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Currying versus partial application (with JavaScript code)

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Currying and partial application are two ways of transforming a function into another function with a generally smaller arity. While they are often confused with each other, they do different things. This post explains the details.


Currying takes a function
f: X × Y → R
and turns it into a function
f': X → (Y → R)
Instead of calling f with two arguments, we invoke f' with the first argument. The result is a function that we then call with the second argument to produce the result. Thus, if the uncurried f is invoked as
f(3, 5)
then the curried f' is invoked as
JavaScript example. The following is the uncurried binary function add().
    function add(x, y) {
        return × + y;
Calling it:
    > add(3, 5)
The curried version of add looks as follows.
    function addC(x) {
        return function (y) {
            return x + y;
Calling it:
    > addC(3)(5)
The algorithm for currying. The function curry curries a given binary function. It has the signature
curry: (X × Y → R) → (X → (Y → R))
Explanation: curry takes a binary function and returns a unary function that returns a unary function. Its JavaScript code is
    function curry(f) {
        return function(x) {
            return function(y) {
                return f(x, y);

Partial application

Partial application takes a function
f: X × Y → R
and a fixed value x for the first argument to produce a new function
f': Y → R
f' does the same as f, but only has to fill in the second parameter which is why its arity is one less than the arity of f. One says that the first argument is bound to x.

JavaScript example. Binding the first argument of function add to 5 produces the function plus5. Compare their definitions to see that we have simply filled in the first argument.

    function plus5(y) {
        return 5 + y;

The algorithm for partial application. The function partApply partially applies binary functions. It has the signature

partApply : ((X × Y → R) × X) → (Y → R)
Explanation: partApply takes a binary function and a value and produces a unary function. Its JavaScript code is
    function partApply(f, x) {
        return function(y) {
            return f(x, y);
General partial application in JavaScript. JavaScript has the built-in method Function.prototype.bind that works on functions with any arity and can bind an arbitrary amount of parameters. Its invocation has the following syntax.
    func.bind(thisValue, [arg1], [arg2], ...)
It turns func into a new function whose implicit this parameter is thisValue and whose initial arguments are always as given. When one invokes the new function, the arguments of such an invocation are appended to what has already been provided via bind. MDN has more details.


    > var plus5 = add.bind(null, 5)
    > plus5(10)
Note that thisValue does not matter for the (non-method) function add which is why it is null above.

Currying versus partial application

The difference between the two is:
  • Currying always produces nested unary (1-ary) functions. The transformed function is still largely the same as the original.
  • Partial application produces functions of arbitrary arity. The transformed function is different from the original – it needs less arguments.
Interestingly, with a curried function and curried invocation, it is easy to achieve the same effect as binding one argument (performing this operation several times yields the general case): To bind the first argument to a value, you simply apply the outer-most of the nested functions to that value, which returns the desired new function.

Related reading

  1. Partial application on Wikipedia [partial source of this post]
From http://www.2ality.com/2011/09/currying-vs-part-eval.html

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