# teaching machines

## CS 330 Lecture 23 – Ways of Making Functions: Composition

Dear students,

There are at least four ways of defining functions in Haskell:

• by defining a named function
• by partially applying parameters to get a new function expecting only the remaining parameters
• by composing a gauntlet of functions together
• by defining unnamed lambdas

We’ve seen the first two. Now let’s turn to composition. You’ve almost certainly worked with composition in your math classes. What is meant by (f . g)(x)? It means g is evaluated at x, and its result is used to further evaluate f.

Composing means to chain operations together, much like pipelining at the shell. The difference between composition and pipelining or Haskell’s $ operator is a matter of precedence. Haskell’s compose operator lets us chain functions together before we evaluate any of them. We can produce new functions by composing old ones together: h = f . g h 5 -- these two calls will f (g 5) -- produce the same result That means we can eliminate a lot of clutter from our code! Those gauntlets of function calls that our data must pass through can be shrunk down to a single meaningful name. Time for some examples: 1. Let’s write rsort which sorts a list in reverse: rsort = Ord a => [a] -> [a] rsort = reverse . sort 2. Let’s write scalebias to map a value using a linear function: scalebias :: Double -> Double -> Double -> Double scalebias scale bias x = scale * x + bias -- without composition scalebias' scale bias = (+ bias) . (* scale) -- with composition Now we write calc2fahren like we should have back in high school: c2f :: Double -> Double c2f = scalebias' 1.8 32 3. Let’s write a function to compute the number of negatives in a list: nnegatives :: [Integer] -> Int nnegatives = length . filter (< 0) 4. Let’s write a function that gives us the second element of a list, the neck: neck :: [a] -> a neck = head . tail 5. Let’s write a function that gives us a list of the unique items within another list: onlies :: Ord a => [a] -> [a] onlies = concat . filter single . group . sort where single list = length list == 1 6. Let’s write a function to create a spiral. As a first stop, let’s write some utility type synonyms and the (big three) functions to transform a 2D point: type Point = (Double, Double) scale :: Double -> Point -> Point scale factor (x, y) = (factor * x, factor * y) translate :: (Double, Double) -> Point -> Point translate (dx, dy) (x, y) = (x + dx, y + dy) rotate :: Double -> Point -> Point rotate degrees (x, y) = (x', y') where radians = degrees * pi / 180 x' = x * cos radians - y * sin radians y' = x * sin radians + y * cos radians We can make a spiral by repeatedly scaling and rotating a point. Each step will be this compound operation: spiral degrees growthRate = rotate degrees . scale growthRate Okay, now it’s time to write a standalone Haskell program that does its own IO instead of operating in the REPL. Sadly, IO is one of those things that doesn’t fit well in a pure functional language. IO is stateful. It embeds a notion of time: first this line, then this line, and then that one… Pure function languages forbid the notion of time and state. They are about mathematical truth, which is timeless. We must leave the world of getters (expressions) and enter the world of doers (statements). Let’s start with a simple function that prints a Point: print2 (x, y) = printf "%.5f,%.5f\n" x y Now let’s write our first main. It is different: main = do let p = (1, 0) print2 p let p' = spiral 1.05 20 p print2 p' return () We’ll talk about Haskell’s IO more another day. Right now let’s just remember these things: we must return () at the end, time sequences must be embedded in a do block, and let bind values to identifiers with let. What if we want to execute multiple steps? We can write a recursive method. main will kick it off with the initial point and number of iterations: main = step (1, 0) 100 Since there’s only one line, we don’t need do. The step function might look this this: step p n = do let p' = spiral 1.05 20 p print2 p' if n > 0 then step p' (n - 1) else return () Now we can execute this script from the command-line: runhaskell foo.hs To make this a little more flexible, let’s accept some command-line parameters: step :: (Double, Double) -> Double -> Double -> Int -> IO () step p rate degrees n = do let p' = spiral rate degrees p print2 p' if n > 0 then step p' rate degrees (n - 1) else return () main = do args <- getArgs let x = read (head args) :: Double let y = read (neck args) :: Double let rate = read (args !! 2) :: Double let degrees = read (args !! 3) :: Double let n = read (args !! 4) :: Int step (x, y) rate degrees n 7. Let’s write a function to transform a rectangle around an arbitrary pivot point. Let’s represent with rectangle with four numbers: the x- and y-coordinates of the bottom-left corner, a width, and a height. Let’s create some helper functions to determine the coordinates of the points: type Rectangle = (Point, Double, Double) bl (p, _, _) = p br ((left, bottom), width, _) = (left + width, bottom) tl ((left, bottom), _, height) = (left, bottom + height) tr ((left, bottom), width, height) = (left + width, bottom + height) To rotate around an arbitrary pivot, we pass a point through this gauntlet: rotateAround :: Point -> Double -> Point -> Point rotateAround pivot@(dx, dy) degrees = translate pivot . rotate degrees . translate (-dx, -dy) Finally, we can write a main to rotate a rectangle by the number of degrees provided at the command-line: main = do args <- getArgs let degrees = read (head args) :: Double let r = (0, 0, 1, 1) print2$ rotateAround (1, 1) degrees (bl r)
print2 $rotateAround (1, 1) degrees (br r) print2$ rotateAround (1, 1) degrees (tr r)
print2 $rotateAround (1, 1) degrees (tl r) print2$ rotateAround (1, 1) degrees (bl r)
return ()

scramble = fry . whisk . crack
vandalize = freeze . throw