What is the difference between currying and partial application?

I quite often see on the Internet various complaints that other peoples examples of currying are not currying, but are actually just partial application.

I've not found a decent explanation of what partial application is, or how it differs from currying. There seems to be a general confusion, with equivalent examples being described as currying in some places, and partial application in others.

Could someone provide me with a definition of both terms, and details of how they differ?

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

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.

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.

I have answered this in another thread https://stackoverflow.com/a/12846865/1685865 . In short, partial function application is about fixing some arguments of a given multivariable function to yield another function with fewer arguments, while Currying is about turning a function of N arguments into a unary function which returns a unary function...[An example of Currying is shown at the end of this post.]

Currying is mostly of theoretical interest: one can express computations using only unary functions (i.e. every function is unary). In practice and as a byproduct, it is a technique which can make many useful (but not all) partial functional applications trivial, if the language has curried functions. Again, it is not the only means to implement partial applications. So you could encounter scenarios where partial application is done in other way, but people are mistaking it as Currying.

(Example of Currying)

In practice one would not just write

lambda x: lambda y: lambda z: x + y + z

or the equivalent javascript

function (x) { return function (y){ return function (z){ return x + y + z }}}

instead of

lambda x, y, z: x + y + z

for the sake of Currying.

Currying is a function of one argument which takes a function f and returns a new function h. Note that h takes an argument from X and returns a function that maps Y to Z:

curry(f) = h 
f: (X x Y) -> Z 
h: X -> (Y -> Z)

Partial application is a function of two(or more) arguments which takes a function f and one or more additional arguments to f and returns a new function g:

part(f, 2) = g
f: (X x Y) -> Z 
g: Y -> Z

The confusion arises because with a two-argument function the following equality holds:

partial(f, a) = curry(f)(a)

Both sides will yield the same one-argument function.

The equality is not true for higher arity functions because in this case currying will return a one-argument function, whereas partial application will return a multiple-argument function.

The difference is also in the behavior, whereas currying transforms the whole original function recursively(once for each argument), partial application is just a one step replacement.

Source: Wikipedia Currying.

Simple answer

Curry: lets you call a function, splitting it in multiple calls, providing one argument per-call.

Partial: lets you call a function, splitting it in multiple calls, providing multiple arguments per-call.

---

Simple hints

Both allow you to call a function providing less arguments (or, better, providing them cumulatively). Actually both of them bind (at each call) a specific value to specific arguments of the function.

The real difference can be seen when the function has more than 2 arguments.

---

Simple e(c)(sample)

(in Javascript)

We want to run the following process function on different subjects (e.g. let's say our subjects are "subject1" and "foobar" strings):

function process(context, successCallback, errorCallback, subject) {...}

why always passing the arguments, like context and the callbacks, if they will be always the same?

Just bind some values for the the function:

processSubject = _.partial(process, my_context, my_success, my_error)
// assign fixed values to the first 3 arguments of the `process` function

and call it on subject1 and foobar, omitting the repetition of the first 3 arguments, with:

processSubject('subject1');
processSubject('foobar');

Comfy, isn't it? 

---

With currying you'd instead need to pass one argument per time

curriedProcess = _.curry(process);   // make the function curry-able
processWithBoundedContext = curriedProcess(my_context);
processWithCallbacks = processWithBoundedContext(my_success)(my_error); // note: these are two sequential calls

result1 = processWithCallbacks('subject1');
// same as: process(my_context, my_success, my_error, 'subject1');

result2 = processWithCallbacks('foobar'); 
// same as: process(my_context, my_success, my_error, 'foobar');

Disclaimer

I skipped all the academic/mathematical explanation. Cause I don't know it. Maybe it helped 

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EDIT:

As added by @basickarl, a further slight difference in use of the two functions (see Lodash for examples) is that:

• partial returns a pre-cooked function that can be called once with the missing argument(s) and return the final result;
• while curry is being called multiple times (one for each argument), returning a pre-cooked function each time; except in the case of calling with the last argument, that will return the actual result from the processing of all the arguments.

---

With ES6:

here's a quick example of how immediate Currying and Partial-application are in ECMAScript 6.

const curriedSum = math => eng => geo => math + eng + geo;
const partialSum = math => (eng, geo) => math + eng + geo;

The difference between curry and partial application can be best illustrated through this following JavaScript example:

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

var partial = f.bind(null, 1);

6 === partial(2, 3);

Partial application results in a function of smaller arity; in the example above, f has an arity of 3 while partial only has an arity of 2. More importantly, a partially applied function would return the result right away upon being invoke, not another function down the currying chain. So if you are seeing something like partial(2)(3), it's not partial application in actuality.

Further reading:

• Functional Programming in 5 minutes
• Currying: Contrast with Partial Function Application

I had this question a lot while learning and have since been asked it many times. The simplest way I can describe the difference is that both are the same :) Let me explain...there are obviously differences.

Both partial application and currying involve supplying arguments to a function, perhaps not all at once. A fairly canonical example is adding two numbers. In pseudocode (actually JS without keywords), the base function may be the following:

add = (x, y) => x + y

If I wanted an "addOne" function, I could partially apply it or curry it:

addOneC = curry(add, 1)
addOneP = partial(add, 1)

Now using them is clear:

addOneC(2) #=> 3
addOneP(2) #=> 3

So what's the difference? Well, it's subtle, but partial application involves supplying some arguments and the returned function will then execute the main function upon next invocation whereas currying will keep waiting till it has all the arguments necessary:

curriedAdd = curry(add) # notice, no args are provided
addOne = curriedAdd(1) # returns a function that can be used to provide the last argument
addOne(2) #=> returns 3, as we want

partialAdd = partial(add) # no args provided, but this still returns a function
addOne = partialAdd(1) # oops! can only use a partially applied function once, so now we're trying to add one to an undefined value (no second argument), and we get an error

In short, use partial application to prefill some values, knowing that the next time you call the method, it will execute, leaving undefined all unprovided arguments; use currying when you want to continually return a partially-applied function as many times as necessary to fulfill the function signature. One final contrived example:

curriedAdd = curry(add)
curriedAdd()()()()()(1)(2) # ugly and dumb, but it works

partialAdd = partial(add)
partialAdd()()()()()(1)(2) # second invocation of those 7 calls fires it off with undefined parameters

Hope this helps!

UPDATE: Some languages or lib implementations will allow you to pass an arity (total number of arguments in final evaluation) to the partial application implementation which may conflate my two descriptions into a confusing mess...but at that point, the two techniques are largely interchangeable.

For me partial application must create a new function where the used arguments are completely integrated into the resulting function.

Most functional languages implement currying by returning a closure: do not evaluate under lambda when partially applied. So, for partial application to be interesting, we need to make a difference between currying and partial application and consider partial application as currying plus evaluation under lambda.

I could be very wrong here, as I don't have a strong background in theoretical mathematics or functional programming, but from my brief foray into FP, it seems that currying tends to turn a function of N arguments into N functions of one argument, whereas partial application [in practice] works better with variadic functions with an indeterminate number of arguments. I know some of the examples in previous answers defy this explanation, but it has helped me the most to separate the concepts. Consider this example (written in CoffeeScript for succinctness, my apologies if it confuses further, but please ask for clarification, if needed):

# partial application
partial_apply = (func) ->
  args = [].slice.call arguments, 1
  -> func.apply null, args.concat [].slice.call arguments

sum_variadic = -> [].reduce.call arguments, (acc, num) -> acc + num

add_to_7_and_5 = partial_apply sum_variadic, 7, 5

add_to_7_and_5 10 # returns 22
add_to_7_and_5 10, 11, 12 # returns 45

# currying
curry = (func) ->
  num_args = func.length
  helper = (prev) ->
    ->
      args = prev.concat [].slice.call arguments
      return if args.length < num_args then helper args else func.apply null, args
  helper []

sum_of_three = (x, y, z) -> x + y + z
curried_sum_of_three = curry sum_of_three
curried_sum_of_three 4 # returns a function expecting more arguments
curried_sum_of_three(4)(5) # still returns a function expecting more arguments
curried_sum_of_three(4)(5)(6) # returns 15
curried_sum_of_three 4, 5, 6 # returns 15

This is obviously a contrived example, but notice that partially applying a function that accepts any number of arguments allows us to execute a function but with some preliminary data. Currying a function is similar but allows us to execute an N-parameter function in pieces until, but only until, all N parameters are accounted for.

Again, this is my take from things I've read. If anyone disagrees, I would appreciate a comment as to why rather than an immediate downvote. Also, if the CoffeeScript is difficult to read, please visit coffeescript.org, click "try coffeescript" and paste in my code to see the compiled version, which may (hopefully) make more sense. Thanks!

I'm going to assume most people who ask this question are already familiar with the basic concepts so their is no need to talk about that. It's the overlap that is the confusing part.

You might be able to fully use the concepts, but you understand them together as this pseudo-atomic amorphous conceptual blur. What is missing is knowing where the boundary between them is.

Instead of defining what each one is, it's easier to highlight just their differences—the boundary.

Currying is when you define the function.

Partial Application is when you call the function.

Application is math-speak for calling a function.

Partial application requires calling a curried function and getting a function as the return type.

A lot of people here do not address this properly, and no one has talked about overlaps.

Simple answer

Currying: Lets you call a function, splitting it in multiple calls, providing one argument per-call.

Partial Application: Lets you call a function, splitting it in multiple calls, providing multiple arguments per-call.

> One of the significant differences between the two is that a call to a > partially applied function returns the result right away, not another > function down the currying chain; this distinction can be illustrated > clearly for functions whose arity is greater than two.

What does that mean? That means that there are max two calls for a partial function. Currying has as many as the amount of arguments. If the currying function only has two arguments, then it is essentially the same as a partial function.

Examples

Partial Application and Currying

function bothPartialAndCurry(firstArgument) {
    return function(secondArgument) {
        return firstArgument + secondArgument;
    }
}

const partialAndCurry = bothPartialAndCurry(1);
const result = partialAndCurry(2);

Partial Application

function partialOnly(firstArgument, secondArgument) {
    return function(thirdArgument, fourthArgument, fifthArgument) {
        return firstArgument + secondArgument + thirdArgument + fourthArgument + fifthArgument;
    }
}

const partial = partialOnly(1, 2);
const result = partial(3, 4, 5);

Currying

function curryOnly(firstArgument) {
    return function(secondArgument) {
        return function(thirdArgument) {
            return function(fourthArgument ) {
                return function(fifthArgument) {
                    return firstArgument + secondArgument + thirdArgument + fourthArgument + fifthArgument;
                }
            }
        }
    }
}

const curryFirst = curryOnly(1);
const currySecond = curryFirst(2);
const curryThird = currySecond(3);
const curryFourth = curryThird(4);
const result = curryFourth(5);

// or...

const result = curryOnly(1)(2)(3)(4)(5);

Naming conventions

I'll write this when I have time, which is soon.

There are other great answers here but I believe this example (as per my understanding) in Java might be of benefit to some people:

public static <A,B,X> Function< B, X > partiallyApply( BiFunction< A, B, X > aBiFunction, A aValue ){
    return b -> aBiFunction.apply( aValue, b );
}

public static <A,X> Supplier< X > partiallyApply( Function< A, X > aFunction, A aValue ){
    return () -> aFunction.apply( aValue );
}

public static <A,B,X> Function<  A, Function< B, X >  > curry( BiFunction< A, B, X > bif ){
    return a -> partiallyApply( bif, a );
}

So currying gives you a one-argument function to create functions, where partial-application creates a wrapper function that hard codes one or more arguments.

If you want to copy&paste, the following is noisier but friendlier to work with since the types are more lenient:

public static <A,B,X> Function< ? super B, ? extends X > partiallyApply( final BiFunction< ? super A, ? super B, X > aBiFunction, final A aValue ){
    return b -> aBiFunction.apply( aValue, b );
}

public static <A,X> Supplier< ? extends X > partiallyApply( final Function< ? super A, X > aFunction, final A aValue ){
    return () -> aFunction.apply( aValue );
}

public static <A,B,X> Function<  ? super A,  Function< ? super B, ? extends X >  > curry( final BiFunction< ? super A, ? super B, ? extends X > bif ){
    return a -> partiallyApply( bif, a );
}

In writing this, I confused currying and uncurrying. They are inverse transformations on functions. It really doesn't matter what you call which, as long as you get what the transformation and its inverse represent.

Uncurrying isn't defined very clearly (or rather, there are "conflicting" definitions that all capture the spirit of the idea). Basically, it means turning a function that takes multiple arguments into a function that takes a single argument. For example,

(+) :: Int -> Int -> Int

Now, how do you turn this into a function that takes a single argument? You cheat, of course!

plus :: (Int, Int) -> Int

Notice that plus now takes a single argument (that is composed of two things). Super!

What's the point of this? Well, if you have a function that takes two arguments, and you have a pair of arguments, it is nice to know that you can apply the function to the arguments, and still get what you expect. And, in fact, the plumbing to do it already exists, so that you don't have to do things like explicit pattern matching. All you have to do is:

(uncurry (+)) (1,2)

So what is partial function application? It is a different way to turn a function in two arguments into a function with one argument. It works differently though. Again, let's take (+) as an example. How might we turn it into a function that takes a single Int as an argument? We cheat!

((+) 0) :: Int -> Int

That's the function that adds zero to any Int.

((+) 1) :: Int -> Int

adds 1 to any Int. Etc. In each of these cases, (+) is "partially applied".

Currying

Wikipedia says

> Currying is the technique of converting a function that takes multiple arguments into a sequence of functions that each takes a single argument.

Example

const add = (a, b) => a + b

const addC = (a) => (b) => a + b // curried function. Where C means curried

Partial application

Article Just Enough FP: Partial Application

> Partial application is the act of applying some, but not all, of the arguments to a function and returning a new function awaiting the rest of the arguments. These applied arguments are stored in closure and remain available to any of the partially applied returned functions in the future.

Example

const add = (a) => (b) => a + b

const add3 = add(3) // add3 is a partially applied function

add3(5) // 8

The difference is

1. currying is a technique (pattern)
2. partial application is a function with some predefined arguments (like add3 from the previous example)