# Higher-order function

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Not to be confused with Functor (category theory).

In mathematics and computer science, a higher-order function (also functional, functional form or functor) is a function that does at least one of the following:

All other functions are first-order functions. In mathematics higher-order functions are also termed operators or functionals. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function.

In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form ${\displaystyle (\tau _{1}\to \tau _{2})\to \tau _{3}}$.

## General examples

The map function, found in many functional programming languages, is one example of a higher-order function. It takes as arguments a function f and a list of elements, and as the result, returns a new list with f applied to each element from the list. Another very common kind of higher-order function in those languages which support them are sorting functions which take a comparison function as a parameter, allowing the programmer to separate the sorting algorithm from the comparisons of the items being sorted. The C standard function qsort is an example of this.

Other examples of higher-order functions include fold, function composition, and integration.

## Support in programming languages

### Direct support

The examples are not intended to compare and contrast programming languages, but to serve as examples of higher-order function syntax

In the following examples, the higher-order function twice takes a function, and applies the function to some value twice. If twice has to be applied several times for the same f it preferably should return a function rather than a value. This is in line with the "don't repeat yourself" principle.

#### Python

Further information: Python (programming language)
>>> def twice(f):
...     return lambda x: f(f(x))

>>> def f(x):
...     return x + 3

>>> g = twice(f)

>>> print g(7)
13


#### Pascal

Further information: Pascal (programming language)
{$mode objfpc} type fun = function(x:integer):integer; function f(x:integer):integer; begin f:= x+3; end; function g( func:fun; x:integer):integer; begin g:= func(x)*func(x); end; begin write(g(@f, 7)); end.  #### F# Further information: F Sharp (programming language) let twice f = f >> f let f = (+) 3 twice f 7 |> printf "%A" // 13  #### Haskell Further information: Haskell (programming language) twice :: (a -> a) -> (a -> a) twice f = f . f f :: Num a => a -> a f = subtract 3 main :: IO () main = print (twice f 7) -- 1  #### Clojure Further information: Clojure (defn twice [function x] (function (function x))) (twice #(+ % 3) 7) ;13  In Clojure, "#" starts a lambda expression, and "%" refers to the next function argument. #### Scheme Further information: Scheme (programming language) (define (add x y) (+ x y)) (define (f x) (lambda (y) (+ x y))) (display ((f 3) 7)) (display (add 3 7))  In this Scheme example, the higher-order function (f x) is used to implement currying. It takes a single argument and returns a function. The evaluation of the expression ((f 3) 7) first returns a function after evaluating (f 3). The returned function is (lambda (y) (+ 3 y)). Then, it evaluates the returned function with 7 as the argument, returning 10. This is equivalent to the expression (add 3 7), since (f x) is equivalent to the curried form of (add x y). #### Erlang Further information: Erlang (programming language) or_else([], _) -> false; or_else([F | Fs], X) -> or_else(Fs, X, F(X)). or_else(Fs, X, false) -> or_else(Fs, X); or_else(Fs, _, {false, Y}) -> or_else(Fs, Y); or_else(_, _, R) -> R. or_else([fun erlang:is_integer/1, fun erlang:is_atom/1, fun erlang:is_list/1],3.23).  In this Erlang example, the higher-order function or_else/2 takes a list of functions (Fs) and argument (X). It evaluates the function F with the argument X as argument. If the function F returns false then the next function in Fs will be evaluated. If the function F returns {false,Y} then the next function in Fs with argument Y will be evaluated. If the function F returns R the higher-order function or_else/2 will return R. Note that X, Y, and R can be functions. The example returns false. #### JavaScript Further information: JavaScript const twice = (f, v) => f(f(v)); const f = v => v + 3; console.log(twice(f, 7)); // 13  #### Go Further information: Go (programming language) func twice(f func(int) int, v int) int { return f(f(v)) } func main() { f := func(v int) int { return v + 3 } twice(f, 7) // returns 13 }  Notice a function literal can be defined either with an identifier (twice) or anonymously (assigned to variable f). Run full program on Go Playground! #### Scala Further information: Scala (programming language) def twice(f:Int=>Int) = f compose f twice(_+3)(7) // 13  #### Java (1.8+) Function<Function<Integer, Integer>, Function<Integer, Integer>> twice = f -> f.andThen(f); twice.apply(x -> x + 3).apply(7); // 13  #### Swift Further information: Swift (programming language) func f(x:Int) -> Int { return x + 3 } func g(function: (x:Int) -> Int, paramX: Int) -> Int { return function(x: paramX) * function(x: paramX) } g(f,paramX: 7)  #### Rust Further information: Rust (programming language) // Take function f(x), return function f(f(x)) fn twice<A, F>(function: F) -> Box<Fn(A) -> A> where F: 'static + Fn(A) -> A { Box::new(move |a| function(function(a))) } // Return x + 3 fn f(x: i32) -> i32 { x + 3 } fn main() { let g = twice(f); println!("{}", g(7)); }  #### C++ Further information: C++ // Generic lambdas provided by C++14. #include <iostream> auto twice = [](auto f, int v) { return f(f(v)); }; auto f = [](int i) { return i + 3; }; int main() { std::cout << twice(f, 7) << std::endl; } // Or, use std::function in C++11 #include <iostream> #include <functional> auto twice = [](const std::function<int(int)>& f, int v) { return f(f(v)); }; auto f = [](int i) { return i + 3; }; int main() { std::cout << twice(f, 7) << std::endl; }  #### ColdFusion Markup Language (CFML) Further information: ColdFusion Markup Language twice = function(f, v) { return f(f(v)); }; f = function(v) { return v + 3; }; writeOutput(twice(f, 7)); // 13  #### PHP Further information: PHP $twice = function($f,$v) {
return $f($f($v)); };$f = function($v) { return$v + 3;
};

echo($twice($f, 7)); // 13


#### R

Further information: R (programming language)
twice <- function(func) {
return(function(x) {
func(func(x))
})
}

f <- function(x) {
return(x + 3)
}

g <- twice(f)

> print(g(7))
[1] 13


### XACML

Further information: XACML

The XACML standard defines higher-order functions in the standard to apply a function to multiple values of attribute bags.

rule allowEntry{
permit
condition anyOfAny(function[stringEqual], citizenships, allowedCitizenships)
}


The list of higher-order functions is can be found here.

### Alternatives

#### Function pointers

Function pointers in languages such as C and C++ allow programmers to pass around references to functions. The following C code computes an approximation of the integral of an arbitrary function:

#include <stdio.h>

double square(double x)
{
return x * x;
}

double cube(double x)
{
return x * x * x;
}

/* Compute the integral of f() within the interval [a,b] */
double integral(double f(double x), double a, double b, int n)
{
int i;
double sum = 0;
double dt = (b - a) / n;
for (i = 0;  i < n;  ++i) {
sum += f(a + (i + 0.5) * dt);
}
return sum * dt;
}

int main()
{
printf("%g\n", integral(square, 0, 1, 100));
printf("%g\n", integral(cube, 0, 1, 100));
return 0;
}


The qsort function from the C standard library uses a function pointer to emulate the behavior of a higher-order function.

#### Macros

Macros can also be used to achieve some of the effects of higher order functions. However, macros cannot easily avoid the problem of variable capture; they may also result in large amounts of duplicated code, which can be more difficult for a compiler to optimize. Macros are generally not strongly typed, although they may produce strongly typed code.

#### Dynamic code evaluation

In other imperative programming languages, it is possible to achieve some of the same algorithmic results as are obtained via higher-order functions by dynamically executing code (sometimes called Eval or Execute operations) in the scope of evaluation. There can be significant drawbacks to this approach:

• The argument code to be executed is usually not statically typed; these languages generally rely on dynamic typing to determine the well-formedness and safety of the code to be executed.
• The argument is usually provided as a string, the value of which may not be known until run-time. This string must either be compiled during program execution (using just-in-time compilation) or evaluated by interpretation, causing some added overhead at run-time, and usually generating less efficient code.

#### Objects

In object-oriented programming languages that do not support higher-order functions, objects can be an effective substitute. An object's methods act in essence like functions, and a method may accept objects as parameters and produce objects as return values. Objects often carry added run-time overhead compared to pure functions, however, and added boilerplate code for defining and instantiating an object and its method(s). Languages that permit stack-based (versus heap-based) objects or structs can provide more flexibility with this method.

An example of using a simple stack based record in Free Pascal with a function that returns a function:

program example;

type
int = integer;
Txy = record x, y: int; end;
Tf = function (xy: Txy): int;

function f(xy: Txy): int;
begin
Result := xy.y + xy.x;
end;

function g(func: Tf): Tf;
begin
result := func;
end;

var
a: Tf;
xy: Txy = (x: 3; y: 7);

begin
a := g(@f);     // return a function to "a"
writeln(a(xy)); // prints 10
end.


The function a() takes a Txy record as input and returns the integer value of the sum of the record's x and y fields (3 + 7).

#### Defunctionalization

Defunctionalization can be used to implement higher-order functions in languages that lack first-class functions:

// Defunctionalized function data structures
template<typename T> struct Add { T value; };
template<typename T> struct DivBy { T value; };
template<typename F, typename G> struct Composition { F f; G g; };

// Defunctionalized function application implementations
template<typename F, typename G, typename X>
auto apply(Composition<F, G> f, X arg) {
return apply(f.f, apply(f.g, arg));
}

template<typename T, typename X>
auto apply(Add<T> f, X arg) {
return arg  + f.value;
}

template<typename T, typename X>
auto apply(DivBy<T> f, X arg) {
return arg / f.value;
}

// Higher-order compose function
template<typename F, typename G>
Composition<F, G> compose(F f, G g) {
return Composition<F, G> {f, g};
}

int main(int argc, const char* argv[]) {
auto f = compose(DivBy<float>{ 2.0f }, Add<int>{ 5 });
apply(f, 3); // 4.0f
apply(f, 9); // 7.0f
return 0;
}


In this case, different types are used to trigger different functions via function overloading. The overloaded function in this example has the signature auto apply.