```{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveTraversable #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Complex
-- Copyright   :  (c) The University of Glasgow 2001
--
-- Stability   :  provisional
-- Portability :  portable
--
-- Complex numbers.
--
-----------------------------------------------------------------------------

module Data.Complex
(
-- * Rectangular form
Complex((:+))

, realPart
, imagPart
-- * Polar form
, mkPolar
, cis
, polar
, magnitude
, phase
-- * Conjugate
, conjugate

)  where

import GHC.Base (Applicative (..))
import GHC.Generics (Generic, Generic1)
import GHC.Float (Floating(..))
import Data.Data (Data)
import Foreign (Storable, castPtr, peek, poke, pokeElemOff, peekElemOff, sizeOf,
alignment)

infix  6  :+

-- -----------------------------------------------------------------------------
-- The Complex type

-- | Complex numbers are an algebraic type.
--
-- For a complex number @z@, @'abs' z@ is a number with the magnitude of @z@,
-- but oriented in the positive real direction, whereas @'signum' z@
-- has the phase of @z@, but unit magnitude.
--
-- The 'Foldable' and 'Traversable' instances traverse the real part first.
--
-- Note that `Complex`'s instances inherit the deficiencies from the type
-- parameter's. For example, @Complex Float@'s 'Ord' instance has similar
-- problems to `Float`'s.
data Complex a
= !a :+ !a    -- ^ forms a complex number from its real and imaginary
-- rectangular components.
deriving ( Eq          -- ^ @since 2.01
, Show        -- ^ @since 2.01
, Read        -- ^ @since 2.01
, Data        -- ^ @since 2.01
, Generic     -- ^ @since 4.9.0.0
, Generic1    -- ^ @since 4.9.0.0
, Functor     -- ^ @since 4.9.0.0
, Foldable    -- ^ @since 4.9.0.0
, Traversable -- ^ @since 4.9.0.0
)

-- -----------------------------------------------------------------------------
-- Functions over Complex

-- | Extracts the real part of a complex number.
realPart :: Complex a -> a
realPart (x :+ _) =  x

-- | Extracts the imaginary part of a complex number.
imagPart :: Complex a -> a
imagPart (_ :+ y) =  y

-- | The conjugate of a complex number.
{-# SPECIALISE conjugate :: Complex Double -> Complex Double #-}
conjugate        :: Num a => Complex a -> Complex a
conjugate (x:+y) =  x :+ (-y)

-- | Form a complex number from polar components of magnitude and phase.
{-# SPECIALISE mkPolar :: Double -> Double -> Complex Double #-}
mkPolar          :: Floating a => a -> a -> Complex a
mkPolar r theta  =  r * cos theta :+ r * sin theta

-- | @'cis' t@ is a complex value with magnitude @1@
-- and phase @t@ (modulo @2*'pi'@).
{-# SPECIALISE cis :: Double -> Complex Double #-}
cis              :: Floating a => a -> Complex a
cis theta        =  cos theta :+ sin theta

-- | The function 'polar' takes a complex number and
-- returns a (magnitude, phase) pair in canonical form:
-- the magnitude is nonnegative, and the phase in the range @(-'pi', 'pi']@;
-- if the magnitude is zero, then so is the phase.
{-# SPECIALISE polar :: Complex Double -> (Double,Double) #-}
polar            :: (RealFloat a) => Complex a -> (a,a)
polar z          =  (magnitude z, phase z)

-- | The nonnegative magnitude of a complex number.
{-# SPECIALISE magnitude :: Complex Double -> Double #-}
magnitude :: (RealFloat a) => Complex a -> a
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

-- -----------------------------------------------------------------------------
-- Instances of Complex

-- | @since 2.01
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

-- | @since 2.01
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'
k   = - max (exponent x') (exponent y')
d   = x'*x'' + y'*y''

fromRational a      =  fromRational a :+ 0

-- | @since 2.01
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

x ** y = case (x,y) of
(_ , (0:+0))  -> 1 :+ 0
((0:+0), (exp_re:+_)) -> case compare exp_re 0 of
GT -> 0 :+ 0
LT -> inf :+ 0
EQ -> nan :+ nan
((re:+im), (exp_re:+_))
| (isInfinite re || isInfinite im) -> case compare exp_re 0 of
GT -> inf :+ 0
LT -> 0 :+ 0
EQ -> nan :+ nan
| otherwise -> exp (log x * y)
where
inf = 1/0
nan = 0/0

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))
-- Take care to allow (-1)::Complex, fixing #8532
acosh z        =  log (z + (sqrt \$ z+1) * (sqrt \$ z-1))
atanh z        =  0.5 * log ((1.0+z) / (1.0-z))

log1p x@(a :+ b)
| abs a < 0.5 && abs b < 0.5
, u <- 2*a + a*a + b*b = log1p (u/(1 + sqrt(u+1))) :+ atan2 (1 + a) b
| otherwise = log (1 + x)
{-# INLINE log1p #-}

expm1 x@(a :+ b)
| a*a + b*b < 1
, u <- expm1 a
, v <- sin (b/2)
, w <- -2*v*v = (u*w + u + w) :+ (u+1)*sin b
| otherwise = exp x - 1
{-# INLINE expm1 #-}

-- | @since 4.8.0.0
instance Storable a => Storable (Complex a) where
sizeOf a       = 2 * sizeOf (realPart a)
alignment a    = alignment (realPart a)
peek p           = do
q <- return \$ castPtr p
r <- peek q
i <- peekElemOff q 1
return (r :+ i)
poke p (r :+ i)  = do
q <-return \$  (castPtr p)
poke q r
pokeElemOff q 1 i

-- | @since 4.9.0.0
instance Applicative Complex where
pure a = a :+ a
f :+ g <*> a :+ b = f a :+ g b
liftA2 f (x :+ y) (a :+ b) = f x a :+ f y b

-- | @since 4.9.0.0
a :+ b >>= f = realPart (f a) :+ imagPart (f b)

-- -----------------------------------------------------------------------------
-- Rules on Complex

{-# RULES

"realToFrac/a->Complex Double"
realToFrac = \x -> realToFrac x :+ (0 :: Double)

"realToFrac/a->Complex Float"
realToFrac = \x -> realToFrac x :+ (0 :: Float)

#-}
```