How does Haskell printf work?

Haskell's type safety is second to none only to dependently-typed languages. But there is some deep magic going on with Text.Printf that seems rather type-wonky.

> printf "%d\n" 3
3
> printf "%s %f %d" "foo" 3.3 3
foo 3.3 3

What is the deep magic behind this? How can the Text.Printf.printf function take variadic arguments like this?

What is the general technique used to allow for variadic arguments in Haskell, and how does it work?

(Side note: some type safety is apparently lost when using this technique.)

> :t printf "%d\n" "foo"
printf "%d\n" "foo" :: (PrintfType ([Char] -> t)) => t

The trick is to use type classes. In the case of printf, the key is the PrintfType type class. It does not expose any methods, but the important part is in the types anyway.

class PrintfType r
printf :: PrintfType r => String -> r

So printf has an overloaded return type. In the trivial case, we have no extra arguments, so we need to be able to instantiate r to IO (). For this, we have the instance

instance PrintfType (IO ())

Next, in order to support a variable number of arguments, we need to use recursion at the instance level. In particular we need an instance so that if r is a PrintfType, a function type x -> r is also a PrintfType.

-- instance PrintfType r => PrintfType (x -> r)

Of course, we only want to support arguments which can actually be formatted. That's where the second type class PrintfArg comes in. So the actual instance is

instance (PrintfArg x, PrintfType r) => PrintfType (x -> r)

Here's a simplified version which takes any number of arguments in the Show class and just prints them:

{-# LANGUAGE FlexibleInstances #-}

foo :: FooType a => a
foo = bar (return ())

class FooType a where
    bar :: IO () -> a

instance FooType (IO ()) where
    bar = id

instance (Show x, FooType r) => FooType (x -> r) where
    bar s x = bar (s >> print x)

Here, bar takes an IO action which is built up recursively until there are no more arguments, at which point we simply execute it.

*Main> foo 3 :: IO ()
3
*Main> foo 3 "hello" :: IO ()
3
"hello"
*Main> foo 3 "hello" True :: IO ()
3
"hello"
True

QuickCheck also uses the same technique, where the Testable class has an instance for the base case Bool, and a recursive one for functions which take arguments in the Arbitrary class.

class Testable a
instance Testable Bool
instance (Arbitrary x, Testable r) => Testable (x -> r)