What is the difference between currying and partial application?

Note: this was taken from F# Basics an excellent introductory article for .NET developers getting into functional programming.

Currying means breaking a function with many arguments into a series of functions that each take one argument and ultimately produce the same result as the original function. Currying is probably the most challenging topic for developers new to functional programming, particularly because it is often confused with partial application. You can see both at work in this example:

let multiply x y = x * y    
let double = multiply 2
let ten = double 5

Right away, you should see behavior that is different from most imperative languages. The second statement creates a new function called double by passing one argument to a function that takes two. The result is a function that accepts one int argument and yields the same output as if you had called multiply with x equal to 2 and y equal to that argument. In terms of behavior, it’s the same as this code:

let double2 z = multiply 2 z

Often, people mistakenly say that multiply is curried to form double. But this is only somewhat true. The multiply function is curried, but that happens when it is defined because functions in F# are curried by default. When the double function is created, it’s more accurate to say that the multiply function is partially applied.

The multiply function is really a series of two functions. The first function takes one int argument and returns another function, effectively binding x to a specific value. This function also accepts an int argument that you can think of as the value to bind to y. After calling this second function, x and y are both bound, so the result is the product of x and y as defined in the body of double.

To create double, the first function in the chain of multiply functions is evaluated to partially apply multiply. The resulting function is given the name double. When double is evaluated, it uses its argument along with the partially applied value to create the result.

Interesting question. After a bit of searching, "Partial Function Application is not currying" gave the best explanation I found. I can't say that the practical difference is particularly obvious to me, but then I'm not an FP expert...

Another useful-looking page (which I confess I haven't fully read yet) is "Currying and Partial Application with Java Closures".

It does look like this is widely-confused pair of terms, mind you.

The easiest way to see how they differ is to consider a real example. Let's assume that we have a function Add which takes 2 numbers as input and returns a number as output, e.g. Add(7, 5) returns 12. In this case:

• Partial applying the function Add with a value 7 will give us a new function as output. That function itself takes 1 number as input and outputs a number. As such:

  Partial(Add, 7); // returns a function f2 as output
  
                   // f2 takes 1 number as input and returns a number as output
  

  So we can do this:

  f2 = Partial(Add, 7);
  f2(5); // returns 12;
         // f2(7)(5) is just a syntactic shortcut
  

• Currying the function Add will give us a new function as output. That function itself takes 1 number as input and outputs yet another new function. That third function then takes 1 number as input and returns a number as output. As such:

  Curry(Add); // returns a function f2 as output
  
              // f2 takes 1 number as input and returns a function f3 as output
              // i.e. f2(number) = f3
  
              // f3 takes 1 number as input and returns a number as output
              // i.e. f3(number) = number
  

  So we can do this:

  f2 = Curry(Add);
  f3 = f2(7);
  f3(5); // returns 12
  

In other words, "currying" and "partial application" are two totally different functions. Currying takes exactly 1 input, whereas partial application takes 2 (or more) inputs.

Even though they both return a function as output, the returned functions are of totally different forms as demonstrated above.

Currying is converting a single function of n arguments into n functions with a single argument each. Given the following function:

function f(x,y,z) { z(x(y));}

When curried, becomes:

function f(x) { lambda(y) { lambda(z) { z(x(y)); } } }

In order to get the full application of f(x,y,z), you need to do this:

f(x)(y)(z);

Many functional languages let you write f x y z. If you only call f x y or f(x)(y) then you get a partially-applied function—the return value is a closure of lambda(z){z(x(y))} with passed-in the values of x and y to f(x,y).

One way to use partial application is to define functions as partial applications of generalized functions, like fold:

function fold(combineFunction, accumulator, list) {/* ... */}
function sum     = curry(fold)(lambda(accum,e){e+accum}))(0);
function length  = curry(fold)(lambda(accum,_){1+accum})(empty-list);
function reverse = curry(fold)(lambda(accum,e){concat(e,accum)})(empty-list);

/* ... */
@list = [1, 2, 3, 4]
sum(list) //returns 10
@f = fold(lambda(accum,e){e+accum}) //f = lambda(accumulator,list) {/*...*/}
f(0,list) //returns 10
@g = f(0) //same as sum
g(list)  //returns 10