\begin{code}
instance (RealFloat a) => Num (Complex a) where
+ {-# SPECIALISE instance Num (Complex Float) #-}
+ {-# SPECIALISE instance Num (Complex Double) #-}
(x:+y) + (x':+y') = (x+x') :+ (y+y')
(x:+y) - (x':+y') = (x-x') :+ (y-y')
(x:+y) * (x':+y') = (x*x'-y*y') :+ (x*y'+y*x')
fromInteger n = fromInteger n :+ 0
instance (RealFloat a) => Fractional (Complex a) where
+ {-# SPECIALISE instance Fractional (Complex Float) #-}
+ {-# SPECIALISE instance Fractional (Complex Double) #-}
(x:+y) / (x':+y') = (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
where x'' = scaleFloat k x'
y'' = scaleFloat k y'
fromRational a = fromRational a :+ 0
instance (RealFloat a) => Floating (Complex a) where
+ {-# SPECIALISE instance Floating (Complex Float) #-}
+ {-# SPECIALISE instance Floating (Complex Double) #-}
pi = pi :+ 0
exp (x:+y) = expx * cos y :+ expx * sin y
where expx = exp x