rational.c

/*
  rational.c: Coded by Tadayoshi Funaba 2008-2011

  This implementation is based on Keiju Ishitsuka's Rational library
  which is written in ruby.
*/

#include "ruby.h"
#include "internal.h"
#include <math.h>
#include <float.h>

#ifdef HAVE_IEEEFP_H
#include <ieeefp.h>
#endif

#define NDEBUG
#include <assert.h>

#define ZERO INT2FIX(0)
#define ONE INT2FIX(1)
#define TWO INT2FIX(2)

VALUE rb_cRational;

static ID id_abs, id_cmp, id_convert, id_eqeq_p, id_expt, id_fdiv,
    id_floor, id_idiv, id_inspect, id_integer_p, id_negate, id_to_f,
    id_to_i, id_to_s, id_truncate;

#define f_boolcast(x) ((x) ? Qtrue : Qfalse)

#define binop(n,op) \
inline static VALUE \
f_##n(VALUE x, VALUE y)\
{\
  return rb_funcall(x, (op), 1, y);\
}

#define fun1(n) \
inline static VALUE \
f_##n(VALUE x)\
{\
    return rb_funcall(x, id_##n, 0);\
}

#define fun2(n) \
inline static VALUE \
f_##n(VALUE x, VALUE y)\
{\
    return rb_funcall(x, id_##n, 1, y);\
}

inline static VALUE
f_add(VALUE x, VALUE y)
{
    if (FIXNUM_P(y) && FIX2LONG(y) == 0)
	return x;
    else if (FIXNUM_P(x) && FIX2LONG(x) == 0)
	return y;
    return rb_funcall(x, '+', 1, y);
}

inline static VALUE
f_cmp(VALUE x, VALUE y)
{
    if (FIXNUM_P(x) && FIXNUM_P(y)) {
	long c = FIX2LONG(x) - FIX2LONG(y);
	if (c > 0)
	    c = 1;
	else if (c < 0)
	    c = -1;
	return INT2FIX(c);
    }
    return rb_funcall(x, id_cmp, 1, y);
}

inline static VALUE
f_div(VALUE x, VALUE y)
{
    if (FIXNUM_P(y) && FIX2LONG(y) == 1)
	return x;
    return rb_funcall(x, '/', 1, y);
}

inline static VALUE
f_gt_p(VALUE x, VALUE y)
{
    if (FIXNUM_P(x) && FIXNUM_P(y))
	return f_boolcast(FIX2LONG(x) > FIX2LONG(y));
    return rb_funcall(x, '>', 1, y);
}

inline static VALUE
f_lt_p(VALUE x, VALUE y)
{
    if (FIXNUM_P(x) && FIXNUM_P(y))
	return f_boolcast(FIX2LONG(x) < FIX2LONG(y));
    return rb_funcall(x, '<', 1, y);
}

binop(mod, '%')

inline static VALUE
f_mul(VALUE x, VALUE y)
{
    if (FIXNUM_P(y)) {
	long iy = FIX2LONG(y);
	if (iy == 0) {
	    if (FIXNUM_P(x) || TYPE(x) == T_BIGNUM)
		return ZERO;
	}
	else if (iy == 1)
	    return x;
    }
    else if (FIXNUM_P(x)) {
	long ix = FIX2LONG(x);
	if (ix == 0) {
	    if (FIXNUM_P(y) || TYPE(y) == T_BIGNUM)
		return ZERO;
	}
	else if (ix == 1)
	    return y;
    }
    return rb_funcall(x, '*', 1, y);
}

inline static VALUE
f_sub(VALUE x, VALUE y)
{
    if (FIXNUM_P(y) && FIX2LONG(y) == 0)
	return x;
    return rb_funcall(x, '-', 1, y);
}

fun1(abs)
fun1(floor)
fun1(inspect)
fun1(integer_p)
fun1(negate)

inline static VALUE
f_to_i(VALUE x)
{
    if (TYPE(x) == T_STRING)
	return rb_str_to_inum(x, 10, 0);
    return rb_funcall(x, id_to_i, 0);
}
inline static VALUE
f_to_f(VALUE x)
{
    if (TYPE(x) == T_STRING)
	return DBL2NUM(rb_str_to_dbl(x, 0));
    return rb_funcall(x, id_to_f, 0);
}

fun1(to_s)
fun1(truncate)

inline static VALUE
f_eqeq_p(VALUE x, VALUE y)
{
    if (FIXNUM_P(x) && FIXNUM_P(y))
	return f_boolcast(FIX2LONG(x) == FIX2LONG(y));
    return rb_funcall(x, id_eqeq_p, 1, y);
}

fun2(expt)
fun2(fdiv)
fun2(idiv)

#define f_expt10(x) f_expt(INT2FIX(10), x)

inline static VALUE
f_negative_p(VALUE x)
{
    if (FIXNUM_P(x))
	return f_boolcast(FIX2LONG(x) < 0);
    return rb_funcall(x, '<', 1, ZERO);
}

#define f_positive_p(x) (!f_negative_p(x))

inline static VALUE
f_zero_p(VALUE x)
{
    switch (TYPE(x)) {
      case T_FIXNUM:
	return f_boolcast(FIX2LONG(x) == 0);
      case T_BIGNUM:
	return Qfalse;
      case T_RATIONAL:
      {
	  VALUE num = RRATIONAL(x)->num;

	  return f_boolcast(FIXNUM_P(num) && FIX2LONG(num) == 0);
      }
    }
    return rb_funcall(x, id_eqeq_p, 1, ZERO);
}

#define f_nonzero_p(x) (!f_zero_p(x))

inline static VALUE
f_one_p(VALUE x)
{
    switch (TYPE(x)) {
      case T_FIXNUM:
	return f_boolcast(FIX2LONG(x) == 1);
      case T_BIGNUM:
	return Qfalse;
      case T_RATIONAL:
      {
	  VALUE num = RRATIONAL(x)->num;
	  VALUE den = RRATIONAL(x)->den;

	  return f_boolcast(FIXNUM_P(num) && FIX2LONG(num) == 1 &&
			    FIXNUM_P(den) && FIX2LONG(den) == 1);
      }
    }
    return rb_funcall(x, id_eqeq_p, 1, ONE);
}

inline static VALUE
f_kind_of_p(VALUE x, VALUE c)
{
    return rb_obj_is_kind_of(x, c);
}

inline static VALUE
k_numeric_p(VALUE x)
{
    return f_kind_of_p(x, rb_cNumeric);
}

inline static VALUE
k_integer_p(VALUE x)
{
    return f_kind_of_p(x, rb_cInteger);
}

inline static VALUE
k_float_p(VALUE x)
{
    return f_kind_of_p(x, rb_cFloat);
}

inline static VALUE
k_rational_p(VALUE x)
{
    return f_kind_of_p(x, rb_cRational);
}

#define k_exact_p(x) (!k_float_p(x))
#define k_inexact_p(x) k_float_p(x)

#define k_exact_zero_p(x) (k_exact_p(x) && f_zero_p(x))
#define k_exact_one_p(x) (k_exact_p(x) && f_one_p(x))

#ifndef NDEBUG
#define f_gcd f_gcd_orig
#endif

inline static long
i_gcd(long x, long y)
{
    if (x < 0)
	x = -x;
    if (y < 0)
	y = -y;

    if (x == 0)
	return y;
    if (y == 0)
	return x;

    while (x > 0) {
	long t = x;
	x = y % x;
	y = t;
    }
    return y;
}

inline static VALUE
f_gcd(VALUE x, VALUE y)
{
    VALUE z;

    if (FIXNUM_P(x) && FIXNUM_P(y))
	return LONG2NUM(i_gcd(FIX2LONG(x), FIX2LONG(y)));

    if (f_negative_p(x))
	x = f_negate(x);
    if (f_negative_p(y))
	y = f_negate(y);

    if (f_zero_p(x))
	return y;
    if (f_zero_p(y))
	return x;

    for (;;) {
	if (FIXNUM_P(x)) {
	    if (FIX2LONG(x) == 0)
		return y;
	    if (FIXNUM_P(y))
		return LONG2NUM(i_gcd(FIX2LONG(x), FIX2LONG(y)));
	}
	z = x;
	x = f_mod(y, x);
	y = z;
    }
    /* NOTREACHED */
}

#ifndef NDEBUG
#undef f_gcd

inline static VALUE
f_gcd(VALUE x, VALUE y)
{
    VALUE r = f_gcd_orig(x, y);
    if (f_nonzero_p(r)) {
	assert(f_zero_p(f_mod(x, r)));
	assert(f_zero_p(f_mod(y, r)));
    }
    return r;
}
#endif

inline static VALUE
f_lcm(VALUE x, VALUE y)
{
    if (f_zero_p(x) || f_zero_p(y))
	return ZERO;
    return f_abs(f_mul(f_div(x, f_gcd(x, y)), y));
}

#define get_dat1(x) \
    struct RRational *dat;\
    dat = ((struct RRational *)(x))

#define get_dat2(x,y) \
    struct RRational *adat, *bdat;\
    adat = ((struct RRational *)(x));\
    bdat = ((struct RRational *)(y))

inline static VALUE
nurat_s_new_internal(VALUE klass, VALUE num, VALUE den)
{
    NEWOBJ(obj, struct RRational);
    OBJSETUP(obj, klass, T_RATIONAL);

    obj->num = num;
    obj->den = den;

    return (VALUE)obj;
}

static VALUE
nurat_s_alloc(VALUE klass)
{
    return nurat_s_new_internal(klass, ZERO, ONE);
}

#define rb_raise_zerodiv() rb_raise(rb_eZeroDivError, "divided by 0")

#if 0
static VALUE
nurat_s_new_bang(int argc, VALUE *argv, VALUE klass)
{
    VALUE num, den;

    switch (rb_scan_args(argc, argv, "11", &num, &den)) {
      case 1:
	if (!k_integer_p(num))
	    num = f_to_i(num);
	den = ONE;
	break;
      default:
	if (!k_integer_p(num))
	    num = f_to_i(num);
	if (!k_integer_p(den))
	    den = f_to_i(den);

	switch (FIX2INT(f_cmp(den, ZERO))) {
	  case -1:
	    num = f_negate(num);
	    den = f_negate(den);
	    break;
	  case 0:
	    rb_raise_zerodiv();
	    break;
	}
	break;
    }

    return nurat_s_new_internal(klass, num, den);
}
#endif

inline static VALUE
f_rational_new_bang1(VALUE klass, VALUE x)
{
    return nurat_s_new_internal(klass, x, ONE);
}

inline static VALUE
f_rational_new_bang2(VALUE klass, VALUE x, VALUE y)
{
    assert(f_positive_p(y));
    assert(f_nonzero_p(y));
    return nurat_s_new_internal(klass, x, y);
}

#ifdef CANONICALIZATION_FOR_MATHN
#define CANON
#endif

#ifdef CANON
static int canonicalization = 0;

RUBY_FUNC_EXPORTED void
nurat_canonicalization(int f)
{
    canonicalization = f;
}
#endif

inline static void
nurat_int_check(VALUE num)
{
    switch (TYPE(num)) {
      case T_FIXNUM:
      case T_BIGNUM:
	break;
      default:
	if (!k_numeric_p(num) || !f_integer_p(num))
	    rb_raise(rb_eTypeError, "not an integer");
    }
}

inline static VALUE
nurat_int_value(VALUE num)
{
    nurat_int_check(num);
    if (!k_integer_p(num))
	num = f_to_i(num);
    return num;
}

inline static VALUE
nurat_s_canonicalize_internal(VALUE klass, VALUE num, VALUE den)
{
    VALUE gcd;

    switch (FIX2INT(f_cmp(den, ZERO))) {
      case -1:
	num = f_negate(num);
	den = f_negate(den);
	break;
      case 0:
	rb_raise_zerodiv();
	break;
    }

    gcd = f_gcd(num, den);
    num = f_idiv(num, gcd);
    den = f_idiv(den, gcd);

#ifdef CANON
    if (f_one_p(den) && canonicalization)
	return num;
#endif
    return nurat_s_new_internal(klass, num, den);
}

inline static VALUE
nurat_s_canonicalize_internal_no_reduce(VALUE klass, VALUE num, VALUE den)
{
    switch (FIX2INT(f_cmp(den, ZERO))) {
      case -1:
	num = f_negate(num);
	den = f_negate(den);
	break;
      case 0:
	rb_raise_zerodiv();
	break;
    }

#ifdef CANON
    if (f_one_p(den) && canonicalization)
	return num;
#endif
    return nurat_s_new_internal(klass, num, den);
}

static VALUE
nurat_s_new(int argc, VALUE *argv, VALUE klass)
{
    VALUE num, den;

    switch (rb_scan_args(argc, argv, "11", &num, &den)) {
      case 1:
	num = nurat_int_value(num);
	den = ONE;
	break;
      default:
	num = nurat_int_value(num);
	den = nurat_int_value(den);
	break;
    }

    return nurat_s_canonicalize_internal(klass, num, den);
}

inline static VALUE
f_rational_new1(VALUE klass, VALUE x)
{
    assert(!k_rational_p(x));
    return nurat_s_canonicalize_internal(klass, x, ONE);
}

inline static VALUE
f_rational_new2(VALUE klass, VALUE x, VALUE y)
{
    assert(!k_rational_p(x));
    assert(!k_rational_p(y));
    return nurat_s_canonicalize_internal(klass, x, y);
}

inline static VALUE
f_rational_new_no_reduce1(VALUE klass, VALUE x)
{
    assert(!k_rational_p(x));
    return nurat_s_canonicalize_internal_no_reduce(klass, x, ONE);
}

inline static VALUE
f_rational_new_no_reduce2(VALUE klass, VALUE x, VALUE y)
{
    assert(!k_rational_p(x));
    assert(!k_rational_p(y));
    return nurat_s_canonicalize_internal_no_reduce(klass, x, y);
}

/*
 * call-seq:
 *    Rational(x[, y])  ->  numeric
 *
 * Returns x/y;
 */
static VALUE
nurat_f_rational(int argc, VALUE *argv, VALUE klass)
{
    return rb_funcall2(rb_cRational, id_convert, argc, argv);
}

/*
 * call-seq:
 *    rat.numerator  ->  integer
 *
 * Returns the numerator.
 *
 * For example:
 *
 *    Rational(7).numerator        #=> 7
 *    Rational(7, 1).numerator     #=> 7
 *    Rational(9, -4).numerator    #=> -9
 *    Rational(-2, -10).numerator  #=> 1
 */
static VALUE
nurat_numerator(VALUE self)
{
    get_dat1(self);
    return dat->num;
}

/*
 * call-seq:
 *    rat.denominator  ->  integer
 *
 * Returns the denominator (always positive).
 *
 * For example:
 *
 *    Rational(7).denominator             #=> 1
 *    Rational(7, 1).denominator          #=> 1
 *    Rational(9, -4).denominator         #=> 4
 *    Rational(-2, -10).denominator       #=> 5
 *    rat.numerator.gcd(rat.denominator)  #=> 1
 */
static VALUE
nurat_denominator(VALUE self)
{
    get_dat1(self);
    return dat->den;
}

#ifndef NDEBUG
#define f_imul f_imul_orig
#endif

inline static VALUE
f_imul(long a, long b)
{
    VALUE r;
    volatile long c;

    if (a == 0 || b == 0)
	return ZERO;
    else if (a == 1)
	return LONG2NUM(b);
    else if (b == 1)
	return LONG2NUM(a);

    c = a * b;
    r = LONG2NUM(c);
    if (NUM2LONG(r) != c || (c / a) != b)
	r = rb_big_mul(rb_int2big(a), rb_int2big(b));
    return r;
}

#ifndef NDEBUG
#undef f_imul

inline static VALUE
f_imul(long x, long y)
{
    VALUE r = f_imul_orig(x, y);
    assert(f_eqeq_p(r, f_mul(LONG2NUM(x), LONG2NUM(y))));
    return r;
}
#endif

inline static VALUE
f_addsub(VALUE self, VALUE anum, VALUE aden, VALUE bnum, VALUE bden, int k)
{
    VALUE num, den;

    if (FIXNUM_P(anum) && FIXNUM_P(aden) &&
	FIXNUM_P(bnum) && FIXNUM_P(bden)) {
	long an = FIX2LONG(anum);
	long ad = FIX2LONG(aden);
	long bn = FIX2LONG(bnum);
	long bd = FIX2LONG(bden);
	long ig = i_gcd(ad, bd);

	VALUE g = LONG2NUM(ig);
	VALUE a = f_imul(an, bd / ig);
	VALUE b = f_imul(bn, ad / ig);
	VALUE c;

	if (k == '+')
	    c = f_add(a, b);
	else
	    c = f_sub(a, b);

	b = f_idiv(aden, g);
	g = f_gcd(c, g);
	num = f_idiv(c, g);
	a = f_idiv(bden, g);
	den = f_mul(a, b);
    }
    else {
	VALUE g = f_gcd(aden, bden);
	VALUE a = f_mul(anum, f_idiv(bden, g));
	VALUE b = f_mul(bnum, f_idiv(aden, g));
	VALUE c;

	if (k == '+')
	    c = f_add(a, b);
	else
	    c = f_sub(a, b);

	b = f_idiv(aden, g);
	g = f_gcd(c, g);
	num = f_idiv(c, g);
	a = f_idiv(bden, g);
	den = f_mul(a, b);
    }
    return f_rational_new_no_reduce2(CLASS_OF(self), num, den);
}

/*
 * call-seq:
 *    rat + numeric  ->  numeric
 *
 * Performs addition.
 *
 * For example:
 *
 *    Rational(2, 3)  + Rational(2, 3)   #=> (4/3)
 *    Rational(900)   + Rational(1)      #=> (900/1)
 *    Rational(-2, 9) + Rational(-9, 2)  #=> (-85/18)
 *    Rational(9, 8)  + 4                #=> (41/8)
 *    Rational(20, 9) + 9.8              #=> 12.022222222222222
 */
static VALUE
nurat_add(VALUE self, VALUE other)
{
    switch (TYPE(other)) {
      case T_FIXNUM:
      case T_BIGNUM:
	{
	    get_dat1(self);

	    return f_addsub(self,
			    dat->num, dat->den,
			    other, ONE, '+');
	}
      case T_FLOAT:
	return f_add(f_to_f(self), other);
      case T_RATIONAL:
	{
	    get_dat2(self, other);

	    return f_addsub(self,
			    adat->num, adat->den,
			    bdat->num, bdat->den, '+');
	}
      default:
	return rb_num_coerce_bin(self, other, '+');
    }
}

/*
 * call-seq:
 *    rat - numeric  ->  numeric
 *
 * Performs subtraction.
 *
 * For example:
 *
 *    Rational(2, 3)  - Rational(2, 3)   #=> (0/1)
 *    Rational(900)   - Rational(1)      #=> (899/1)
 *    Rational(-2, 9) - Rational(-9, 2)  #=> (77/18)
 *    Rational(9, 8)  - 4                #=> (23/8)
 *    Rational(20, 9) - 9.8              #=> -7.577777777777778
 */
static VALUE
nurat_sub(VALUE self, VALUE other)
{
    switch (TYPE(other)) {
      case T_FIXNUM:
      case T_BIGNUM:
	{
	    get_dat1(self);

	    return f_addsub(self,
			    dat->num, dat->den,
			    other, ONE, '-');
	}
      case T_FLOAT:
	return f_sub(f_to_f(self), other);
      case T_RATIONAL:
	{
	    get_dat2(self, other);

	    return f_addsub(self,
			    adat->num, adat->den,
			    bdat->num, bdat->den, '-');
	}
      default:
	return rb_num_coerce_bin(self, other, '-');
    }
}

inline static VALUE
f_muldiv(VALUE self, VALUE anum, VALUE aden, VALUE bnum, VALUE bden, int k)
{
    VALUE num, den;

    if (k == '/') {
	VALUE t;

	if (f_negative_p(bnum)) {
	    anum = f_negate(anum);
	    bnum = f_negate(bnum);
	}
	t = bnum;
	bnum = bden;
	bden = t;
    }

    if (FIXNUM_P(anum) && FIXNUM_P(aden) &&
	FIXNUM_P(bnum) && FIXNUM_P(bden)) {
	long an = FIX2LONG(anum);
	long ad = FIX2LONG(aden);
	long bn = FIX2LONG(bnum);
	long bd = FIX2LONG(bden);
	long g1 = i_gcd(an, bd);
	long g2 = i_gcd(ad, bn);

	num = f_imul(an / g1, bn / g2);
	den = f_imul(ad / g2, bd / g1);
    }
    else {
	VALUE g1 = f_gcd(anum, bden);
	VALUE g2 = f_gcd(aden, bnum);

	num = f_mul(f_idiv(anum, g1), f_idiv(bnum, g2));
	den = f_mul(f_idiv(aden, g2), f_idiv(bden, g1));
    }
    return f_rational_new_no_reduce2(CLASS_OF(self), num, den);
}

/*
 * call-seq:
 *    rat * numeric  ->  numeric
 *
 * Performs multiplication.
 *
 * For example:
 *
 *    Rational(2, 3)  * Rational(2, 3)   #=> (4/9)
 *    Rational(900)   * Rational(1)      #=> (900/1)
 *    Rational(-2, 9) * Rational(-9, 2)  #=> (1/1)
 *    Rational(9, 8)  * 4                #=> (9/2)
 *    Rational(20, 9) * 9.8              #=> 21.77777777777778
 */
static VALUE
nurat_mul(VALUE self, VALUE other)
{
    switch (TYPE(other)) {
      case T_FIXNUM:
      case T_BIGNUM:
	{
	    get_dat1(self);

	    return f_muldiv(self,
			    dat->num, dat->den,
			    other, ONE, '*');
	}
      case T_FLOAT:
	return f_mul(f_to_f(self), other);
      case T_RATIONAL:
	{
	    get_dat2(self, other);

	    return f_muldiv(self,
			    adat->num, adat->den,
			    bdat->num, bdat->den, '*');
	}
      default:
	return rb_num_coerce_bin(self, other, '*');
    }
}

/*
 * call-seq:
 *    rat / numeric     ->  numeric
 *    rat.quo(numeric)  ->  numeric
 *
 * Performs division.
 *
 * For example:
 *
 *    Rational(2, 3)  / Rational(2, 3)   #=> (1/1)
 *    Rational(900)   / Rational(1)      #=> (900/1)
 *    Rational(-2, 9) / Rational(-9, 2)  #=> (4/81)
 *    Rational(9, 8)  / 4                #=> (9/32)
 *    Rational(20, 9) / 9.8              #=> 0.22675736961451246
 */
static VALUE
nurat_div(VALUE self, VALUE other)
{
    switch (TYPE(other)) {
      case T_FIXNUM:
      case T_BIGNUM:
	if (f_zero_p(other))
	    rb_raise_zerodiv();
	{
	    get_dat1(self);

	    return f_muldiv(self,
			    dat->num, dat->den,
			    other, ONE, '/');
	}
      case T_FLOAT:
	{
	    double x = RFLOAT_VALUE(other), den;
	    get_dat1(self);

	    if (isnan(x)) return DBL2NUM(NAN);
	    if (isinf(x)) return INT2FIX(0);
	    if (x != 0.0 && modf(x, &den) == 0.0) {
		return rb_rational_raw2(dat->num, f_mul(rb_dbl2big(den), dat->den));
	    }
	}
	return rb_funcall(f_to_f(self), '/', 1, other);
      case T_RATIONAL:
	if (f_zero_p(other))
	    rb_raise_zerodiv();
	{
	    get_dat2(self, other);

	    if (f_one_p(self))
		return f_rational_new_no_reduce2(CLASS_OF(self),
						 bdat->den, bdat->num);

	    return f_muldiv(self,
			    adat->num, adat->den,
			    bdat->num, bdat->den, '/');
	}
      default:
	return rb_num_coerce_bin(self, other, '/');
    }
}

/*
 * call-seq:
 *    rat.fdiv(numeric)  ->  float
 *
 * Performs division and returns the value as a float.
 *
 * For example:
 *
 *    Rational(2, 3).fdiv(1)       #=> 0.6666666666666666
 *    Rational(2, 3).fdiv(0.5)     #=> 1.3333333333333333
 *    Rational(2).fdiv(3)          #=> 0.6666666666666666
 */
static VALUE
nurat_fdiv(VALUE self, VALUE other)
{
    if (f_zero_p(other))
	return f_div(self, f_to_f(other));
    return f_to_f(f_div(self, other));
}

/*
 * call-seq:
 *    rat ** numeric  ->  numeric
 *
 * Performs exponentiation.
 *
 * For example:
 *
 *    Rational(2)    ** Rational(3)    #=> (8/1)
 *    Rational(10)   ** -2             #=> (1/100)
 *    Rational(10)   ** -2.0           #=> 0.01
 *    Rational(-4)   ** Rational(1,2)  #=> (1.2246063538223773e-16+2.0i)
 *    Rational(1, 2) ** 0              #=> (1/1)
 *    Rational(1, 2) ** 0.0            #=> 1.0
 */
static VALUE
nurat_expt(VALUE self, VALUE other)
{
    if (k_numeric_p(other) && k_exact_zero_p(other))
	return f_rational_new_bang1(CLASS_OF(self), ONE);

    if (k_rational_p(other)) {
	get_dat1(other);

	if (f_one_p(dat->den))
	    other = dat->num; /* c14n */
    }

    switch (TYPE(other)) {
      case T_FIXNUM:
	{
	    VALUE num, den;

	    get_dat1(self);

	    switch (FIX2INT(f_cmp(other, ZERO))) {
	      case 1:
		num = f_expt(dat->num, other);
		den = f_expt(dat->den, other);
		break;
	      case -1:
		num = f_expt(dat->den, f_negate(other));
		den = f_expt(dat->num, f_negate(other));
		break;
	      default:
		num = ONE;
		den = ONE;
		break;
	    }
	    return f_rational_new2(CLASS_OF(self), num, den);
	}
      case T_BIGNUM:
	rb_warn("in a**b, b may be too big");
	/* fall through */
      case T_FLOAT:
      case T_RATIONAL:
	return f_expt(f_to_f(self), other);
      default:
	return rb_num_coerce_bin(self, other, id_expt);
    }
}

/*
 * call-seq:
 *    rat <=> numeric  ->  -1, 0, +1 or nil
 *
 * Performs comparison and returns -1, 0, or +1.
 *
 * For example:
 *
 *    Rational(2, 3)  <=> Rational(2, 3)  #=> 0
 *    Rational(5)     <=> 5               #=> 0
 *    Rational(2,3)   <=> Rational(1,3)   #=> 1
 *    Rational(1,3)   <=> 1               #=> -1
 *    Rational(1,3)   <=> 0.3             #=> 1
 */
static VALUE
nurat_cmp(VALUE self, VALUE other)
{
    switch (TYPE(other)) {
      case T_FIXNUM:
      case T_BIGNUM:
	{
	    get_dat1(self);

	    if (FIXNUM_P(dat->den) && FIX2LONG(dat->den) == 1)
		return f_cmp(dat->num, other); /* c14n */
	    return f_cmp(self, f_rational_new_bang1(CLASS_OF(self), other));
	}
      case T_FLOAT:
	return f_cmp(f_to_f(self), other);
      case T_RATIONAL:
	{
	    VALUE num1, num2;

	    get_dat2(self, other);

	    if (FIXNUM_P(adat->num) && FIXNUM_P(adat->den) &&
		FIXNUM_P(bdat->num) && FIXNUM_P(bdat->den)) {
		num1 = f_imul(FIX2LONG(adat->num), FIX2LONG(bdat->den));
		num2 = f_imul(FIX2LONG(bdat->num), FIX2LONG(adat->den));
	    }
	    else {
		num1 = f_mul(adat->num, bdat->den);
		num2 = f_mul(bdat->num, adat->den);
	    }
	    return f_cmp(f_sub(num1, num2), ZERO);
	}
      default:
	return rb_num_coerce_cmp(self, other, id_cmp);
    }
}

/*
 * call-seq:
 *    rat == object  ->  true or false
 *
 * Returns true if rat equals object numerically.
 *
 * For example:
 *
 *    Rational(2, 3)  == Rational(2, 3)   #=> true
 *    Rational(5)     == 5                #=> true
 *    Rational(0)     == 0.0              #=> true
 *    Rational('1/3') == 0.33             #=> false
 *    Rational('1/2') == '1/2'            #=> false
 */
static VALUE
nurat_eqeq_p(VALUE self, VALUE other)
{
    switch (TYPE(other)) {
      case T_FIXNUM:
      case T_BIGNUM:
	{
	    get_dat1(self);

	    if (f_zero_p(dat->num) && f_zero_p(other))
		return Qtrue;

	    if (!FIXNUM_P(dat->den))
		return Qfalse;
	    if (FIX2LONG(dat->den) != 1)
		return Qfalse;
	    if (f_eqeq_p(dat->num, other))
		return Qtrue;
	    return Qfalse;
	}
      case T_FLOAT:
	return f_eqeq_p(f_to_f(self), other);
      case T_RATIONAL:
	{
	    get_dat2(self, other);

	    if (f_zero_p(adat->num) && f_zero_p(bdat->num))
		return Qtrue;

	    return f_boolcast(f_eqeq_p(adat->num, bdat->num) &&
			      f_eqeq_p(adat->den, bdat->den));
	}
      default:
	return f_eqeq_p(other, self);
    }
}

/* :nodoc: */
static VALUE
nurat_coerce(VALUE self, VALUE other)
{
    switch (TYPE(other)) {
      case T_FIXNUM:
      case T_BIGNUM:
	return rb_assoc_new(f_rational_new_bang1(CLASS_OF(self), other), self);
      case T_FLOAT:
	return rb_assoc_new(other, f_to_f(self));
      case T_RATIONAL:
	return rb_assoc_new(other, self);
      case T_COMPLEX:
	if (k_exact_zero_p(RCOMPLEX(other)->imag))
	    return rb_assoc_new(f_rational_new_bang1
				(CLASS_OF(self), RCOMPLEX(other)->real), self);
	else
	    return rb_assoc_new(other, rb_Complex(self, INT2FIX(0)));
    }

    rb_raise(rb_eTypeError, "%s can't be coerced into %s",
	     rb_obj_classname(other), rb_obj_classname(self));
    return Qnil;
}

#if 0
/* :nodoc: */
static VALUE
nurat_idiv(VALUE self, VALUE other)
{
    return f_idiv(self, other);
}

/* :nodoc: */
static VALUE
nurat_quot(VALUE self, VALUE other)
{
    return f_truncate(f_div(self, other));
}

/* :nodoc: */
static VALUE
nurat_quotrem(VALUE self, VALUE other)
{
    VALUE val = f_truncate(f_div(self, other));
    return rb_assoc_new(val, f_sub(self, f_mul(other, val)));
}
#endif

#if 0
/* :nodoc: */
static VALUE
nurat_true(VALUE self)
{
    return Qtrue;
}
#endif

static VALUE
nurat_floor(VALUE self)
{
    get_dat1(self);
    return f_idiv(dat->num, dat->den);
}

static VALUE
nurat_ceil(VALUE self)
{
    get_dat1(self);
    return f_negate(f_idiv(f_negate(dat->num), dat->den));
}

/*
 * call-seq:
 *    rat.to_i  ->  integer
 *
 * Returns the truncated value as an integer.
 *
 * Equivalent to
 *    rat.truncate.
 *
 * For example:
 *
 *    Rational(2, 3).to_i   #=> 0
 *    Rational(3).to_i      #=> 3
 *    Rational(300.6).to_i  #=> 300
 *    Rational(98,71).to_i  #=> 1
 *    Rational(-30,2).to_i  #=> -15
 */
static VALUE
nurat_truncate(VALUE self)
{
    get_dat1(self);
    if (f_negative_p(dat->num))
	return f_negate(f_idiv(f_negate(dat->num), dat->den));
    return f_idiv(dat->num, dat->den);
}

static VALUE
nurat_round(VALUE self)
{
    VALUE num, den, neg;

    get_dat1(self);

    num = dat->num;
    den = dat->den;
    neg = f_negative_p(num);

    if (neg)
	num = f_negate(num);

    num = f_add(f_mul(num, TWO), den);
    den = f_mul(den, TWO);
    num = f_idiv(num, den);

    if (neg)
	num = f_negate(num);

    return num;
}

static VALUE
f_round_common(int argc, VALUE *argv, VALUE self, VALUE (*func)(VALUE))
{
    VALUE n, b, s;

    if (argc == 0)
	return (*func)(self);

    rb_scan_args(argc, argv, "01", &n);

    if (!k_integer_p(n))
	rb_raise(rb_eTypeError, "not an integer");

    b = f_expt10(n);
    s = f_mul(self, b);

    s = (*func)(s);

    s = f_div(f_rational_new_bang1(CLASS_OF(self), s), b);

    if (f_lt_p(n, ONE))
	s = f_to_i(s);

    return s;
}

/*
 * call-seq:
 *    rat.floor               ->  integer
 *    rat.floor(precision=0)  ->  rational
 *
 * Returns the truncated value (toward negative infinity).
 *
 * For example:
 *
 *    Rational(3).floor      #=> 3
 *    Rational(2, 3).floor   #=> 0
 *    Rational(-3, 2).floor  #=> -1
 *
 *           decimal      -  1  2  3 . 4  5  6
 *                          ^  ^  ^  ^   ^  ^
 *          precision      -3 -2 -1  0  +1 +2
 *
 *    '%f' % Rational('-123.456').floor(+1)  #=> "-123.500000"
 *    '%f' % Rational('-123.456').floor(-1)  #=> "-130.000000"
 */
static VALUE
nurat_floor_n(int argc, VALUE *argv, VALUE self)
{
    return f_round_common(argc, argv, self, nurat_floor);
}

/*
 * call-seq:
 *    rat.ceil               ->  integer
 *    rat.ceil(precision=0)  ->  rational
 *
 * Returns the truncated value (toward positive infinity).
 *
 * For example:
 *
 *    Rational(3).ceil      #=> 3
 *    Rational(2, 3).ceil   #=> 1
 *    Rational(-3, 2).ceil  #=> -1
 *
 *           decimal      -  1  2  3 . 4  5  6
 *                          ^  ^  ^  ^   ^  ^
 *          precision      -3 -2 -1  0  +1 +2
 *
 *    '%f' % Rational('-123.456').ceil(+1)  #=> "-123.400000"
 *    '%f' % Rational('-123.456').ceil(-1)  #=> "-120.000000"
 */
static VALUE
nurat_ceil_n(int argc, VALUE *argv, VALUE self)
{
    return f_round_common(argc, argv, self, nurat_ceil);
}

/*
 * call-seq:
 *    rat.truncate               ->  integer
 *    rat.truncate(precision=0)  ->  rational
 *
 * Returns the truncated value (toward zero).
 *
 * For example:
 *
 *    Rational(3).truncate      #=> 3
 *    Rational(2, 3).truncate   #=> 0
 *    Rational(-3, 2).truncate  #=> -1
 *
 *           decimal      -  1  2  3 . 4  5  6
 *                          ^  ^  ^  ^   ^  ^
 *          precision      -3 -2 -1  0  +1 +2
 *
 *    '%f' % Rational('-123.456').truncate(+1)  #=>  "-123.400000"
 *    '%f' % Rational('-123.456').truncate(-1)  #=>  "-120.000000"
 */
static VALUE
nurat_truncate_n(int argc, VALUE *argv, VALUE self)
{
    return f_round_common(argc, argv, self, nurat_truncate);
}

/*
 * call-seq:
 *    rat.round               ->  integer
 *    rat.round(precision=0)  ->  rational
 *
 * Returns the truncated value (toward the nearest integer;
 * 0.5 => 1; -0.5 => -1).
 *
 * For example:
 *
 *    Rational(3).round      #=> 3
 *    Rational(2, 3).round   #=> 1
 *    Rational(-3, 2).round  #=> -2
 *
 *           decimal      -  1  2  3 . 4  5  6
 *                          ^  ^  ^  ^   ^  ^
 *          precision      -3 -2 -1  0  +1 +2
 *
 *    '%f' % Rational('-123.456').round(+1)  #=> "-123.500000"
 *    '%f' % Rational('-123.456').round(-1)  #=> "-120.000000"
 */
static VALUE
nurat_round_n(int argc, VALUE *argv, VALUE self)
{
    return f_round_common(argc, argv, self, nurat_round);
}

/*
 * call-seq:
 *    rat.to_f  ->  float
 *
 * Return the value as a float.
 *
 * For example:
 *
 *    Rational(2).to_f      #=> 2.0
 *    Rational(9, 4).to_f   #=> 2.25
 *    Rational(-3, 4).to_f  #=> -0.75
 *    Rational(20, 3).to_f  #=> 6.666666666666667
 */
static VALUE
nurat_to_f(VALUE self)
{
    get_dat1(self);
    return f_fdiv(dat->num, dat->den);
}

/*
 * call-seq:
 *    rat.to_r  ->  self
 *
 * Returns self.
 *
 * For example:
 *
 *    Rational(2).to_r      #=> (2/1)
 *    Rational(-8, 6).to_r  #=> (-4/3)
 */
static VALUE
nurat_to_r(VALUE self)
{
    return self;
}

#define id_ceil rb_intern("ceil")
#define f_ceil(x) rb_funcall((x), id_ceil, 0)

#define id_quo rb_intern("quo")
#define f_quo(x,y) rb_funcall((x), id_quo, 1, (y))

#define f_reciprocal(x) f_quo(ONE, (x))

/*
  The algorithm here is the method described in CLISP.  Bruno Haible has
  graciously given permission to use this algorithm.  He says, "You can use
  it, if you present the following explanation of the algorithm."

  Algorithm (recursively presented):
    If x is a rational number, return x.
    If x = 0.0, return 0.
    If x < 0.0, return (- (rationalize (- x))).
    If x > 0.0:
      Call (integer-decode-float x). It returns a m,e,s=1 (mantissa,
      exponent, sign).
      If m = 0 or e >= 0: return x = m*2^e.
      Search a rational number between a = (m-1/2)*2^e and b = (m+1/2)*2^e
      with smallest possible numerator and denominator.
      Note 1: If m is a power of 2, we ought to take a = (m-1/4)*2^e.
        But in this case the result will be x itself anyway, regardless of
        the choice of a. Therefore we can simply ignore this case.
      Note 2: At first, we need to consider the closed interval [a,b].
        but since a and b have the denominator 2^(|e|+1) whereas x itself
        has a denominator <= 2^|e|, we can restrict the search to the open
        interval (a,b).
      So, for given a and b (0 < a < b) we are searching a rational number
      y with a <= y <= b.
      Recursive algorithm fraction_between(a,b):
        c := (ceiling a)
        if c < b
          then return c       ; because a <= c < b, c integer
          else
            ; a is not integer (otherwise we would have had c = a < b)
            k := c-1          ; k = floor(a), k < a < b <= k+1
            return y = k + 1/fraction_between(1/(b-k), 1/(a-k))
                              ; note 1 <= 1/(b-k) < 1/(a-k)

  You can see that we are actually computing a continued fraction expansion.

  Algorithm (iterative):
    If x is rational, return x.
    Call (integer-decode-float x). It returns a m,e,s (mantissa,
      exponent, sign).
    If m = 0 or e >= 0, return m*2^e*s. (This includes the case x = 0.0.)
    Create rational numbers a := (2*m-1)*2^(e-1) and b := (2*m+1)*2^(e-1)
    (positive and already in lowest terms because the denominator is a
    power of two and the numerator is odd).
    Start a continued fraction expansion
      p[-1] := 0, p[0] := 1, q[-1] := 1, q[0] := 0, i := 0.
    Loop
      c := (ceiling a)
      if c >= b
        then k := c-1, partial_quotient(k), (a,b) := (1/(b-k),1/(a-k)),
             goto Loop
    finally partial_quotient(c).
    Here partial_quotient(c) denotes the iteration
      i := i+1, p[i] := c*p[i-1]+p[i-2], q[i] := c*q[i-1]+q[i-2].
    At the end, return s * (p[i]/q[i]).
    This rational number is already in lowest terms because
    p[i]*q[i-1]-p[i-1]*q[i] = (-1)^i.
*/

static void
nurat_rationalize_internal(VALUE a, VALUE b, VALUE *p, VALUE *q)
{
    VALUE c, k, t, p0, p1, p2, q0, q1, q2;

    p0 = ZERO;
    p1 = ONE;
    q0 = ONE;
    q1 = ZERO;

    while (1) {
	c = f_ceil(a);
	if (f_lt_p(c, b))
	    break;
	k = f_sub(c, ONE);
	p2 = f_add(f_mul(k, p1), p0);
	q2 = f_add(f_mul(k, q1), q0);
	t = f_reciprocal(f_sub(b, k));
	b = f_reciprocal(f_sub(a, k));
	a = t;
	p0 = p1;
	q0 = q1;
	p1 = p2;
	q1 = q2;
    }
    *p = f_add(f_mul(c, p1), p0);
    *q = f_add(f_mul(c, q1), q0);
}

/*
 * call-seq:
 *    rat.rationalize       ->  self
 *    rat.rationalize(eps)  ->  rational
 *
 * Returns a simpler approximation of the value if an optional
 * argument eps is given (rat-|eps| <= result <= rat+|eps|), self
 * otherwise.
 *
 * For example:
 *
 *    r = Rational(5033165, 16777216)
 *    r.rationalize                    #=> (5033165/16777216)
 *    r.rationalize(Rational('0.01'))  #=> (3/10)
 *    r.rationalize(Rational('0.1'))   #=> (1/3)
 */
static VALUE
nurat_rationalize(int argc, VALUE *argv, VALUE self)
{
    VALUE e, a, b, p, q;

    if (argc == 0)
	return self;

    if (f_negative_p(self))
	return f_negate(nurat_rationalize(argc, argv, f_abs(self)));

    rb_scan_args(argc, argv, "01", &e);
    e = f_abs(e);
    a = f_sub(self, e);
    b = f_add(self, e);

    if (f_eqeq_p(a, b))
	return self;

    nurat_rationalize_internal(a, b, &p, &q);
    return f_rational_new2(CLASS_OF(self), p, q);
}

/* :nodoc: */
static VALUE
nurat_hash(VALUE self)
{
    st_index_t v, h[2];
    VALUE n;

    get_dat1(self);
    n = rb_hash(dat->num);
    h[0] = NUM2LONG(n);
    n = rb_hash(dat->den);
    h[1] = NUM2LONG(n);
    v = rb_memhash(h, sizeof(h));
    return LONG2FIX(v);
}

static VALUE
f_format(VALUE self, VALUE (*func)(VALUE))
{
    VALUE s;
    get_dat1(self);

    s = (*func)(dat->num);
    rb_str_cat2(s, "/");
    rb_str_concat(s, (*func)(dat->den));

    return s;
}

/*
 * call-seq:
 *    rat.to_s  ->  string
 *
 * Returns the value as a string.
 *
 * For example:
 *
 *    Rational(2).to_s      #=> "2/1"
 *    Rational(-8, 6).to_s  #=> "-4/3"
 *    Rational('0.5').to_s  #=> "1/2"
 */
static VALUE
nurat_to_s(VALUE self)
{
    return f_format(self, f_to_s);
}

/*
 * call-seq:
 *    rat.inspect  ->  string
 *
 * Returns the value as a string for inspection.
 *
 * For example:
 *
 *    Rational(2).inspect      #=> "(2/1)"
 *    Rational(-8, 6).inspect  #=> "(-4/3)"
 *    Rational('0.5').inspect  #=> "(1/2)"
 */
static VALUE
nurat_inspect(VALUE self)
{
    VALUE s;

    s = rb_usascii_str_new2("(");
    rb_str_concat(s, f_format(self, f_inspect));
    rb_str_cat2(s, ")");

    return s;
}

/* :nodoc: */
static VALUE
nurat_marshal_dump(VALUE self)
{
    VALUE a;
    get_dat1(self);

    a = rb_assoc_new(dat->num, dat->den);
    rb_copy_generic_ivar(a, self);
    return a;
}

/* :nodoc: */
static VALUE
nurat_marshal_load(VALUE self, VALUE a)
{
    get_dat1(self);
    Check_Type(a, T_ARRAY);
    dat->num = RARRAY_PTR(a)[0];
    dat->den = RARRAY_PTR(a)[1];
    rb_copy_generic_ivar(self, a);

    if (f_zero_p(dat->den))
	rb_raise_zerodiv();

    return self;
}

/* --- */

VALUE
rb_rational_reciprocal(VALUE x)
{
    get_dat1(x);
    return f_rational_new_no_reduce2(CLASS_OF(x), dat->den, dat->num);
}

/*
 * call-seq:
 *    int.gcd(int2)  ->  integer
 *
 * Returns the greatest common divisor (always positive).  0.gcd(x)
 * and x.gcd(0) return abs(x).
 *
 * For example:
 *
 *    2.gcd(2)                    #=> 2
 *    3.gcd(-7)                   #=> 1
 *    ((1<<31)-1).gcd((1<<61)-1)  #=> 1
 */
VALUE
rb_gcd(VALUE self, VALUE other)
{
    other = nurat_int_value(other);
    return f_gcd(self, other);
}

/*
 * call-seq:
 *    int.lcm(int2)  ->  integer
 *
 * Returns the least common multiple (always positive).  0.lcm(x) and
 * x.lcm(0) return zero.
 *
 * For example:
 *
 *    2.lcm(2)                    #=> 2
 *    3.lcm(-7)                   #=> 21
 *    ((1<<31)-1).lcm((1<<61)-1)  #=> 4951760154835678088235319297
 */
VALUE
rb_lcm(VALUE self, VALUE other)
{
    other = nurat_int_value(other);
    return f_lcm(self, other);
}

/*
 * call-seq:
 *    int.gcdlcm(int2)  ->  array
 *
 * Returns an array; [int.gcd(int2), int.lcm(int2)].
 *
 * For example:
 *
 *    2.gcdlcm(2)                    #=> [2, 2]
 *    3.gcdlcm(-7)                   #=> [1, 21]
 *    ((1<<31)-1).gcdlcm((1<<61)-1)  #=> [1, 4951760154835678088235319297]
 */
VALUE
rb_gcdlcm(VALUE self, VALUE other)
{
    other = nurat_int_value(other);
    return rb_assoc_new(f_gcd(self, other), f_lcm(self, other));
}

VALUE
rb_rational_raw(VALUE x, VALUE y)
{
    return nurat_s_new_internal(rb_cRational, x, y);
}

VALUE
rb_rational_new(VALUE x, VALUE y)
{
    return nurat_s_canonicalize_internal(rb_cRational, x, y);
}

static VALUE nurat_s_convert(int argc, VALUE *argv, VALUE klass);

VALUE
rb_Rational(VALUE x, VALUE y)
{
    VALUE a[2];
    a[0] = x;
    a[1] = y;
    return nurat_s_convert(2, a, rb_cRational);
}

#define id_numerator rb_intern("numerator")
#define f_numerator(x) rb_funcall((x), id_numerator, 0)

#define id_denominator rb_intern("denominator")
#define f_denominator(x) rb_funcall((x), id_denominator, 0)

#define id_to_r rb_intern("to_r")
#define f_to_r(x) rb_funcall((x), id_to_r, 0)

/*
 * call-seq:
 *    num.numerator  ->  integer
 *
 * Returns the numerator.
 */
static VALUE
numeric_numerator(VALUE self)
{
    return f_numerator(f_to_r(self));
}

/*
 * call-seq:
 *    num.denominator  ->  integer
 *
 * Returns the denominator (always positive).
 */
static VALUE
numeric_denominator(VALUE self)
{
    return f_denominator(f_to_r(self));
}

/*
 * call-seq:
 *    int.numerator  ->  self
 *
 * Returns self.
 */
static VALUE
integer_numerator(VALUE self)
{
    return self;
}

/*
 * call-seq:
 *    int.denominator  ->  1
 *
 * Returns 1.
 */
static VALUE
integer_denominator(VALUE self)
{
    return INT2FIX(1);
}

/*
 * call-seq:
 *    flo.numerator  ->  integer
 *
 * Returns the numerator.  The result is machine dependent.
 *
 * For example:
 *
 *    n = 0.3.numerator    #=> 5404319552844595
 *    d = 0.3.denominator  #=> 18014398509481984
 *    n.fdiv(d)            #=> 0.3
 */
static VALUE
float_numerator(VALUE self)
{
    double d = RFLOAT_VALUE(self);
    if (isinf(d) || isnan(d))
	return self;
    return rb_call_super(0, 0);
}

/*
 * call-seq:
 *    flo.denominator  ->  integer
 *
 * Returns the denominator (always positive).  The result is machine
 * dependent.
 *
 * See numerator.
 */
static VALUE
float_denominator(VALUE self)
{
    double d = RFLOAT_VALUE(self);
    if (isinf(d) || isnan(d))
	return INT2FIX(1);
    return rb_call_super(0, 0);
}

/*
 * call-seq:
 *    nil.to_r  ->  (0/1)
 *
 * Returns zero as a rational.
 */
static VALUE
nilclass_to_r(VALUE self)
{
    return rb_rational_new1(INT2FIX(0));
}

/*
 * call-seq:
 *    nil.rationalize([eps])  ->  (0/1)
 *
 * Returns zero as a rational.  An optional argument eps is always
 * ignored.
 */
static VALUE
nilclass_rationalize(int argc, VALUE *argv, VALUE self)
{
    rb_scan_args(argc, argv, "01", NULL);
    return nilclass_to_r(self);
}

/*
 * call-seq:
 *    int.to_r  ->  rational
 *
 * Returns the value as a rational.
 *
 * For example:
 *
 *    1.to_r        #=> (1/1)
 *    (1<<64).to_r  #=> (18446744073709551616/1)
 */
static VALUE
integer_to_r(VALUE self)
{
    return rb_rational_new1(self);
}

/*
 * call-seq:
 *    int.rationalize([eps])  ->  rational
 *
 * Returns the value as a rational.  An optional argument eps is
 * always ignored.
 */
static VALUE
integer_rationalize(int argc, VALUE *argv, VALUE self)
{
    rb_scan_args(argc, argv, "01", NULL);
    return integer_to_r(self);
}

static void
float_decode_internal(VALUE self, VALUE *rf, VALUE *rn)
{
    double f;
    int n;

    f = frexp(RFLOAT_VALUE(self), &n);
    f = ldexp(f, DBL_MANT_DIG);
    n -= DBL_MANT_DIG;
    *rf = rb_dbl2big(f);
    *rn = INT2FIX(n);
}

#if 0
static VALUE
float_decode(VALUE self)
{
    VALUE f, n;

    float_decode_internal(self, &f, &n);
    return rb_assoc_new(f, n);
}
#endif

#define id_lshift rb_intern("<<")
#define f_lshift(x,n) rb_funcall((x), id_lshift, 1, (n))

/*
 * call-seq:
 *    flt.to_r  ->  rational
 *
 * Returns the value as a rational.
 *
 * NOTE: 0.3.to_r isn't the same as '0.3'.to_r.  The latter is
 * equivalent to '3/10'.to_r, but the former isn't so.
 *
 * For example:
 *
 *    2.0.to_r    #=> (2/1)
 *    2.5.to_r    #=> (5/2)
 *    -0.75.to_r  #=> (-3/4)
 *    0.0.to_r    #=> (0/1)
 */
static VALUE
float_to_r(VALUE self)
{
    VALUE f, n;

    float_decode_internal(self, &f, &n);
#if FLT_RADIX == 2
    {
	long ln = FIX2LONG(n);

	if (ln == 0)
	    return f_to_r(f);
	if (ln > 0)
	    return f_to_r(f_lshift(f, n));
	ln = -ln;
	return rb_rational_new2(f, f_lshift(ONE, INT2FIX(ln)));
    }
#else
    return f_to_r(f_mul(f, f_expt(INT2FIX(FLT_RADIX), n)));
#endif
}

/*
 * call-seq:
 *    flt.rationalize([eps])  ->  rational
 *
 * Returns a simpler approximation of the value (flt-|eps| <= result
 * <= flt+|eps|).  if eps is not given, it will be chosen
 * automatically.
 *
 * For example:
 *
 *    0.3.rationalize          #=> (3/10)
 *    1.333.rationalize        #=> (1333/1000)
 *    1.333.rationalize(0.01)  #=> (4/3)
 */
static VALUE
float_rationalize(int argc, VALUE *argv, VALUE self)
{
    VALUE e, a, b, p, q;

    if (f_negative_p(self))
	return f_negate(float_rationalize(argc, argv, f_abs(self)));

    rb_scan_args(argc, argv, "01", &e);

    if (argc != 0) {
	e = f_abs(e);
	a = f_sub(self, e);
	b = f_add(self, e);
    }
    else {
	VALUE f, n;

	float_decode_internal(self, &f, &n);
	if (f_zero_p(f) || f_positive_p(n))
	    return rb_rational_new1(f_lshift(f, n));

#if FLT_RADIX == 2
	a = rb_rational_new2(f_sub(f_mul(TWO, f), ONE),
			     f_lshift(ONE, f_sub(ONE, n)));
	b = rb_rational_new2(f_add(f_mul(TWO, f), ONE),
			     f_lshift(ONE, f_sub(ONE, n)));
#else
	a = rb_rational_new2(f_sub(f_mul(INT2FIX(FLT_RADIX), f),
				   INT2FIX(FLT_RADIX - 1)),
			     f_expt(INT2FIX(FLT_RADIX), f_sub(ONE, n)));
	b = rb_rational_new2(f_add(f_mul(INT2FIX(FLT_RADIX), f),
				   INT2FIX(FLT_RADIX - 1)),
			     f_expt(INT2FIX(FLT_RADIX), f_sub(ONE, n)));
#endif
    }

    if (f_eqeq_p(a, b))
	return f_to_r(self);

    nurat_rationalize_internal(a, b, &p, &q);
    return rb_rational_new2(p, q);
}

static VALUE rat_pat, an_e_pat, a_dot_pat, underscores_pat, an_underscore;

#define WS "\\s*"
#define DIGITS "(?:[0-9](?:_[0-9]|[0-9])*)"
#define NUMERATOR "(?:" DIGITS "?\\.)?" DIGITS "(?:[eE][-+]?" DIGITS ")?"
#define DENOMINATOR DIGITS
#define PATTERN "\\A" WS "([-+])?(" NUMERATOR ")(?:\\/(" DENOMINATOR "))?" WS

static void
make_patterns(void)
{
    static const char rat_pat_source[] = PATTERN;
    static const char an_e_pat_source[] = "[eE]";
    static const char a_dot_pat_source[] = "\\.";
    static const char underscores_pat_source[] = "_+";

    if (rat_pat) return;

    rat_pat = rb_reg_new(rat_pat_source, sizeof rat_pat_source - 1, 0);
    rb_gc_register_mark_object(rat_pat);

    an_e_pat = rb_reg_new(an_e_pat_source, sizeof an_e_pat_source - 1, 0);
    rb_gc_register_mark_object(an_e_pat);

    a_dot_pat = rb_reg_new(a_dot_pat_source, sizeof a_dot_pat_source - 1, 0);
    rb_gc_register_mark_object(a_dot_pat);

    underscores_pat = rb_reg_new(underscores_pat_source,
				 sizeof underscores_pat_source - 1, 0);
    rb_gc_register_mark_object(underscores_pat);

    an_underscore = rb_usascii_str_new2("_");
    rb_gc_register_mark_object(an_underscore);
}

#define id_match rb_intern("match")
#define f_match(x,y) rb_funcall((x), id_match, 1, (y))

#define id_split rb_intern("split")
#define f_split(x,y) rb_funcall((x), id_split, 1, (y))

#include <ctype.h>

static VALUE
string_to_r_internal(VALUE self)
{
    VALUE s, m;

    s = self;

    if (RSTRING_LEN(s) == 0)
	return rb_assoc_new(Qnil, self);

    m = f_match(rat_pat, s);

    if (!NIL_P(m)) {
	VALUE v, ifp, exp, ip, fp;
	VALUE si = rb_reg_nth_match(1, m);
	VALUE nu = rb_reg_nth_match(2, m);
	VALUE de = rb_reg_nth_match(3, m);
	VALUE re = rb_reg_match_post(m);

	{
	    VALUE a;

	    if (!strpbrk(RSTRING_PTR(nu), "eE")) {
		ifp = nu; /* not a copy */
		exp = Qnil;
	    }
	    else {
		a = f_split(nu, an_e_pat);
		ifp = RARRAY_PTR(a)[0];
		if (RARRAY_LEN(a) != 2)
		    exp = Qnil;
		else
		    exp = RARRAY_PTR(a)[1];
	    }

	    if (!strchr(RSTRING_PTR(ifp), '.')) {
		ip = ifp; /* not a copy */
		fp = Qnil;
	    }
	    else {
		a = f_split(ifp, a_dot_pat);
		ip = RARRAY_PTR(a)[0];
		if (RARRAY_LEN(a) != 2)
		    fp = Qnil;
		else
		    fp = RARRAY_PTR(a)[1];
	    }
	}

	v = rb_rational_new1(f_to_i(ip));

	if (!NIL_P(fp)) {
	    char *p = RSTRING_PTR(fp);
	    long count = 0;
	    VALUE l;

	    while (*p) {
		if (rb_isdigit(*p))
		    count++;
		p++;
	    }
	    l = f_expt10(LONG2NUM(count));
	    v = f_mul(v, l);
	    v = f_add(v, f_to_i(fp));
	    v = f_div(v, l);
	}
	if (!NIL_P(si) && *RSTRING_PTR(si) == '-')
	    v = f_negate(v);
	if (!NIL_P(exp))
	    v = f_mul(v, f_expt10(f_to_i(exp)));
#if 0
	if (!NIL_P(de) && (!NIL_P(fp) || !NIL_P(exp)))
	    return rb_assoc_new(v, rb_usascii_str_new2("dummy"));
#endif
	if (!NIL_P(de))
	    v = f_div(v, f_to_i(de));

	return rb_assoc_new(v, re);
    }
    return rb_assoc_new(Qnil, self);
}

static VALUE
string_to_r_strict(VALUE self)
{
    VALUE a = string_to_r_internal(self);
    if (NIL_P(RARRAY_PTR(a)[0]) || RSTRING_LEN(RARRAY_PTR(a)[1]) > 0) {
	VALUE s = f_inspect(self);
	rb_raise(rb_eArgError, "invalid value for convert(): %s",
		 StringValuePtr(s));
    }
    return RARRAY_PTR(a)[0];
}

#define id_gsub rb_intern("gsub")
#define f_gsub(x,y,z) rb_funcall((x), id_gsub, 2, (y), (z))

/*
 * call-seq:
 *    str.to_r  ->  rational
 *
 * Returns a rational which denotes the string form.  The parser
 * ignores leading whitespaces and trailing garbage.  Any digit
 * sequences can be separated by an underscore.  Returns zero for null
 * or garbage string.
 *
 * NOTE: '0.3'.to_r isn't the same as 0.3.to_r.  The former is
 * equivalent to '3/10'.to_r, but the latter isn't so.
 *
 * For example:
 *
 *    '  2  '.to_r       #=> (2/1)
 *    '300/2'.to_r       #=> (150/1)
 *    '-9.2'.to_r        #=> (-46/5)
 *    '-9.2e2'.to_r      #=> (-920/1)
 *    '1_234_567'.to_r   #=> (1234567/1)
 *    '21 june 09'.to_r  #=> (21/1)
 *    '21/06/09'.to_r    #=> (7/2)
 *    'bwv 1079'.to_r    #=> (0/1)
 */
static VALUE
string_to_r(VALUE self)
{
    VALUE s, a, a1, backref;

    backref = rb_backref_get();
    rb_match_busy(backref);

    s = f_gsub(self, underscores_pat, an_underscore);
    a = string_to_r_internal(s);

    rb_backref_set(backref);

    a1 = RARRAY_PTR(a)[0];
    if (!NIL_P(a1)) {
	if (TYPE(a1) == T_FLOAT)
	    rb_raise(rb_eFloatDomainError, "Infinity");
	return a1;
    }
    return rb_rational_new1(INT2FIX(0));
}

#define id_to_r rb_intern("to_r")
#define f_to_r(x) rb_funcall((x), id_to_r, 0)

static VALUE
nurat_s_convert(int argc, VALUE *argv, VALUE klass)
{
    VALUE a1, a2, backref;

    rb_scan_args(argc, argv, "11", &a1, &a2);

    if (NIL_P(a1) || (argc == 2 && NIL_P(a2)))
	rb_raise(rb_eTypeError, "can't convert nil into Rational");

    switch (TYPE(a1)) {
      case T_COMPLEX:
	if (k_exact_zero_p(RCOMPLEX(a1)->imag))
	    a1 = RCOMPLEX(a1)->real;
    }

    switch (TYPE(a2)) {
      case T_COMPLEX:
	if (k_exact_zero_p(RCOMPLEX(a2)->imag))
	    a2 = RCOMPLEX(a2)->real;
    }

    backref = rb_backref_get();
    rb_match_busy(backref);

    switch (TYPE(a1)) {
      case T_FIXNUM:
      case T_BIGNUM:
	break;
      case T_FLOAT:
	a1 = f_to_r(a1);
	break;
      case T_STRING:
	a1 = string_to_r_strict(a1);
	break;
    }

    switch (TYPE(a2)) {
      case T_FIXNUM:
      case T_BIGNUM:
	break;
      case T_FLOAT:
	a2 = f_to_r(a2);
	break;
      case T_STRING:
	a2 = string_to_r_strict(a2);
	break;
    }

    rb_backref_set(backref);

    switch (TYPE(a1)) {
      case T_RATIONAL:
	if (argc == 1 || (k_exact_one_p(a2)))
	    return a1;
    }

    if (argc == 1) {
	if (!(k_numeric_p(a1) && k_integer_p(a1)))
	    return rb_convert_type(a1, T_RATIONAL, "Rational", "to_r");
    }
    else {
	if ((k_numeric_p(a1) && k_numeric_p(a2)) &&
	    (!f_integer_p(a1) || !f_integer_p(a2)))
	    return f_div(a1, a2);
    }

    {
	VALUE argv2[2];
	argv2[0] = a1;
	argv2[1] = a2;
	return nurat_s_new(argc, argv2, klass);
    }
}

/*
 * A rational number can be represented as a paired integer number;
 * a/b (b>0).  Where a is numerator and b is denominator.  Integer a
 * equals rational a/1 mathematically.
 *
 * In ruby, you can create rational object with Rational, to_r or
 * rationalize method.  The return values will be irreducible.
 *
 *    Rational(1)      #=> (1/1)
 *    Rational(2, 3)   #=> (2/3)
 *    Rational(4, -6)  #=> (-2/3)
 *    3.to_r           #=> (3/1)
 *
 * You can also create rational object from floating-point numbers or
 * strings.
 *
 *    Rational(0.3)    #=> (5404319552844595/18014398509481984)
 *    Rational('0.3')  #=> (3/10)
 *    Rational('2/3')  #=> (2/3)
 *
 *    0.3.to_r         #=> (5404319552844595/18014398509481984)
 *    '0.3'.to_r       #=> (3/10)
 *    '2/3'.to_r       #=> (2/3)
 *    0.3.rationalize  #=> (3/10)
 *
 * A rational object is an exact number, which helps you to write
 * program without any rounding errors.
 *
 *    10.times.inject(0){|t,| t + 0.1}              #=> 0.9999999999999999
 *    10.times.inject(0){|t,| t + Rational('0.1')}  #=> (1/1)
 *
 * However, when an expression has inexact factor (numerical value or
 * operation), will produce an inexact result.
 *
 *    Rational(10) / 3   #=> (10/3)
 *    Rational(10) / 3.0 #=> 3.3333333333333335
 *
 *    Rational(-8) ** Rational(1, 3)
 *                       #=> (1.0000000000000002+1.7320508075688772i)
 */
void
Init_Rational(void)
{
#undef rb_intern
#define rb_intern(str) rb_intern_const(str)

    assert(fprintf(stderr, "assert() is now active\n"));

    id_abs = rb_intern("abs");
    id_cmp = rb_intern("<=>");
    id_convert = rb_intern("convert");
    id_eqeq_p = rb_intern("==");
    id_expt = rb_intern("**");
    id_fdiv = rb_intern("fdiv");
    id_floor = rb_intern("floor");
    id_idiv = rb_intern("div");
    id_inspect = rb_intern("inspect");
    id_integer_p = rb_intern("integer?");
    id_negate = rb_intern("-@");
    id_to_f = rb_intern("to_f");
    id_to_i = rb_intern("to_i");
    id_to_s = rb_intern("to_s");
    id_truncate = rb_intern("truncate");

    rb_cRational = rb_define_class("Rational", rb_cNumeric);

    rb_define_alloc_func(rb_cRational, nurat_s_alloc);
    rb_undef_method(CLASS_OF(rb_cRational), "allocate");

#if 0
    rb_define_private_method(CLASS_OF(rb_cRational), "new!", nurat_s_new_bang, -1);
    rb_define_private_method(CLASS_OF(rb_cRational), "new", nurat_s_new, -1);
#else
    rb_undef_method(CLASS_OF(rb_cRational), "new");
#endif

    rb_define_global_function("Rational", nurat_f_rational, -1);

    rb_define_method(rb_cRational, "numerator", nurat_numerator, 0);
    rb_define_method(rb_cRational, "denominator", nurat_denominator, 0);

    rb_define_method(rb_cRational, "+", nurat_add, 1);
    rb_define_method(rb_cRational, "-", nurat_sub, 1);
    rb_define_method(rb_cRational, "*", nurat_mul, 1);
    rb_define_method(rb_cRational, "/", nurat_div, 1);
    rb_define_method(rb_cRational, "quo", nurat_div, 1);
    rb_define_method(rb_cRational, "fdiv", nurat_fdiv, 1);
    rb_define_method(rb_cRational, "**", nurat_expt, 1);

    rb_define_method(rb_cRational, "<=>", nurat_cmp, 1);
    rb_define_method(rb_cRational, "==", nurat_eqeq_p, 1);
    rb_define_method(rb_cRational, "coerce", nurat_coerce, 1);

#if 0 /* NUBY */
    rb_define_method(rb_cRational, "//", nurat_idiv, 1);
#endif

#if 0
    rb_define_method(rb_cRational, "quot", nurat_quot, 1);
    rb_define_method(rb_cRational, "quotrem", nurat_quotrem, 1);
#endif

#if 0
    rb_define_method(rb_cRational, "rational?", nurat_true, 0);
    rb_define_method(rb_cRational, "exact?", nurat_true, 0);
#endif

    rb_define_method(rb_cRational, "floor", nurat_floor_n, -1);
    rb_define_method(rb_cRational, "ceil", nurat_ceil_n, -1);
    rb_define_method(rb_cRational, "truncate", nurat_truncate_n, -1);
    rb_define_method(rb_cRational, "round", nurat_round_n, -1);

    rb_define_method(rb_cRational, "to_i", nurat_truncate, 0);
    rb_define_method(rb_cRational, "to_f", nurat_to_f, 0);
    rb_define_method(rb_cRational, "to_r", nurat_to_r, 0);
    rb_define_method(rb_cRational, "rationalize", nurat_rationalize, -1);

    rb_define_method(rb_cRational, "hash", nurat_hash, 0);

    rb_define_method(rb_cRational, "to_s", nurat_to_s, 0);
    rb_define_method(rb_cRational, "inspect", nurat_inspect, 0);

    rb_define_method(rb_cRational, "marshal_dump", nurat_marshal_dump, 0);
    rb_define_method(rb_cRational, "marshal_load", nurat_marshal_load, 1);

    /* --- */

    rb_define_method(rb_cInteger, "gcd", rb_gcd, 1);
    rb_define_method(rb_cInteger, "lcm", rb_lcm, 1);
    rb_define_method(rb_cInteger, "gcdlcm", rb_gcdlcm, 1);

    rb_define_method(rb_cNumeric, "numerator", numeric_numerator, 0);
    rb_define_method(rb_cNumeric, "denominator", numeric_denominator, 0);

    rb_define_method(rb_cInteger, "numerator", integer_numerator, 0);
    rb_define_method(rb_cInteger, "denominator", integer_denominator, 0);

    rb_define_method(rb_cFloat, "numerator", float_numerator, 0);
    rb_define_method(rb_cFloat, "denominator", float_denominator, 0);

    rb_define_method(rb_cNilClass, "to_r", nilclass_to_r, 0);
    rb_define_method(rb_cNilClass, "rationalize", nilclass_rationalize, -1);
    rb_define_method(rb_cInteger, "to_r", integer_to_r, 0);
    rb_define_method(rb_cInteger, "rationalize", integer_rationalize, -1);
    rb_define_method(rb_cFloat, "to_r", float_to_r, 0);
    rb_define_method(rb_cFloat, "rationalize", float_rationalize, -1);

    make_patterns();

    rb_define_method(rb_cString, "to_r", string_to_r, 0);

    rb_define_private_method(CLASS_OF(rb_cRational), "convert", nurat_s_convert, -1);
}

/*
Local variables:
c-file-style: "ruby"
End:
*/