Higher-order Functions

Functions are values just like any other value in OCaml. What does that mean exactly? This means that we can pass functions around as arguments to other functions, that we can store functions in data structures, that we can return functions as a result from other functions.

Let us look at why it is useful to have higher-order functions. The first reason is that it allows you to write general, reusable code. Consider these functions double and square on integers:

let double x = 2 * x
let square x = x * x

Let's use these functions to write other functions that quadruple and raise a number to the fourth power:

let quad x   = double (double x)
let fourth x = square (square x)

There is an obvious similarity between these two functions: what they do is apply a given function twice to a value. By passing in the function to another function twice as an argument, we can abstract this functionality:

let twice f x = f (f x)
(* twice : ('a -> 'a) -> 'a -> 'a *)

Using twice, we can implement quad and fourth in a uniform way:

let quad   x = twice double x
let fourth x = twice square x

Higher-order functions either take other functions as input or return other functions as output (or both). The function twice is higher-order: its input f is a function. And—recalling that all OCaml functions really take only a single argument—its output is technically fun x -> f (f x), so twice returns a function hence is also higher-order in that way. Higher-order functions are also known as functionals, and programming with them could be called functional programming—indicating what the heart of programming in languages like OCaml is all about.

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