rational_raw1(x), rb_intern("**"), 1, y);
  
  	if (b == 0) return INT2FIX(1);
  	if (b == 1) return x;
  	if (a == 0) {
  	    if (b > 0) return INT2FIX(0);
  	    return DBL2NUM(INFINITY);
  	}
  	return int_pow(a, b);
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	if (a == 1) return INT2FIX(1);
  	if (a == -1) {
  	    if (int_even_p(y)) return INT2FIX(1);
  	    else return INT2FIX(-1);
  	}
  	if (negative_int_p(y))
  	    return rb_funcall(rb_rational_raw1(x), rb_intern("**"), 1, y);
  	if (a == 0) return INT2FIX(0);
  	x = rb_int2big(FIX2LONG(x));
  	return rb_big_pow(x, y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	if (RFLOAT_VALUE(y) == 0.0) return DBL2NUM(1.0);
  	if (a == 0) {
  	    return DBL2NUM(RFLOAT_VALUE(y) < 0 ? INFINITY : 0.0);
  	}
  	if (a == 1) return DBL2NUM(1.0);
  	{
  	    double dy = RFLOAT_VALUE(y);
  	    if (a < 0 && dy != round(dy))
  		return rb_funcall(rb_complex_raw1(x), rb_intern("**"), 1, y);
  	    return DBL2NUM(pow((double)a, dy));
  	}
      }
      else {
  	return rb_num_coerce_bin(x, y, rb_intern("**"));
      }
  }
  
  /*
   * call-seq:
   *   fix == other  ->  true or false
   *
   * Return +true+ if +fix+ equals +other+ numerically.
   *
   *   1 == 2      #=> false
   *   1 == 1.0    #=> true
   */
  
  static VALUE
  fix_equal(VALUE x, VALUE y)
  {
      if (x == y) return Qtrue;
      if (FIXNUM_P(y)) return Qfalse;
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_eq(y, x);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
          return rb_integer_float_eq(x, y);
      }
      else {
  	return num_equal(x, y);
      }
  }
  
  /*
   *  call-seq:
   *     fix <=> numeric  ->  -1, 0, +1 or nil
   *
   *  Comparison---Returns +-1+, +0+, ++1+ or +nil+ depending on whether +fix+ is
   *  less than, equal to, or greater than +numeric+.
   *
   *  This is the basis for the tests in the Comparable module.
   *
   *  +nil+ is returned if the two values are incomparable.
   */
  
  static VALUE
  fix_cmp(VALUE x, VALUE y)
  {
      if (x == y) return INT2FIX(0);
      if (FIXNUM_P(y)) {
  	if (FIX2LONG(x) > FIX2LONG(y)) return INT2FIX(1);
  	return INT2FIX(-1);
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_cmp(rb_int2big(FIX2LONG(x)), y);
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
          return rb_integer_float_cmp(x, y);
      }
      else {
  	return rb_num_coerce_cmp(x, y, id_cmp);
      }
  }
  
  /*
   * call-seq:
   *   fix > real  ->  true or false
   *
   * Returns +true+ if the value of +fix+ is greater than that of +real+.
   */
  
  static VALUE
  fix_gt(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	if (FIX2LONG(x) > FIX2LONG(y)) return Qtrue;
  	return Qfalse;
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return FIX2INT(rb_big_cmp(rb_int2big(FIX2LONG(x)), y)) > 0 ? Qtrue : Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
          return rb_integer_float_cmp(x, y) == INT2FIX(1) ? Qtrue : Qfalse;
      }
      else {
  	return rb_num_coerce_relop(x, y, '>');
      }
  }
  
  /*
   * call-seq:
   *   fix >= real  ->  true or false
   *
   * Returns +true+ if the value of +fix+ is greater than or equal to that of
   * +real+.
   */
  
  static VALUE
  fix_ge(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	if (FIX2LONG(x) >= FIX2LONG(y)) return Qtrue;
  	return Qfalse;
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return FIX2INT(rb_big_cmp(rb_int2big(FIX2LONG(x)), y)) >= 0 ? Qtrue : Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	VALUE rel = rb_integer_float_cmp(x, y);
  	return rel == INT2FIX(1) || rel == INT2FIX(0) ? Qtrue : Qfalse;
      }
      else {
  	return rb_num_coerce_relop(x, y, rb_intern(">="));
      }
  }
  
  /*
   * call-seq:
   *   fix < real  ->  true or false
   *
   * Returns +true+ if the value of +fix+ is less than that of +real+.
   */
  
  static VALUE
  fix_lt(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	if (FIX2LONG(x) < FIX2LONG(y)) return Qtrue;
  	return Qfalse;
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return FIX2INT(rb_big_cmp(rb_int2big(FIX2LONG(x)), y)) < 0 ? Qtrue : Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
          return rb_integer_float_cmp(x, y) == INT2FIX(-1) ? Qtrue : Qfalse;
      }
      else {
  	return rb_num_coerce_relop(x, y, '<');
      }
  }
  
  /*
   * call-seq:
   *   fix <= real  ->  true or false
   *
   * Returns +true+ if the value of +fix+ is less than or equal to that of
   * +real+.
   */
  
  static VALUE
  fix_le(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	if (FIX2LONG(x) <= FIX2LONG(y)) return Qtrue;
  	return Qfalse;
      }
      else if (RB_TYPE_P(y, T_BIGNUM)) {
  	return FIX2INT(rb_big_cmp(rb_int2big(FIX2LONG(x)), y)) <= 0 ? Qtrue : Qfalse;
      }
      else if (RB_TYPE_P(y, T_FLOAT)) {
  	VALUE rel = rb_integer_float_cmp(x, y);
  	return rel == INT2FIX(-1) || rel == INT2FIX(0) ? Qtrue : Qfalse;
      }
      else {
  	return rb_num_coerce_relop(x, y, rb_intern("<="));
      }
  }
  
  /*
   * call-seq:
   *   ~fix  ->  integer
   *
   * One's complement: returns a number where each bit is flipped.
   */
  
  static VALUE
  fix_rev(VALUE num)
  {
      return ~num | FIXNUM_FLAG;
  }
  
  static int
  bit_coerce(VALUE *x, VALUE *y, int err)
  {
      if (!FIXNUM_P(*y) && !RB_TYPE_P(*y, T_BIGNUM)) {
  	do_coerce(x, y, err);
  	if (!FIXNUM_P(*x) && !RB_TYPE_P(*x, T_BIGNUM)
  	    && !FIXNUM_P(*y) && !RB_TYPE_P(*y, T_BIGNUM)) {
  	    if (!err) return FALSE;
  	    coerce_failed(*x, *y);
  	}
      }
      return TRUE;
  }
  
  VALUE
  rb_num_coerce_bit(VALUE x, VALUE y, ID func)
  {
      bit_coerce(&x, &y, TRUE);
      return rb_funcall(x, func, 1, y);
  }
  
  /*
   * call-seq:
   *   fix & integer  ->  integer_result
   *
   * Bitwise AND.
   */
  
  static VALUE
  fix_and(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	long val = FIX2LONG(x) & FIX2LONG(y);
  	return LONG2NUM(val);
      }
  
      if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_and(y, x);
      }
  
      bit_coerce(&x, &y, TRUE);
      return rb_funcall(x, rb_intern("&"), 1, y);
  }
  
  /*
   * call-seq:
   *   fix | integer  ->  integer_result
   *
   * Bitwise OR.
   */
  
  static VALUE
  fix_or(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	long val = FIX2LONG(x) | FIX2LONG(y);
  	return LONG2NUM(val);
      }
  
      if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_or(y, x);
      }
  
      bit_coerce(&x, &y, TRUE);
      return rb_funcall(x, rb_intern("|"), 1, y);
  }
  
  /*
   * call-seq:
   *   fix ^ integer  ->  integer_result
   *
   * Bitwise EXCLUSIVE OR.
   */
  
  static VALUE
  fix_xor(VALUE x, VALUE y)
  {
      if (FIXNUM_P(y)) {
  	long val = FIX2LONG(x) ^ FIX2LONG(y);
  	return LONG2NUM(val);
      }
  
      if (RB_TYPE_P(y, T_BIGNUM)) {
  	return rb_big_xor(y, x);
      }
  
      bit_coerce(&x, &y, TRUE);
      return rb_funcall(x, rb_intern("^"), 1, y);
  }
  
  static VALUE fix_lshift(long, unsigned long);
  static VALUE fix_rshift(long, unsigned long);
  
  /*
   * call-seq:
   *   fix << count  ->  integer
   *
   * Shifts +fix+ left +count+ positions, or right if +count+ is negative.
   */
  
  static VALUE
  rb_fix_lshift(VALUE x, VALUE y)
  {
      long val, width;
  
      val = NUM2LONG(x);
      if (!FIXNUM_P(y))
  	return rb_big_lshift(rb_int2big(val), y);
      width = FIX2LONG(y);
      if (width < 0)
  	return fix_rshift(val, (unsigned long)-width);
      return fix_lshift(val, width);
  }
  
  static VALUE
  fix_lshift(long val, unsigned long width)
  {
      if (width > (SIZEOF_LONG*CHAR_BIT-1)
  	|| ((unsigned long)val)>>(SIZEOF_LONG*CHAR_BIT-1-width) > 0) {
  	return rb_big_lshift(rb_int2big(val), ULONG2NUM(width));
      }
      val = val << width;
      return LONG2NUM(val);
  }
  
  /*
   * call-seq:
   *   fix >> count  ->  integer
   *
   * Shifts +fix+ right +count+ positions, or left if +count+ is negative.
   */
  
  static VALUE
  rb_fix_rshift(VALUE x, VALUE y)
  {
      long i, val;
  
      val = FIX2LONG(x);
      if (!FIXNUM_P(y))
  	return rb_big_rshift(rb_int2big(val), y);
      i = FIX2LONG(y);
      if (i == 0) return x;
      if (i < 0)
  	return fix_lshift(val, (unsigned long)-i);
      return fix_rshift(val, i);
  }
  
  static VALUE
  fix_rshift(long val, unsigned long i)
  {
      if (i >= sizeof(long)*CHAR_BIT-1) {
  	if (val < 0) return INT2FIX(-1);
  	return INT2FIX(0);
      }
      val = RSHIFT(val, i);
      return LONG2FIX(val);
  }
  
  /*
   *  call-seq:
   *     fix[n]  ->  0, 1
   *
   *  Bit Reference---Returns the +n+th bit in the binary representation of
   *  +fix+, where <code>fix[0]</code> is the least significant bit.
   *
   *  For example:
   *
   *     a = 0b11001100101010
   *     30.downto(0) do |n| print a[n] end
   *     #=> 0000000000000000011001100101010
   */
  
  static VALUE
  fix_aref(VALUE fix, VALUE idx)
  {
      long val = FIX2LONG(fix);
      long i;
  
      idx = rb_to_int(idx);
      if (!FIXNUM_P(idx)) {
  	idx = rb_big_norm(idx);
  	if (!FIXNUM_P(idx)) {
  	    if (!RBIGNUM_SIGN(idx) || val >= 0)
  		return INT2FIX(0);
  	    return INT2FIX(1);
  	}
      }
      i = FIX2LONG(idx);
  
      if (i < 0) return INT2FIX(0);
      if (SIZEOF_LONG*CHAR_BIT-1 <= i) {
  	if (val < 0) return INT2FIX(1);
  	return INT2FIX(0);
      }
      if (val & (1L<<i))
  	return INT2FIX(1);
      return INT2FIX(0);
  }
  
  /*
   *  call-seq:
   *     fix.to_f  ->  float
   *
   *  Converts +fix+ to a Float.
   *
   */
  
  static VALUE
  fix_to_f(VALUE num)
  {
      double val;
  
      val = (double)FIX2LONG(num);
  
      return DBL2NUM(val);
  }
  
  /*
   *  call-seq:
   *     fix.abs        ->  integer
   *     fix.magnitude  ->  integer
   *
   *  Returns the absolute value of +fix+.
   *
   *     -12345.abs   #=> 12345
   *     12345.abs    #=> 12345
   *
   */
  
  static VALUE
  fix_abs(VALUE fix)
  {
      long i = FIX2LONG(fix);
  
      if (i < 0) i = -i;
  
      return LONG2NUM(i);
  }
  
  
  
  /*
   *  call-seq:
   *     fix.size  ->  fixnum
   *
   *  Returns the number of bytes in the machine representation of +fix+.
   *
   *     1.size            #=> 4
   *     -1.size           #=> 4
   *     2147483647.size   #=> 4
   */
  
  static VALUE
  fix_size(VALUE fix)
  {
      return INT2FIX(sizeof(long));
  }
  
  /*
   *  call-seq:
   *     int.bit_length -> integer
   *
   *  Returns the number of bits of the value of <i>int</i>.
   *
   *  "the number of bits" means that
   *  the bit position of the highest bit which is different to the sign bit.
   *  (The bit position of the bit 2**n is n+1.)
   *  If there is no such bit (zero or minus one), zero is returned.
   *
   *  I.e. This method returns ceil(log2(int < 0 ? -int : int+1)).
   *
   *     (-2**12-1).bit_length     #=> 13
   *     (-2**12).bit_length       #=> 12
   *     (-2**12+1).bit_length     #=> 12
   *     -0x101.bit_length         #=> 9
   *     -0x100.bit_length         #=> 8
   *     -0xff.bit_length          #=> 8
   *     -2.bit_length             #=> 1
   *     -1.bit_length             #=> 0
   *     0.bit_length              #=> 0
   *     1.bit_length              #=> 1
   *     0xff.bit_length           #=> 8
   *     0x100.bit_length          #=> 9
   *     (2**12-1).bit_length      #=> 12
   *     (2**12).bit_length        #=> 13
   *     (2**12+1).bit_length      #=> 13
   */
  
  static VALUE
  rb_fix_bit_length(VALUE fix)
  {
      long v = FIX2LONG(fix);
      if (v < 0)
          v = ~v;
      return LONG2FIX(bit_length(v));
  }
  
  static VALUE
  int_upto_size(VALUE from, VALUE args, VALUE eobj)
  {
      return ruby_num_interval_step_size(from, RARRAY_AREF(args, 0), INT2FIX(1), FALSE);
  }
  
  /*
   *  call-seq:
   *     int.upto(limit) {|i| block }  ->  self
   *     int.upto(limit)               ->  an_enumerator
   *
   *  Iterates the given block, passing in integer values from +int+ up to and
   *  including +limit+.
   *
   *  If no block is given, an Enumerator is returned instead.
   *
   *  For example:
   *
   *     5.upto(10) { |i| print i, " " }
   *     #=> 5 6 7 8 9 10
   */
  
  static VALUE
  int_upto(VALUE from, VALUE to)
  {
      RETURN_SIZED_ENUMERATOR(from, 1, &to, int_upto_size);
      if (FIXNUM_P(from) && FIXNUM_P(to)) {
  	long i, end;
  
  	end = FIX2LONG(to);
  	for (i = FIX2LONG(from); i <= end; i++) {
  	    rb_yield(LONG2FIX(i));
  	}
      }
      else {
  	VALUE i = from, c;
  
  	while (!(c = rb_funcall(i, '>', 1, to))) {
  	    rb_yield(i);
  	    i = rb_funcall(i, '+', 1, INT2FIX(1));
  	}
  	if (NIL_P(c)) rb_cmperr(i, to);
      }
      return from;
  }
  
  static VALUE
  int_downto_size(VALUE from, VALUE args, VALUE eobj)
  {
      return ruby_num_interval_step_size(from, RARRAY_AREF(args, 0), INT2FIX(-1), FALSE);
  }
  
  /*
   *  call-seq:
   *     int.downto(limit) {|i| block }  ->  self
   *     int.downto(limit)               ->  an_enumerator
   *
   *  Iterates the given block, passing decreasing values from +int+ down to and
   *  including +limit+.
   *
   *  If no block is given, an Enumerator is returned instead.
   *
   *     5.downto(1) { |n| print n, ".. " }
   *     print "  Liftoff!\n"
   *     #=> "5.. 4.. 3.. 2.. 1..   Liftoff!"
   */
  
  static VALUE
  int_downto(VALUE from, VALUE to)
  {
      RETURN_SIZED_ENUMERATOR(from, 1, &to, int_downto_size);
      if (FIXNUM_P(from) && FIXNUM_P(to)) {
  	long i, end;
  
  	end = FIX2LONG(to);
  	for (i=FIX2LONG(from); i >= end; i--) {
  	    rb_yield(LONG2FIX(i));
  	}
      }
      else {
  	VALUE i = from, c;
  
  	while (!(c = rb_funcall(i, '<', 1, to))) {
  	    rb_yield(i);
  	    i = rb_funcall(i, '-', 1, INT2FIX(1));
  	}
  	if (NIL_P(c)) rb_cmperr(i, to);
      }
      return from;
  }
  
  static VALUE
  int_dotimes_size(VALUE num, VALUE args, VALUE eobj)
  {
      if (FIXNUM_P(num)) {
  	if (NUM2LONG(num) <= 0) return INT2FIX(0);
      }
      else {
  	if (RTEST(rb_funcall(num, '<', 1, INT2FIX(0)))) return INT2FIX(0);
      }
      return num;
  }
  
  /*
   *  call-seq:
   *     int.times {|i| block }  ->  self
   *     int.times               ->  an_enumerator
   *
   *  Iterates the given block +int+ times, passing in values from zero to
   *  <code>int - 1</code>.
   *
   *  If no block is given, an Enumerator is returned instead.
   *
   *     5.times do |i|
   *       print i, " "
   *     end
   *     #=> 0 1 2 3 4
   */
  
  static VALUE
  int_dotimes(VALUE num)
  {
      RETURN_SIZED_ENUMERATOR(num, 0, 0, int_dotimes_size);
  
      if (FIXNUM_P(num)) {
  	long i, end;
  
  	end = FIX2LONG(num);
  	for (i=0; i<end; i++) {
  	    rb_yield(LONG2FIX(i));
  	}
      }
      else {
  	VALUE i = INT2FIX(0);
  
  	for (;;) {
  	    if (!RTEST(rb_funcall(i, '<', 1, num))) break;
  	    rb_yield(i);
  	    i = rb_funcall(i, '+', 1, INT2FIX(1));
  	}
      }
      return num;
  }
  
  /*
   *  call-seq:
   *     int.round([ndigits])  ->  integer or float
   *
   *  Rounds +int+ to a given precision in decimal digits (default 0 digits).
   *
   *  Precision may be negative.  Returns a floating point number when +ndigits+
   *  is positive, +self+ for zero, and round down for negative.
   *
   *     1.round        #=> 1
   *     1.round(2)     #=> 1.0
   *     15.round(-1)   #=> 20
   */
  
  static VALUE
  int_round(int argc, VALUE* argv, VALUE num)
  {
      VALUE n;
      int ndigits;
  
      if (argc == 0) return num;
      rb_scan_args(argc, argv, "1", &n);
      ndigits = NUM2INT(n);
      if (ndigits > 0) {
  	return rb_Float(num);
      }
      if (ndigits == 0) {
  	return num;
      }
      return int_round_0(num, ndigits);
  }
  
  /*
   *  call-seq:
   *     fix.zero?  ->  true or false
   *
   *  Returns +true+ if +fix+ is zero.
   *
   */
  
  static VALUE
  fix_zero_p(VALUE num)
  {
      if (FIX2LONG(num) == 0) {
  	return Qtrue;
      }
      return Qfalse;
  }
  
  /*
   *  call-seq:
   *     fix.odd?  ->  true or false
   *
   *  Returns +true+ if +fix+ is an odd number.
   */
  
  static VALUE
  fix_odd_p(VALUE num)
  {
      if (num & 2) {
  	return Qtrue;
      }
      return Qfalse;
  }
  
  /*
   *  call-seq:
   *     fix.even?  ->  true or false
   *
   *  Returns +true+ if +fix+ is an even number.
   */
  
  static VALUE
  fix_even_p(VALUE num)
  {
      if (num & 2) {
  	return Qfalse;
      }
      return Qtrue;
  }
  
  /*
   *  Document-class: ZeroDivisionError
   *
   *  Raised when attempting to divide an integer by 0.
   *
   *     42 / 0
   *     #=> ZeroDivisionError: divided by 0
   *
   *  Note that only division by an exact 0 will raise the exception:
   *
   *     42 /  0.0 #=> Float::INFINITY
   *     42 / -0.0 #=> -Float::INFINITY
   *     0  /  0.0 #=> NaN
   */
  
  /*
   *  Document-class: FloatDomainError
   *
   *  Raised when attempting to convert special float values (in particular
   *  +infinite+ or +NaN+) to numerical classes which don't support them.
   *
   *     Float::INFINITY.to_r
   *     #=> FloatDomainError: Infinity
   */
  
  /*
   *  The top-level number class.
   */
  void
  Init_Numeric(void)
  {
  #undef rb_intern
  #define rb_intern(str) rb_intern_const(str)
  
  #if defined(__FreeBSD__) && __FreeBSD__ < 4
      /* allow divide by zero -- Inf */
      fpsetmask(fpgetmask() & ~(FP_X_DZ|FP_X_INV|FP_X_OFL));
  #elif defined(_UNICOSMP)
      /* Turn off floating point exceptions for divide by zero, etc. */
      _set_Creg(0, 0);
  #elif defined(__BORLANDC__)
      /* Turn off floating point exceptions for overflow, etc. */
      _control87(MCW_EM, MCW_EM);
      _control87(_control87(0,0),0x1FFF);
  #endif
      id_coerce = rb_intern("coerce");
      id_to_i = rb_intern("to_i");
      id_eq = rb_intern("==");
      id_div = rb_intern("div");
      id_cmp = rb_intern("<=>");
  
      rb_eZeroDivError = rb_define_class("ZeroDivisionError", rb_eStandardError);
      rb_eFloatDomainError = rb_define_class("FloatDomainError", rb_eRangeError);
      rb_cNumeric = rb_define_class("Numeric", rb_cObject);
  
      rb_define_method(rb_cNumeric, "singleton_method_added", num_sadded, 1);
      rb_include_module(rb_cNumeric, rb_mComparable);
      rb_define_method(rb_cNumeric, "initialize_copy", num_init_copy, 1);
      rb_define_method(rb_cNumeric, "coerce", num_coerce, 1);
  
      rb_define_method(rb_cNumeric, "i", num_imaginary, 0);
      rb_define_method(rb_cNumeric, "+@", num_uplus, 0);
      rb_define_method(rb_cNumeric, "-@", num_uminus, 0);
      rb_define_method(rb_cNumeric, "<=>", num_cmp, 1);
      rb_define_method(rb_cNumeric, "eql?", num_eql, 1);
      rb_define_method(rb_cNumeric, "fdiv", num_fdiv, 1);
      rb_define_method(rb_cNumeric, "div", num_div, 1);
      rb_define_method(rb_cNumeric, "divmod", num_divmod, 1);
      rb_define_method(rb_cNumeric, "%", num_modulo, 1);
      rb_define_method(rb_cNumeric, "modulo", num_modulo, 1);
      rb_define_method(rb_cNumeric, "remainder", num_remainder, 1);
      rb_define_method(rb_cNumeric, "abs", num_abs, 0);
      rb_define_method(rb_cNumeric, "magnitude", num_abs, 0);
      rb_define_method(rb_cNumeric, "to_int", num_to_int, 0);
  
      rb_define_method(rb_cNumeric, "real?", num_real_p, 0);
      rb_define_method(rb_cNumeric, "integer?", num_int_p, 0);
      rb_define_method(rb_cNumeric, "zero?", num_zero_p, 0);
      rb_define_method(rb_cNumeric, "nonzero?", num_nonzero_p, 0);
  
      rb_define_method(rb_cNumeric, "floor", num_floor, 0);
      rb_define_method(rb_cNumeric, "ceil", num_ceil, 0);
      rb_define_method(rb_cNumeric, "round", num_round, -1);
      rb_define_method(rb_cNumeric, "truncate", num_truncate, 0);
      rb_define_method(rb_cNumeric, "step", num_step, -1);
  
      rb_cInteger = rb_define_class("Integer", rb_cNumeric);
      rb_undef_alloc_func(rb_cInteger);
      rb_undef_method(CLASS_OF(rb_cInteger), "new");
  
      rb_define_method(rb_cInteger, "integer?", int_int_p, 0);
      rb_define_method(rb_cInteger, "odd?", int_odd_p, 0);
      rb_define_method(rb_cInteger, "even?", int_even_p, 0);
      rb_define_method(rb_cInteger, "upto", int_upto, 1);
      rb_define_method(rb_cInteger, "downto", int_downto, 1);
      rb_define_method(rb_cInteger, "times", int_dotimes, 0);
      rb_define_method(rb_cInteger, "succ", int_succ, 0);
      rb_define_method(rb_cInteger, "next", int_succ, 0);
      rb_define_method(rb_cInteger, "pred", int_pred, 0);
      rb_define_method(rb_cInteger, "chr", int_chr, -1);
      rb_define_method(rb_cInteger, "ord", int_ord, 0);
      rb_define_method(rb_cInteger, "to_i", int_to_i, 0);
      rb_define_method(rb_cInteger, "to_int", int_to_i, 0);
      rb_define_method(rb_cInteger, "floor", int_to_i, 0);
      rb_define_method(rb_cInteger, "ceil", int_to_i, 0);
      rb_define_method(rb_cInteger, "truncate", int_to_i, 0);
      rb_define_method(rb_cInteger, "round", int_round, -1);
  
      rb_cFixnum = rb_define_class("Fixnum", rb_cInteger);
  
      rb_define_method(rb_cFixnum, "to_s", fix_to_s, -1);
      rb_define_alias(rb_cFixnum, "inspect", "to_s");
  
      rb_define_method(rb_cFixnum, "-@", fix_uminus, 0);
      rb_define_method(rb_cFixnum, "+", fix_plus, 1);
      rb_define_method(rb_cFixnum, "-", fix_minus, 1);
      rb_define_method(rb_cFixnum, "*", fix_mul, 1);
      rb_define_method(rb_cFixnum, "/", fix_div, 1);
      rb_define_method(rb_cFixnum, "div", fix_idiv, 1);
      rb_define_method(rb_cFixnum, "%", fix_mod, 1);
      rb_define_method(rb_cFixnum, "modulo", fix_mod, 1);
      rb_define_method(rb_cFixnum, "divmod", fix_divmod, 1);
      rb_define_method(rb_cFixnum, "fdiv", fix_fdiv, 1);
      rb_define_method(rb_cFixnum, "**", fix_pow, 1);
  
      rb_define_method(rb_cFixnum, "abs", fix_abs, 0);
      rb_define_method(rb_cFixnum, "magnitude", fix_abs, 0);
  
      rb_define_method(rb_cFixnum, "==", fix_equal, 1);
      rb_define_method(rb_cFixnum, "===", fix_equal, 1);
      rb_define_method(rb_cFixnum, "<=>", fix_cmp, 1);
      rb_define_method(rb_cFixnum, ">",  fix_gt, 1);
      rb_define_method(rb_cFixnum, ">=", fix_ge, 1);
      rb_define_method(rb_cFixnum, "<",  fix_lt, 1);
      rb_define_method(rb_cFixnum, "<=", fix_le, 1);
  
      rb_define_method(rb_cFixnum, "~", fix_rev, 0);
      rb_define_method(rb_cFixnum, "&", fix_and, 1);
      rb_define_method(rb_cFixnum, "|", fix_or,  1);
      rb_define_method(rb_cFixnum, "^", fix_xor, 1);
      rb_define_method(rb_cFixnum, "[]", fix_aref, 1);
  
      rb_define_method(rb_cFixnum, "<<", rb_fix_lshift, 1);
      rb_define_method(rb_cFixnum, ">>", rb_fix_rshift, 1);
  
      rb_define_method(rb_cFixnum, "to_f", fix_to_f, 0);
      rb_define_method(rb_cFixnum, "size", fix_size, 0);
      rb_define_method(rb_cFixnum, "bit_length", rb_fix_bit_length, 0);
      rb_define_method(rb_cFixnum, "zero?", fix_zero_p, 0);
      rb_define_method(rb_cFixnum, "odd?", fix_odd_p, 0);
      rb_define_method(rb_cFixnum, "even?", fix_even_p, 0);
      rb_define_method(rb_cFixnum, "succ", fix_succ, 0);
  
      rb_cFloat  = rb_define_class("Float", rb_cNumeric);
  
      rb_undef_alloc_func(rb_cFloat);
      rb_undef_method(CLASS_OF(rb_cFloat), "new");
  
      /*
       *  Represents the rounding mode for floating point addition.
       *
       *  Usually defaults to 1, rounding to the nearest number.
       *
       *  Other modes include:
       *
       *  -1::	Indeterminable
       *	0::	Rounding towards zero
       *	1::	Rounding to the nearest number
       *	2::	Rounding towards positive infinity
       *	3::	Rounding towards negative infinity
       */
      rb_define_const(rb_cFloat, "ROUNDS", INT2FIX(FLT_ROUNDS));
      /*
       *	The base of the floating point, or number of unique digits used to
       *	represent the number.
       *
       *  Usually defaults to 2 on most systems, which would represent a base-10 decimal.
       */
      rb_define_const(rb_cFloat, "RADIX", INT2FIX(FLT_RADIX));
      /*
       * The number of base digits for the +double+ data type.
       *
       * Usually defaults to 53.
       */
      rb_define_const(rb_cFloat, "MANT_DIG", INT2FIX(DBL_MANT_DIG));
      /*
       *	The number of decimal digits in a double-precision floating point.
       *
       *	Usually defaults to 15.
       */
      rb_define_const(rb_cFloat, "DIG", INT2FIX(DBL_DIG));
      /*
       *	The smallest posable exponent value in a double-precision floating
       *	point.
       *
       *	Usually defaults to -1021.
       */
      rb_define_const(rb_cFloat, "MIN_EXP", INT2FIX(DBL_MIN_EXP));
      /*
       *	The largest possible exponent value in a double-precision floating
       *	point.
       *
       *	Usually defaults to 1024.
       */
      rb_define_const(rb_cFloat, "MAX_EXP", INT2FIX(DBL_MAX_EXP));
      /*
       *	The smallest negative exponent in a double-precision floating point
       *	where 10 raised to this power minus 1.
       *
       *	Usually defaults to -307.
       */
      rb_define_const(rb_cFloat, "MIN_10_EXP", INT2FIX(DBL_MIN_10_EXP));
      /*
       *	The largest positive exponent in a double-precision floating point where
       *	10 raised to this power minus 1.
       *
       *	Usually defaults to 308.
       */
      rb_define_const(rb_cFloat, "MAX_10_EXP", INT2FIX(DBL_MAX_10_EXP));
      /*
       *	The smallest positive integer in a double-precision floating point.
       *
       *	Usually defaults to 2.2250738585072014e-308.
       */
      rb_define_const(rb_cFloat, "MIN", DBL2NUM(DBL_MIN));
      /*
       *	The largest possible integer in a double-precision floating point number.
       *
       *	Usually defaults to 1.7976931348623157e+308.
       */
      rb_define_const(rb_cFloat, "MAX", DBL2NUM(DBL_MAX));
      /*
       *	The difference between 1 and the smallest double-precision floating
       *	point number.
       *
       *	Usually defaults to 2.2204460492503131e-16.
       */
      rb_define_const(rb_cFloat, "EPSILON", DBL2NUM(DBL_EPSILON));
      /*
       *	An expression representing positive infinity.
       */
      rb_define_const(rb_cFloat, "INFINITY", DBL2NUM(INFINITY));
      /*
       *	An expression representing a value which is "not a number".
       */
      rb_define_const(rb_cFloat, "NAN", DBL2NUM(NAN));
  
      rb_define_method(rb_cFloat, "to_s", flo_to_s, 0);
      rb_define_alias(rb_cFloat, "inspect", "to_s");
      rb_define_method(rb_cFloat, "coerce", flo_coerce, 1);
      rb_define_method(rb_cFloat, "-@", flo_uminus, 0);
      rb_define_method(rb_cFloat, "+", flo_plus, 1);
      rb_define_method(rb_cFloat, "-", flo_minus, 1);
      rb_define_method(rb_cFloat, "*", flo_mul, 1);
      rb_define_method(rb_cFloat, "/", flo_div, 1);
      rb_define_method(rb_cFloat, "quo", flo_quo, 1);
      rb_define_method(rb_cFloat, "fdiv", flo_quo, 1);
      rb_define_method(rb_cFloat, "%", flo_mod, 1);
      rb_define_method(rb_cFloat, "modulo", flo_mod, 1);
      rb_define_method(rb_cFloat, "divmod", flo_divmod, 1);
      rb_define_method(rb_cFloat, "**", flo_pow, 1);
      rb_define_method(rb_cFloat, "==", flo_eq, 1);
      rb_define_method(rb_cFloat, "===", flo_eq, 1);
      rb_define_method(rb_cFloat, "<=>", flo_cmp, 1);
      rb_define_method(rb_cFloat, ">",  flo_gt, 1);
      rb_define_method(rb_cFloat, ">=", flo_ge, 1);
      rb_define_method(rb_cFloat, "<",  flo_lt, 1);
      rb_define_method(rb_cFloat, "<=", flo_le, 1);
      rb_define_method(rb_cFloat, "eql?", flo_eql, 1);
      rb_define_method(rb_cFloat, "hash", flo_hash, 0);
      rb_define_method(rb_cFloat, "to_f", flo_to_f, 0);
      rb_define_method(rb_cFloat, "abs", flo_abs, 0);
      rb_define_method(rb_cFloat, "magnitude", flo_abs, 0);
      rb_define_method(rb_cFloat, "zero?", flo_zero_p, 0);
  
      rb_define_method(rb_cFloat, "to_i", flo_truncate, 0);
      rb_define_method(rb_cFloat, "to_int", flo_truncate, 0);
      rb_define_method(rb_cFloat, "floor", flo_floor, 0);
      rb_define_method(rb_cFloat, "ceil", flo_ceil, 0);
      rb_define_method(rb_cFloat, "round", flo_round, -1);
      rb_define_method(rb_cFloat, "truncate", flo_truncate, 0);
  
      rb_define_method(rb_cFloat, "nan?",      flo_is_nan_p, 0);
      rb_define_method(rb_cFloat, "infinite?", flo_is_infinite_p, 0);
      rb_define_method(rb_cFloat, "finite?",   flo_is_finite_p, 0);
  
      id_to = rb_intern("to");
      id_by = rb_intern("by");
  }
  
  #undef rb_float_value
  double
  rb_float_value(VALUE v)
  {
      return rb_float_value_inline(v);
  }
  
  #undef rb_float_new
  VALUE
  rb_float_new(double d)
  {
      return rb_float_new_inline(d);
  }
  