text: /*
    complex.c: Coded by Tadayoshi Funaba 2008-2012
  
    This implementation is based on Keiju Ishitsuka's Complex library
    which is written in ruby.
  */
  
  #include "ruby/config.h"
  #if defined _MSC_VER
  /* Microsoft Visual C does not define M_PI and others by default */
  # define _USE_MATH_DEFINES 1
  #endif
  #include <math.h>
  #include "internal.h"
  #include "id.h"
  
  #define NDEBUG
  #include "ruby_assert.h"
  
  #define ZERO INT2FIX(0)
  #define ONE INT2FIX(1)
  #define TWO INT2FIX(2)
  #if USE_FLONUM
  #define RFLOAT_0 DBL2NUM(0)
  #else
  static VALUE RFLOAT_0;
  #endif
  #if defined(HAVE_SIGNBIT) && defined(__GNUC__) && defined(__sun) && \
      !defined(signbit)
  extern int signbit(double);
  #endif
  
  VALUE rb_cComplex;
  
  static ID id_abs, id_arg,
      id_denominator, id_numerator,
      id_real_p, id_i_real, id_i_imag,
      id_finite_p, id_infinite_p, id_rationalize,
      id_PI;
  #define id_to_i idTo_i
  #define id_to_r idTo_r
  #define id_negate idUMinus
  #define id_expt idPow
  #define id_to_f idTo_f
  #define id_quo idQuo
  #define id_fdiv idFdiv
  
  #define f_boolcast(x) ((x) ? Qtrue : Qfalse)
  
  #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);\
  }
  
  #define PRESERVE_SIGNEDZERO
  
  inline static VALUE
  f_add(VALUE x, VALUE y)
  {
      if (RB_INTEGER_TYPE_P(x) &&
          LIKELY(rb_method_basic_definition_p(rb_cInteger, idPLUS))) {
          if (FIXNUM_ZERO_P(x))
              return y;
          if (FIXNUM_ZERO_P(y))
              return x;
          return rb_int_plus(x, y);
      }
      else if (RB_FLOAT_TYPE_P(x) &&
               LIKELY(rb_method_basic_definition_p(rb_cFloat, idPLUS))) {
          if (FIXNUM_ZERO_P(y))
              return x;
          return rb_float_plus(x, y);
      }
      else if (RB_TYPE_P(x, T_RATIONAL) &&
               LIKELY(rb_method_basic_definition_p(rb_cRational, idPLUS))) {
          if (FIXNUM_ZERO_P(y))
              return x;
          return rb_rational_plus(x, y);
      }
  
      return rb_funcall(x, '+', 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 int
  f_gt_p(VALUE x, VALUE y)
  {
      if (RB_INTEGER_TYPE_P(x)) {
          if (FIXNUM_P(x) && FIXNUM_P(y))
              return (SIGNED_VALUE)x > (SIGNED_VALUE)y;
          return RTEST(rb_int_gt(x, y));
      }
      else if (RB_FLOAT_TYPE_P(x))
          return RTEST(rb_float_gt(x, y));
      else if (RB_TYPE_P(x, T_RATIONAL)) {
          int const cmp = rb_cmpint(rb_rational_cmp(x, y), x, y);
          return cmp > 0;
      }
      return RTEST(rb_funcall(x, '>', 1, y));
  }
  
  inline static VALUE
  f_mul(VALUE x, VALUE y)
  {
      if (RB_INTEGER_TYPE_P(x) &&
          LIKELY(rb_method_basic_definition_p(rb_cInteger, idMULT))) {
          if (FIXNUM_ZERO_P(y))
              return ZERO;
          if (FIXNUM_ZERO_P(x) && RB_INTEGER_TYPE_P(y))
              return ZERO;
          if (x == ONE) return y;
          if (y == ONE) return x;
          return rb_int_mul(x, y);
      }
      else if (RB_FLOAT_TYPE_P(x) &&
               LIKELY(rb_method_basic_definition_p(rb_cFloat, idMULT))) {
          if (y == ONE) return x;
          return rb_float_mul(x, y);
      }
      else if (RB_TYPE_P(x, T_RATIONAL) &&
               LIKELY(rb_method_basic_definition_p(rb_cRational, idMULT))) {
          if (y == ONE) return x;
          return rb_rational_mul(x, y);
      }
      else if (LIKELY(rb_method_basic_definition_p(CLASS_OF(x), idMULT))) {
          if (y == ONE) return x;
      }
      return rb_funcall(x, '*', 1, y);
  }
  
  inline static VALUE
  f_sub(VALUE x, VALUE y)
  {
      if (FIXNUM_ZERO_P(y) &&
          LIKELY(rb_method_basic_definition_p(CLASS_OF(x), idMINUS))) {
  	return x;
      }
      return rb_funcall(x, '-', 1, y);
  }
  
  inline static VALUE
  f_abs(VALUE x)
  {
      if (RB_INTEGER_TYPE_P(x)) {
          return rb_int_abs(x);
      }
      else if (RB_FLOAT_TYPE_P(x)) {
          return rb_float_abs(x);
      }
      else if (RB_TYPE_P(x, T_RATIONAL)) {
          return rb_rational_abs(x);
      }
      else if (RB_TYPE_P(x, T_COMPLEX)) {
          return rb_complex_abs(x);
      }
      return rb_funcall(x, id_abs, 0);
  }
  
  static VALUE numeric_arg(VALUE self);
  static VALUE float_arg(VALUE self);
  
  inline static VALUE
  f_arg(VALUE x)
  {
      if (RB_INTEGER_TYPE_P(x)) {
          return numeric_arg(x);
      }
      else if (RB_FLOAT_TYPE_P(x)) {
          return float_arg(x);
      }
      else if (RB_TYPE_P(x, T_RATIONAL)) {
          return numeric_arg(x);
      }
      else if (RB_TYPE_P(x, T_COMPLEX)) {
          return rb_complex_arg(x);
      }
      return rb_funcall(x, id_arg, 0);
  }
  
  inline static VALUE
  f_numerator(VALUE x)
  {
      if (RB_TYPE_P(x, T_RATIONAL)) {
          return RRATIONAL(x)->num;
      }
      if (RB_FLOAT_TYPE_P(x)) {
          return rb_float_numerator(x);
      }
      return x;
  }
  
  inline static VALUE
  f_denominator(VALUE x)
  {
      if (RB_TYPE_P(x, T_RATIONAL)) {
          return RRATIONAL(x)->den;
      }
      if (RB_FLOAT_TYPE_P(x)) {
          return rb_float_denominator(x);
      }
      return INT2FIX(1);
  }
  
  inline static VALUE
  f_negate(VALUE x)
  {
      if (RB_INTEGER_TYPE_P(x)) {
          return rb_int_uminus(x);
      }
      else if (RB_FLOAT_TYPE_P(x)) {
          return rb_float_uminus(x);
      }
      else if (RB_TYPE_P(x, T_RATIONAL)) {
          return rb_rational_uminus(x);
      }
      else if (RB_TYPE_P(x, T_COMPLEX)) {
          return rb_complex_uminus(x);
      }
      return rb_funcall(x, id_negate, 0);
  }
  
  static bool nucomp_real_p(VALUE self);
  
  static inline bool
  f_real_p(VALUE x)
  {
      if (RB_INTEGER_TYPE_P(x)) {
          return true;
      }
      else if (RB_FLOAT_TYPE_P(x)) {
          return true;
      }
      else if (RB_TYPE_P(x, T_RATIONAL)) {
          return true;
      }
      else if (RB_TYPE_P(x, T_COMPLEX)) {
          return nucomp_real_p(x);
      }
      return rb_funcall(x, id_real_p, 0);
  }
  
  inline static VALUE
  f_to_i(VALUE x)
  {
      if (RB_TYPE_P(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 (RB_TYPE_P(x, T_STRING))
  	return DBL2NUM(rb_str_to_dbl(x, 0));
      return rb_funcall(x, id_to_f, 0);
  }
  
  fun1(to_r)
  
  inline static int
  f_eqeq_p(VALUE x, VALUE y)
  {
      if (FIXNUM_P(x) && FIXNUM_P(y))
  	return x == y;
      else if (RB_FLOAT_TYPE_P(x) || RB_FLOAT_TYPE_P(y))
  	return NUM2DBL(x) == NUM2DBL(y);
      return (int)rb_equal(x, y);
  }
  
  fun2(expt)
  fun2(fdiv)
  
  static VALUE
  f_quo(VALUE x, VALUE y)
  {
      if (RB_INTEGER_TYPE_P(x))
          return rb_numeric_quo(x, y);
      if (RB_FLOAT_TYPE_P(x))
          return rb_float_div(x, y);
      if (RB_TYPE_P(x, T_RATIONAL))
          return rb_numeric_quo(x, y);
  
      return rb_funcallv(x, id_quo, 1, &y);
  }
  
  inline static int
  f_negative_p(VALUE x)
  {
      if (RB_INTEGER_TYPE_P(x))
          return INT_NEGATIVE_P(x);
      else if (RB_FLOAT_TYPE_P(x))
          return RFLOAT_VALUE(x) < 0.0;
      else if (RB_TYPE_P(x, T_RATIONAL))
          return INT_NEGATIVE_P(RRATIONAL(x)->num);
      return rb_num_negative_p(x);
  }
  
  #define f_positive_p(x) (!f_negative_p(x))
  
  inline static int
  f_zero_p(VALUE x)
  {
      if (RB_FLOAT_TYPE_P(x)) {
          return FLOAT_ZERO_P(x);
      }
      else if (RB_INTEGER_TYPE_P(x)) {
          return FIXNUM_ZERO_P(x);
      }
      else if (RB_TYPE_P(x, T_RATIONAL)) {
          const VALUE num = RRATIONAL(x)->num;
          return FIXNUM_ZERO_P(num);
      }
      return (int)rb_equal(x, ZERO);
  }
  
  #define f_nonzero_p(x) (!f_zero_p(x))
  
  VALUE rb_flo_is_finite_p(VALUE num);
  inline static int
  f_finite_p(VALUE x)
  {
      if (RB_INTEGER_TYPE_P(x)) {
          return TRUE;
      }
      else if (RB_FLOAT_TYPE_P(x)) {
  	return (int)rb_flo_is_finite_p(x);
      }
      else if (RB_TYPE_P(x, T_RATIONAL)) {
  	return TRUE;
      }
      return RTEST(rb_funcallv(x, id_finite_p, 0, 0));
  }
  
  VALUE rb_flo_is_infinite_p(VALUE num);
  inline static VALUE
  f_infinite_p(VALUE x)
  {
      if (RB_INTEGER_TYPE_P(x)) {
          return Qnil;
      }
      else if (RB_FLOAT_TYPE_P(x)) {
  	return rb_flo_is_infinite_p(x);
      }
      else if (RB_TYPE_P(x, T_RATIONAL)) {
          return Qnil;
      }
      return rb_funcallv(x, id_infinite_p, 0, 0);
  }
  
  inline static int
  f_kind_of_p(VALUE x, VALUE c)
  {
      return (int)rb_obj_is_kind_of(x, c);
  }
  
  inline static int
  k_numeric_p(VALUE x)
  {
      return f_kind_of_p(x, rb_cNumeric);
  }
  
  #define k_exact_p(x) (!RB_FLOAT_TYPE_P(x))
  
  #define k_exact_zero_p(x) (k_exact_p(x) && f_zero_p(x))
  
  #define get_dat1(x) \
      struct RComplex *dat = RCOMPLEX(x)
  
  #define get_dat2(x,y) \
      struct RComplex *adat = RCOMPLEX(x), *bdat = RCOMPLEX(y)
  
  inline static VALUE
  nucomp_s_new_internal(VALUE klass, VALUE real, VALUE imag)
  {
      NEWOBJ_OF(obj, struct RComplex, klass, T_COMPLEX | (RGENGC_WB_PROTECTED_COMPLEX ? FL_WB_PROTECTED : 0));
  
      RCOMPLEX_SET_REAL(obj, real);
      RCOMPLEX_SET_IMAG(obj, imag);
      OBJ_FREEZE_RAW(obj);
  
      return (VALUE)obj;
  }
  
  static VALUE
  nucomp_s_alloc(VALUE klass)
  {
      return nucomp_s_new_internal(klass, ZERO, ZERO);
  }
  
  inline static VALUE
  f_complex_new_bang1(VALUE klass, VALUE x)
  {
      assert(!RB_TYPE_P(x, T_COMPLEX));
      return nucomp_s_new_internal(klass, x, ZERO);
  }
  
  inline static VALUE
  f_complex_new_bang2(VALUE klass, VALUE x, VALUE y)
  {
      assert(!RB_TYPE_P(x, T_COMPLEX));
      assert(!RB_TYPE_P(y, T_COMPLEX));
      return nucomp_s_new_internal(klass, x, y);
  }
  
  #ifdef CANONICALIZATION_FOR_MATHN
  static int canonicalization = 0;
  
  RUBY_FUNC_EXPORTED void
  nucomp_canonicalization(int f)
  {
      canonicalization = f;
  }
  #else
  #define canonicalization 0
  #endif
  
  inline static void
  nucomp_real_check(VALUE num)
  {
      if (!RB_INTEGER_TYPE_P(num) &&
  	!RB_FLOAT_TYPE_P(num) &&
  	!RB_TYPE_P(num, T_RATIONAL)) {
  	if (!k_numeric_p(num) || !f_real_p(num))
  	    rb_raise(rb_eTypeError, "not a real");
      }
  }
  
  inline static VALUE
  nucomp_s_canonicalize_internal(VALUE klass, VALUE real, VALUE imag)
  {
      int complex_r, complex_i;
  #ifdef CANONICALIZATION_FOR_MATHN
      if (k_exact_zero_p(imag) && canonicalization)
  	return real;
  #endif
      complex_r = RB_TYPE_P(real, T_COMPLEX);
      complex_i = RB_TYPE_P(imag, T_COMPLEX);
      if (!complex_r && !complex_i) {
  	return nucomp_s_new_internal(klass, real, imag);
      }
      else if (!complex_r) {
  	get_dat1(imag);
  
  	return nucomp_s_new_internal(klass,
  				     f_sub(real, dat->imag),
  				     f_add(ZERO, dat->real));
      }
      else if (!complex_i) {
  	get_dat1(real);
  
  	return nucomp_s_new_internal(klass,
  				     dat->real,
  				     f_add(dat->imag, imag));
      }
      else {
  	get_dat2(real, imag);
  
  	return nucomp_s_new_internal(klass,
  				     f_sub(adat->real, bdat->imag),
  				     f_add(adat->imag, bdat->real));
      }
  }
  
  /*
   * call-seq:
   *    Complex.rect(real[, imag])         ->  complex
   *    Complex.rectangular(real[, imag])  ->  complex
   *
   * Returns a complex object which denotes the given rectangular form.
   *
   *    Complex.rectangular(1, 2)  #=> (1+2i)
   */
  static VALUE
  nucomp_s_new(int argc, VALUE *argv, VALUE klass)
  {
      VALUE real, imag;
  
      switch (rb_scan_args(argc, argv, "11", &real, &imag)) {
        case 1:
  	nucomp_real_check(real);
  	imag = ZERO;
  	break;
        default:
  	nucomp_real_check(real);
  	nucomp_real_check(imag);
  	break;
      }
  
      return nucomp_s_canonicalize_internal(klass, real, imag);
  }
  
  inline static VALUE
  f_complex_new2(VALUE klass, VALUE x, VALUE y)
  {
      assert(!RB_TYPE_P(x, T_COMPLEX));
      return nucomp_s_canonicalize_internal(klass, x, y);
  }
  
  static VALUE nucomp_convert(VALUE klass, VALUE a1, VALUE a2, int raise);
  static VALUE nucomp_s_convert(int argc, VALUE *argv, VALUE klass);
  
  /*
   * call-seq:
   *    Complex(x[, y], exception: true)  ->  numeric or nil
   *
   * Returns x+i*y;
   *
   *    Complex(1, 2)    #=> (1+2i)
   *    Complex('1+2i')  #=> (1+2i)
   *    Complex(nil)     #=> TypeError
   *    Complex(1, nil)  #=> TypeError
   *
   *    Complex(1, nil, exception: false)  #=> nil
   *    Complex('1+2', exception: false)   #=> nil
   *
   * Syntax of string form:
   *
   *   string form = extra spaces , complex , extra spaces ;
   *   complex = real part | [ sign ] , imaginary part
   *           | real part , sign , imaginary part
   *           | rational , "@" , rational ;
   *   real part = rational ;
   *   imaginary part = imaginary unit | unsigned rational , imaginary unit ;
   *   rational = [ sign ] , unsigned rational ;
   *   unsigned rational = numerator | numerator , "/" , denominator ;
   *   numerator = integer part | fractional part | integer part , fractional part ;
   *   denominator = digits ;
   *   integer part = digits ;
   *   fractional part = "." , digits , [ ( "e" | "E" ) , [ sign ] , digits ] ;
   *   imaginary unit = "i" | "I" | "j" | "J" ;
   *   sign = "-" | "+" ;
   *   digits = digit , { digit | "_" , digit };
   *   digit = "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" ;
   *   extra spaces = ? \s* ? ;
   *
   * See String#to_c.
   */
  static VALUE
  nucomp_f_complex(int argc, VALUE *argv, VALUE klass)
  {
      VALUE a1, a2, opts = Qnil;
      int raise = TRUE;
  
      if (rb_scan_args(argc, argv, "11:", &a1, &a2, &opts) == 1) {
          a2 = Qundef;
      }
      if (!NIL_P(opts)) {
          raise = rb_opts_exception_p(opts, raise);
      }
      if (argc > 0 && CLASS_OF(a1) == rb_cComplex && a2 == Qundef) {
          return a1;
      }
      return nucomp_convert(rb_cComplex, a1, a2, raise);
  }
  
  #define imp1(n) \
  inline static VALUE \
  m_##n##_bang(VALUE x)\
  {\
      return rb_math_##n(x);\
  }
  
  imp1(cos)
  imp1(cosh)
  imp1(exp)
  
  static VALUE
  m_log_bang(VALUE x)
  {
      return rb_math_log(1, &x);
  }
  
  imp1(sin)
  imp1(sinh)
  
  static VALUE
  m_cos(VALUE x)
  {
      if (!RB_TYPE_P(x, T_COMPLEX))
  	return m_cos_bang(x);
      {
  	get_dat1(x);
  	return f_complex_new2(rb_cComplex,
  			      f_mul(m_cos_bang(dat->real),
  				    m_cosh_bang(dat->imag)),
  			      f_mul(f_negate(m_sin_bang(dat->real)),
  				    m_sinh_bang(dat->imag)));
      }
  }
  
  static VALUE
  m_sin(VALUE x)
  {
      if (!RB_TYPE_P(x, T_COMPLEX))
  	return m_sin_bang(x);
      {
  	get_dat1(x);
  	return f_complex_new2(rb_cComplex,
  			      f_mul(m_sin_bang(dat->real),
  				    m_cosh_bang(dat->imag)),
  			      f_mul(m_cos_bang(dat->real),
  				    m_sinh_bang(dat->imag)));
      }
  }
  
  static VALUE
  f_complex_polar(VALUE klass, VALUE x, VALUE y)
  {
      assert(!RB_TYPE_P(x, T_COMPLEX));
      assert(!RB_TYPE_P(y, T_COMPLEX));
      if (f_zero_p(x) || f_zero_p(y)) {
  	if (canonicalization) return x;
  	return nucomp_s_new_internal(klass, x, RFLOAT_0);
      }
      if (RB_FLOAT_TYPE_P(y)) {
  	const double arg = RFLOAT_VALUE(y);
  	if (arg == M_PI) {
  	    x = f_negate(x);
  	    if (canonicalization) return x;
  	    y = RFLOAT_0;
  	}
  	else if (arg == M_PI_2) {
  	    y = x;
  	    x = RFLOAT_0;
  	}
  	else if (arg == M_PI_2+M_PI) {
  	    y = f_negate(x);
  	    x = RFLOAT_0;
  	}
  	else if (RB_FLOAT_TYPE_P(x)) {
  	    const double abs = RFLOAT_VALUE(x);
  	    const double real = abs * cos(arg), imag = abs * sin(arg);
  	    x = DBL2NUM(real);
  	    if (canonicalization && imag == 0.0) return x;
  	    y = DBL2NUM(imag);
  	}
  	else {
  	    y = f_mul(x, DBL2NUM(sin(arg)));
  	    x = f_mul(x, DBL2NUM(cos(arg)));
  	    if (canonicalization && f_zero_p(y)) return x;
  	}
  	return nucomp_s_new_internal(klass, x, y);
      }
      return nucomp_s_canonicalize_internal(klass,
  					  f_mul(x, m_cos(y)),
  					  f_mul(x, m_sin(y)));
  }
  
  /* returns a Complex or Float of ang*PI-rotated abs */
  VALUE
  rb_dbl_complex_new_polar_pi(double abs, double ang)
  {
      double fi;
      const double fr = modf(ang, &fi);
      int pos = fr == +0.5;
  
      if (pos || fr == -0.5) {
  	if ((modf(fi / 2.0, &fi) != fr) ^ pos) abs = -abs;
  	return rb_complex_new(RFLOAT_0, DBL2NUM(abs));
      }
      else if (fr == 0.0) {
  	if (modf(fi / 2.0, &fi) != 0.0) abs = -abs;
  	return DBL2NUM(abs);
      }
      else {
  	ang *= M_PI;
  	return rb_complex_new(DBL2NUM(abs * cos(ang)), DBL2NUM(abs * sin(ang)));
      }
  }
  
  /*
   * call-seq:
   *    Complex.polar(abs[, arg])  ->  complex
   *
   * Returns a complex object which denotes the given polar form.
   *
   *    Complex.polar(3, 0)            #=> (3.0+0.0i)
   *    Complex.polar(3, Math::PI/2)   #=> (1.836909530733566e-16+3.0i)
   *    Complex.polar(3, Math::PI)     #=> (-3.0+3.673819061467132e-16i)
   *    Complex.polar(3, -Math::PI/2)  #=> (1.836909530733566e-16-3.0i)
   */
  static VALUE
  nucomp_s_polar(int argc, VALUE *argv, VALUE klass)
  {
      VALUE abs, arg;
  
      switch (rb_scan_args(argc, argv, "11", &abs, &arg)) {
        case 1:
  	nucomp_real_check(abs);
  	if (canonicalization) return abs;
  	return nucomp_s_new_internal(klass, abs, ZERO);
        default:
  	nucomp_real_check(abs);
  	nucomp_real_check(arg);
  	break;
      }
      return f_complex_polar(klass, abs, arg);
  }
  
  /*
   * call-seq:
   *    cmp.real  ->  real
   *
   * Returns the real part.
   *
   *    Complex(7).real      #=> 7
   *    Complex(9, -4).real  #=> 9
   */
  VALUE
  rb_complex_real(VALUE self)
  {
      get_dat1(self);
      return dat->real;
  }
  
  /*
   * call-seq:
   *    cmp.imag       ->  real
   *    cmp.imaginary  ->  real
   *
   * Returns the imaginary part.
   *
   *    Complex(7).imaginary      #=> 0
   *    Complex(9, -4).imaginary  #=> -4
   */
  VALUE
  rb_complex_imag(VALUE self)
  {
      get_dat1(self);
      return dat->imag;
  }
  
  /*
   * call-seq:
   *    -cmp  ->  complex
   *
   * Returns negation of the value.
   *
   *    -Complex(1, 2)  #=> (-1-2i)
   */
  VALUE
  rb_complex_uminus(VALUE self)
  {
      get_dat1(self);
      return f_complex_new2(CLASS_OF(self),
  			  f_negate(dat->real), f_negate(dat->imag));
  }
  
  /*
   * call-seq:
   *    cmp + numeric  ->  complex
   *
   * Performs addition.
   *
   *    Complex(2, 3)  + Complex(2, 3)   #=> (4+6i)
   *    Complex(900)   + Complex(1)      #=> (901+0i)
   *    Complex(-2, 9) + Complex(-9, 2)  #=> (-11+11i)
   *    Complex(9, 8)  + 4               #=> (13+8i)
   *    Complex(20, 9) + 9.8             #=> (29.8+9i)
   */
  VALUE
  rb_complex_plus(VALUE self, VALUE other)
  {
      if (RB_TYPE_P(other, T_COMPLEX)) {
  	VALUE real, imag;
  
  	get_dat2(self, other);
  
  	real = f_add(adat->real, bdat->real);
  	imag = f_add(adat->imag, bdat->imag);
  
  	return f_complex_new2(CLASS_OF(self), real, imag);
      }
      if (k_numeric_p(other) && f_real_p(other)) {
  	get_dat1(self);
  
  	return f_complex_new2(CLASS_OF(self),
  			      f_add(dat->real, other), dat->imag);
      }
      return rb_num_coerce_bin(self, other, '+');
  }
  
  /*
   * call-seq:
   *    cmp - numeric  ->  complex
   *
   * Performs subtraction.
   *
   *    Complex(2, 3)  - Complex(2, 3)   #=> (0+0i)
   *    Complex(900)   - Complex(1)      #=> (899+0i)
   *    Complex(-2, 9) - Complex(-9, 2)  #=> (7+7i)
   *    Complex(9, 8)  - 4               #=> (5+8i)
   *    Complex(20, 9) - 9.8             #=> (10.2+9i)
   */
  VALUE
  rb_complex_minus(VALUE self, VALUE other)
  {
      if (RB_TYPE_P(other, T_COMPLEX)) {
  	VALUE real, imag;
  
  	get_dat2(self, other);
  
  	real = f_sub(adat->real, bdat->real);
  	imag = f_sub(adat->imag, bdat->imag);
  
  	return f_complex_new2(CLASS_OF(self), real, imag);
      }
      if (k_numeric_p(other) && f_real_p(other)) {
  	get_dat1(self);
  
  	return f_complex_new2(CLASS_OF(self),
  			      f_sub(dat->real, other), dat->imag);
      }
      return rb_num_coerce_bin(self, other, '-');
  }
  
  static VALUE
  safe_mul(VALUE a, VALUE b, int az, int bz)
  {
      double v;
      if (!az && bz && RB_FLOAT_TYPE_P(a) && (v = RFLOAT_VALUE(a), !isnan(v))) {
  	a = signbit(v) ? DBL2NUM(-1.0) : DBL2NUM(1.0);
      }
      if (!bz && az && RB_FLOAT_TYPE_P(b) && (v = RFLOAT_VALUE(b), !isnan(v))) {
  	b = signbit(v) ? DBL2NUM(-1.0) : DBL2NUM(1.0);
      }
      return f_mul(a, b);
  }
  
  static void
  comp_mul(VALUE areal, VALUE aimag, VALUE breal, VALUE bimag, VALUE *real, VALUE *imag)
  {
      int arzero = f_zero_p(areal);
      int aizero = f_zero_p(aimag);
      int brzero = f_zero_p(breal);
      int bizero = f_zero_p(bimag);
      *real = f_sub(safe_mul(areal, breal, arzero, brzero),
                    safe_mul(aimag, bimag, aizero, bizero));
      *imag = f_add(safe_mul(areal, bimag, arzero, bizero),
                    safe_mul(aimag, breal, aizero, brzero));
  }
  
  /*
   * call-seq:
   *    cmp * numeric  ->  complex
   *
   * Performs multiplication.
   *
   *    Complex(2, 3)  * Complex(2, 3)   #=> (-5+12i)
   *    Complex(900)   * Complex(1)      #=> (900+0i)
   *    Complex(-2, 9) * Complex(-9, 2)  #=> (0-85i)
   *    Complex(9, 8)  * 4               #=> (36+32i)
   *    Complex(20, 9) * 9.8             #=> (196.0+88.2i)
   */
  VALUE
  rb_complex_mul(VALUE self, VALUE other)
  {
      if (RB_TYPE_P(other, T_COMPLEX)) {
  	VALUE real, imag;
  	get_dat2(self, other);
  
          comp_mul(adat->real, adat->imag, bdat->real, bdat->imag, &real, &imag);
  
  	return f_complex_new2(CLASS_OF(self), real, imag);
      }
      if (k_numeric_p(other) && f_real_p(other)) {
  	get_dat1(self);
  
  	return f_complex_new2(CLASS_OF(self),
  			      f_mul(dat->real, other),
  			      f_mul(dat->imag, other));
      }
      return rb_num_coerce_bin(self, other, '*');
  }
  
  inline static VALUE
  f_divide(VALUE self, VALUE other,
  	 VALUE (*func)(VALUE, VALUE), ID id)
  {
      if (RB_TYPE_P(other, T_COMPLEX)) {
          VALUE r, n, x, y;
  	int flo;
  	get_dat2(self, other);
  
  	flo = (RB_FLOAT_TYPE_P(adat->real) || RB_FLOAT_TYPE_P(adat->imag) ||
  	       RB_FLOAT_TYPE_P(bdat->real) || RB_FLOAT_TYPE_P(bdat->imag));
  
  	if (f_gt_p(f_abs(bdat->real), f_abs(bdat->imag))) {
  	    r = (*func)(bdat->imag, bdat->real);
  	    n = f_mul(bdat->real, f_add(ONE, f_mul(r, r)));
              x = (*func)(f_add(adat->real, f_mul(adat->imag, r)), n);
              y = (*func)(f_sub(adat->imag, f_mul(adat->real, r)), n);
  	}
  	else {
  	    r = (*func)(bdat->real, bdat->imag);
  	    n = f_mul(bdat->imag, f_add(ONE, f_mul(r, r)));
              x = (*func)(f_add(f_mul(adat->real, r), adat->imag), n);
              y = (*func)(f_sub(f_mul(adat->imag, r), adat->real), n);
  	}
          if (!flo) {
              x = rb_rational_canonicalize(x);
              y = rb_rational_canonicalize(y);
          }
          return f_complex_new2(CLASS_OF(self), x, y);
      }
      if (k_numeric_p(other) && f_real_p(other)) {
          VALUE x, y;
  	get_dat1(self);
          x = rb_rational_canonicalize((*func)(dat->real, other));
          y = rb_rational_canonicalize((*func)(dat->imag, other));
          return f_complex_new2(CLASS_OF(self), x, y);
      }
      return rb_num_coerce_bin(self, other, id);
  }
  
  #define rb_raise_zerodiv() rb_raise(rb_eZeroDivError, "divided by 0")
  
  /*
   * call-seq:
   *    cmp / numeric     ->  complex
   *    cmp.quo(numeric)  ->  complex
   *
   * Performs division.
   *
   *    Complex(2, 3)  / Complex(2, 3)   #=> ((1/1)+(0/1)*i)
   *    Complex(900)   / Complex(1)      #=> ((900/1)+(0/1)*i)
   *    Complex(-2, 9) / Complex(-9, 2)  #=> ((36/85)-(77/85)*i)
   *    Complex(9, 8)  / 4               #=> ((9/4)+(2/1)*i)
   *    Complex(20, 9) / 9.8             #=> (2.0408163265306123+0.9183673469387754i)
   */
  VALUE
  rb_complex_div(VALUE self, VALUE other)
  {
      return f_divide(self, other, f_quo, id_quo);
  }
  
  #define nucomp_quo rb_complex_div
  
  /*
   * call-seq:
   *    cmp.fdiv(numeric)  ->  complex
   *
   * Performs division as each part is a float, never returns a float.
   *
   *    Complex(11, 22).fdiv(3)  #=> (3.6666666666666665+7.333333333333333i)
   */
  static VALUE
  nucomp_fdiv(VALUE self, VALUE other)
  {
      return f_divide(self, other, f_fdiv, id_fdiv);
  }
  
  inline static VALUE
  f_reciprocal(VALUE x)
  {
      return f_quo(ONE, x);
  }
  
  /*
   * call-seq:
   *    cmp ** numeric  ->  complex
   *
   * Performs exponentiation.
   *
   *    Complex('i') ** 2              #=> (-1+0i)
   *    Complex(-8) ** Rational(1, 3)  #=> (1.0000000000000002+1.7320508075688772i)
   */
  VALUE
  rb_complex_pow(VALUE self, VALUE other)
  {
      if (k_numeric_p(other) && k_exact_zero_p(other))
  	return f_complex_new_bang1(CLASS_OF(self), ONE);
  
      if (RB_TYPE_P(other, T_RATIONAL) && RRATIONAL(other)->den == LONG2FIX(1))
  	other = RRATIONAL(other)->num; /* c14n */
  
      if (RB_TYPE_P(other, T_COMPLEX)) {
  	get_dat1(other);
  
  	if (k_exact_zero_p(dat->imag))
  	    other = dat->real; /* c14n */
      }
  
      if (RB_TYPE_P(other, T_COMPLEX)) {
  	VALUE r, theta, nr, ntheta;
  
  	get_dat1(other);
  
  	r = f_abs(self);
  	theta = f_arg(self);
  
  	nr = m_exp_bang(f_sub(f_mul(dat->real, m_log_bang(r)),
  			      f_mul(dat->imag, theta)));
  	ntheta = f_add(f_mul(theta, dat->real),
  		       f_mul(dat->imag, m_log_bang(r)));
  	return f_complex_polar(CLASS_OF(self), nr, ntheta);
      }
      if (FIXNUM_P(other)) {
          long n = FIX2LONG(other);
          if (n == 0) {
              return nucomp_s_new_internal(CLASS_OF(self), ONE, ZERO);
          }
          if (n < 0) {
              self = f_reciprocal(self);
              other = rb_int_uminus(other);
              n = -n;
          }
          {
              get_dat1(self);
              VALUE xr = dat->real, xi = dat->imag, zr = xr, zi = xi;
  
              if (f_zero_p(xi)) {
                  zr = rb_num_pow(zr, other);
              }
              else if (f_zero_p(xr)) {
                  zi = rb_num_pow(zi, other);
                  if (n & 2) zi = f_negate(zi);
                  if (!(n & 1)) {
                      VALUE tmp = zr;
                      zr = zi;
                      zi = tmp;
                  }
              }
              else {
                  while (--n) {
                      long q, r;
  
                      for (; q = n / 2, r = n % 2, r == 0; n = q) {
                          VALUE tmp = f_sub(f_mul(xr, xr), f_mul(xi, xi));
                          xi = f_mul(f_mul(TWO, xr), xi);
                          xr = tmp;
                      }
                      comp_mul(zr, zi, xr, xi, &zr, &zi);
                  }
              }
              return nucomp_s_new_internal(CLASS_OF(self), zr, zi);
  	}
      }
      if (k_numeric_p(other) && f_real_p(other)) {
  	VALUE r, theta;
  
  	if (RB_TYPE_P(other, T_BIGNUM))
  	    rb_warn("in a**b, b may be too big");
  
  	r = f_abs(self);
  	theta = f_arg(self);
  
  	return f_complex_polar(CLASS_OF(self), f_expt(r, other),
  			       f_mul(theta, other));
      }
      return rb_num_coerce_bin(self, other, id_expt);
  }
  
  /*
   * call-seq:
   *    cmp == object  ->  true or false
   *
   * Returns true if cmp equals object numerically.
   *
   *    Complex(2, 3)  == Complex(2, 3)   #=> true
   *    Complex(5)     == 5               #=> true
   *    Complex(0)     == 0.0             #=> true
   *    Complex('1/3') == 0.33            #=> false
   *    Complex('1/2') == '1/2'           #=> false
   */
  static VALUE
  nucomp_eqeq_p(VALUE self, VALUE other)
  {
      if (RB_TYPE_P(other, T_COMPLEX)) {
  	get_dat2(self, other);
  
  	return f_boolcast(f_eqeq_p(adat->real, bdat->real) &&
  			  f_eqeq_p(adat->imag, bdat->imag));
      }
      if (k_numeric_p(other) && f_real_p(other)) {
  	get_dat1(self);
  
  	return f_boolcast(f_eqeq_p(dat->real, other) && f_zero_p(dat->imag));
      }
      return f_boolcast(f_eqeq_p(other, self));
  }
  
  static bool
  nucomp_real_p(VALUE self)
  {
      get_dat1(self);
      return(f_zero_p(dat->imag) ? true : false);
  }
  
  /*
   * call-seq:
   *    cmp <=> object  ->  0, 1, -1, or nil
   *
   * If +cmp+'s imaginary part is zero, and +object+ is also a
   * real number (or a Complex number where the imaginary part is zero),
   * compare the real part of +cmp+ to object.  Otherwise, return nil.
   *
   *    Complex(2, 3)  <=> Complex(2, 3)   #=> nil
   *    Complex(2, 3)  <=> 1               #=> nil
   *    Complex(2)     <=> 1               #=> 1
   *    Complex(2)     <=> 2               #=> 0
   *    Complex(2)     <=> 3               #=> -1
   */
  static VALUE
  nucomp_cmp(VALUE self, VALUE other)
  {
      if (nucomp_real_p(self) && k_numeric_p(other)) {
          if (RB_TYPE_P(other, T_COMPLEX) && nucomp_real_p(other)) {
              get_dat2(self, other);
              return rb_funcall(adat->real, idCmp, 1, bdat->real);
          }
          else if (f_real_p(other)) {
              get_dat1(self);
              return rb_funcall(dat->real, idCmp, 1, other);
          }
      }
      return Qnil;
  }
  
  /* :nodoc: */
  static VALUE
  nucomp_coerce(VALUE self, VALUE other)
  {
      if (RB_TYPE_P(other, T_COMPLEX))
  	return rb_assoc_new(other, self);
      if (k_numeric_p(other) && f_real_p(other))
          return rb_assoc_new(f_complex_new_bang1(CLASS_OF(self), other), self);
  
      rb_raise(rb_eTypeError, "%"PRIsVALUE" can't be coerced into %"PRIsVALUE,
  	     rb_obj_class(other), rb_obj_class(self));
      return Qnil;
  }
  
  /*
   * call-seq:
   *    cmp.abs        ->  real
   *    cmp.magnitude  ->  real
   *
   * Returns the absolute part of its polar form.
   *
   *    Complex(-1).abs         #=> 1
   *    Complex(3.0, -4.0).abs  #=> 5.0
   */
  VALUE
  rb_complex_abs(VALUE self)
  {
      get_dat1(self);
  
      if (f_zero_p(dat->real)) {
  	VALUE a = f_abs(dat->imag);
  	if (RB_FLOAT_TYPE_P(dat->real) && !RB_FLOAT_TYPE_P(dat->imag))
  	    a = f_to_f(a);
  	return a;
      }
      if (f_zero_p(dat->imag)) {
  	VALUE a = f_abs(dat->real);
  	if (!RB_FLOAT_TYPE_P(dat->real) && RB_FLOAT_TYPE_P(dat->imag))
  	    a = f_to_f(a);
  	return a;
      }
      return rb_math_hypot(dat->real, dat->imag);
  }
  
  /*
   * call-seq:
   *    cmp.abs2  ->  real
   *
   * Returns square of the absolute value.
   *
   *    Complex(-1).abs2         #=> 1
   *    Complex(3.0, -4.0).abs2  #=> 25.0
   */
  static VALUE
  nucomp_abs2(VALUE self)
  {
      get_dat1(self);
      return f_add(f_mul(dat->real, dat->real),
  		 f_mul(dat->imag, dat->imag));
  }
  
  /*
   * call-seq:
   *    cmp.arg    ->  float
   *    cmp.angle  ->  float
   *    cmp.phase  ->  float
   *
   * Returns the angle part of its polar form.
   *
   *    Complex.polar(3, Math::PI/2).arg  #=> 1.5707963267948966
   */
  VALUE
  rb_complex_arg(VALUE self)
  {
      get_dat1(self);
      return rb_math_atan2(dat->imag, dat->real);
  }
  
  /*
   * call-seq:
   *    cmp.rect         ->  array
   *    cmp.rectangular  ->  array
   *
   * Returns an array; [cmp.real, cmp.imag].
   *
   *    Complex(1, 2).rectangular  #=> [1, 2]
   */
  static VALUE
  nucomp_rect(VALUE self)
  {
      get_dat1(self);
      return rb_assoc_new(dat->real, dat->imag);
  }
  
  /*
   * call-seq:
   *    cmp.polar  ->  array
   *
   * Returns an array; [cmp.abs, cmp.arg].
   *
   *    Complex(1, 2).polar  #=> [2.23606797749979, 1.1071487177940904]
   */
  static VALUE
  nucomp_polar(VALUE self)
  {
      return rb_assoc_new(f_abs(self), f_arg(self));
  }
  
  /*
   * call-seq:
   *    cmp.conj       ->  complex
   *    cmp.conjugate  ->  complex
   *
   * Returns the complex conjugate.
   *
   *    Complex(1, 2).conjugate  #=> (1-2i)
   */
  VALUE
  rb_complex_conjugate(VALUE self)
  {
      get_dat1(self);
      return f_complex_new2(CLASS_OF(self), dat->real, f_negate(dat->imag));
  }
  
  /*
   * call-seq:
   *    Complex(1).real?     ->  false
   *    Complex(1, 2).real?  ->  false
   *
   * Returns false, even if the complex number has no imaginary part.
   */
  static VALUE
  nucomp_false(VALUE self)
  {
      return Qfalse;
  }
  
  /*
   * call-seq:
   *    cmp.denominator  ->  integer
   *
   * Returns the denominator (lcm of both denominator - real and imag).
   *
   * See numerator.
   */
  static VALUE
  nucomp_denominator(VALUE self)
  {
      get_dat1(self);
      return rb_lcm(f_denominator(dat->real), f_denominator(dat->imag));
  }
  
  /*
   * call-seq:
   *    cmp.numerator  ->  numeric
   *
   * Returns the numerator.
   *
   *        1   2       3+4i  <-  numerator
   *        - + -i  ->  ----
   *        2   3        6    <-  denominator
   *
   *    c = Complex('1/2+2/3i')  #=> ((1/2)+(2/3)*i)
   *    n = c.numerator          #=> (3+4i)
   *    d = c.denominator        #=> 6
   *    n / d                    #=> ((1/2)+(2/3)*i)
   *    Complex(Rational(n.real, d), Rational(n.imag, d))
   *                             #=> ((1/2)+(2/3)*i)
   * See denominator.
   */
  static VALUE
  nucomp_numerator(VALUE self)
  {
      VALUE cd;
  
      get_dat1(self);
  
      cd = nucomp_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->imag),
  				f_div(cd, f_denominator(dat->imag))));
  }
  
  /* :nodoc: */
  static VALUE
  nucomp_hash(VALUE self)
  {
      st_index_t v, h[2];
      VALUE n;
  
      get_dat1(self);
      n = rb_hash(dat->real);
      h[0] = NUM2LONG(n);
      n = rb_hash(dat->imag);
      h[1] = NUM2LONG(n);
      v = rb_memhash(h, sizeof(h));
      return ST2FIX(v);
  }
  
  /* :nodoc: */
  static VALUE
  nucomp_eql_p(VALUE self, VALUE other)
  {
      if (RB_TYPE_P(other, T_COMPLEX)) {
  	get_dat2(self, other);
  
  	return f_boolcast((CLASS_OF(adat->real) == CLASS_OF(bdat->real)) &&
  			  (CLASS_OF(adat->imag) == CLASS_OF(bdat->imag)) &&
  			  f_eqeq_p(self, other));
  
      }
      return Qfalse;
  }
  
  inline static int
  f_signbit(VALUE x)
  {
      if (RB_FLOAT_TYPE_P(x)) {
  	double f = RFLOAT_VALUE(x);
  	return !isnan(f) && signbit(f);
      }
      return f_negative_p(x);
  }
  
  inline static int
  f_tpositive_p(VALUE x)
  {
      return !f_signbit(x);
  }
  
  static VALUE
  f_format(VALUE self, VALUE (*func)(VALUE))
  {
      VALUE s;
      int impos;
  
      get_dat1(self);
  
      impos = f_tpositive_p(dat->imag);
  
      s = (*func)(dat->real);
      rb_str_cat2(s, !impos ? "-" : "+");
  
      rb_str_concat(s, (*func)(f_abs(dat->imag)));
      if (!rb_isdigit(RSTRING_PTR(s)[RSTRING_LEN(s) - 1]))
  	rb_str_cat2(s, "*");
      rb_str_cat2(s, "i");
  
      return s;
  }
  
  /*
   * call-seq:
   *    cmp.to_s  ->  string
   *
   * Returns the value as a string.
   *
   *    Complex(2).to_s                       #=> "2+0i"
   *    Complex('-8/6').to_s                  #=> "-4/3+0i"
   *    Complex('1/2i').to_s                  #=> "0+1/2i"
   *    Complex(0, Float::INFINITY).to_s      #=> "0+Infinity*i"
   *    Complex(Float::NAN, Float::NAN).to_s  #=> "NaN+NaN*i"
   */
  static VALUE
  nucomp_to_s(VALUE self)
  {
      return f_format(self, rb_String);
  }
  
  /*
   * call-seq:
   *    cmp.inspect  ->  string
   *
   * Returns the value as a string for inspection.
   *
   *    Complex(2).inspect                       #=> "(2+0i)"
   *    Complex('-8/6').inspect                  #=> "((-4/3)+0i)"
   *    Complex('1/2i').inspect                  #=> "(0+(1/2)*i)"
   *    Complex(0, Float::INFINITY).inspect      #=> "(0+Infinity*i)"
   *    Complex(Float::NAN, Float::NAN).inspect  #=> "(NaN+NaN*i)"
   */
  static VALUE
  nucomp_inspect(VALUE self)
  {
      VALUE s;
  
      s = rb_usascii_str_new2("(");
      rb_str_concat(s, f_format(self, rb_inspect));
      rb_str_cat2(s, ")");
  
      return s;
  }
  
  #define FINITE_TYPE_P(v) (RB_INTEGER_TYPE_P(v) || RB_TYPE_P(v, T_RATIONAL))
  
  /*
   * call-seq:
   *    cmp.finite?  ->  true or false
   *
   * Returns +true+ if +cmp+'s real and imaginary parts are both finite numbers,
   * otherwise returns +false+.
   */
  static VALUE
  rb_complex_finite_p(VALUE self)
  {
      get_dat1(self);
  
      if (f_finite_p(dat->real) && f_finite_p(dat->imag)) {
  	return Qtrue;
      }
      return Qfalse;
  }
  
  /*
   * call-seq:
   *    cmp.infinite?  ->  nil or 1
   *
   * Returns +1+ if +cmp+'s real or imaginary part is an infinite number,
   * otherwise returns +nil+.
   *
   *  For example:
   *
   *     (1+1i).infinite?                   #=> nil
   *     (Float::INFINITY + 1i).infinite?   #=> 1
   */
  static VALUE
  rb_complex_infinite_p(VALUE self)
  {
      get_dat1(self);
  
      if (NIL_P(f_infinite_p(dat->real)) && NIL_P(f_infinite_p(dat->imag))) {
  	return Qnil;
      }
      return ONE;
  }
  
  /* :nodoc: */
  static VALUE
  nucomp_dumper(VALUE self)
  {
      return self;
  }
  
  /* :nodoc: */
  static VALUE
  nucomp_loader(VALUE self, VALUE a)
  {
      get_dat1(self);
  
      RCOMPLEX_SET_REAL(dat, rb_ivar_get(a, id_i_real));
      RCOMPLEX_SET_IMAG(dat, rb_ivar_get(a, id_i_imag));
      OBJ_FREEZE_RAW(self);
  
      return self;
  }
  
  /* :nodoc: */
  static VALUE
  nucomp_marshal_dump(VALUE self)
  {
      VALUE a;
      get_dat1(self);
  
      a = rb_assoc_new(dat->real, dat->imag);
      rb_copy_generic_ivar(a, self);
      return a;
  }
  
  /* :nodoc: */
  static VALUE
  nucomp_marshal_load(VALUE self, VALUE a)
  {
      Check_Type(a, T_ARRAY);
      if (RARRAY_LEN(a) != 2)
  	rb_raise(rb_eArgError, "marshaled complex must have an array whose length is 2 but %ld", RARRAY_LEN(a));
      rb_ivar_set(self, id_i_real, RARRAY_AREF(a, 0));
      rb_ivar_set(self, id_i_imag, RARRAY_AREF(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);
  }
  
  VALUE
  rb_complex_new_polar(VALUE x, VALUE y)
  {
      return f_complex_polar(rb_cComplex, x, y);
  }
  
  VALUE
  rb_complex_polar(VALUE x, VALUE y)
  {
      return rb_complex_new_polar(x, y);
  }
  
  VALUE
  rb_Complex(VALUE x, VALUE y)
  {
      VALUE a[2];
      a[0] = x;
      a[1] = y;
      return nucomp_s_convert(2, a, rb_cComplex);
  }
  
  /*!
   * Creates a Complex object.
   *
   * \param real    real part value
   * \param imag    imaginary part value
   * \return        a new Complex object
   */
  VALUE
  rb_dbl_complex_new(double real, double imag)
  {
      return rb_complex_raw(DBL2NUM(real), DBL2NUM(imag));
  }
  
  /*
   * call-seq:
   *    cmp.to_i  ->  integer
   *
   * Returns the value as an integer if possible (the imaginary part
   * should be exactly zero).
   *
   *    Complex(1, 0).to_i    #=> 1
   *    Complex(1, 0.0).to_i  # RangeError
   *    Complex(1, 2).to_i    # RangeError
   */
  static VALUE
  nucomp_to_i(VALUE self)
  {
      get_dat1(self);
  
      if (!k_exact_zero_p(dat->imag)) {
  	rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Integer",
  		 self);
      }
      return f_to_i(dat->real);
  }
  
  /*
   * call-seq:
   *    cmp.to_f  ->  float
   *
   * Returns the value as a float if possible (the imaginary part should
   * be exactly zero).
   *
   *    Complex(1, 0).to_f    #=> 1.0
   *    Complex(1, 0.0).to_f  # RangeError
   *    Complex(1, 2).to_f    # RangeError
   */
  static VALUE
  nucomp_to_f(VALUE self)
  {
      get_dat1(self);
  
      if (!k_exact_zero_p(dat->imag)) {
  	rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Float",
  		 self);
      }
      return f_to_f(dat->real);
  }
  
  /*
   * call-seq:
   *    cmp.to_r  ->  rational
   *
   * Returns the value as a rational if possible (the imaginary part
   * should be exactly zero).
   *
   *    Complex(1, 0).to_r    #=> (1/1)
   *    Complex(1, 0.0).to_r  # RangeError
   *    Complex(1, 2).to_r    # RangeError
   *
   * See rationalize.
   */
  static VALUE
  nucomp_to_r(VALUE self)
  {
      get_dat1(self);
  
      if (!k_exact_zero_p(dat->imag)) {
  	rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Rational",
  		 self);
      }
      return f_to_r(dat->real);
  }
  
  /*
   * call-seq:
   *    cmp.rationalize([eps])  ->  rational
   *
   * Returns the value as a rational if possible (the imaginary part
   * should be exactly zero).
   *
   *    Complex(1.0/3, 0).rationalize  #=> (1/3)
   *    Complex(1, 0.0).rationalize    # RangeError
   *    Complex(1, 2).rationalize      # RangeError
   *
   * See to_r.
   */
  static VALUE
  nucomp_rationalize(int argc, VALUE *argv, VALUE self)
  {
      get_dat1(self);
  
      rb_check_arity(argc, 0, 1);
  
      if (!k_exact_zero_p(dat->imag)) {
         rb_raise(rb_eRangeError, "can't convert %"PRIsVALUE" into Rational",
                  self);
      }
      return rb_funcallv(dat->real, id_rationalize, argc, argv);
  }
  
  /*
   * call-seq:
   *    complex.to_c  ->  self
   *
   * Returns self.
   *
   *    Complex(2).to_c      #=> (2+0i)
   *    Complex(-8, 6).to_c  #=> (-8+6i)
   */
  static VALUE
  nucomp_to_c(VALUE self)
  {
      return self;
  }
  
  /*
   * call-seq:
   *    nil.to_c  ->  (0+0i)
   *
   * Returns zero as a complex.
   */
  static VALUE
  nilclass_to_c(VALUE self)
  {
      return rb_complex_new1(INT2FIX(0));
  }
  
  /*
   * call-seq:
   *    num.to_c  ->  complex
   *
   * Returns the value as a complex.
   */
  static VALUE
  numeric_to_c(VALUE self)
  {
      return rb_complex_new1(self);
  }
  
  #include <ctype.h>
  
  inline static int
  issign(int c)
  {
      return (c == '-' || c == '+');
  }
  
  static int
  read_sign(const char **s,
  	  char **b)
  {
      int sign = '?';
  
      if (issign(**s)) {
  	sign = **b = **s;
  	(*s)++;
  	(*b)++;
      }
      return sign;
  }
  
  inline static int
  isdecimal(int c)
  {
      return isdigit((unsigned char)c);
  }
  
  static int
  read_digits(const char **s, int strict,
  	    char **b)
  {
      int us = 1;
  
      if (!isdecimal(**s))
  	return 0;
  
      while (isdecimal(**s) || **s == '_') {
  	if (**s == '_') {
  	    if (strict) {
  		if (us)
  		    return 0;
  	    }
  	    us = 1;
  	}
  	else {
  	    **b = **s;
  	    (*b)++;
  	    us = 0;
  	}
  	(*s)++;
      }
      if (us)
  	do {
  	    (*s)--;
  	} while (**s == '_');
      return 1;
  }
  
  inline static int
  islettere(int c)
  {
      return (c == 'e' || c == 'E');
  }
  
  static int
  read_num(const char **s, int strict,
  	 char **b)
  {
      if (**s != '.') {
  	if (!read_digits(s, strict, b))
  	    return 0;
      }
  
      if (**s == '.') {
  	**b = **s;
  	(*s)++;
  	(*b)++;
  	if (!read_digits(s, strict, b)) {
  	    (*b)--;
  	    return 0;
  	}
      }
  
      if (islettere(**s)) {
  	**b = **s;
  	(*s)++;
  	(*b)++;
  	read_sign(s, b);
  	if (!read_digits(s, strict, b)) {
  	    (*b)--;
  	    return 0;
  	}
      }
      return 1;
  }
  
  inline static int
  read_den(const char **s, int strict,
  	 char **b)
  {
      if (!read_digits(s, strict, b))
  	return 0;
      return 1;
  }
  
  static int
  read_rat_nos(const char **s, int strict,
  	     char **b)
  {
      if (!read_num(s, strict, b))
  	return 0;
      if (**s == '/') {
  	**b = **s;
  	(*s)++;
  	(*b)++;
  	if (!read_den(s, strict, b)) {
  	    (*b)--;
  	    return 0;
  	}
      }
      return 1;
  }
  
  static int
  read_rat(const char **s, int strict,
  	 char **b)
  {
      read_sign(s, b);
      if (!read_rat_nos(s, strict, b))
  	return 0;
      return 1;
  }
  
  inline static int
  isimagunit(int c)
  {
      return (c == 'i' || c == 'I' ||
  	    c == 'j' || c == 'J');
  }
  
  static VALUE
  str2num(char *s)
  {
      if (strchr(s, '/'))
  	return rb_cstr_to_rat(s, 0);
      if (strpbrk(s, ".eE"))
  	return DBL2NUM(rb_cstr_to_dbl(s, 0));
      return rb_cstr_to_inum(s, 10, 0);
  }
  
  static int
  read_comp(const char **s, int strict,
  	  VALUE *ret, char **b)
  {
      char *bb;
      int sign;
      VALUE num, num2;
  
      bb = *b;
  
      sign = read_sign(s, b);
  
      if (isimagunit(**s)) {
  	(*s)++;
  	num = INT2FIX((sign == '-') ? -1 : + 1);
  	*ret = rb_complex_new2(ZERO, num);
  	return 1; /* e.g. "i" */
      }
  
      if (!read_rat_nos(s, strict, b)) {
  	**b = '\0';
  	num = str2num(bb);
  	*ret = rb_complex_new2(num, ZERO);
  	return 0; /* e.g. "-" */
      }
      **b = '\0';
      num = str2num(bb);
  
      if (isimagunit(**s)) {
  	(*s)++;
  	*ret = rb_complex_new2(ZERO, num);
  	return 1; /* e.g. "3i" */
      }
  
      if (**s == '@') {
  	int st;
  
  	(*s)++;
  	bb = *b;
  	st = read_rat(s, strict, b);
  	**b = '\0';
  	if (strlen(bb) < 1 ||
  	    !isdecimal(*(bb + strlen(bb) - 1))) {
  	    *ret = rb_complex_new2(num, ZERO);
  	    return 0; /* e.g. "1@-" */
  	}
  	num2 = str2num(bb);
  	*ret = rb_complex_new_polar(num, num2);
  	if (!st)
  	    return 0; /* e.g. "1@2." */
  	else
  	    return 1; /* e.g. "1@2" */
      }
  
      if (issign(**s)) {
  	bb = *b;
  	sign = read_sign(s, b);
  	if (isimagunit(**s))
  	    num2 = INT2FIX((sign == '-') ? -1 : + 1);
  	else {
  	    if (!read_rat_nos(s, strict, b)) {
  		*ret = rb_complex_new2(num, ZERO);
  		return 0; /* e.g. "1+xi" */
  	    }
  	    **b = '\0';
  	    num2 = str2num(bb);
  	}
  	if (!isimagunit(**s)) {
  	    *ret = rb_complex_new2(num, ZERO);
  	    return 0; /* e.g. "1+3x" */
  	}
  	(*s)++;
  	*ret = rb_complex_new2(num, num2);
  	return 1; /* e.g. "1+2i" */
      }
      /* !(@, - or +) */
      {
  	*ret = rb_complex_new2(num, ZERO);
  	return 1; /* e.g. "3" */
      }
  }
  
  inline static void
  skip_ws(const char **s)
  {
      while (isspace((unsigned char)**s))
  	(*s)++;
  }
  
  static int
  parse_comp(const char *s, int strict, VALUE *num)
  {
      char *buf, *b;
      VALUE tmp;
      int ret = 1;
  
      buf = ALLOCV_N(char, tmp, strlen(s) + 1);
      b = buf;
  
      skip_ws(&s);
      if (!read_comp(&s, strict, num, &b)) {
          ret = 0;
      }
      else {
          skip_ws(&s);
  
          if (strict)
              if (*s != '\0')
                  ret = 0;
      }
      ALLOCV_END(tmp);
  
      return ret;
  }
  
  static VALUE
  string_to_c_strict(VALUE self, int raise)
  {
      char *s;
      VALUE num;
  
      rb_must_asciicompat(self);
  
      s = RSTRING_PTR(self);
  
      if (!s || memchr(s, '\0', RSTRING_LEN(self))) {
          if (!raise) return Qnil;
  	rb_raise(rb_eArgError, "string contains null byte");
      }
  
      if (s && s[RSTRING_LEN(self)]) {
  	rb_str_modify(self);
  	s = RSTRING_PTR(self);
  	s[RSTRING_LEN(self)] = '\0';
      }
  
      if (!s)
  	s = (char *)"";
  
      if (!parse_comp(s, 1, &num)) {
          if (!raise) return Qnil;
  	rb_raise(rb_eArgError, "invalid value for convert(): %+"PRIsVALUE,
  		 self);
      }
  
      return num;
  }
  
  /*
   * call-seq:
   *    str.to_c  ->  complex
   *
   * Returns a complex 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.
   *
   *    '9'.to_c           #=> (9+0i)
   *    '2.5'.to_c         #=> (2.5+0i)
   *    '2.5/1'.to_c       #=> ((5/2)+0i)
   *    '-3/2'.to_c        #=> ((-3/2)+0i)
   *    '-i'.to_c          #=> (0-1i)
   *    '45i'.to_c         #=> (0+45i)
   *    '3-4i'.to_c        #=> (3-4i)
   *    '-4e2-4e-2i'.to_c  #=> (-400.0-0.04i)
   *    '-0.0-0.0i'.to_c   #=> (-0.0-0.0i)
   *    '1/2+3/4i'.to_c    #=> ((1/2)+(3/4)*i)
   *    'ruby'.to_c        #=> (0+0i)
   *
   * See Kernel.Complex.
   */
  static VALUE
  string_to_c(VALUE self)
  {
      char *s;
      VALUE num;
  
      rb_must_asciicompat(self);
  
      s = RSTRING_PTR(self);
  
      if (s && s[RSTRING_LEN(self)]) {
  	rb_str_modify(self);
  	s = RSTRING_PTR(self);
  	s[RSTRING_LEN(self)] = '\0';
      }
  
      if (!s)
  	s = (char *)"";
  
      (void)parse_comp(s, 0, &num);
  
      return num;
  }
  
  static VALUE
  to_complex(VALUE val)
  {
      return rb_convert_type(val, T_COMPLEX, "Complex", "to_c");
  }
  
  static VALUE
  nucomp_convert(VALUE klass, VALUE a1, VALUE a2, int raise)
  {
      if (NIL_P(a1) || NIL_P(a2)) {
          if (!raise) return Qnil;
  	rb_raise(rb_eTypeError, "can't convert nil into Complex");
      }
  
      if (RB_TYPE_P(a1, T_STRING)) {
  	a1 = string_to_c_strict(a1, raise);
          if (NIL_P(a1)) return Qnil;
      }
  
      if (RB_TYPE_P(a2, T_STRING)) {
  	a2 = string_to_c_strict(a2, raise);
          if (NIL_P(a2)) return Qnil;
      }
  
      if (RB_TYPE_P(a1, T_COMPLEX)) {
  	{
  	    get_dat1(a1);
  
  	    if (k_exact_zero_p(dat->imag))
  		a1 = dat->real;
  	}
      }
  
      if (RB_TYPE_P(a2, T_COMPLEX)) {
  	{
  	    get_dat1(a2);
  
  	    if (k_exact_zero_p(dat->imag))
  		a2 = dat->real;
  	}
      }
  
      if (RB_TYPE_P(a1, T_COMPLEX)) {
  	if (a2 == Qundef || (k_exact_zero_p(a2)))
  	    return a1;
      }
  
      if (a2 == Qundef) {
  	if (k_numeric_p(a1) && !f_real_p(a1))
  	    return a1;
  	/* should raise exception for consistency */
  	if (!k_numeric_p(a1)) {
              if (!raise)
                  return rb_protect(to_complex, a1, NULL);
  	    return to_complex(a1);
          }
      }
      else {
  	if ((k_numeric_p(a1) && k_numeric_p(a2)) &&
  	    (!f_real_p(a1) || !f_real_p(a2)))
  	    return f_add(a1,
  			 f_mul(a2,
  			       f_complex_new_bang2(rb_cComplex, ZERO, ONE)));
      }
  
      {
          int argc;
  	VALUE argv2[2];
  	argv2[0] = a1;
          if (a2 == Qundef) {
              argv2[1] = Qnil;
              argc = 1;
          }
          else {
              if (!raise && !RB_INTEGER_TYPE_P(a2) && !RB_FLOAT_TYPE_P(a2) && !RB_TYPE_P(a2, T_RATIONAL))
                  return Qnil;
              argv2[1] = a2;
              argc = 2;
          }
  	return nucomp_s_new(argc, argv2, klass);
      }
  }
  
  static VALUE
  nucomp_s_convert(int argc, VALUE *argv, VALUE klass)
  {
      VALUE a1, a2;
  
      if (rb_scan_args(argc, argv, "11", &a1, &a2) == 1) {
          a2 = Qundef;
      }
  
      return nucomp_convert(klass, a1, a2, TRUE);
  }
  
  /* --- */
  
  /*
   * call-seq:
   *    num.real  ->  self
   *
   * Returns self.
   */
  static VALUE
  numeric_real(VALUE self)
  {
      return self;
  }
  
  /*
   * call-seq:
   *    num.imag       ->  0
   *    num.imaginary  ->  0
   *
   * Returns zero.
   */
  static VALUE
  numeric_imag(VALUE self)
  {
      return INT2FIX(0);
  }
  
  /*
   * call-seq:
   *    num.abs2  ->  real
   *
   * Returns square of self.
   */
  static VALUE
  numeric_abs2(VALUE self)
  {
      return f_mul(self, self);
  }
  
  /*
   * call-seq:
   *    num.arg    ->  0 or float
   *    num.angle  ->  0 or float
   *    num.phase  ->  0 or float
   *
   * Returns 0 if the value is positive, pi otherwise.
   */
  static VALUE
  numeric_arg(VALUE self)
  {
      if (f_positive_p(self))
          return INT2FIX(0);
      return DBL2NUM(M_PI);
  }
  
  /*
   * call-seq:
   *    num.rect  ->  array
   *    num.rectangular  ->  array
   *
   * Returns an array; [num, 0].
   */
  static VALUE
  numeric_rect(VALUE self)
  {
      return rb_assoc_new(self, INT2FIX(0));
  }
  
  static VALUE float_arg(VALUE self);
  
  /*
   * call-seq:
   *    num.polar  ->  array
   *
   * Returns an array; [num.abs, num.arg].
   */
  static VALUE
  numeric_polar(VALUE self)
  {
      VALUE abs, arg;
  
      if (RB_INTEGER_TYPE_P(self)) {
          abs = rb_int_abs(self);
          arg = numeric_arg(self);
      }
      else if (RB_FLOAT_TYPE_P(self)) {
          abs = rb_float_abs(self);
          arg = float_arg(self);
      }
      else if (RB_TYPE_P(self, T_RATIONAL)) {
          abs = rb_rational_abs(self);
          arg = numeric_arg(self);
      }
      else {
          abs = f_abs(self);
          arg = f_arg(self);
      }
      return rb_assoc_new(abs, arg);
  }
  
  /*
   * call-seq:
   *    num.conj       ->  self
   *    num.conjugate  ->  self
   *
   * Returns self.
   */
  static VALUE
  numeric_conj(VALUE self)
  {
      return self;
  }
  
  /*
   * call-seq:
   *    flo.arg    ->  0 or float
   *    flo.angle  ->  0 or float
   *    flo.phase  ->  0 or float
   *
   * Returns 0 if the value is positive, pi otherwise.
   */
  static VALUE
  float_arg(VALUE self)
  {
      if (isnan(RFLOAT_VALUE(self)))
  	return self;
      if (f_tpositive_p(self))
  	return INT2FIX(0);
      return rb_const_get(rb_mMath, id_PI);
  }
  
  /*
   * A complex number can be represented as a paired real number with
   * imaginary unit; a+bi.  Where a is real part, b is imaginary part
   * and i is imaginary unit.  Real a equals complex a+0i
   * mathematically.
   *
   * Complex object can be created as literal, and also by using
   * Kernel#Complex, Complex::rect, Complex::polar or to_c method.
   *
   *    2+1i                 #=> (2+1i)
   *    Complex(1)           #=> (1+0i)
   *    Complex(2, 3)        #=> (2+3i)
   *    Complex.polar(2, 3)  #=> (-1.9799849932008908+0.2822400161197344i)
   *    3.to_c               #=> (3+0i)
   *
   * You can also create complex object from floating-point numbers or
   * strings.
   *
   *    Complex(0.3)         #=> (0.3+0i)
   *    Complex('0.3-0.5i')  #=> (0.3-0.5i)
   *    Complex('2/3+3/4i')  #=> ((2/3)+(3/4)*i)
   *    Complex('1@2')       #=> (-0.4161468365471424+0.9092974268256817i)
   *
   *    0.3.to_c             #=> (0.3+0i)
   *    '0.3-0.5i'.to_c      #=> (0.3-0.5i)
   *    '2/3+3/4i'.to_c      #=> ((2/3)+(3/4)*i)
   *    '1@2'.to_c           #=> (-0.4161468365471424+0.9092974268256817i)
   *
   * A complex object is either an exact or an inexact number.
   *
   *    Complex(1, 1) / 2    #=> ((1/2)+(1/2)*i)
   *    Complex(1, 1) / 2.0  #=> (0.5+0.5i)
   */
  void
  Init_Complex(void)
  {
      VALUE compat;
  #undef rb_intern
  #define rb_intern(str) rb_intern_const(str)
  
      id_abs = rb_intern("abs");
      id_arg = rb_intern("arg");
      id_denominator = rb_intern("denominator");
      id_numerator = rb_intern("numerator");
      id_real_p = rb_intern("real?");
      id_i_real = rb_intern("@real");
      id_i_imag = rb_intern("@image"); /* @image, not @imag */
      id_finite_p = rb_intern("finite?");
      id_infinite_p = rb_intern("infinite?");
      id_rationalize = rb_intern("rationalize");
      id_PI = rb_intern("PI");
  
      rb_cComplex = rb_define_class("Complex", rb_cNumeric);
  
      rb_define_alloc_func(rb_cComplex, nucomp_s_alloc);
      rb_undef_method(CLASS_OF(rb_cComplex), "allocate");
  
      rb_undef_method(CLASS_OF(rb_cComplex), "new");
  
      rb_define_singleton_method(rb_cComplex, "rectangular", nucomp_s_new, -1);
      rb_define_singleton_method(rb_cComplex, "rect", nucomp_s_new, -1);
      rb_define_singleton_method(rb_cComplex, "polar", nucomp_s_polar, -1);
  
      rb_define_global_function("Complex", nucomp_f_complex, -1);
  
      rb_undef_methods_from(rb_cComplex, rb_mComparable);
      rb_undef_method(rb_cComplex, "%");
      rb_undef_method(rb_cComplex, "div");
      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, "remainder");
      rb_undef_method(rb_cComplex, "round");
      rb_undef_method(rb_cComplex, "step");
      rb_undef_method(rb_cComplex, "truncate");
      rb_undef_method(rb_cComplex, "i");
  
      rb_define_method(rb_cComplex, "real", rb_complex_real, 0);
      rb_define_method(rb_cComplex, "imaginary", rb_complex_imag, 0);
      rb_define_method(rb_cComplex, "imag", rb_complex_imag, 0);
  
      rb_define_method(rb_cComplex, "-@", rb_complex_uminus, 0);
      rb_define_method(rb_cComplex, "+", rb_complex_plus, 1);
      rb_define_method(rb_cComplex, "-", rb_complex_minus, 1);
      rb_define_method(rb_cComplex, "*", rb_complex_mul, 1);
      rb_define_method(rb_cComplex, "/", rb_complex_div, 1);
      rb_define_method(rb_cComplex, "quo", nucomp_quo, 1);
      rb_define_method(rb_cComplex, "fdiv", nucomp_fdiv, 1);
      rb_define_method(rb_cComplex, "**", rb_complex_pow, 1);
  
      rb_define_method(rb_cComplex, "==", nucomp_eqeq_p, 1);
      rb_define_method(rb_cComplex, "<=>", nucomp_cmp, 1);
      rb_define_method(rb_cComplex, "coerce", nucomp_coerce, 1);
  
      rb_define_method(rb_cComplex, "abs", rb_complex_abs, 0);
      rb_define_method(rb_cComplex, "magnitude", rb_complex_abs, 0);
      rb_define_method(rb_cComplex, "abs2", nucomp_abs2, 0);
      rb_define_method(rb_cComplex, "arg", rb_complex_arg, 0);
      rb_define_method(rb_cComplex, "angle", rb_complex_arg, 0);
      rb_define_method(rb_cComplex, "phase", rb_complex_arg, 0);
      rb_define_method(rb_cComplex, "rectangular", nucomp_rect, 0);
      rb_define_method(rb_cComplex, "rect", nucomp_rect, 0);
      rb_define_method(rb_cComplex, "polar", nucomp_polar, 0);
      rb_define_method(rb_cComplex, "conjugate", rb_complex_conjugate, 0);
      rb_define_method(rb_cComplex, "conj", rb_complex_conjugate, 0);
  
      rb_define_method(rb_cComplex, "real?", nucomp_false, 0);
  
      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, "eql?", nucomp_eql_p, 1);
  
      rb_define_method(rb_cComplex, "to_s", nucomp_to_s, 0);
      rb_define_method(rb_cComplex, "inspect", nucomp_inspect, 0);
  
      rb_undef_method(rb_cComplex, "positive?");
      rb_undef_method(rb_cComplex, "negative?");
  
      rb_define_method(rb_cComplex, "finite?", rb_complex_finite_p, 0);
      rb_define_method(rb_cComplex, "infinite?", rb_complex_infinite_p, 0);
  
      rb_define_private_method(rb_cComplex, "marshal_dump", nucomp_marshal_dump, 0);
      /* :nodoc: */
      compat = rb_define_class_under(rb_cComplex, "compatible", rb_cObject);
      rb_define_private_method(compat, "marshal_load", nucomp_marshal_load, 1);
      rb_marshal_define_compat(rb_cComplex, compat, nucomp_dumper, nucomp_loader);
  
      /* --- */
  
      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_cComplex, "rationalize", nucomp_rationalize, -1);
      rb_define_method(rb_cComplex, "to_c", nucomp_to_c, 0);
      rb_define_method(rb_cNilClass, "to_c", nilclass_to_c, 0);
      rb_define_method(rb_cNumeric, "to_c", numeric_to_c, 0);
  
      rb_define_method(rb_cString, "to_c", string_to_c, 0);
  
      rb_define_private_method(CLASS_OF(rb_cComplex), "convert", nucomp_s_convert, -1);
  
      /* --- */
  
      rb_define_method(rb_cNumeric, "real", numeric_real, 0);
      rb_define_method(rb_cNumeric, "imaginary", numeric_imag, 0);
      rb_define_method(rb_cNumeric, "imag", numeric_imag, 0);
      rb_define_method(rb_cNumeric, "abs2", numeric_abs2, 0);
      rb_define_method(rb_cNumeric, "arg", numeric_arg, 0);
      rb_define_method(rb_cNumeric, "angle", numeric_arg, 0);
      rb_define_method(rb_cNumeric, "phase", numeric_arg, 0);
      rb_define_method(rb_cNumeric, "rectangular", numeric_rect, 0);
      rb_define_method(rb_cNumeric, "rect", numeric_rect, 0);
      rb_define_method(rb_cNumeric, "polar", numeric_polar, 0);
      rb_define_method(rb_cNumeric, "conjugate", numeric_conj, 0);
      rb_define_method(rb_cNumeric, "conj", numeric_conj, 0);
  
      rb_define_method(rb_cFloat, "arg", float_arg, 0);
      rb_define_method(rb_cFloat, "angle", float_arg, 0);
      rb_define_method(rb_cFloat, "phase", float_arg, 0);
  
      /*
       * The imaginary unit.
       */
      rb_define_const(rb_cComplex, "I",
  		    f_complex_new_bang2(rb_cComplex, ZERO, ONE));
  
  #if !USE_FLONUM
      rb_gc_register_mark_object(RFLOAT_0 = DBL2NUM(0.0));
  #endif
  
      rb_provide("complex.so");	/* for backward compatibility */
  }
  