content: module Data.Complex(
      Complex((:+)),
      realPart,
      imagPart,
      conjugate,
      mkPolar,
      cis,
      polar,
      magnitude,
      phase
      )  where
  
  import Jhc.Num
  import Jhc.Class.Real
  
  infix  6  :+
  
  -- the standard says this should have a RealFloat constraint, but that is silly.
  data  Complex a = !a :+ !a  deriving (Eq,Read,Show)
  
  realPart, imagPart :: (RealFloat a) => Complex a -> a
  realPart (x:+y)  =  x
  imagPart (x:+y)  =  y
  
  conjugate  :: (RealFloat a) => Complex a -> Complex a
  conjugate (x:+y) =  x :+ (-y)
  
  mkPolar  :: (RealFloat a) => a -> a -> Complex a
  mkPolar r theta  =  r * cos theta :+ r * sin theta
  
  cis  :: (RealFloat a) => a -> Complex a
  cis theta  =  cos theta :+ sin theta
  
  polar  :: (RealFloat a) => Complex a -> (a,a)
  polar z  =  (magnitude z, phase z)
  
  magnitude :: (RealFloat a) => Complex a -> a
  magnitude (x:+y) =  scaleFloat k
       (sqrt ((scaleFloat mk x)^(2::Int) + (scaleFloat mk y)^(2::Int)))
      where k  = max (exponent x) (exponent y)
            mk = - k
  
  phase :: (RealFloat a) => Complex a -> a
  phase (0 :+ 0) = 0
  phase (x :+ y) = atan2 y x
  
  instance  (RealFloat a) => Num (Complex a)  where
      (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')
      negate (x:+y) =  negate x :+ negate y
      abs z =  magnitude z :+ 0
      signum (0 :+ 0) =  0
      signum z@(x:+y) =  x/r :+ y/r  where r = magnitude z
      fromInteger n =  fromInteger n :+ 0
      fromInt n =  fromInt n :+ 0
  
  instance  (RealFloat a) => Fractional (Complex a)  where
      (x:+y) / (x':+y') =  (x*x''+y*y'') / d :+ (y*x''-x*y'') / d
         where x'' = scaleFloat k x'
               y'' = scaleFloat k y'
               k   = - max (exponent x') (exponent y')
               d   = x'*x'' + y'*y''
      fromRational a =  fromRational a :+ 0
      fromDouble a = fromDouble a :+ 0
  
  instance  (RealFloat a) => Floating (Complex a) where
      pi             =  pi :+ 0
      exp (x:+y)     =  expx * cos y :+ expx * sin y
                        where expx = exp x
      log z          =  log (magnitude z) :+ phase z
  
      sqrt (0 :+ 0)  =  0
      sqrt z@(x:+y)  =  u :+ (if y < 0 then -v else v)
                        where (u,v) = if x < 0 then (v',u') else (u',v')
                              v'    = abs y / (u'*2)
                              u'    = sqrt ((magnitude z + abs x) / 2)
  
      sin (x:+y)     =  sin x * cosh y :+ cos x * sinh y
      cos (x:+y)     =  cos x * cosh y :+ (- sin x * sinh y)
      tan (x:+y)     =  (sinx*coshy:+cosx*sinhy)/(cosx*coshy:+(-sinx*sinhy))
                        where sinx  = sin x
                              cosx  = cos x
                              sinhy = sinh y
                              coshy = cosh y
  
      sinh (x:+y)    =  cos y * sinh x :+ sin  y * cosh x
      cosh (x:+y)    =  cos y * cosh x :+ sin y * sinh x
      tanh (x:+y)    =  (cosy*sinhx:+siny*coshx)/(cosy*coshx:+siny*sinhx)
                        where siny  = sin y
                              cosy  = cos y
                              sinhx = sinh x
                              coshx = cosh x
  
      asin z@(x:+y)  =  y':+(-x')
                        where  (x':+y') = log (((-y):+x) + sqrt (1 - z*z))
      acos z@(x:+y)  =  y'':+(-x'')
                        where (x'':+y'') = log (z + ((-y'):+x'))
                              (x':+y')   = sqrt (1 - z*z)
      atan z@(x:+y)  =  y':+(-x')
                        where (x':+y') = log (((1-y):+x) / sqrt (1+z*z))
  
      asinh z        =  log (z + sqrt (1+z*z))
      acosh z        =  log (z + (z+1) * sqrt ((z-1)/(z+1)))
      atanh z        =  log ((1+z) / sqrt (1-z*z))
  