both complex and rational are now builtin classes.

git-svn-id: svn+ssh://ci.ruby-lang.org/ruby/trunk@15783 b2dd03c8-39d4-4d8f-98ff-823fe69b080e

1 files changed, 1533 insertions, 0 deletions

diff --git a/complex.c b/complex.c
new file mode 100644
index 0000000000..9982db0c90
--- /dev/null
+++ b/complex.c
@@ -0,0 +1,1533 @@
+/*
+  nucomp_core.c: Coded by Tadayoshi Funaba 2008
+
+  This implementation is based on Keiju Ishitsuka's Complex library
+  which is written in ruby.
+*/
+
+#include "ruby.h"
+#include <math.h>
+
+#define NDEBUG
+#include <assert.h>
+
+#ifndef COMPLEX_NAME
+#define COMPLEX_NAME "Complex"
+#endif
+
+#define ZERO INT2FIX(0)
+#define ONE INT2FIX(1)
+#define TWO INT2FIX(2)
+
+VALUE rb_cComplex;
+
+static ID id_Unify, id_abs, id_abs2, id_arg, id_atan2_bang, id_cmp,
+  id_coerce, id_conjugate, id_convert, id_cos, id_denominator, id_divmod,
+  id_equal_p, id_exact_p, id_exp_bang, id_expt, id_floor, id_format,
+  id_hypot, id_idiv, id_inspect, id_log_bang, id_negate, id_new, id_new_bang,
+  id_numerator, id_polar, id_quo, id_scalar_p, id_sin, id_sqrt, id_to_f,
+  id_to_i, id_to_r, id_to_s, id_truncate;
+
+#define f_add(x,y) rb_funcall(x, '+', 1, y)
+#define f_div(x,y) rb_funcall(x, '/', 1, y)
+#define f_gt_p(x,y) rb_funcall(x, '>', 1, y)
+#define f_lt_p(x,y) rb_funcall(x, '<', 1, y)
+#define f_mod(x,y) rb_funcall(x, '%', 1, y)
+#define f_mul(x,y) rb_funcall(x, '*', 1, y)
+#define f_sub(x,y) rb_funcall(x, '-', 1, y)
+#define f_xor(x,y) rb_funcall(x, '^', 1, y)
+
+#define f_abs(x) rb_funcall(x, id_abs, 0)
+#define f_abs2(x) rb_funcall(x, id_abs2, 0)
+#define f_arg(x) rb_funcall(x, id_arg, 0)
+#define f_conjugate(x) rb_funcall(x, id_conjugate, 0)
+#define f_denominator(x) rb_funcall(x, id_denominator, 0)
+#define f_exact_p(x) rb_funcall(x, id_exact_p, 0)
+#define f_floor(x) rb_funcall(x, id_floor, 0)
+#define f_negate(x) rb_funcall(x, id_negate, 0)
+#define f_numerator(x) rb_funcall(x, id_numerator, 0)
+#define f_polar(x) rb_funcall(x, id_polar, 0)
+#define f_scalar_p(x) rb_funcall(x, id_scalar_p, 0)
+#define f_to_f(x) rb_funcall(x, id_to_f, 0)
+#define f_to_i(x) rb_funcall(x, id_to_i, 0)
+#define f_to_r(x) rb_funcall(x, id_to_r, 0)
+#define f_to_s(x) rb_funcall(x, id_to_s, 0)
+#define f_truncate(x) rb_funcall(x, id_truncate, 0)
+#define f_cmp(x,y) rb_funcall(x, id_cmp, 1, y)
+#define f_coerce(x,y) rb_funcall(x, id_coerce, 1, y)
+#define f_divmod(x,y) rb_funcall(x, id_divmod, 1, y)
+#define f_equal_p(x,y) rb_funcall(x, id_equal_p, 1, y)
+#define f_expt(x,y) rb_funcall(x, id_expt, 1, y)
+#define f_idiv(x,y) rb_funcall(x, id_idiv, 1, y)
+#define f_inspect(x) rb_funcall(x, id_inspect, 0)
+#define f_quo(x,y) rb_funcall(x, id_quo, 1, y)
+
+#if 0
+#define m_cos(x) rb_funcall(rb_mMath, id_cos, 1, x)
+#define m_exp_bang(x) rb_funcall(rb_mMath, id_exp_bang, 1, x)
+#define m_log_bang(x) rb_funcall(rb_mMath, id_log_bang, 1, x)
+#define m_sin(x) rb_funcall(rb_mMath, id_sin, 1, x)
+#define m_sqrt(x) rb_funcall(rb_mMath, id_sqrt, 1, x)
+#define m_atan2_bang(x,y) rb_funcall(rb_mMath, id_atan2_bang, 2, x, y)
+#define m_hypot(x,y) rb_funcall(rb_mMath, id_hypot, 2, x, y)
+#endif
+
+#define f_negative_p(x) f_lt_p(x, ZERO)
+#define f_zero_p(x) f_equal_p(x, ZERO)
+#define f_one_p(x) f_equal_p(x, ONE)
+#define f_kind_of_p(x,c) rb_obj_is_kind_of(x, c)
+#define k_numeric_p(x) f_kind_of_p(x, rb_cNumeric)
+#define k_integer_p(x) f_kind_of_p(x, rb_cInteger)
+#define k_float_p(x) f_kind_of_p(x, rb_cFloat)
+#define k_rational_p(x) f_kind_of_p(x, rb_cRational)
+#define k_complex_p(x) f_kind_of_p(x, rb_cComplex)
+
+#define f_boolcast(x) ((x) ? Qtrue : Qfalse)
+
+inline static VALUE
+f_generic_p(VALUE x)
+{
+  switch (TYPE(x)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+  case T_RATIONAL:
+    return Qtrue;
+  default:
+    return Qfalse;
+  }
+}
+
+static VALUE
+nucomp_s_generic_p(VALUE klass, VALUE x)
+{
+  return f_generic_p(x);
+}
+
+#define get_dat1(x) \
+  struct RComplex *dat;\
+  dat = ((struct RComplex *)(x))
+
+#define get_dat2(x,y) \
+  struct RComplex *adat, *bdat;\
+  adat = ((struct RComplex *)(x));\
+  bdat = ((struct RComplex *)(y))
+
+inline static VALUE
+nucomp_s_new_internal(VALUE klass, VALUE real, VALUE image)
+{
+  NEWOBJ(obj, struct RComplex);
+  OBJSETUP(obj, klass, T_COMPLEX);
+
+  obj->real = real;
+  obj->image = image;
+
+  return (VALUE)obj;
+}
+
+static VALUE
+nucomp_s_alloc(VALUE klass)
+{
+  return nucomp_s_new_internal(klass, ZERO, ZERO);
+}
+
+static VALUE
+nucomp_s_new_bang(int argc, VALUE *argv, VALUE klass)
+{
+  VALUE real, image;
+
+  switch (rb_scan_args(argc, argv, "11", &real, &image)) {
+  case 1:
+    if (!k_numeric_p(real))
+      real = f_to_i(real);
+    image = ZERO;
+    break;
+  default:
+    if (!k_numeric_p(real))
+      real = f_to_i(real);
+    if (!k_numeric_p(image))
+      image = f_to_i(image);
+    break;
+  }
+
+  return nucomp_s_new_internal(klass, real, image);
+}
+
+inline static VALUE
+f_complex_new_bang1(VALUE klass, VALUE x)
+{
+  return nucomp_s_new_internal(klass, x, ZERO);
+}
+
+inline static VALUE
+f_complex_new_bang2(VALUE klass, VALUE x, VALUE y)
+{
+  return nucomp_s_new_internal(klass, x, y);
+}
+
+#define f_unify_p(klass) rb_const_defined(klass, id_Unify)
+
+inline static VALUE
+nucomp_s_canonicalize_internal(VALUE klass, VALUE real, VALUE image)
+{
+#define CL_CANON
+#ifdef CL_CANON
+  if (f_zero_p(image) && f_unify_p(klass) &&
+      !k_float_p(real) && !k_float_p(image))
+    return real;
+#else
+  if (f_zero_p(image) && f_unify_p(klass))
+    return real;
+#endif
+  else if (f_scalar_p(real) && f_scalar_p(image))
+    return nucomp_s_new_internal(klass, real, image);
+  else if (f_scalar_p(real)) {
+    get_dat1(image);
+
+    return nucomp_s_new_internal(klass,
+				 f_sub(real, dat->image),
+				 f_add(ZERO, dat->real));
+  } else if (f_scalar_p(image)) {
+    get_dat1(real);
+
+    return nucomp_s_new_internal(klass,
+				 dat->real,
+				 f_add(dat->image, image));
+  } else {
+    get_dat2(real, image);
+
+    return nucomp_s_new_internal(klass,
+				 f_sub(adat->real, bdat->image),
+				 f_add(adat->image, bdat->real));
+  }
+}
+
+static VALUE
+nucomp_s_canonicalize(int argc, VALUE *argv, VALUE klass)
+{
+  VALUE real, image;
+
+  switch (rb_scan_args(argc, argv, "11", &real, &image)) {
+  case 1:
+    image = ZERO;
+    break;
+  }
+
+  switch (TYPE(real)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+    break;
+  default:
+    if (!k_rational_p(real))
+      rb_raise(rb_eArgError, "not a real");
+  }
+
+  switch (TYPE(image)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+    break;
+  default:
+    if (!k_rational_p(image))
+      rb_raise(rb_eArgError, "not a real");
+  }
+
+  return nucomp_s_canonicalize_internal(klass, real, image);
+}
+
+static VALUE
+nucomp_s_new(int argc, VALUE *argv, VALUE klass)
+{
+  VALUE real, image;
+
+  switch (rb_scan_args(argc, argv, "11", &real, &image)) {
+  case 1:
+    image = ZERO;
+    break;
+  }
+
+  switch (TYPE(real)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+    break;
+  default:
+    if (!k_rational_p(real))
+      rb_raise(rb_eArgError, "not a real");
+  }
+
+  switch (TYPE(image)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+    break;
+  default:
+    if (!k_rational_p(image))
+      rb_raise(rb_eArgError, "not a real");
+  }
+
+  return nucomp_s_canonicalize_internal(klass, real, image);
+}
+
+inline static VALUE
+f_complex_new1(VALUE klass, VALUE x)
+{
+  assert(!k_complex_p(x));
+  return nucomp_s_canonicalize_internal(klass, x, ZERO);
+}
+
+inline static VALUE
+f_complex_new2(VALUE klass, VALUE x, VALUE y)
+{
+  assert(!k_complex_p(x));
+  return nucomp_s_canonicalize_internal(klass, x, y);
+}
+
+static VALUE
+nucomp_f_complex(int argc, VALUE *argv, VALUE klass)
+{
+  return rb_funcall2(rb_cComplex, id_convert, argc, argv);
+}
+
+#if 1
+/* the following code is copied from math.c */
+
+#include <errno.h>
+
+#define Need_Float(x) (x) = rb_Float(x)
+#define Need_Float2(x,y) do {\
+    Need_Float(x);\
+    Need_Float(y);\
+} while (0)
+
+static void
+domain_check(double x, char *msg)
+{
+  while(1) {
+    if (errno) {
+      rb_sys_fail(msg);
+    }
+    if (isnan(x)) {
+#if defined(EDOM)
+      errno = EDOM;
+#elif defined(ERANGE)
+      errno = ERANGE;
+#endif
+      continue;
+    }
+    break;
+  }
+}
+
+static VALUE
+m_cos_bang(VALUE x)
+{
+  Need_Float(x);
+  return DOUBLE2NUM(cos(RFLOAT_VALUE(x)));
+}
+
+static VALUE m_cos_bang(VALUE);
+static VALUE m_cosh_bang(VALUE);
+static VALUE m_sin_bang(VALUE);
+static VALUE m_sinh_bang(VALUE);
+
+static VALUE
+m_cos(VALUE x)
+{
+  get_dat1(x);
+
+  if (f_generic_p(x))
+    return m_cos_bang(x);
+  else
+    return f_complex_new2(rb_cComplex,
+			  f_mul(m_cos_bang(dat->real),
+				m_cosh_bang(dat->image)),
+			  f_mul(f_negate(m_sin_bang(dat->real)),
+				m_sinh_bang(dat->image)));
+}
+
+#ifndef HAVE_COSH
+double
+cosh(double x)
+{
+  return (exp(x) + exp(-x)) / 2;
+}
+#endif
+
+static VALUE
+m_cosh_bang(VALUE x)
+{
+  Need_Float(x);
+  return DOUBLE2NUM(cosh(RFLOAT_VALUE(x)));
+}
+
+static VALUE
+m_exp_bang(VALUE x)
+{
+  Need_Float(x);
+  return DOUBLE2NUM(exp(RFLOAT_VALUE(x)));
+}
+
+static VALUE
+m_log_bang(VALUE x)
+{
+  double d;
+
+  Need_Float(x);
+  errno = 0;
+  d = log(RFLOAT_VALUE(x));
+  domain_check(d, "log");
+  return DOUBLE2NUM(d);
+}
+
+static VALUE
+m_sin_bang(VALUE x)
+{
+  Need_Float(x);
+  return DOUBLE2NUM(sin(RFLOAT_VALUE(x)));
+}
+
+static VALUE
+m_sin(VALUE x)
+{
+  get_dat1(x);
+
+  if (f_generic_p(x))
+    return m_sin_bang(x);
+  else
+    return f_complex_new2(rb_cComplex,
+			  f_mul(m_sin_bang(dat->real),
+				m_cosh_bang(dat->image)),
+			  f_mul(m_cos_bang(dat->real),
+				m_sinh_bang(dat->image)));
+}
+
+#ifndef HAVE_SINH
+double
+sinh(double x)
+{
+  return (exp(x) - exp(-x)) / 2;
+}
+#endif
+
+static VALUE
+m_sinh_bang(VALUE x)
+{
+  Need_Float(x);
+  return DOUBLE2NUM(sinh(RFLOAT_VALUE(x)));
+}
+
+static VALUE
+m_sqrt_bang(VALUE x)
+{
+  double d;
+
+  Need_Float(x);
+  errno = 0;
+  d = sqrt(RFLOAT_VALUE(x));
+  domain_check(d, "sqrt");
+  return DOUBLE2NUM(d);
+}
+
+static VALUE
+m_sqrt(VALUE x)
+{
+  if (f_generic_p(x)) {
+    if (!f_negative_p(x))
+      return m_sqrt_bang(x);
+    else
+      return f_complex_new2(rb_cComplex, ZERO, m_sqrt_bang(f_negate(x)));
+  } else {
+    get_dat1(x);
+
+    if (f_negative_p(dat->image))
+      return f_conjugate(m_sqrt(f_conjugate(x)));
+    else {
+      VALUE a = f_abs(x);
+      return f_complex_new2(rb_cComplex,
+			    m_sqrt_bang(f_div(f_add(a, dat->real), TWO)),
+			    m_sqrt_bang(f_div(f_sub(a, dat->real), TWO)));
+    }
+  }
+}
+
+static VALUE
+m_atan2_bang(VALUE y, VALUE x)
+{
+  Need_Float2(y, x);
+  return DOUBLE2NUM(atan2(RFLOAT_VALUE(y), RFLOAT_VALUE(x)));
+}
+
+static VALUE
+m_hypot(VALUE x, VALUE y)
+{
+  Need_Float2(x, y);
+  return DOUBLE2NUM(hypot(RFLOAT_VALUE(x), RFLOAT_VALUE(y)));
+}
+#endif
+
+static VALUE
+nucomp_s_polar(VALUE klass, VALUE abs, VALUE arg)
+{
+  return f_complex_new2(klass,
+			f_mul(abs, m_cos(arg)),
+			f_mul(abs, m_sin(arg)));
+}
+
+static VALUE
+nucomp_real(VALUE self)
+{
+  get_dat1(self);
+  return dat->real;
+}
+
+static VALUE
+nucomp_image(VALUE self)
+{
+  get_dat1(self);
+  return dat->image;
+}
+
+static VALUE
+nucomp_add(VALUE self, VALUE other)
+{
+  switch (TYPE(other)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+  case T_RATIONAL:
+    {
+      get_dat1(self);
+
+      return f_complex_new2(CLASS_OF(self),
+			    f_add(dat->real, other), dat->image);
+    }
+  case T_COMPLEX:
+    {
+      VALUE real, image;
+
+      get_dat2(self, other);
+
+      real = f_add(adat->real, bdat->real);
+      image = f_add(adat->image, bdat->image);
+
+      return f_complex_new2(CLASS_OF(self), real, image);
+    }
+  default:
+    {
+      VALUE a = f_coerce(other, self);
+      return f_add(RARRAY_PTR(a)[0], RARRAY_PTR(a)[1]);
+    }
+  }
+}
+
+static VALUE
+nucomp_sub(VALUE self, VALUE other)
+{
+  switch (TYPE(other)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+  case T_RATIONAL:
+    {
+      get_dat1(self);
+
+      return f_complex_new2(CLASS_OF(self),
+			    f_sub(dat->real, other), dat->image);
+    }
+  case T_COMPLEX:
+    {
+      VALUE real, image;
+
+      get_dat2(self, other);
+
+      real = f_sub(adat->real, bdat->real);
+      image = f_sub(adat->image, bdat->image);
+
+      return f_complex_new2(CLASS_OF(self), real, image);
+    }
+  default:
+    {
+      VALUE a = f_coerce(other, self);
+      return f_add(RARRAY_PTR(a)[0], RARRAY_PTR(a)[1]);
+    }
+  }
+}
+
+static VALUE
+nucomp_mul(VALUE self, VALUE other)
+{
+  switch (TYPE(other)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+  case T_RATIONAL:
+    {
+      get_dat1(self);
+
+      return f_complex_new2(CLASS_OF(self),
+			    f_mul(dat->real, other),
+			    f_mul(dat->image, other));
+    }
+  case T_COMPLEX:
+    {
+      VALUE real, image;
+
+      get_dat2(self, other);
+
+      real = f_sub(f_mul(adat->real, bdat->real),
+		   f_mul(adat->image, bdat->image));
+      image = f_add(f_mul(adat->real, bdat->image),
+		    f_mul(adat->image, bdat->real));
+
+      return f_complex_new2(CLASS_OF(self), real, image);
+    }
+  default:
+    {
+      VALUE a = f_coerce(other, self);
+      return f_mul(RARRAY_PTR(a)[0], RARRAY_PTR(a)[1]);
+    }
+  }
+}
+
+static VALUE
+nucomp_div(VALUE self, VALUE other)
+{
+  switch (TYPE(other)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+  case T_RATIONAL:
+    {
+      get_dat1(self);
+
+      return f_complex_new2(CLASS_OF(self),
+			    f_div(dat->real, other),
+			    f_div(dat->image, other));
+    }
+  case T_COMPLEX:
+    return f_div(f_mul(self, f_conjugate(other)), f_abs2(other));
+  default:
+    {
+      VALUE a = f_coerce(other, self);
+      return f_div(RARRAY_PTR(a)[0], RARRAY_PTR(a)[1]);
+    }
+  }
+}
+
+static VALUE
+nucomp_rdiv(VALUE self, VALUE other)
+{
+  get_dat1(self);
+
+  return f_div(f_complex_new2(CLASS_OF(self),
+			      f_to_r(dat->real),
+			      f_to_r(dat->image)), other);
+}
+
+static VALUE
+nucomp_fdiv(VALUE self, VALUE other)
+{
+  get_dat1(self);
+
+  return f_div(f_complex_new2(CLASS_OF(self),
+			      f_to_f(dat->real),
+			      f_to_f(dat->image)), other);
+}
+
+static VALUE
+nucomp_expt(VALUE self, VALUE other)
+{
+  if (f_zero_p(other))
+    return f_complex_new_bang1(CLASS_OF(self), ONE);
+
+  if (k_rational_p(other) && f_one_p(f_denominator(other)))
+    other = f_numerator(other); /* good? */
+
+  switch (TYPE(other)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+    if (f_gt_p(other, ZERO)) {
+      VALUE x, z, n;
+
+      x = self;
+      z = x;
+      n = f_sub(other, ONE);
+
+      while (!f_zero_p(n)) {
+	VALUE a;
+
+	while (a = f_divmod(n, TWO),
+	       f_zero_p(RARRAY_PTR(a)[1])) {
+	  get_dat1(x);
+
+	  x = f_complex_new2(CLASS_OF(self),
+			     f_sub(f_mul(dat->real, dat->real),
+				   f_mul(dat->image, dat->image)),
+			     f_mul(f_mul(TWO, dat->real), dat->image));
+	  n = RARRAY_PTR(a)[0];
+	}
+	z = f_mul(z, x);
+	n = f_sub(n, ONE);
+      }
+      return z;
+    } else {
+      return f_expt(f_div(f_to_r(ONE), self), f_negate(other));
+    }
+  case T_FLOAT:
+  case T_RATIONAL:
+    {
+      VALUE a, r, theta;
+
+      a = f_polar(self);
+      r = RARRAY_PTR(a)[0];
+      theta = RARRAY_PTR(a)[1];
+      return nucomp_s_polar(CLASS_OF(self), f_expt(r, other),
+			    f_mul(theta, other));
+    }
+  case T_COMPLEX:
+    {
+      VALUE a, r, theta, ore, oim, nr, ntheta;
+
+      get_dat1(other);
+
+      a = f_polar(self);
+      r = RARRAY_PTR(a)[0];
+      theta = RARRAY_PTR(a)[1];
+
+      ore = dat->real;
+      oim = dat->image;
+      nr = m_exp_bang(f_sub(f_mul(ore, m_log_bang(r)),
+			    f_mul(oim, theta)));
+      ntheta = f_add(f_mul(theta, ore), f_mul(oim, m_log_bang(r)));
+      return nucomp_s_polar(CLASS_OF(self), nr, ntheta);
+    }
+  default:
+    {
+      VALUE a = f_coerce(other, self);
+      return f_div(RARRAY_PTR(a)[0], RARRAY_PTR(a)[1]);
+    }
+  }
+}
+
+static VALUE
+nucomp_equal_p(VALUE self, VALUE other)
+{
+  switch (TYPE(other)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+  case T_RATIONAL:
+    {
+      get_dat1(self);
+
+      return f_boolcast(f_equal_p(dat->real, other) && f_zero_p(dat->image));
+    }
+  case T_COMPLEX:
+    {
+      get_dat2(self, other);
+
+      return f_boolcast(f_equal_p(adat->real, bdat->real) &&
+			f_equal_p(adat->image, bdat->image));
+    }
+  default:
+    return f_equal_p(other, self);
+  }
+}
+
+static VALUE
+nucomp_coerce(VALUE self, VALUE other)
+{
+  switch (TYPE(other)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+  case T_RATIONAL:
+    return rb_assoc_new(f_complex_new_bang1(CLASS_OF(self), other), self);
+  }
+
+  rb_raise(rb_eTypeError, "%s can't be coerced into %s",
+	   rb_obj_classname(other), rb_obj_classname(self));
+  return Qnil;
+}
+
+static VALUE
+nucomp_abs(VALUE self)
+{
+  get_dat1(self);
+  return m_sqrt(f_add(f_mul(dat->real, dat->real),
+		      f_mul(dat->image, dat->image)));
+}
+
+static VALUE
+nucomp_abs2(VALUE self)
+{
+  get_dat1(self);
+  return f_add(f_mul(dat->real, dat->real),
+	       f_mul(dat->image, dat->image));
+}
+
+static VALUE
+nucomp_arg(VALUE self)
+{
+  get_dat1(self);
+  return m_atan2_bang(dat->image, dat->real);
+}
+
+static VALUE
+nucomp_polar(VALUE self)
+{
+  return rb_assoc_new(f_abs(self), f_arg(self));
+}
+
+static VALUE
+nucomp_conjugate(VALUE self)
+{
+  get_dat1(self);
+  return f_complex_new2(CLASS_OF(self), dat->real, f_negate(dat->image));
+}
+
+static VALUE
+nucomp_real_p(VALUE self)
+{
+  return Qfalse;
+}
+
+static VALUE
+nucomp_complex_p(VALUE self)
+{
+  return Qtrue;
+}
+
+static VALUE
+nucomp_exact_p(VALUE self)
+{
+  get_dat1(self);
+  return f_boolcast(f_exact_p(dat->real) && f_exact_p(dat->image));
+}
+
+static VALUE
+nucomp_inexact_p(VALUE self)
+{
+  return f_boolcast(!nucomp_exact_p(self));
+}
+
+inline static long
+i_gcd(long x, long y)
+{
+  long b;
+
+  if (x < 0)
+    x = -x;
+  if (y < 0)
+    y = -y;
+
+  if (x == 0)
+    return y;
+  if (y == 0)
+    return x;
+
+  b = 0;
+  while ((x & 1) == 0 && (y & 1) == 0) {
+    b += 1;
+    x >>= 1;
+    y >>= 1;
+  }
+
+  while ((x & 1) == 0)
+    x >>= 1;
+
+  while ((y & 1) == 0)
+    y >>= 1;
+
+  while (x != y) {
+    if (y > x) {
+      long t;
+      t = x;
+      x = y;
+      y = t;
+    }
+    x -= y;
+    while ((x & 1) == 0)
+      x >>= 1;
+  }
+
+  return x << b;
+}
+
+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 (FIX2INT(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 */
+}
+
+static VALUE
+f_lcm(VALUE x, VALUE y)
+{
+  if (f_zero_p(x) || f_zero_p(y))
+    return ZERO;
+  else
+    return f_abs(f_mul(f_div(x, f_gcd(x, y)), y));
+}
+
+static VALUE
+nucomp_denominator(VALUE self)
+{
+  get_dat1(self);
+  return f_lcm(f_denominator(dat->real), f_denominator(dat->image));
+}
+
+static VALUE
+nucomp_numerator(VALUE self)
+{
+  VALUE cd;
+
+  get_dat1(self);
+
+  cd = f_denominator(self);
+  return f_complex_new2(CLASS_OF(self),
+			f_mul(f_numerator(dat->real),
+			      f_div(cd, f_denominator(dat->real))),
+			f_mul(f_numerator(dat->image),
+			      f_div(cd, f_denominator(dat->image))));
+}
+
+static VALUE
+nucomp_hash(VALUE self)
+{
+  get_dat1(self);
+  return f_xor(dat->real, dat->image);
+}
+
+#ifndef HAVE_SIGNBIT
+#ifdef signbit
+#define HAVE_SIGNBIT 1
+#endif
+#endif
+
+inline static VALUE
+f_signbit(VALUE x)
+{
+  switch (TYPE(x)) {
+  case T_FLOAT:
+#ifdef HAVE_SIGNBIT
+    return f_boolcast(signbit(RFLOAT_VALUE(x)));
+#else
+    {
+      char s[2];
+
+      (void)snprintf(s, sizeof s, "%.0f", RFLOAT_VALUE(x));
+
+      return f_boolcast(s[0] == '-');
+    }
+#endif
+  }
+  return f_negative_p(x);
+}
+
+inline static VALUE
+f_tzero_p(VALUE x)
+{
+  return f_boolcast(f_zero_p(x) && !f_signbit(x));
+}
+
+inline static VALUE
+f_tpositive_p(VALUE x)
+{
+  return f_boolcast(!f_signbit(x));
+}
+
+static VALUE
+nucomp_to_s(VALUE self)
+{
+  VALUE s, rezero, impos;
+
+  get_dat1(self);
+
+  rezero = f_tzero_p(dat->real);
+  impos = f_tpositive_p(dat->image);
+
+  if (rezero)
+    s = rb_str_new2("");
+  else {
+    s = f_to_s(dat->real);
+    rb_str_concat(s, rb_str_new2(!impos ? "-" : "+"));
+  }
+
+  if (k_rational_p(dat->image) &&
+      !f_one_p(f_denominator(dat->image))) {
+    rb_str_concat(s, rb_str_new2("("));
+    rb_str_concat(s, f_to_s(rezero ? dat->image : f_abs(dat->image)));
+    rb_str_concat(s, rb_str_new2(")i"));
+  } else {
+    rb_str_concat(s, f_to_s(rezero ? dat->image : f_abs(dat->image)));
+    rb_str_concat(s, rb_str_new2("i"));
+  }
+
+  return s;
+}
+
+static VALUE
+nucomp_inspect(VALUE self)
+{
+  VALUE s;
+
+  get_dat1(self);
+
+  s = rb_str_new2("Complex(");
+  rb_str_concat(s, f_inspect(dat->real));
+  rb_str_concat(s, rb_str_new2(", "));
+  rb_str_concat(s, f_inspect(dat->image));
+  rb_str_concat(s, rb_str_new2(")"));
+
+  return s;
+}
+
+static VALUE
+nucomp_marshal_dump(VALUE self)
+{
+  get_dat1(self);
+  return rb_assoc_new(dat->real, dat->image);
+}
+
+static VALUE
+nucomp_marshal_load(VALUE self, VALUE a)
+{
+  get_dat1(self);
+  dat->real = RARRAY_PTR(a)[0];
+  dat->image = RARRAY_PTR(a)[1];
+  return self;
+}
+
+/* --- */
+
+VALUE
+rb_complex_raw(VALUE x, VALUE y)
+{
+  return nucomp_s_new_internal(rb_cComplex, x, y);
+}
+
+VALUE
+rb_complex_new(VALUE x, VALUE y)
+{
+  return nucomp_s_canonicalize_internal(rb_cComplex, x, y);
+}
+
+static VALUE nucomp_s_convert(int argc, VALUE *argv, VALUE klass);
+
+VALUE
+rb_Complex(VALUE x, VALUE y)
+{
+  VALUE a[2];
+  a[0] = x;
+  a[1] = y;
+  return nucomp_s_convert(2, a, rb_cComplex);
+}
+
+static VALUE
+nucomp_scalar_p(VALUE self)
+{
+  return Qfalse;
+}
+
+static VALUE
+nucomp_to_i(VALUE self)
+{
+  get_dat1(self);
+
+  if (k_float_p(dat->image) || !f_zero_p(dat->image)) {
+    VALUE s = f_to_s(self);
+    rb_raise(rb_eRangeError, "can't convert %s into Integer",
+	     StringValuePtr(s));
+  }
+  return f_to_i(dat->real);
+}
+
+static VALUE
+nucomp_to_f(VALUE self)
+{
+  get_dat1(self);
+
+  if (k_float_p(dat->image) || !f_zero_p(dat->image)) {
+    VALUE s = f_to_s(self);
+    rb_raise(rb_eRangeError, "can't convert %s into Integer",
+	     StringValuePtr(s));
+  }
+  return f_to_f(dat->real);
+}
+
+static VALUE
+nucomp_to_r(VALUE self)
+{
+  get_dat1(self);
+
+  if (k_float_p(dat->image) || !f_zero_p(dat->image)) {
+    VALUE s = f_to_s(self);
+    rb_raise(rb_eRangeError, "can't convert %s into Integer",
+	     StringValuePtr(s));
+  }
+  return f_to_r(dat->real);
+}
+
+static VALUE
+nilclass_to_c(VALUE self)
+{
+  return rb_complex_new1(INT2FIX(0));
+}
+
+static VALUE
+numeric_to_c(VALUE self)
+{
+  return rb_complex_new1(self);
+}
+
+static VALUE comp_pat1, comp_pat2, a_slash, a_dot_and_an_e,
+  image_garbages_pat, null_string, underscores_pat, an_underscore;
+
+#define DIGITS "(?:\\d(?:_\\d|\\d)*)"
+#define NUMERATOR "(?:" DIGITS "?\\.)?" DIGITS "(?:[eE][-+]?" DIGITS ")?"
+#define DENOMINATOR "[-+]?" DIGITS
+#define NUMBER "[-+]?" NUMERATOR "(?:\\/" DENOMINATOR ")?"
+#define NUMBERNOS NUMERATOR "(?:\\/" DENOMINATOR ")?"
+#define PATTERN1 "\\A(" NUMBER "|\\(" NUMBER "\\))[iIjJ]"
+#define PATTERN2 "\\A(" NUMBER ")([-+](?:" NUMBERNOS "|\\(" NUMBER "\\))[iIjJ])?"
+
+static void
+make_patterns(void)
+{
+  static char *comp_pat1_source = PATTERN1;
+  static char *comp_pat2_source = PATTERN2;
+  static char *image_garbages_pat_source = "[+\\(\\)iIjJ]";
+  static char *underscores_pat_source = "_+";
+
+  comp_pat1 = rb_reg_new(comp_pat1_source, strlen(comp_pat1_source), 0);
+  rb_global_variable(&comp_pat1);
+
+  comp_pat2 = rb_reg_new(comp_pat2_source, strlen(comp_pat2_source), 0);
+  rb_global_variable(&comp_pat2);
+
+  a_slash = rb_str_new2("/");
+  rb_global_variable(&a_slash);
+
+  a_dot_and_an_e = rb_str_new2(".eE");
+  rb_global_variable(&a_dot_and_an_e);
+
+  image_garbages_pat = rb_reg_new(image_garbages_pat_source,
+				  strlen(image_garbages_pat_source), 0);
+  rb_global_variable(&image_garbages_pat);
+
+  null_string = rb_str_new2("");
+  rb_global_variable(&null_string);
+
+  underscores_pat = rb_reg_new(underscores_pat_source,
+			       strlen(underscores_pat_source), 0);
+  rb_global_variable(&underscores_pat);
+
+  an_underscore = rb_str_new2("_");
+  rb_global_variable(&an_underscore);
+}
+
+#define id_strip rb_intern("strip")
+#define f_strip(x) rb_funcall(x, id_strip, 0)
+
+#define id_match rb_intern("match")
+#define f_match(x,y) rb_funcall(x, id_match, 1, y)
+
+#define id_aref rb_intern("[]")
+#define f_aref(x,y) rb_funcall(x, id_aref, 1, y)
+
+#define id_post_match rb_intern("post_match")
+#define f_post_match(x) rb_funcall(x, id_post_match, 0)
+
+#define id_split rb_intern("split")
+#define f_split(x,y) rb_funcall(x, id_split, 1, y)
+
+#define id_include_p rb_intern("include?")
+#define f_include_p(x,y) rb_funcall(x, id_include_p, 1, y)
+
+#define id_count rb_intern("count")
+#define f_count(x,y) rb_funcall(x, id_count, 1, y)
+
+#define id_gsub_bang rb_intern("gsub!")
+#define f_gsub_bang(x,y,z) rb_funcall(x, id_gsub_bang, 2, y, z)
+
+static VALUE
+string_to_c_internal(VALUE self)
+{
+  VALUE s;
+
+  s = f_strip(self);
+
+  if (RSTRING_LEN(s) == 0)
+    return rb_assoc_new(Qnil, self);
+
+  {
+    VALUE m, sr, si, re, r, i;
+
+    m = f_match(comp_pat1, s);
+    if (!NIL_P(m)) {
+      sr = Qnil;
+      si = f_aref(m, INT2FIX(1));
+      re = f_post_match(m);
+    }
+    if (NIL_P(m)) {
+      m = f_match(comp_pat2, s);
+      if (NIL_P(m))
+	return rb_assoc_new(Qnil, self);
+      sr = f_aref(m, INT2FIX(1));
+      si = f_aref(m, INT2FIX(2));
+      re = f_post_match(m);
+    }
+    r = INT2FIX(0);
+    i = INT2FIX(0);
+    if (!NIL_P(sr)) {
+      if (f_include_p(sr, a_slash))
+	r = f_to_r(sr);
+      else if (f_gt_p(f_count(sr, a_dot_and_an_e), INT2FIX(0)))
+	r = f_to_f(sr);
+      else
+	r = f_to_i(sr);
+    }
+    if (!NIL_P(si)) {
+      f_gsub_bang(si, image_garbages_pat, null_string);
+      if (f_include_p(si, a_slash))
+	i = f_to_r(si);
+      else if (f_gt_p(f_count(si, a_dot_and_an_e), INT2FIX(0)))
+	i = f_to_f(si);
+      else
+	i = f_to_i(si);
+    }
+    return rb_assoc_new(rb_complex_new2(r, i), re);
+  }
+}
+
+static VALUE
+string_to_c_strict(VALUE self)
+{
+  VALUE a = string_to_c_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 Complex: %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)
+
+static VALUE
+string_to_c(VALUE self)
+{
+  VALUE s = f_gsub(self, underscores_pat, an_underscore);
+  VALUE a = string_to_c_internal(s);
+  if (!NIL_P(RARRAY_PTR(a)[0]))
+    return RARRAY_PTR(a)[0];
+  return rb_complex_new1(INT2FIX(0));
+}
+
+static VALUE
+nucomp_s_convert(int argc, VALUE *argv, VALUE klass)
+{
+  VALUE a1, a2;
+
+  a1 = Qnil;
+  a2 = Qnil;
+  rb_scan_args(argc, argv, "02", &a1, &a2);
+
+  switch (TYPE(a1)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+    break;
+  case T_STRING:
+    a1 = string_to_c_strict(a1);
+    break;
+  }
+
+  switch (TYPE(a2)) {
+  case T_FIXNUM:
+  case T_BIGNUM:
+  case T_FLOAT:
+    break;
+  case T_STRING:
+    a2 = string_to_c_strict(a2);
+    break;
+  }
+
+  switch (TYPE(a1)) {
+  case T_COMPLEX:
+    {
+      get_dat1(a1);
+
+      if (!k_float_p(dat->image) && f_zero_p(dat->image))
+	a1 = dat->real;
+    }
+  }
+
+  switch (TYPE(a2)) {
+  case T_COMPLEX:
+    {
+      get_dat1(a2);
+
+      if (!k_float_p(dat->image) && f_zero_p(dat->image))
+	a2 = dat->real;
+    }
+  }
+
+  switch (TYPE(a1)) {
+  case T_COMPLEX:
+    if (NIL_P(a2) || f_zero_p(a2))
+      return a1;
+  }
+
+  {
+    VALUE argv2[2];
+    argv2[0] = a1;
+    argv2[1] = a2;
+    return nucomp_s_new(argc, argv2, klass);
+  }
+}
+
+/* --- */
+
+#define id_Complex rb_intern("Complex")
+
+static VALUE
+numeric_re(VALUE self)
+{
+  return rb_Complex1(self);
+}
+
+static VALUE
+numeric_im(VALUE self)
+{
+  return rb_Complex2(ZERO, self);
+}
+
+static VALUE
+numeric_real(VALUE self)
+{
+  return self;
+}
+
+static VALUE
+numeric_image(VALUE self)
+{
+  return INT2FIX(0);
+}
+
+#define id_PI rb_intern("PI")
+
+static VALUE
+numeric_arg(VALUE self)
+{
+  if (!f_negative_p(self))
+    return INT2FIX(0);
+  return rb_const_get(rb_mMath, id_PI);
+}
+
+static VALUE
+numeric_polar(VALUE self)
+{
+  return rb_assoc_new(f_abs(self), f_arg(self));
+}
+
+static VALUE
+numeric_conjugate(VALUE self)
+{
+  return self;
+}
+
+void
+Init_Complex(void)
+{
+  assert(fprintf(stderr, "assert() is now active\n"));
+
+  id_Unify = rb_intern("Unify");
+  id_abs = rb_intern("abs");
+  id_abs2 = rb_intern("abs2");
+  id_arg = rb_intern("arg");
+  id_atan2_bang = rb_intern("atan2!");
+  id_cmp = rb_intern("<=>");
+  id_coerce = rb_intern("coerce");
+  id_conjugate = rb_intern("conjugate");
+  id_convert = rb_intern("convert");
+  id_cos = rb_intern("cos");
+  id_denominator = rb_intern("denominator");
+  id_divmod = rb_intern("divmod");
+  id_equal_p = rb_intern("==");
+  id_exact_p = rb_intern("exact?");
+  id_exp_bang = rb_intern("exp!");
+  id_expt = rb_intern("**");
+  id_floor = rb_intern("floor");
+  id_format = rb_intern("format");
+  id_hypot = rb_intern("hypot");
+  id_idiv = rb_intern("div");
+  id_inspect = rb_intern("inspect");
+  id_log_bang = rb_intern("log!");
+  id_negate = rb_intern("-@");
+  id_new = rb_intern("new");
+  id_new_bang = rb_intern("new!");
+  id_numerator = rb_intern("numerator");
+  id_polar = rb_intern("polar");
+  id_quo = rb_intern("quo");
+  id_scalar_p = rb_intern("scalar?");
+  id_sin = rb_intern("sin");
+  id_sqrt = rb_intern("sqrt");
+  id_to_f = rb_intern("to_f");
+  id_to_i = rb_intern("to_i");
+  id_to_r = rb_intern("to_r");
+  id_to_s = rb_intern("to_s");
+  id_truncate = rb_intern("truncate");
+
+  rb_cComplex = rb_define_class(COMPLEX_NAME, rb_cNumeric);
+
+  rb_define_alloc_func(rb_cComplex, nucomp_s_alloc);
+  rb_funcall(rb_cComplex, rb_intern("private_class_method"), 1,
+	     ID2SYM(rb_intern("allocate")));
+
+  rb_define_singleton_method(rb_cComplex, "generic?", nucomp_s_generic_p, 1);
+
+  rb_define_singleton_method(rb_cComplex, "new!", nucomp_s_new_bang, -1);
+  rb_funcall(rb_cComplex, rb_intern("private_class_method"), 1,
+	     ID2SYM(rb_intern("new!")));
+
+  rb_define_singleton_method(rb_cComplex, "new", nucomp_s_new, -1);
+  rb_funcall(rb_cComplex, rb_intern("private_class_method"), 1,
+	     ID2SYM(rb_intern("new")));
+
+#if 0
+  rb_define_singleton_method(rb_cComplex, "rect", nucomp_s_new, -1);
+  rb_define_singleton_method(rb_cComplex, "rectangular", nucomp_s_new, -1);
+#endif
+  rb_define_singleton_method(rb_cComplex, "polar", nucomp_s_polar, 2);
+
+  rb_define_global_function(COMPLEX_NAME, nucomp_f_complex, -1);
+
+  rb_undef_method(rb_cComplex, "<");
+  rb_undef_method(rb_cComplex, "<=");
+  rb_undef_method(rb_cComplex, "<=>");
+  rb_undef_method(rb_cComplex, ">");
+  rb_undef_method(rb_cComplex, ">=");
+  rb_undef_method(rb_cComplex, "between?");
+  rb_undef_method(rb_cComplex, "divmod");
+  rb_undef_method(rb_cComplex, "floor");
+  rb_undef_method(rb_cComplex, "ceil");
+  rb_undef_method(rb_cComplex, "modulo");
+  rb_undef_method(rb_cComplex, "round");
+  rb_undef_method(rb_cComplex, "step");
+  rb_undef_method(rb_cComplex, "truncate");
+
+#if NUBY
+  rb_undef_method(rb_cComplex, "//");
+#endif
+
+  rb_define_method(rb_cComplex, "real", nucomp_real, 0);
+  rb_define_method(rb_cComplex, "image", nucomp_image, 0);
+  rb_define_method(rb_cComplex, "imag", nucomp_image, 0);
+
+  rb_define_method(rb_cComplex, "+", nucomp_add, 1);
+  rb_define_method(rb_cComplex, "-", nucomp_sub, 1);
+  rb_define_method(rb_cComplex, "*", nucomp_mul, 1);
+  rb_define_method(rb_cComplex, "/", nucomp_div, 1);
+  rb_define_method(rb_cComplex, "quo", nucomp_rdiv, 1);
+  rb_define_method(rb_cComplex, "rdiv", nucomp_rdiv, 1);
+  rb_define_method(rb_cComplex, "fdiv", nucomp_rdiv, 1);
+  rb_define_method(rb_cComplex, "**", nucomp_expt, 1);
+
+  rb_define_method(rb_cComplex, "==", nucomp_equal_p, 1);
+  rb_define_method(rb_cComplex, "coerce", nucomp_coerce, 1);
+
+  rb_define_method(rb_cComplex, "abs", nucomp_abs, 0);
+#if 0
+  rb_define_method(rb_cComplex, "magnitude", nucomp_abs, 0);
+#endif
+  rb_define_method(rb_cComplex, "abs2", nucomp_abs2, 0);
+  rb_define_method(rb_cComplex, "arg", nucomp_arg, 0);
+  rb_define_method(rb_cComplex, "angle", nucomp_arg, 0);
+  rb_define_method(rb_cComplex, "polar", nucomp_polar, 0);
+  rb_define_method(rb_cComplex, "conjugate", nucomp_conjugate, 0);
+  rb_define_method(rb_cComplex, "conj", nucomp_conjugate, 0);
+#if 0
+  rb_define_method(rb_cComplex, "~", nucomp_conjugate, 0); /* gcc */
+#endif
+
+#if 0
+  rb_define_method(rb_cComplex, "real?", nucomp_real_p, 0);
+  rb_define_method(rb_cComplex, "complex?", nucomp_complex_p, 0);
+  rb_define_method(rb_cComplex, "exact?", nucomp_exact_p, 0);
+  rb_define_method(rb_cComplex, "inexact?", nucomp_inexact_p, 0);
+#endif
+
+  rb_define_method(rb_cComplex, "numerator", nucomp_numerator, 0);
+  rb_define_method(rb_cComplex, "denominator", nucomp_denominator, 0);
+
+  rb_define_method(rb_cComplex, "hash", nucomp_hash, 0);
+
+  rb_define_method(rb_cComplex, "to_s", nucomp_to_s, 0);
+  rb_define_method(rb_cComplex, "inspect", nucomp_inspect, 0);
+
+  rb_define_method(rb_cComplex, "marshal_dump", nucomp_marshal_dump, 0);
+  rb_define_method(rb_cComplex, "marshal_load", nucomp_marshal_load, 1);
+
+  /* --- */
+
+  rb_define_method(rb_cComplex, "scalar?", nucomp_scalar_p, 0);
+  rb_define_method(rb_cComplex, "to_i", nucomp_to_i, 0);
+  rb_define_method(rb_cComplex, "to_f", nucomp_to_f, 0);
+  rb_define_method(rb_cComplex, "to_r", nucomp_to_r, 0);
+  rb_define_method(rb_cNilClass, "to_c", nilclass_to_c, 0);
+  rb_define_method(rb_cNumeric, "to_c", numeric_to_c, 0);
+
+  make_patterns();
+
+  rb_define_method(rb_cString, "to_c", string_to_c, 0);
+
+  rb_define_singleton_method(rb_cComplex, "convert", nucomp_s_convert, -1);
+  rb_funcall(rb_cComplex, rb_intern("private_class_method"), 1,
+	     ID2SYM(rb_intern("convert")));
+
+  /* --- */
+
+  rb_define_method(rb_cNumeric, "re", numeric_re, 0);
+  rb_define_method(rb_cNumeric, "im", numeric_im, 0);
+  rb_define_method(rb_cNumeric, "real", numeric_real, 0);
+  rb_define_method(rb_cNumeric, "image", numeric_image, 0);
+  rb_define_method(rb_cNumeric, "imag", numeric_image, 0);
+  rb_define_method(rb_cNumeric, "arg", numeric_arg, 0);
+  rb_define_method(rb_cNumeric, "angle", numeric_arg, 0);
+  rb_define_method(rb_cNumeric, "polar", numeric_polar, 0);
+  rb_define_method(rb_cNumeric, "conjugate", numeric_conjugate, 0);
+  rb_define_method(rb_cNumeric, "conj", numeric_conjugate, 0);
+
+  rb_define_const(rb_cComplex, "I",
+		  f_complex_new_bang2(rb_cComplex, ZERO, ONE));
+}