# Higher-order function

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In mathematics and computer science, a higher-order function 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. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (disambiguation).

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

• 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.
• 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.
• fold
• Function composition
• Integration
• Callback
• Tree traversal

## 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.

#### OCAML

Explicitly

let add3 x = x + 3

let twice f x = f (f x)

print_int (twice add3 7) (* 13 *)


One-Line

print_int ((fun f x -> f (f x)) ((+)3) 7) (* 13 *)


#### APL

      twice←{⍺⍺ ⍺⍺ ⍵}

plusthree←{⍵+3}

g←{plusthree twice ⍵}

g 7
13


Or in a tacit manner:

      twice←⍣2

plusthree←+∘3

g←plusthree twice

g 7
13


#### J

Explicitly,

   twice=.     adverb : 'u u y'
plusthree=. verb   : 'y + 3'

g=. plusthree twice

g 7
13


or tacitly,

   twice=. ^:2
plusthree=. +&3

g=. plusthree twice

g 7
13


or point-free style,

   +&3(^:2) 7
13


#### Python

>>> def twice(f):
...   def result(a):
...     return f(f(a))
...   return result

>>> plusthree = lambda x: x+3

>>> g = twice(plusthree)

>>> g(7)
13


#### Wolfram Language

In[1]:= Nest[# + 3 &, 7, 2]
Out[1]:= 13



#### 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

(define (add x y) (+ x y))
(define (f x)
(lambda (y) (+ x y)))
(display ((f 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

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.

#### Elixir

In Elixir, you can mix module definitions and anonymous functions

defmodule Hop do
def twice(f, v) do
f.(f.(v))
end
end

add3 = fn(v) -> 3 + v end



Alternatively, we can also compose using pure anonymous functions.

twice = fn(f, v) -> f.(f.(v)) end
add3 = fn(v) -> 3 + v end



#### JavaScript

const twice = (f, v) => f(f(v));
const add3 = v => v + 3;



#### Go

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

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


#### Kotlin

fun <T> twice(f: (T)->T): (T)->T = {f(f(it))}
fun f(x:Int) = x + 3
println(twice(::f)(7)) // 13


#### Lua

local twice = function(f,v)
return f(f(v))
end

local f = function(v)
return v + 3
end

print(twice(f,7)) -- 13


#### Swift

// generic function
func twice<T>(_ v: @escaping (T) -> T) -> (T) -> T {
return { v(v($0)) } } // inferred closure let f = {$0 + 3 }

twice(f)(10) // 16


#### Rust

// Take function f(x), return function f(f(x))
fn twice<A>(function: impl Fn(A) -> A) -> impl Fn(A) -> A
{
move |a| function(function(a))
}

// Return x + 3
fn f(x: i32) -> i32 {
x + 3
}

fn main() {
let g = twice(f);
println!("{}", g(7));
}


#### Ruby

def twice(f, x)
f.call f.call(x)
end

add3 = ->(x) { x + 3 }


#### C++

With 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, using 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;
}


#### D

import std.stdio : writeln;

alias twice = (f, i) => f(f(i));
alias f = (int i) => i + 3;

void main()
{
writeln(twice(f, 7));
}


#### ColdFusion Markup Language (CFML)

twice = function(f, v) {
return f(f(v));
};

f = function(v) {
return v + 3;
};

writeOutput(twice(f, 7)); // 13


#### PHP

$twice = function($f, $v) { return$f($f($v));
};

$f = function($v) {
return $v + 3; }; echo($twice($f, 7)); // 13  #### R twice <- function(func) { return(function(x) { func(func(x)) }) } f <- function(x) { return(x + 3) } g <- twice(f) > print(g(7)) [1] 13  #### Perl 6 sub twice(Callable:D$c) {
return sub { $c($c($^x)) }; } sub f(Int:D$x) {
return $x + 3; } my$g = twice(&f);

set f {{v} {return [expr $v + 3]}} # result: 13 puts [apply$twice $f 7]  Tcl uses apply command to apply an anonymous function (since 8.6). #### XQuery declare function local:twice($f, $x) {$f($f($x))
};

declare function local:f($x) {$x + 3
};

local:twice(local:f#1, 7) (: 13 :)


### 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.