Learn you a Haskell - In a nutshell

This is a summary of the "Learn You A Haskell" online book under http://learnyouahaskell.com/chapters.

1. Introduction

Haskell is a functional programming language.
"what it is" over "what to do"
Haskell is lazy - no calculation until a result is used
Statically typed - errors are caught on compile time
Type inference - it auto-detects the right type e.g. for a = 5 + 4
GHC is the most widely used Haskell compiler
To load a file you do:

l: myfunctions.hs

2. Starting Out

Interactive GHC is started with ghci
Simple arithmetic:

2 + 15
49 * 100
1892 - 1472
5 / 2
50 * 100 - 4999
50 * (100 - 4999)

Surround negative numbers with brackets:

5 * (-3)

Boolean algebra with True, False, not, &&and ||

not (True && True)

Test for equality with == and inequality with /=
infix functions like * stand between the operators
Most functions are prefix functions:

succ 8 : successor (9)
min 3 4 : minimum of 2 values (3)
max 4 5 : maximum of 2 values (5)
div 13 6 : integral division of 2 integers (2)
odd 5 : wether number is odd (True)

Prefix functions can be written as infix with backticks:

div 92 10
92 `div` 10

Functions are defined with =

doubleMe x = x + x
doubleUs x y = x*2 + y*2

if have a then and always require a else

   doubleSmallNumber x = if x > 100
       then x
       else x*2

if is also an expression
Lists are collections of homogenous elements: all elements have the same type

lostNumbers = [4,8,15,16,23,42]
someString = "Some string"

Use ++ to put two lists together (goes through the complete list!)

[1,2,3,4] ++ [6,7,8]
"Hello" ++ " " ++ "world"

Use : to prepend LHS to RHS list

'A':" SMALL CAT"
1:[2,3,4,5]
1:2:3:[]

Use !! to get an element by index (0 based, index must exist)

"Steve Buscemi" !! 6

Use <, <=, > and >= to lexographically compare lists
List ranges are defined with ..:

   [1..20]
   ['a'..'z']

Ranges can define a step size

   [2,4..20]   ([2,4,6,8,10,12,16,18,20])
   [3,6..20]   ([3,6,9,12,15,18])
   [20,19..15] ([20,19,18,17,16,15])

List related functions

head [5,4,3] : first element of a list (5)
tail [5,4,3] : tail without head ([4,3])
last [5,4,3] : last element of a list (3)
init [5,4,3] : list without tail ([5,4])
length [5,4,3] : number of elements (3)
null [5,4,3] : wether list is empty (False)
reverse [5,4,3] : reverse list ([3,4,5])
take 3 [5,4,3,2,1] : extract number of elements from list start ([5,4,3])
drop 3 [5,4,3,2,1] : drop first elements of a list ([2,1])
maximum [5,4,3,2,1] : maximum of orderable list (5)
minimum [5,4,3,2,1] : minimum of orderable list (1)
sum [5,4,3] : sum of number list (12)
product [5,4,3] : product of number list (60)
4 `elem` [5,4,3] : wether element is in list, usually infixed (True)
take 10 (cycle [1,2,3]) : repeat the list elements ([1,2,3,1,2,3,1,2,3,1])
take 10 (repeat 5) : repeat the element ([5,5,5,5,5,5,5,5,5,5])
replicate 3 10 : repeat the element a number of times ([10,10,10])

List comprehensions are similar to mathematical equations

   [x*2 | x <- [1..5]]    ([2,4,6,8,10])

Predicates are conditions for list comprehensions

   [x*2 | x <- [1..5], x*2 >= 5]   ([6,8,10])

There can be more than one predicates

   [ x | x <- [10..20], x /= 10, x /= 15, x /= 19]

Comprehensions can be put inside a function

   boomBangs xs = [if x < 10 then "BOOM!" else "BANG!" | x <- xs, odd x]

Comprehensions can draw from several lists, which multiplies the lengths

   [ x*y | x <- [2,3], y <- [3,4,5]]    ([6,8,10,9,12,15])

_ is a dummy placeholder for a unused value

   length' xs = sum [1 | _ <- xs]

List comprehensions can be nested

   let xxs = [[1,2,3],[2,3,4],[4,5]]
   [ [x | x <- xs, even x] | xs <- xxs]    ([[2],[2,4],[4]]

Tuples can contain several types, but for the type of two Tuples to be the same, the number and types of their elements must match

   (1,2)
   [(1,2), (3,2), (4,9)]
   [("Johnny", "Walker", 38), ("Kate", "Middleton", 27)]

Tuple related functions

fst (8,11) : returns first component of a pair (8)
snd (9,"Hey") : returns second component of a pair ("Hey")
zip [1..3] ['a'..'c'] : combine two lists to a list of tuples ( [(1,'a'), (2,'b'), (3,'c')] )

3. Types and Typeclasses

Types always start with an uppercase letter
Use :t to get a type of something

   :t 'a'     ('a' :: Char)
   :t "HELLO"  ("HELLO" :: [Char])
   :t (True, 'a')   ( (True,'a') :: (Bool, Char) )

All functions should use a type declaration with :: ("has type of"). Multiple arguments are also separeted with -> just like the type declaration itself.

   removeUppercase :: [Char] -> [Char]
   removeUppercase :: String -> String   (same as above)
   addThree :: Int -> Int -> Int -> Int
   addThree x y z = x + y + z

Common types

Int : Integer
Integer : Integer (big)
Float : Floating point
Double : Floating point with double precision
Bool : Boolean
Char : Character
Tuples, as mentioned in chapter 2
Ordering : Can be GT, LT or EQ

Type variables can be used in function declarations. They stand for a type. Without a class constraint they mean "any type"

   :t head    (head :: [a] -> a)
   :t fst     (fst :: (a, b) -> a)

Typeclasses are like interfaces for types. If a type is part of a typeclass it implements that class' behavior. The => symbol separates the class constraint from the declaration:

   :t (==)    ( (==) :: (Eq a) => a -> a -> Bool )

Eq : supports equality testing
Ord : can have ordering
Show : can be presented as string
Read : can be read from a string
Enum : ordered type which can be enumerated
Bounded : have an upper and lower bound
Num : can act like a number
Integral : can act like an integral number
Floating : can act like a floating point number

Related functions

5 `compare` 3 : takes two Ord members of same type and returns a Ordering (GT)
show 3 : takes a Show and returns a String ("3")
read "True" : takes a Read and returns a Read (True)
succ LT : takes a Enum and returns next element (GT)
pred 'b' : takes a Enum and returns previous element ('a')
minBound :: Bool : takes a Bounded and returns lower bound (False)
maxBound :: Bool : takes a Bounded and returns lower bound (True)
fromIntegral 5 : takes a Integral and returns a Num

Type annotations define the type of ambiguous expressions

   (read "5" :: Float) * 4

4. Syntax in Functions

Functions can be defined with pattern matching. Patterns make sure that the input matches a specified pattern. The first matching pattern is executed. There should always be a catch-all pattern at the end

   lucky :: (Integral a) => a -> String  
   lucky 7 = "LUCKY NUMBER SEVEN!"  
   lucky x = "Sorry, you're out of luck, pal!"

   factorial :: (Integral a) => a -> a  
   factorial 0 = 1  
   factorial n = n * factorial (n - 1)

   addVectors :: (Num a) => (a, a) -> (a, a) -> (a, a)  
   addVectors (x1, y1) (x2, y2) = (x1 + x2, y1 + y2)

   head' :: [a] -> a  
   head' [] = error "Can't call head on an empty list, dummy!"  
   head' (x:_) = x

   tell :: (Show a) => [a] -> String  
   tell [] = "The list is empty"  
   tell (x:[]) = "The list has one element: " ++ show x  
   tell (x:y:[]) = "The list has two elements: " ++ show x ++ " and " ++ show y  
   tell (x:y:_) = "This list is long. The first two elements are: " ++ show x ++ " and " ++ show y

   length' :: (Num b) => [a] -> b  
   length' [] = 0  
   length' (_:xs) = 1 + length' xs

   sum' :: (Num a) => [a] -> a  
   sum' [] = 0  
   sum' (x:xs) = x + sum' xs

The as pattern is used to reference the "whole thing": Put a name followed by @ in front of the pattern:

   capital :: String -> String  
   capital "" = "Empty string, whoops!"  
   capital all@(x:xs) = "The first letter of " ++ all ++ " is " ++ [x]

Guards are similar to if statements and check for boolean conditions. There's no = after the function name:

   bmiTell :: (RealFloat a) => a -> String  
   bmiTell bmi  
       | bmi <= 18.5 = "You're underweight, you emo, you!"  
       | bmi <= 25.0 = "You're supposedly normal. Pffft, I bet you're ugly!"  
       | bmi <= 30.0 = "You're fat! Lose some weight, fatty!"  
       | otherwise   = "You're a whale, congratulations!"

Guards can be combined with patterns: If all guards of a function evaluate to False, evaluation falls through to the next pattern
Guards can have as many parameters as we want

   bmiTell :: (RealFloat a) => a -> a -> String  
   bmiTell weight height  
       | weight / height ^ 2 <= 18.5 = "You're underweight, you emo, you!"  
       | weight / height ^ 2 <= 25.0 = "You're supposedly normal. Pffft, I bet you're ugly!"  
       | weight / height ^ 2 <= 30.0 = "You're fat! Lose some weight, fatty!"  
       | otherwise                 = "You're a whale, congratulations!"

   max' :: (Ord a) => a -> a -> a  
   max' a b   
       | a > b     = a  
       | otherwise = b

   myCompare :: (Ord a) => a -> a -> Ordering  
   a `myCompare` b  
       | a > b     = GT  
       | a == b    = EQ  
       | otherwise = LT

Guards can use a where block to define functions that are only visible inside the guard function

   bmiTell :: (RealFloat a) => a -> a -> String  
   bmiTell weight height  
       | bmi <= skinny = "You're underweight, you emo, you!"  
       | bmi <= normal = "You're supposedly normal. Pffft, I bet you're ugly!"  
       | bmi <= fat    = "You're fat! Lose some weight, fatty!"  
       | otherwise     = "You're a whale, congratulations!"  
       where bmi = weight / height ^ 2  
             skinny = 18.5  
             normal = 25.0  
             fat = 30.0

Combined with a patteren match

   ...  
   where bmi = weight / height ^ 2  
         (skinny, normal, fat) = (18.5, 25.0, 30.0)

More Examples

   initials :: String -> String -> String  
   initials firstname lastname = [f] ++ ". " ++ [l] ++ "."  
       where (f:_) = firstname  
             (l:_) = lastname

   calcBmis :: (RealFloat a) => [(a, a)] -> [a]  
   calcBmis xs = [bmi w h | (w, h) <- xs]  
       where bmi weight height = weight / height ^ 2

Let bindings are similar to where bindings. Their form is let <bindings> in <expression>.

   cylinder :: (RealFloat a) => a -> a -> a  
   cylinder r h = 
       let sideArea = 2 * pi * r * h  
           topArea = pi * r ^2  
       in  sideArea + 2 * topArea

The difference between let bindings and where bindings is that let bindings are expressions, just like if statements:

   4 * (if 10 > 5 then 10 else 0) + 2     (42)

   4 * (let a = 9 in a + 1) + 2    (42)

Let bindings can be used to introduce functions in a local scope

   [let square x = x * x in (square 5, square 3, square 2)]     ( [(25,9,4)] )

To let bind several variables they get separated by a semicolon

   (let a = 100; b = 200; c = 300 in a*b*c, let foo="Hey "; bar = "there!" in foo ++ bar)

Let bindings can be used with pattern matching

   (let (a,b,c) = (1,2,3) in a+b+c) * 100      (600)

Let bindings can be used in list comprehensions

   calcBmis :: (RealFloat a) => [(a, a)] -> [a]  
   calcBmis xs = [bmi | (w, h) <- xs, let bmi = w / h ^ 2]

Predicates come after the let binding

   calcBmis :: (RealFloat a) => [(a, a)] -> [a]  
   calcBmis xs = [bmi | (w, h) <- xs, let bmi = w / h ^ 2, bmi >= 25.0]

Case expressions are very similar to pattern matching:

   head' :: [a] -> a  
   head' [] = error "No head for empty lists!"  
   head' (x:_) = x

   head' :: [a] -> a  
   head' xs = case xs of [] -> error "No head for empty lists!"  
                         (x:_) -> x

   case expression of pattern -> result  
                      pattern -> result  
                      pattern -> result  
                      ...

Pattern matching can only be used with function definitions. Cases expressions work everywhere

   describeList :: [a] -> String  
   describeList xs = "The list is " ++ case xs of [] -> "empty."  
                                                  [x] -> "a singleton list."   
                                                  xs -> "a longer list."

5. Recursion

Try to start with the edge cases

   maximum' :: (Ord a) => [a] -> a  
   maximum' [] = error "maximum of empty list"  
   maximum' [x] = x  
   maximum' (x:xs)   
       | x > maxTail = x  
       | otherwise = maxTail  
       where maxTail = maximum' xs

   maximum' :: (Ord a) => [a] -> a  
   maximum' [] = error "maximum of empty list"  
   maximum' [x] = x  
   maximum' (x:xs) = max x (maximum' xs)

   replicate' :: (Num i, Ord i) => i -> a -> [a]
   replicate' n x
       | n <= 0     = []
       | otherwhise = x:replicate' (n-1) x

   take' :: (Num i, Ord i) => i -> [a] -> [a]  
   take' n _  
       | n <= 0   = []  
   take' _ []     = []  
   take' n (x:xs) = x : take' (n-1) xs

   reverse' :: [a] -> [a]
   reverse' [] = []
   reverse' (x:xs) = reverse' xs ++ [x]

   repeat' :: a -> [a]  
   repeat' x = x:repeat' x

   zip' :: [a] -> [b] -> [(a,b)]  
   zip' _ [] = []  
   zip' [] _ = []  
   zip' (x:xs) (y:ys) = (x,y):zip' xs ys

   elem' :: (Eq a) => a -> [a] -> Bool  
   elem' a [] = False  
   elem' a (x:xs)  
       | a == x    = True  
       | otherwise = a `elem'` xs

Quicksort can be implemented very elegantly. The main algorithm is

A sorted list is a list that has all the values smaller than (or equal to) the head of the list in front (and those values are sorted), then comes the head of the list in the middle and then come all the values that are bigger than the head (they're also sorted)

   quicksort :: (Ord a) => [a] -> [a]  
   quicksort [] = []  
   quicksort (x:xs) =   
       let smallerSorted = quicksort [a | a <- xs, a <= x]  
           biggerSorted = quicksort [a | a <- xs, a > x]  
       in  smallerSorted ++ [x] ++ biggerSorted

6. Higher order functions

Higher order functions take other functions as parameters and return functions themselves (which is what makes up the ultimate haskell experience)
Functions in haskell only take one argument
Curried functions are used to give the impression that a function can have more than one argument: Calling max 4 5 creates a function which takes one argument and returns 4 if the argument is smaller and the argument itself if it is bigger than 4. These calls are equivalent:

   max 4 5
   (max 4) 5

A space between two things ist simply function application. The type of max can be written in two equivalent ways:

max :: (Ord a) => a -> a -> a
max :: (Ord a) => a -> (a -> a)

It takes an a and returns a function, which takes another a and returns an a.

If a function is callaed with not all parameters we get back a partially applied function.

multThree :: (Num a) => a -> a -> a -> a  
multThree x y z = x * y * z

When we do multThree 3 5 9 it's actually ((multThree 3) 5) 9 or multThree :: (Num a) => a -> (a -> (a -> a)). First multThree 3 is applied, which returns a function. This function takes a single argument and returns another function. This other function again takes a single argument and returns the parameter multiplied by 15. This allows us to do things like

multTwoWithNine = multThree 9  
multTwoWithNine 2 3    (54)
multWithEighteen = multTwoWithNine 2  
multWithEighteen 10    (180)

A function that compares the argument with 100 could be written like:

compareWithHundred :: (Num a, Ord a) => a -> Ordering  
compareWithHundred x = compare 100 x

The part compare 100 returns a function that takes a number and compares it with 100. The above thus can be rewritten as

compareWithHundred :: (Num a, Ord a) => a -> Ordering  
compareWithHundred = compare 100

More examples

divideByTen :: (Floating a) => a -> a  
divideByTen = (/10)

isUpperAlphanum :: Char -> Bool  
isUpperAlphanum = (`elem` ['A'..'Z'])

The type declaration of functions that take functions look different

applyTwice :: (a -> a) -> a -> a  
applyTwice f x = f (f x)

applyTwice (+3) 10               (13)
applyTwice (++ " HAHA") "HEY"    ("HEY HAHA HAHA")
applyTwice ("HAHA " ++) "HEY"    ("HAHA HAHA HEY")
applyTwice (multThree 2 2) 9     (144)
applyTwice (3:) [1]              ([3,3,1])

zipWith takes a function and two lists and applies the function on each two items of both lists to get a new list

zipWith' :: (a -> b -> c) -> [a] -> [b] -> [c]  
zipWith' _ [] _ = []  
zipWith' _ _ [] = []  
zipWith' f (x:xs) (y:ys) = f x y : zipWith' f xs ys

zipWith' (+) [4,2,5,6] [2,6,2,3]     ([6,8,7,9])
zipWith' max [6,3,2,1] [7,3,1,5]     ([7,3,2,5])
zipWith' (++) ["foo ", "bar "] ["fighters", "hoppers"]    (["foo fighters", "bar hoppers"])
zipWith' (*) (replicate 5 2) [1..]   ([2,4,6,8,10])
zipWith' (zipWith'(*)) [ [1,2], [3,4] ] [ [2,3], [4,5] ]  ([[2,6],[12,20]])

flip takes a function and returns a function where the first two parameters are flipped

   flip' :: (a -> b -> c) -> (b -> a -> c)  
   flip' f = g  
       where g x y = f y x

Or even shorter

   flip' :: (a -> b -> c) -> b -> a -> c  
   flip' f y x = f x y

zipWith (flip' div) [2,2..] [10,8,6,4,2]    ( [5,4,3,2,1] )

The map function takes a function and a list and returns a list with the function applied to every element

map :: (a -> b) -> [a] -> [b]  
map _ [] = []  
map f (x:xs) = f x : map f xs

map (+3) [1,5,3,1,6]             ([4,8,6,4,9])
map (++ "!") ["BIFF", "BANG"]    (["BIFF!","BANG!"])
map (replicate 2) [3..6]         ([[3,3],[4,4],[5,5],[6,6]])
map (map (^2)) [[1,2],[3,4,5]]   ([[1,4],[9,16,25]])
map fst [(1,2),(3,5)]            ([1,3])

The filter function takes a function that filters elements from a list

filter :: (a -> bool) -> [a] -> [a]
filter _ [] = []
filter p (x:xs)
   | p x = x : filter p xs
   | otherwhise = filter p xs

filter (>3) [1,5,3,6]    ([5,6])
filter even [1..10]      ([1,2,4,6,8,10])
let notNull x = not (null x) in filter notNull [[1,2,3],[],[2,2]]   ([[1,2,3],[2,2]])

Alternative quicksort implementation

   quicksort :: (Ord a) => [a] -> [a]    
   quicksort [] = []    
   quicksort (x:xs) =     
       let smallerSorted = quicksort (filter (<=x) xs)  
           biggerSorted = quicksort (filter (>x) xs)   
       in  smallerSorted ++ [x] ++ biggerSorted

The takeWhile function takes a predicate function and a list. It returns elements from the list until the predicate becomes False.

Sum of all odd squares that are smaller than 10000

sum (takeWhile (<10000) (filter odd (map (^2) [1..])))

Alternative with list comprehension

sum (takeWhile (<10000) [n^2 | n <- [1..], odd (n^2)])

Example: The Collatz sequence starts at any number. If the number is even, it's divided by 2. If it is odd it is multiplied by 3 and 1 is added. In any case the chain continues at this new number until the sequence reaches 1. Question: For all starting numbers between 1 and 100, how many chains have a length greater than 15?

   chain :: (Integral a) => a -> [a]  
   chain 1 = [1]  
   chain n  
       | even n =  n:chain (n `div` 2)  
       | odd n  =  n:chain (n*3 + 1)

   numLongChains :: Int  
   numLongChains = length (filter isLong (map chain [1..100]))  
       where isLong xs = length xs > 15

Lambdas are anonymous functions. They are defined with a \, followed by space separated parameters, and a -> that points at the function body:

   numLongChains :: Int  
   numLongChains = length (filter (\xs -> length xs > 15) (map chain [1..100]))  

   zipWith (\a b -> (a * 30 + 3) / b) [5,4,3,2,1] [1,2,3,4,5]

   map (\(a,b) -> a + b) [(1,2),(3,5),(6,3),(2,6),(2,5)]

   flip' f = \x y -> f y x

Sometimes lambdas are used even if that is not neccessary, like map (\x -> x + 3) [1,6,3,2] instead of more readable map (+3) [1,6,3,2].

Folds take a binary function a start value and a list as parameter. The binary function is called with the accumulator and the first (or last) element from the list and produces a new accumulator. Then it's called again with this new accumulator and the next list element. Whenever you want to traverse a list to return something, chances are you want a fold.
foldl or left fold folds up a list from the left side.

sum' :: (Num a) => [a] -> a  
sum' xs = foldl (\acc x -> acc + x) 0 xs

In the beginning the binary function is called with acc value 0 and the first list element. The lambda function (\acc x -> acc + x) is the same as (+) and we can omit xs because foo a = bar b a can be written as foo = bar b:

sum' :: (Num a) => [a] -> a  
sum' = foldl (+) 0

elem' :: (Eq a) => a -> [a] -> Bool  
elem' y ys = foldl (\acc x -> if x == y then True else acc) False ys

foldr does a right fold. Here the parameters to the binary function are flipped.

map' :: (a -> b) -> [a] -> [b]  
map' f xs = foldr (\x acc -> f x : acc) [] xs

We could also use foldl with the function \acc x -> acc ++ [f x]. But ++ is much more expensive than :. Right folds are usually used to build up lists. Right folds also work with infinite lists whereas left folds don't.

foldl1 and foldr1 work like foldl and foldr but use the first (or last) list element as starting value. They don't work with emtpy lists.

sum = foldl1 (+)

Fold examples

   maximum' :: (Ord a) => [a] -> a  
   maximum' = foldr1 (\x acc -> if x > acc then x else acc)  

   reverse' :: [a] -> [a]  
   reverse' = foldl (\acc x -> x : acc) []  
   -- alternative
   reverse' = foldl (flip (:)) []
 
   product' :: (Num a) => [a] -> a  
   product' = foldr1 (*)  
 
   filter' :: (a -> Bool) -> [a] -> [a]  
   filter' p = foldr (\x acc -> if p x then x : acc else acc) []  
 
   head' :: [a] -> a  
   head' = foldr1 (\x _ -> x)  
 
   last' :: [a] -> a  
   last' = foldl1 (\_ x -> x)

scanl and scanr are like foldl and foldr but they also give all intermediate results back

scanl (+) 0 [3,5,2,1]    ([0,3,8,10,11])
scanr (+) 0 [3,5,2,1]    ([11,8,3,1,0])
scanl1 (\acc x -> if x > acc then x else acc) [3,2,9,2,1]    ([3,3,9,9,9])
scanl (flip (:)) [] [3,2,1]     ([[],[3],[2,3],[1,2,3]])

How many elements does it take for the sum of the roots of all natural numbers to exceed 1000?

sqrtSums :: Int  
sqrtSums = length (takeWhile (<1000) (scanl1 (+) (map sqrt [1..]))) + 1

The function application function $ looks like

($) :: (a -> b) -> a -> b  
f $ x = f x

It has lowest precedencde and thus can often be used instead of parentheses: sum $ map sqrt [1..130] instead of sum (map sqrt [1..130]) or sum $ filter (> 10) $ map (*2) [2..10] instead of sum (filter (> 10) (map (*2) [2..10])).

It also allows to map function application to a list of functions like map ($ 3) [(4+), (10*), (^2), sqrt].

Function composition is done with the . function. If called with two functions f and g it returns a new function. This function is like calling f with argument x and then g with that result: f( g(x) ).

(.) :: (b -> c) -> (a -> b) -> a -> c  
f . g = \x -> f (g x)

map (\x -> negate (abs x)) [5,-3,-6,7,-3,2,-19,24]
map (\xs -> negate (sum (tail xs))) [[1..5],[3..6],[1..7]]
sum (replicate 5 (max 6.7 8.9))
replicate 100 (product (map (*3) (zipWith max [1,2,3,4,5] [4,5,6,7,8])))
-- same with function composition
map (negate . abs) [5,-3,-6,7,-3,2,-19,24]
map (negate . sum . tail) [[1..5],[3..6],[1..7]]
sum . replicate 5 . max 6.7 $ 8.9
replicate 100 . product . map (*3) . zipWith max [1,2,3,4,5] $ [4,5,6,7,8]

A common use case for function composition is defining function in so called point free style. It's when we write sum' = foldl (+) 0 instead of sum' xs = foldl (+) 0 xs (omitting the xs).

fn x = ceiling (negate (tan (cos (max 50 x))))
-- in point free style
fn = ceiling . negate . tan . cos . max 50

djoudi/gist:56850d078f777501886e1e79317d3561