-- Simple functions

square :: Integer -> Integer
square x = x * x

add :: Integer -> Integer -> Integer
add x y = x + y

(????!) :: Integer -> Integer -> Integer
(????!) x y = x * y - x - y

-- Calculations

fact :: Integer -> Integer
fact 0 = 1
fact n = n * fact (n - 1)

fib :: Integer -> Integer
fib 1 = 1
fib 2 = 1
fib n = fib(n-1) + fib(n-2)

-- Factorial with optimization

fact1 :: Integer -> Integer
fact1 n = fact1a n 1

fact1a :: Integer -> Integer -> Integer
fact1a 0 f = f
fact1a n f = fact1a (n-1) $! (n*f)

-- Finnonachi with optimization

fib1 :: Integer -> Integer
fib1 n = fib1a 1 1 n

fib1a :: Integer -> Integer -> Integer -> Integer
fib1a a b 1 = a
fib1a a b 2 = b
fib1a a b n = fib1a b (a+b) (n-1)

-- Simple pattern matching

anda :: Bool -> Bool -> Bool
anda False False = False
anda _ _ = True

-- Constant

eps ::Float
eps = 0.000001

-- High-order functions

twice :: (Integer -> Integer) -> Integer -> Integer
twice f x = f (f x)

applyn :: Int -> (Integer -> Integer) -> Integer -> Integer
applyn 0 _ x = x
applyn n f x = f (applyn (n-1) f x)

diff :: (Float -> Float) -> Float -> Float
diff f x = (f (x+eps) - f (x-eps)) / (2*eps)

-- Branching

mina :: Float -> Float -> Float
mina a b = if a < b then a else b

minb :: Float -> Float -> Float
minb a b = case a < b of
                True -> a
                False -> b

minc :: Float -> Float -> Float
minc a b | a <= b = a
         | a >  b = b

-- Where and let

minRoot :: Float -> Float -> Float -> Float
minRoot a b c = min ((-b-sd)/n) ((-b+sd)/n)
                where sd = sqrt(b*b - 4*a*c)
                      n = 2*a

minRootL a b c = let sd = sqrt(b*b - 4*a*c)
                     n = 2*a
                 in min ((-b-sd)/n) ((-b+sd)/n)