ref: b4a7a0d4c04ad0fb96d0b279d35da2658d58a0e1
dir: /lib/Data/Complex.hs/
module Data.Complex(module Data.Complex) where
import Data.Typeable
infix 6 :+
data Complex a = !a :+ !a
deriving(Typeable)
instance forall a . Eq a => Eq (Complex a) where
(x :+ y) == (x' :+ y') = x == x' && y == y' -- parser bug
instance forall a . Show a => Show (Complex a) where
show (x :+ y) = show x ++ " :+ " ++ show y
realPart :: forall a . Complex a -> a
realPart (x :+ _) = x
imagPart :: forall a . Complex a -> a
imagPart (_ :+ y) = y
conjugate :: forall a . Num a => Complex a -> Complex a
conjugate (x:+y) = x :+ (- y)
mkPolar :: forall a . Floating a => a -> a -> Complex a
mkPolar r theta = r * cos theta :+ r * sin theta
cis :: forall a . Floating a => a -> Complex a
cis theta = cos theta :+ sin theta
polar :: forall a . (RealFloat a) => Complex a -> (a,a)
polar z = (magnitude z, phase z)
magnitude :: forall a . (Ord a, Floating a) => Complex a -> a
magnitude (x:+y) =
-- slightly contorted to avoid overflow
let ax = abs x
ay = abs y
mx = max ax ay
mn = min ax ay
r = mn / mx
in mx * sqrt(1 + r*r)
phase :: forall a . (RealFloat a) => Complex a -> a
-- XXX phase (0 :+ 0) = 0
phase (x:+y) | x==0 && y==0 = 0
| otherwise = atan2 y x
instance forall a . (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==0 && y==0 = 0
| otherwise = x/r :+ y/r where r = magnitude z
fromInteger n = fromInteger n :+ 0
instance forall a . (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
instance forall a . (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
{-
x ** y = case (x,y) of
{- XXX
(_ , (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 z@(x:+y) | x==0 && y==0 = 0
| otherwise = 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 + (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)
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