content: {-# LANGUAGE CPP, TypeOperators, TypeFamilies #-}
  {-# LANGUAGE DeriveFunctor, DeriveDataTypeable #-}
  {-# OPTIONS_GHC -Wall #-}
  
  -- {-# OPTIONS_GHC -fno-warn-unused-imports #-} -- TEMP
  -- {-# OPTIONS_GHC -fno-warn-unused-binds   #-} -- TEMP
  
  #define CustomComplex
  
  #if !defined CustomComplex
  {-# OPTIONS_GHC -fno-warn-orphans #-} -- for HasRep
  #endif
  
  ----------------------------------------------------------------------
  -- |
  -- Module      :  ConCat.Complex
  -- Copyright   :  (c) 2015 Conal Elliott and David Banas
  -- License     :  BSD3
  --
  -- Maintainer  :  conal@conal.net
  -- Stability   :  experimental
  --
  -- Simplified Complex type
  ----------------------------------------------------------------------
  
  module ConCat.Complex (Complex(..),cis) where
  
  #if defined CustomComplex
  import Control.Applicative (liftA2)
  import Data.Typeable
  import Data.Data
  import Test.QuickCheck (Arbitrary(..),CoArbitrary(..))
  #else
  import Data.Complex
  #endif
  
  #if defined CustomComplex
  infixl 1 :+
  data Complex a = a :+ a deriving (Functor,Eq,Show,Typeable,Data,Ord)
  
  -- | The nonnegative magnitude of a complex number.
  -- {-# SPECIALISE magnitude :: Complex Double -> Double #-}
  magnitude :: (RealFloat a) => Complex a -> a
  
  magnitude (x:+y) = sqrt (x*x + y*y)
  
  -- From Data.Complex
  
  -- magnitude (x:+y) =  scaleFloat k
  --                      (sqrt (sqr (scaleFloat mk x) + sqr (scaleFloat mk y)))
  --                     where k  = max (exponent x) (exponent y)
  --                           mk = - k
  --                           sqr z = z * z
  
  -- | The phase of a complex number, in the range @(-'pi', 'pi']@.
  -- If the magnitude is zero, then so is the phase.
  -- {-# SPECIALISE phase :: Complex Double -> Double #-}
  phase :: (RealFloat a) => Complex a -> a
  phase (0 :+ 0)   = 0            -- SLPJ July 97 from John Peterson
  phase (x:+y)     = atan2 y x
  
  -- | Complex Identity Scaler
  -- Returns a point on the unit circle, such that the angle between the
  -- vector to that point and the positive x-axis is equal to the function
  -- argument, given in radians.
  cis :: (RealFloat a) => a -> Complex a
  cis w = cos w :+ sin w
  
  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')
      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
      {-# INLINE (+)         #-}
      {-# INLINE (-)         #-}
      {-# INLINE (*)         #-}
      {-# INLINE negate      #-}
      {-# INLINE abs         #-}
      {-# INLINE signum      #-}
      {-# INLINE fromInteger #-}
  
  instance  (RealFloat a) => Fractional (Complex a)  where
      -- {-# SPECIALISE instance Fractional (Complex Float) #-}
      -- {-# SPECIALISE instance Fractional (Complex Double) #-}
      recip z@(x:+y) = x/mag :+ (-y)/mag where mag = magnitude z
  --     (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
  
  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
      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         =  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        =  0.5 * log ((1.0+z) / (1.0-z))
  
  instance Arbitrary a => Arbitrary (Complex a) where
    arbitrary = liftA2 (:+) arbitrary arbitrary
    shrink (x :+ y) = map (uncurry (:+)) (shrink (x,y))
  
  instance CoArbitrary a => CoArbitrary (Complex a) where
    coarbitrary (x :+ y) = coarbitrary x . coarbitrary y
  
  #endif
  