First-class functions

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Task
First-class functions
You are encouraged to solve this task according to the task description, using any language you may know.

A language has first-class functions if it can do each of the following without recursively invoking a compiler or interpreter or otherwise metaprogramming:

  • Create new functions from preexisting functions at run-time
  • Store functions in collections
  • Use functions as arguments to other functions
  • Use functions as return values of other functions

Write a program to create an ordered collection A of functions of a real number. At least one function should be built-in and at least one should be user-defined; try using the sine, cosine, and cubing functions. Fill another collection B with the inverse of each function in A. Implement function composition as in Functional Composition. Finally, demonstrate that the result of applying the composition of each function in A and its inverse in B to a value, is the original value. (Within the limits of computational accuracy).

(A solution need not actually call the collections "A" and "B". These names are only used in the preceding paragraph for clarity.)

C.f. First-class Numbers

Contents

[edit] ActionScript

Translation of: JavaScript
 
var cube:Function = function(x) {
return Math.pow(x, 3);
};
var cuberoot:Function = function(x) {
return Math.pow(x, 1/3);
};
 
function compose(f:Function, g:Function):Function {
return function(x:Number) {return f(g(x));};
}
var functions:Array = [Math.cos, Math.tan, cube];
var inverse:Array = [Math.acos, Math.atan, cuberoot];
 
function test() {
for (var i:uint = 0; i < functions.length; i++) {
// Applying the composition to 0.5
trace(compose(functions[i], inverse[i])(0.5));
}
}
 
test();

Output:

0.5000000000000001
0.5000000000000001
0.5000000000000001

[edit] Ada

Even if the example below solves the task, there are some limitations to how dynamically you can create, store and use functions in Ada, so it is debatable if Ada really has first class functions.

with Ada.Float_Text_IO,
Ada.Integer_Text_IO,
Ada.Text_IO,
Ada.Numerics.Elementary_Functions;
 
procedure First_Class_Functions is
use Ada.Float_Text_IO,
Ada.Integer_Text_IO,
Ada.Text_IO,
Ada.Numerics.Elementary_Functions;
 
function Sqr (X : Float) return Float is
begin
return X ** 2;
end Sqr;
 
type A_Function is access function (X : Float) return Float;
 
generic
F, G : A_Function;
function Compose (X : Float) return Float;
 
function Compose (X : Float) return Float is
begin
return F (G (X));
end Compose;
 
Functions : array (Positive range <>) of A_Function := (Sin'Access,
Cos'Access,
Sqr'Access);
Inverses  : array (Positive range <>) of A_Function := (Arcsin'Access,
Arccos'Access,
Sqrt'Access);
begin
for I in Functions'Range loop
declare
function Identity is new Compose (Functions (I), Inverses (I));
Test_Value : Float := 0.5;
Result  : Float;
begin
Result := Identity (Test_Value);
 
if Result = Test_Value then
Put ("Example ");
Put (I, Width => 0);
Put_Line (" is perfect for the given test value.");
else
Put ("Example ");
Put (I, Width => 0);
Put (" is off by");
Put (abs (Result - Test_Value));
Put_Line (" for the given test value.");
end if;
end;
end loop;
end First_Class_Functions;

It is bad style (but an explicit requirement in the task description) to put the functions and their inverses in separate arrays rather than keeping each pair in a record and then having an array of that record type.

[edit] Aikido

This example is incomplete. Fails to demonstrate that the result of applying the composition of each function in A and its inverse in B to a value, is the original value Please ensure that it meets all task requirements and remove this message.
Translation of: Javascript
 
import math
 
function compose (f, g) {
return function (x) { return f(g(x)) }
}
 
var fn = [Math.sin, Math.cos, function(x) { return x*x*x }]
var inv = [Math.asin, Math.acos, function(x) { return Math.pow(x, 1.0/3) }]
 
for (var i=0; i<3; i++) {
var f = compose(inv[i], fn[i])
println(f(0.5)) // 0.5
}
 
 

[edit] ALGOL 68

Translation of: Python
Works with: ALGOL 68 version Standard - no extensions to language used
Works with: ALGOL 68G version Any - tested with release mk15-0.8b.fc9.i386
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8.8d.fc9.i386 using non-standard compose

Note: Returning PROC (REAL x)REAL: f1(f2(x)) from a function apparently violates standard ALGOL 68's scoping rules. ALGOL 68G warns about this during parsing, and then - if run out of scope - rejects during runtime.

MODE F = PROC (REAL)REAL;
OP ** = (REAL x, power)REAL: exp(ln(x)*power);
 
# Add a user defined function and its inverse #
PROC cube = (REAL x)REAL: x * x * x;
PROC cube root = (REAL x)REAL: x ** (1/3);
 
# First class functions allow run-time creation of functions from functions #
# return function compose(f,g)(x) == f(g(x)) #
PROC non standard compose = (F f1, f2)F: (REAL x)REAL: f1(f2(x)); # eg ELLA ALGOL 68RS #
PROC compose = (F f, g)F: ((F f2, g2, REAL x)REAL: f2(g2(x)))(f, g, );
 
# Or the classic "o" functional operator #
PRIO O = 5;
OP (F,F)F O = compose;
 
# first class functions should be able to be members of collection types #
[]F func list = (sin, cos, cube);
[]F arc func list = (arc sin, arc cos, cube root);
 
# Apply functions from lists as easily as integers #
FOR index TO UPB func list DO
STRUCT(F f, inverse f) this := (func list[index], arc func list[index]);
print(((inverse f OF this O f OF this)(.5), new line))
OD

Output:

+.500000000000000e +0
+.500000000000000e +0
+.500000000000000e +0

[edit] AppleScript

AppleScript does not have built-in functions like sine or cosine.

-- Compose two functions, where each function is
-- a script object with a call(x) handler.
on compose(f, g)
script
on call(x)
f's call(g's call(x))
end call
end script
end compose
 
script increment
on call(n)
n + 1
end call
end script
 
script decrement
on call(n)
n - 1
end call
end script
 
script twice
on call(x)
x * 2
end call
end script
 
script half
on call(x)
x / 2
end call
end script
 
script cube
on call(x)
x ^ 3
end call
end script
 
script cuberoot
on call(x)
x ^ (1 / 3)
end call
end script
 
set functions to {increment, twice, cube}
set inverses to {decrement, half, cuberoot}
set answers to {}
repeat with i from 1 to 3
set end of answers to ¬
compose(item i of inverses, ¬
item i of functions)'s ¬
call(0.5)
end repeat
answers -- Result: {0.5, 0.5, 0.5}

[edit] AutoHotkey

By just me. Forum Post

#NoEnv
; Set the floating-point precision
SetFormat, Float, 0.15
; Super-global variables for function objects
Global F, G
; User-defined functions
Cube(X) {
Return X ** 3
}
CubeRoot(X) {
Return X ** (1/3)
}
; Function arrays, Sin/ASin and Cos/ACos are built-in
FuncArray1 := [Func("Sin"), Func("Cos"), Func("Cube")]
FuncArray2 := [Func("ASin"), Func("ACos"), Func("CubeRoot")]
; Compose
Compose(FN1, FN2) {
Static FG := Func("ComposedFunction")
F := FN1, G:= FN2
Return FG
}
ComposedFunction(X) {
Return F.(G.(X))
}
; Run
X := 0.5 + 0
Result := "Input:`n" . X . "`n`nOutput:"
For Index In FuncArray1
Result .= "`n" . Compose(FuncArray1[Index], FuncArray2[Index]).(X)
MsgBox, 0, First-Class Functions, % Result
ExitApp
Output:
Input:
0.500000000000000

Output:
0.500000000000000
0.500000000000000
0.500000000000001

[edit] Axiom

Using the interpreter:

fns := [sin$Float, cos$Float, (x:Float):Float +-> x^3]
inv := [asin$Float, acos$Float, (x:Float):Float +-> x^(1/3)]
[(f*g) 0.5 for f in fns for g in inv]

Using the Spad compiler:

)abbrev package TESTP TestPackage
TestPackage(T:SetCategory) : with
_*: (List((T->T)),List((T->T))) -> (T -> List T)
== add
import MappingPackage3(T,T,T)
fs * gs ==
((x:T):(List T) +-> [(f*g) x for f in fs for g in gs])

This would be called using:

(fns * inv) 0.5

Output:

[0.5,0.5,0.5]

[edit] BBC BASIC

Strictly speaking you cannot return a function, but you can return a function pointer which allows the task to be implemented.

      REM Create some functions and their inverses:
DEF FNsin(a) = SIN(a)
DEF FNasn(a) = ASN(a)
DEF FNcos(a) = COS(a)
DEF FNacs(a) = ACS(a)
DEF FNcube(a) = a^3
DEF FNroot(a) = a^(1/3)
 
dummy = FNsin(1)
 
REM Create the collections (here structures are used):
DIM cA{Sin%, Cos%, Cube%}
DIM cB{Asn%, Acs%, Root%}
cA.Sin% = ^FNsin() : cA.Cos% = ^FNcos() : cA.Cube% = ^FNcube()
cB.Asn% = ^FNasn() : cB.Acs% = ^FNacs() : cB.Root% = ^FNroot()
 
REM Create some function compositions:
AsnSin% = FNcompose(cB.Asn%, cA.Sin%)
AcsCos% = FNcompose(cB.Acs%, cA.Cos%)
RootCube% = FNcompose(cB.Root%, cA.Cube%)
 
REM Test applying the compositions:
x = 1.234567 : PRINT x, FN(AsnSin%)(x)
x = 2.345678 : PRINT x, FN(AcsCos%)(x)
x = 3.456789 : PRINT x, FN(RootCube%)(x)
END
 
DEF FNcompose(f%,g%)
LOCAL f$, p%
f$ = "(x)=" + CHR$&A4 + "(&" + STR$~f% + ")(" + \
\ CHR$&A4 + "(&" + STR$~g% + ")(x))"
DIM p% LEN(f$) + 4
$(p%+4) = f$ : !p% = p%+4
= p%

Output:

  1.234567  1.234567
  2.345678  2.345678
  3.456789  3.456789

[edit] Bori

double acos (double d)	{ return Math.acos(d); }
double asin (double d) { return Math.asin(d); }
double cos (double d) { return Math.cos(d); }
double sin (double d) { return Math.sin(d); }
double croot (double d) { return Math.pow(d, 1/3); }
double cube (double x) { return x * x * x; }
 
Var compose (Var f, Var g, double x)
{
Func ff = f;
Func fg = g;
return ff(fg(x));
}
 
void button1_onClick (Widget widget)
{
Array arr1 = [ sin, cos, cube ];
Array arr2 = [ asin, acos, croot ];
 
str s;
for (int i = 1; i <= 3; i++)
{
s << compose(arr1.get(i), arr2.get(i), 0.5) << str.newline;
}
label1.setText(s);
}

Output on Android phone:

0.5
0.4999999999999999
0.5000000000000001

[edit] Bracmat

Bracmat has no built-in functions of real values. To say the truth, Bracmat has no real values. The only pair of currently defined built-in functions for which inverse functions exist are d2x and x2d for decimal to hexadecimal conversion and vice versa. These functions also happen to be each other's inverse. Because these two functions only take non-negative integer arguments, the example uses the argument 3210 for each pair of functions.

The lists A and B contain a mix of function names and function definitions, which illustrates that they always can take each other's role, except when a function definition is assigned to a function name, as for example in the first and second lines.

The compose function uses macro substitution.

( (sqrt=.!arg^1/2)
& (log=.e\L!arg)
& (A=x2d (=.!arg^2) log (=.!arg*pi))
& ( B
= d2x sqrt (=.e^!arg) (=.!arg*pi^-1)
)
& ( compose
= f g
.  !arg:(?f.?g)
& '(.($f)$(($g)$!arg))
)
& whl
' ( !A:%?F ?A
& !B:%?G ?B
& out$((compose$(!F.!G))$3210)
)
)

Output:

3210
3210
3210
3210

[edit] C

Since one can't create new functions dynamically within a C program, C doesn't have first class functions. But you can pass references to functions as parameters and return values and you can have a list of function references, so I guess you can say C has second class functions.

Here goes.

#include <stdlib.h>
#include <stdio.h>
#include <math.h>
 
/* declare a typedef for a function pointer */
typedef double (*Class2Func)(double);
 
/*A couple of functions with the above prototype */
double functionA( double v)
{
return v*v*v;
}
double functionB(double v)
{
return exp(log(v)/3);
}
 
/* A function taking a function as an argument */
double Function1( Class2Func f2, double val )
{
return f2(val);
}
 
/*A function returning a function */
Class2Func WhichFunc( int idx)
{
return (idx < 4) ? &functionA : &functionB;
}
 
/* A list of functions */
Class2Func funcListA[] = {&functionA, &sin, &cos, &tan };
Class2Func funcListB[] = {&functionB, &asin, &acos, &atan };
 
/* Composing Functions */
double InvokeComposed( Class2Func f1, Class2Func f2, double val )
{
return f1(f2(val));
}
 
typedef struct sComposition {
Class2Func f1;
Class2Func f2;
} *Composition;
 
Composition Compose( Class2Func f1, Class2Func f2)
{
Composition comp = malloc(sizeof(struct sComposition));
comp->f1 = f1;
comp->f2 = f2;
return comp;
}
 
double CallComposed( Composition comp, double val )
{
return comp->f1( comp->f2(val) );
}
/** * * * * * * * * * * * * * * * * * * * * * * * * * * */
 
int main(int argc, char *argv[])
{
int ix;
Composition c;
 
printf("Function1(functionA, 3.0) = %f\n", Function1(WhichFunc(0), 3.0));
 
for (ix=0; ix<4; ix++) {
c = Compose(funcListA[ix], funcListB[ix]);
printf("Compostion %d(0.9) = %f\n", ix, CallComposed(c, 0.9));
}
 
return 0;
}

[edit] Non-portable function body duplication

Following code generates true functions at run time. Extremely unportable, and should be considered harmful in general, but it's one (again, harmful) way for the truly desperate (or perhaps for people supporting only one platform -- and note that some other languages only work on one platform).

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
 
typedef double (*f_dbl)(double);
#define TAGF (f_dbl)0xdeadbeef
#define TAGG (f_dbl)0xbaddecaf
 
double dummy(double x)
{
f_dbl f = TAGF;
f_dbl g = TAGG;
return f(g(x));
}
 
f_dbl composite(f_dbl f, f_dbl g)
{
size_t len = (void*)composite - (void*)dummy;
f_dbl ret = malloc(len);
char *ptr;
memcpy(ret, dummy, len);
for (ptr = (char*)ret; ptr < (char*)ret + len - sizeof(f_dbl); ptr++) {
if (*(f_dbl*)ptr == TAGF) *(f_dbl*)ptr = f;
else if (*(f_dbl*)ptr == TAGG) *(f_dbl*)ptr = g;
}
return ret;
}
 
double cube(double x)
{
return x * x * x;
}
 
/* uncomment next line if your math.h doesn't have cbrt() */
/* double cbrt(double x) { return pow(x, 1/3.); } */
 
int main()
{
int i;
double x;
 
f_dbl A[3] = { cube, exp, sin };
f_dbl B[3] = { cbrt, log, asin}; /* not sure about availablity of cbrt() */
f_dbl C[3];
 
for (i = 0; i < 3; i++)
C[i] = composite(A[i], B[i]);
 
for (i = 0; i < 3; i++) {
for (x = .2; x <= 1; x += .2)
printf("C%d(%g) = %g\n", i, x, C[i](x));
printf("\n");
}
return 0;
}
(Boring) output
C0(0.2) = 0.2
C0(0.4) = 0.4
C0(0.6) = 0.6
C0(0.8) = 0.8
C0(1) = 1
 
C1(0.2) = 0.2
C1(0.4) = 0.4
C1(0.6) = 0.6
C1(0.8) = 0.8
C1(1) = 1
 
C2(0.2) = 0.2
C2(0.4) = 0.4
C2(0.6) = 0.6
C2(0.8) = 0.8
C2(1) = 1

[edit] C#

using System;
 
class Program
{
delegate Func<A,C> Composer<A,B,C>(Func<B,C> f, Func<A,B> g);
 
static void Main(string[] args)
{
Func<double, double> cube = x => Math.Pow(x, 3);
Func<double, double> croot = x => Math.Pow(x, (double)1/3);
 
var fun = new[] { Math.Sin, Math.Cos, cube };
var inv = new[] { Math.Asin, Math.Acos, croot };
Composer<double, double, double> compose = (f, g) => delegate(double x) { return f(g(x)); };
 
for (var i = 0; i < fun.Length; ++i)
{
Console.WriteLine(compose(fun[i],inv[i])(0.5));
}
}
}
 

Output:

0.5
0.5
0.5

[edit] C++

Works with: C++11
 
#include <functional>
#include <algorithm>
#include <iostream>
#include <vector>
#include <cmath>
 
using std::cout;
using std::endl;
using std::vector;
using std::function;
using std::transform;
using std::back_inserter;
 
typedef function<double(double)> FunType;
 
vector<FunType> A = {sin, cos, tan, [](double x) { return x*x*x; } };
vector<FunType> B = {asin, acos, atan, [](double x) { return exp(log(x)/3); } };
 
template <typename A, typename B, typename C>
function<C(A)> compose(function<C(B)> f, function<B(A)> g) {
return [f,g](A x) { return f(g(x)); };
}
 
int main() {
vector<FunType> composedFuns;
auto exNums = {0.0, 0.2, 0.4, 0.6, 0.8, 1.0};
 
transform(B.begin(), B.end(),
A.begin(),
back_inserter(composedFuns),
compose<double, double, double>);
 
for (auto num: exNums)
for (auto fun: composedFuns)
cout << u8"f\u207B\u00B9.f(" << num << ") = " << fun(num) << endl;
 
return 0;
}
 

[edit] Clojure

 
(use 'clojure.contrib.math)
(let [fns [#(Math/sin %) #(Math/cos %) (fn [x] (* x x x))]
inv [#(Math/asin %) #(Math/acos %) #(expt % 1/3)]]
(map #(% 0.5) (map #(comp %1 %2) fns inv)))
 

Output:

(0.5 0.4999999999999999 0.5000000000000001)

[edit] CoffeeScript

Translation of: JavaScript
# Functions as values of a variable
cube = (x) -> Math.pow x, 3
cuberoot = (x) -> Math.pow x, 1 / 3
 
# Higher order function
compose = (f, g) -> (x) -> f g(x)
 
# Storing functions in a array
fun = [Math.sin, Math.cos, cube]
inv = [Math.asin, Math.acos, cuberoot]
 
# Applying the composition to 0.5
console.log compose(inv[i], fun[i])(0.5) for i in [0..2]​​​​​​​

Output:

0.5
0.4999999999999999
0.5

[edit] Common Lisp

(defun compose (f g) (lambda (x) (funcall f (funcall g x))))
(defun cube (x) (expt x 3))
(defun cube-root (x) (expt x (/ 3)))
 
(loop with value = 0.5
for function in (list #'sin #'cos #'cube )
for inverse in (list #'asin #'acos #'cube-root)
for composed = (compose inverse function)
do (format t "~&(~A ∘ ~A)(~A) = ~A~%"
inverse
function
value
(funcall composed value)))

Output:

(#<FUNCTION ASIN> ∘ #<FUNCTION SIN>)(0.5) = 0.5
(#<FUNCTION ACOS> ∘ #<FUNCTION COS>)(0.5) = 0.5
(#<FUNCTION CUBE-ROOT> ∘ #<FUNCTION CUBE>)(0.5) = 0.5

[edit] D

[edit] Using Standard Compose

void main() {
import std.stdio, std.math, std.typetuple, std.functional;
 
alias dir = TypeTuple!(sin, cos, x => x ^^ 3);
alias inv = TypeTuple!(asin, acos, cbrt);
// foreach (f, g; staticZip!(dir, inv))
foreach (immutable i, f; dir)
writefln("%6.3f", compose!(f, inv[i])(0.5));
}
Output:
 0.500
 0.500
 0.500

[edit] Defining Compose

Here we need wrappers because the standard functions have different signatures (eg pure/nothrow). Same output.

void main() {
import std.stdio, std.math, std.range;
 
static T delegate(S) compose(T, U, S)(in T function(in U) f,
in U function(in S) g) {
return s => f(g(s));
}
 
immutable sin = (in real x) pure nothrow => x.sin,
asin = (in real x) pure nothrow => x.asin,
cos = (in real x) pure nothrow => x.cos,
acos = (in real x) pure nothrow => x.acos,
cube = (in real x) pure nothrow => x ^^ 3,
cbrt = (in real x) /*pure*/ nothrow => x.cbrt;
 
foreach (f, g; [sin, cos, cube].zip([asin, acos, cbrt]))
writefln("%6.3f", compose(f, g)(0.5));
}

[edit] Dart

import 'dart:math' as Math;
cube(x) => x*x*x;
cuberoot(x) => Math.pow(x, 1/3);
compose(f,g) => ((x)=>f(g(x)));
main(){
var functions = [Math.sin, Math.exp, cube];
var inverses = [Math.asin, Math.log, cuberoot];
for (int i = 0; i < 3; i++){
print(compose(functions[i], inverses[i])(0.5));
}
}
Output:
0.49999999999999994
0.5
0.5000000000000001

[edit] Déjà Vu

negate:
- 0
 
set :A [ @++ $ @negate @-- ]
 
set :B [ @-- $ @++ @negate ]
 
test n:
for i range 0 -- len A:
if /= n call compose @B! i @A! i n:
return false
true
 
test to-num !prompt "Enter a number: "
if:
!print "f^-1(f(x)) = x"
else:
!print "Something went wrong."
 
Output:
Enter a number: 23
f^-1(f(x)) = x

[edit] E

First, a brief summary of the relevant semantics: In E, every value, including built-in and user-defined functions, "is an object" — it has methods which respond to messages. Methods are distinguished by the given name (verb) and the number of parameters (arity). By convention and syntactic sugar, a function is an object which has a method whose verb is "run".

The relevant mathematical operations are provided as methods on floats, so the first thing we must do is define them as functions.

def sin(x)  { return x.sin() }
def cos(x) { return x.cos() }
def asin(x) { return x.asin() }
def acos(x) { return x.acos() }
def cube(x) { return x ** 3 }
def curt(x) { return x ** (1/3) }
 
def forward := [sin, cos, cube]
def reverse := [asin, acos, curt]

There are no built-in functions in this list, since the original author couldn't easily think of any which had one parameter and were inverses of each other, but composition would work just the same with them.

Defining composition. fn params { expr } is shorthand for an anonymous function returning a value.

def compose(f, g) {
return fn x { f(g(x)) }
}
? def x := 0.5  \
> for i => f in forward {
> def g := reverse[i]
> println(`x = $x, f = $f, g = $g, compose($f, $g)($x) = ${compose(f, g)(x)}`)
> }
 
x = 0.5, f = <sin>, g = <asin>, compose(<sin>, <asin>)(0.5) = 0.5
x = 0.5, f = <cos>, g = <acos>, compose(<cos>, <acos>)(0.5) = 0.4999999999999999
x = 0.5, f = <cube>, g = <curt>, compose(<cube>, <curt>)(0.5) = 0.5000000000000001

Note: def g := reverse[i] is needed here because E as yet has no defined protocol for iterating over collections in parallel. Page for this issue.

[edit] Ela

Translation of Haskell:

open number //sin,cos,asin,acos
open list //zipWith
 
cube x = x ** 3
croot x = x ** (1/3)
 
funclist = [sin, cos, cube]
funclisti = [asin, acos, croot]
 
zipWith (\f inversef -> (inversef << f) 0.5) funclist funclisti

Function (<<) is defined in standard prelude as:

(<<) f g x = f (g x)

Output (calculations are performed on 32-bit floats):

[0.5,0.5,0.499999989671302]

[edit] Erlang

 
-module( first_class_functions ).
 
-export( [task/0] ).
 
task() ->
As = [fun math:sin/1, fun math:cos/1, fun cube/1],
Bs = [fun math:asin/1, fun math:acos/1, fun square_inverse/1],
[io:fwrite( "Value: 1.5 Result: ~p~n", [functional_composition([A, B], 1.5)]) || {A, B} <- lists:zip(As, Bs)].
 
 
 
functional_composition( Funs, X ) -> lists:foldl( fun(F, Acc) -> F(Acc) end, X, Funs ).
 
square( X ) -> math:pow( X, 2 ).
 
square_inverse( X ) -> math:sqrt( X ).
 
Output:
93> first_class_functions:task().
Value: 1.5 Result: 1.5000000000000002
Value: 1.5 Result: 1.5
Value: 1.5 Result: 1.5

[edit] F#

open System
 
let cube x = x ** 3.0
let croot x = x ** (1.0/3.0)
 
let funclist = [Math.Sin; Math.Cos; cube]
let funclisti = [Math.Asin; Math.Acos; croot]
let composed = List.map2 (<<) funclist funclisti
 
let main() = for f in composed do printfn "%f" (f 0.5)
 
main()

Output:

0.500000
0.500000
0.500000

[edit] Factor

The constants A and B consist of arrays containing quotations (aka anonymous functions).

USING: assocs combinators kernel math.functions prettyprint sequences ;
IN: rosettacode.first-class-functions
 
CONSTANT: A { [ sin ] [ cos ] [ 3 ^ ] }
CONSTANT: B { [ asin ] [ acos ] [ 1/3 ^ ] }
 
: compose-all ( seq1 seq2 -- seq ) [ compose ] 2map ;
 
: test-fcf ( -- )
0.5 A B compose-all
[ call( x -- y ) ] with map . ;
Output:
{ 0.5 0.4999999999999999 0.5 }

[edit] Fantom

Methods defined for classes can be pulled out into functions, e.g. "Float#sin.func" pulls the sine method for floats out into a function accepting a single argument. This function is then a first-class value.

 
class FirstClassFns
{
static |Obj -> Obj| compose (|Obj -> Obj| fn1, |Obj -> Obj| fn2)
{
return |Obj x -> Obj| { fn2 (fn1 (x)) }
}
 
public static Void main ()
{
cube := |Float a -> Float| { a * a * a }
cbrt := |Float a -> Float| { a.pow(1/3f) }
 
|Float->Float|[] fns := [Float#sin.func, Float#cos.func, cube]
|Float->Float|[] inv := [Float#asin.func, Float#acos.func, cbrt]
|Float->Float|[] composed := fns.map |fn, i| { compose(fn, inv[i]) }
 
composed.each |fn| { echo (fn(0.5f)) }
}
}
 

Output:

0.5
0.4999999999999999
0.5

[edit] Forth

: compose ( xt1 xt2 -- xt3 )
>r >r :noname
r> compile,
r> compile,
postpone ;
;
 
: cube fdup fdup f* f* ;
: cuberoot 1e 3e f/ f** ;
 
: table create does> swap cells + @ ;
 
table fn ' fsin , ' fcos , ' cube ,
table inverse ' fasin , ' facos , ' cuberoot ,
 
: main
3 0 do
i fn i inverse compose ( xt )
0.5e execute f.
loop ;
 
main \ 0.5 0.5 0.5

[edit] GAP

# Function composition
Composition := function(f, g)
local h;
h := function(x)
return f(g(x));
end;
return h;
end;
 
# Apply each function in list u, to argument x
ApplyList := function(u, x)
local i, n, v;
n := Size(u);
v := [ ];
for i in [1 .. n] do
v[i] := u[i](x);
od;
return v;
end;
 
# Inverse and Sqrt are in the built-in library. Note that GAP doesn't have real numbers nor floating point numbers. Therefore, Sqrt yields values in cyclotomic fields.
# For example,
# gap> Sqrt(7);
# E(28)^3-E(28)^11-E(28)^15+E(28)^19-E(28)^23+E(28)^27
# where E(n) is a primitive n-th root of unity
a := [ i -> i + 1, Inverse, Sqrt ];
# [ function( i ) ... end, <Operation "InverseImmutable">, <Operation "Sqrt"> ]
b := [ i -> i - 1, Inverse, x -> x*x ];
# [ function( i ) ... end, <Operation "InverseImmutable">, function( x ) ... end ]
 
# Compose each couple
z := ListN(a, b, Composition);
 
# Now a test
ApplyList(z, 3);
[ 3, 3, 3 ]

[edit] Go

package main
 
import "math"
import "fmt"
 
// user-defined function, per task. Other math functions used are built-in.
func cube(x float64) float64 { return math.Pow(x, 3) }
 
// ffType and compose function taken from Function composition task
type ffType func(float64) float64
 
func compose(f, g ffType) ffType {
return func(x float64) float64 {
return f(g(x))
}
}
 
func main() {
// collection A
funclist := []ffType{math.Sin, math.Cos, cube}
// collection B
funclisti := []ffType{math.Asin, math.Acos, math.Cbrt}
for i := 0; i < 3; i++ {
// apply composition and show result
fmt.Println(compose(funclisti[i], funclist[i])(.5))
}
}

Output:

0.49999999999999994
0.5
0.5

[edit] Groovy

Solution:

def compose = { f, g -> { x -> f(g(x)) } }

Test program:

def cube = { it * it * it }
def cubeRoot = { it ** (1/3) }
 
funcList = [ Math.&sin, Math.&cos, cube ]
inverseList = [ Math.&asin, Math.&acos, cubeRoot ]
 
println ([funcList, inverseList].transpose().collect { f, finv -> compose(f, finv) }.collect{ it(0.5) })
println ([inverseList, funcList].transpose().collect { finv, f -> compose(finv, f) }.collect{ it(0.5) })

Output:

[0.5, 0.4999999999999999, 0.5000000000346574]
[0.5, 0.4999999999999999, 0.5000000000346574]

[edit] Haskell

Prelude> let cube x = x ^ 3
Prelude> let croot x = x ** (1/3)
Prelude> let compose f g = \x -> f (g x) -- this is already implemented in Haskell as the "." operator
Prelude> -- we could have written "let compose f g x = f (g x)" but we show this for clarity
Prelude> let funclist = [sin, cos, cube]
Prelude> let funclisti = [asin, acos, croot]
Prelude> zipWith (\f inversef -> (compose inversef f) 0.5) funclist funclisti
[0.5,0.4999999999999999,0.5]

[edit] Icon and Unicon

The Unicon solution can be modified to work in Icon. See Function_composition#Icon_and_Unicon.

link compose 
procedure main(arglist)
 
fun := [sin,cos,cube]
inv := [asin,acos,cuberoot]
x := 0.5
every i := 1 to *inv do
write("f(",x,") := ", compose(inv[i],fun[i])(x))
end
 
procedure cube(x)
return x*x*x
end
 
procedure cuberoot(x)
return x ^ (1./3)
end

Please refer to See Function_composition#Icon_and_Unicon for 'compose'.

Sample Output:

f(0.5) := 0.5
f(0.5) := 0.4999999999999999
f(0.5) := 0.5

[edit] J

J has some subtleties which are not addressed in this specification (J functions have grammatical character and their gerundial form may be placed in data structures where the spec sort of implies that there be no such distinction - for those uncomfortable with this terminology it is best to think of these as type distinctions - the type which appears in data structures and the type which may be applied are distinct though each may be directly derived from the other).

However, here are the basics which were requested:

   sin=:  1&o.
cos=: 2&o.
cube=: ^&3
square=: *:
unqo=: `:6
unqcol=: `:0
quot=: 1 :'{.u`'''''
A=: sin`cos`cube`square
B=: monad def'y unqo inv quot'"0 A
BA=. A dyad def'x unqo@(y unqo) quot'"0 B
   A unqcol 0.5
0.479426 0.877583 0.125 0.25
BA unqcol 0.5
0.5 0.5 0.5 0.5

[edit] Java

Java doesn't technically have first-class functions. Java can simulate first-class functions to a certain extent, with anonymous classes and generic function interface.

Works with: Java version 1.5+
import java.util.ArrayList;
 
public class FirstClass{
 
public interface Function<A,B>{
B apply(A x);
}
 
public static <A,B,C> Function<A, C> compose(
final Function<B, C> f, final Function<A, B> g) {
return new Function<A, C>() {
@Override public C apply(A x) {
return f.apply(g.apply(x));
}
};
}
 
public static void main(String[] args){
ArrayList<Function<Double, Double>> functions =
new ArrayList<Function<Double,Double>>();
 
functions.add(
new Function<Double, Double>(){
@Override public Double apply(Double x){
return Math.cos(x);
}
});
functions.add(
new Function<Double, Double>(){
@Override public Double apply(Double x){
return Math.tan(x);
}
});
functions.add(
new Function<Double, Double>(){
@Override public Double apply(Double x){
return x * x;
}
});
 
ArrayList<Function<Double, Double>> inverse = new ArrayList<Function<Double,Double>>();
 
inverse.add(
new Function<Double, Double>(){
@Override public Double apply(Double x){
return Math.acos(x);
}
});
inverse.add(
new Function<Double, Double>(){
@Override public Double apply(Double x){
return Math.atan(x);
}
});
inverse.add(
new Function<Double, Double>(){
@Override public Double apply(Double x){
return Math.sqrt(x);
}
});
System.out.println("Compositions:");
for(int i = 0; i < functions.size(); i++){
System.out.println(compose(functions.get(i), inverse.get(i)).apply(0.5));
}
System.out.println("Hard-coded compositions:");
System.out.println(Math.cos(Math.acos(0.5)));
System.out.println(Math.tan(Math.atan(0.5)));
System.out.println(Math.pow(Math.sqrt(0.5), 2));
}
}

Output:

Compositions:
0.4999999999999999
0.49999999999999994
0.5000000000000001
Hard-coded compositions:
0.4999999999999999
0.49999999999999994
0.5000000000000001
Works with: Java version 8+
import java.util.ArrayList;
import java.util.function.Function;
 
public class FirstClass{
 
public static void main(String... arguments){
ArrayList<Function<Double, Double>> functions = new ArrayList<>();
 
functions.add(Math::cos);
functions.add(Math::tan);
functions.add(x -> x * x);
 
ArrayList<Function<Double, Double>> inverse = new ArrayList<>();
 
inverse.add(Math::acos);
inverse.add(Math::atan);
inverse.add(Math::sqrt);
System.out.println("Compositions:");
for (int i = 0; i < functions.size(); i++){
System.out.println(functions.get(i).compose(inverse.get(i)).apply(0.5));
}
System.out.println("Hard-coded compositions:");
System.out.println(Math.cos(Math.acos(0.5)));
System.out.println(Math.tan(Math.atan(0.5)));
System.out.println(Math.pow(Math.sqrt(0.5), 2));
}
}

[edit] JavaScript

[edit] ES5

// Functions as values of a variable
var cube = function (x) {
return Math.pow(x, 3);
};
var cuberoot = function (x) {
return Math.pow(x, 1 / 3);
};
 
// Higher order function
var compose = function (f, g) {
return function (x) {
return f(g(x));
};
};
 
// Storing functions in a array
var fun = [Math.sin, Math.cos, cube];
var inv = [Math.asin, Math.acos, cuberoot];
 
for (var i = 0; i < 3; i++) {
// Applying the composition to 0.5
console.log(compose(inv[i], fun[i])(0.5));
}

[edit] ES6

// Functions as values of a variable
var cube = x => Math.pow(x, 3);
 
var cuberoot = x => Math.pow(x, 1 / 3);
 
 
// Higher order function
var compose = (f, g) => (x => f(g(x)));
 
// Storing functions in a array
var fun = [ Math.sin, Math.cos, cube ];
var inv = [ Math.asin, Math.acos, cuberoot ];
 
for (var i = 0; i < 3; i++) {
// Applying the composition to 0.5
console.log(compose(inv[i], fun[i])(0.5));
}


Result is always:

0.5
0.4999999999999999
0.5

[edit] Lasso

#!/usr/bin/lasso9
 
define cube(x::decimal) => {
return #x -> pow(3.0)
}
 
define cuberoot(x::decimal) => {
return #x -> pow(1.0/3.0)
}
 
define compose(f, g, v) => {
return {
return #f -> detach -> invoke(#g -> detach -> invoke(#1))
} -> detach -> invoke(#v)
}
 
 
local(functions = array({return #1 -> sin}, {return #1 -> cos}, {return cube(#1)}))
local(inverse = array({return #1 -> asin}, {return #1 -> acos}, {return cuberoot(#1)}))
 
loop(3)
stdoutnl(
compose(
#functions -> get(loop_count),
#inverse -> get(loop_count),
0.5
)
)
 
/loop

Output:

0.500000
0.500000
0.500000

[edit] Lua

function compose(f,g) return function(...) return f(g(...)) end end
 
fn = {math.sin, math.cos, function(x) return x^3 end}
inv = {math.asin, math.acos, function(x) return x^(1/3) end}
 
for i, v in ipairs(fn) do
local f = compose(v, inv[i])
print(f(0.5))
end

Output:

0.5
0.5
0.5

[edit] Maple

The composition operator in Maple is denoted by "@". We use "zip" to produce the list of compositions. The cubing procedure and its inverse are each computed.

 
> A := [ sin, cos, x -> x^3 ]:
> B := [ arcsin, arccos, rcurry( surd, 3 ) ]:
> zip( `@`, A, B )( 2/3 );
[2/3, 2/3, 2/3]
 
> zip( `@`, B, A )( 2/3 );
[2/3, 2/3, 2/3]
 

[edit] Mathematica

The built-in function Composition can do composition, a custom function that does the same would be compose[f_,g_]:=f[g[#]]&. However the latter only works with 2 arguments, Composition works with any number of arguments.

funcs = {Sin, Cos, #^3 &};
funcsi = {ArcSin, ArcCos, #^(1/3) &};
compositefuncs = Composition @@@ Transpose[{funcs, funcsi}];
Table[i[0.666], {i, compositefuncs}]

gives back:

{0.666, 0.666, 0.666}

Note that I implemented cube and cube-root as pure functions. This shows that Mathematica is fully able to handle functions as variables, functions can return functions, and functions can be given as an argument. Composition can be done in more than 1 way:

Composition[f,g,h][x]
f@g@h@x
x//h//g//f

all give back:

f[g[h[x]]]

[edit] Maxima

a: [sin, cos, lambda([x], x^3)]$
b: [asin, acos, lambda([x], x^(1/3))]$
compose(f, g) := buildq([f, g], lambda([x], f(g(x))))$
map(lambda([fun], fun(x)), map(compose, a, b));
[x, x, x]

[edit] Mercury

This solution uses the compose/3 function defined in std_util (part of the Mercury standard library) to demonstrate the use of first-class functions. The following process is followed:

  1. A list of "forward" functions is provided (sin, cosine and a lambda that calls ln).
  2. A list of "reverse" functions is provided (asin, acosine and a lambda that calls exp).
  3. The lists are mapped in corresponding members through an anonymous function that composes the resulting pairs of functions and applies them to the value 0.5.
  4. The results are returned and printed when all function pairs have been processed.

[edit] firstclass.m

 
:- module firstclass.
 
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
 
:- implementation.
:- import_module exception, list, math, std_util.
 
main(!IO) :-
Forward = [sin, cos, (func(X) = ln(X))],
Reverse = [asin, acos, (func(X) = exp(X))],
Results = map_corresponding(
(func(F, R) = compose(R, F, 0.5)),
Forward, Reverse),
write_list(Results, ", ", write_float, !IO),
write_string("\n", !IO).
 

[edit] Use and output

 $ mmc -E firstclass.m && ./firstclass 
0.5, 0.4999999999999999, 0.5

(Limitations of the IEEE floating point representation make the cos/acos pairing lose a little bit of accuracy.)

[edit] Nemerle

Translation of: Python
using System;
using System.Console;
using System.Math;
using Nemerle.Collections.NCollectionsExtensions;
 
module FirstClassFunc
{
Main() : void
{
def cube = fun (x) {x * x * x};
def croot = fun (x) {Pow(x, 1.0/3.0)};
def compose = fun(f, g) {fun (x) {f(g(x))}};
def funcs = [Sin, Cos, cube];
def ifuncs = [Asin, Acos, croot];
WriteLine($[compose(f, g)(0.5) | (f, g) in ZipLazy(funcs, ifuncs)]);
}
}

[edit] Use and Output

C:\Rosetta>ncc -o:FirstClassFunc FirstClassFunc.n

C:Rosetta>FirstClassFunc
[0.5, 0.5, 0.5]

[edit] newLISP

> (define (compose f g) (expand (lambda (x) (f (g x))) 'f 'g))
(lambda (f g) (expand (lambda (x) (f (g x))) 'f 'g))
> (define (cube x) (pow x 3))
(lambda (x) (pow x 3))
> (define (cube-root x) (pow x (div 1 3)))
(lambda (x) (pow x (div 1 3)))
> (define functions '(sin cos cube))
(sin cos cube)
> (define inverses '(asin acos cube-root))
(asin acos cube-root)
> (map (fn (f g) ((compose f g) 0.5)) functions inverses)
(0.5 0.5 0.5)
 

[edit] Nimrod

proc printer(x: int): proc =
proc y() =
echo "hello " & $x
return y
 
proc callMe(p) =
for i in 1..3:
p()
 
var printer10 = printer(10)
printer10()
 
echo ""
 
callMe(printer10)
 
echo ""
 
var printers = @[printer10]
printers.add(printer(9))
printers.add(printer(8))
 
for p in printers:
p()

Output:

hello 10

hello 10
hello 10
hello 10

hello 10
hello 9
hello 8

[edit] OCaml

# let cube x = x ** 3. ;;
val cube : float -> float = <fun>
 
# let croot x = x ** (1. /. 3.) ;;
val croot : float -> float = <fun>
 
# let compose f g = fun x -> f (g x) ;; (* we could have written "let compose f g x = f (g x)" but we show this for clarity *)
val compose : ('a -> 'b) -> ('c -> 'a) -> 'c -> 'b = <fun>
 
# let funclist = [sin; cos; cube] ;;
val funclist : (float -> float) list = [<fun>; <fun>; <fun>]
 
# let funclisti = [asin; acos; croot] ;;
val funclisti : (float -> float) list = [<fun>; <fun>; <fun>]
 
# List.map2 (fun f inversef -> (compose inversef f) 0.5) funclist funclisti ;;
- : float list = [0.5; 0.499999999999999889; 0.5]

[edit] Octave

function r = cube(x)
r = x.^3;
endfunction
 
function r = croot(x)
r = x.^(1/3);
endfunction
 
compose = @(f,g) @(x) f(g(x));
 
f1 = {@sin, @cos, @cube};
f2 = {@asin, @acos, @croot};
 
for i = 1:3
disp(compose(f1{i}, f2{i})(.5))
endfor
Output:
 0.50000
 0.50000
 0.50000

[edit] Oz

This example is incomplete. Fails to demonstrate that the result of applying the composition of each function in A and its inverse in B to a value, is the original value Please ensure that it meets all task requirements and remove this message.

To be executed in the REPL.

declare
 
fun {Compose F G}
fun {$ X}
{F {G X}}
end
end
 
fun {Cube X} X*X*X end
 
fun {CubeRoot X} {Number.pow X 1.0/3.0} end
 
in
 
for
F in [Float.sin Float.cos Cube]
I in [Float.asin Float.acos CubeRoot]
do
{Show {{Compose I F} 0.5}}
end
 

[edit] PARI/GP

Works with: PARI/GP version 2.4.2 and above
compose(f,g)={
x -> f(g(x))
};
 
fcf()={
my(A,B);
A=[x->sin(x), x->cos(x), x->x^2];
B=[x->asin(x), x->acos(x), x->sqrt(x)];
for(i=1,#A,
print(compose(A[i],B[i])(.5))
)
};

Usage note: In Pari/GP 2.4.3 the vectors can be written as

  A=[sin, cos, x->x^2];
B=[asin, acos, x->sqrt(x)];

Output:

0.5000000000000000000000000000
0.5000000000000000000000000000
0.5000000000000000000000000000

[edit] Perl

use Math::Complex ':trig';
 
sub compose {
my ($f, $g) = @_;
 
sub {
$f -> ($g -> (@_));
};
}
 
my $cube = sub { $_[0] ** (3) };
my $croot = sub { $_[0] ** (1/3) };
 
my @flist1 = ( \&Math::Complex::sin, \&Math::Complex::cos, $cube );
my @flist2 = ( \&asin, \&acos, $croot );
 
print join "\n", map {
compose($flist1[$_], $flist2[$_]) -> (0.5)
} 0..2;

Output:

0.5
0.5
0.5

[edit] Perl 6

Here we use the Z ("zipwith") metaoperator to zip the 𝐴 and 𝐵 lists with a user-defined compose function, expressed as an infix operator, . The .() construct invokes the function contained in the $_ (current topic) variable.

sub infix:<> (&𝑔, &𝑓) { -> \x { 𝑔 𝑓 x } }
 
my \𝐴 = &sin, &cos, { $_ ** <3/1> }
my \𝐵 = &asin, &acos, { $_ ** <1/3> }
 
say .(.5) for 𝐴 Z∘ 𝐵

Output:

0.5
0.5
0.5

[edit] PHP

Translation of: JavaScript
Works with: PHP version 5.3+

Non-anonymous functions can only be passed around by name, but the syntax for calling them is identical in both cases. Object or class methods require a different syntax involving array pseudo-types and call_user_func. So PHP could be said to have some first class functionality.

$compose = function ($f, $g) {
return function ($x) use ($f, $g) {
return $f($g($x));
};
};
 
$fn = array('sin', 'cos', function ($x) { return pow($x, 3); });
$inv = array('asin', 'acos', function ($x) { return pow($x, 1/3); });
 
for ($i = 0; $i < 3; $i++) {
$f = $compose($inv[$i], $fn[$i]);
echo $f(0.5), PHP_EOL;
}

Output:

0.5
0.5
0.5

[edit] PicoLisp

(load "@lib/math.l")
 
(de compose (F G)
(curry (F G) (X)
(F (G X)) ) )
 
(de cube (X)
(pow X 3.0) )
 
(de cubeRoot (X)
(pow X 0.3333333) )
 
(mapc
'((Fun Inv)
(prinl (format ((compose Inv Fun) 0.5) *Scl)) )
'(sin cos cube)
'(asin acos cubeRoot) )

Output:

0.500001
0.499999
0.500000

[edit] Prolog

Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found here: http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl

:- use_module(library(lambda)).
 
 
compose(F,G, FG) :-
FG = \X^Z^(call(G,X,Y), call(F,Y,Z)).
 
cube(X, Y) :-
Y is X ** 3.
 
cube_root(X, Y) :-
Y is X ** (1/3).
 
first_class :-
L = [sin, cos, cube],
IL = [asin, acos, cube_root],
 
% we create the composed functions
maplist(compose, L, IL, Lst),
 
% we call the functions
maplist(call, Lst, [0.5,0.5,0.5], R),
 
% we display the results
maplist(writeln, R).
 

Output :

 ?- first_class.
0.5
0.4999999999999999
0.5000000000000001
true.

[edit] Python

>>> # Some built in functions and their inverses
>>> from math import sin, cos, acos, asin
>>> # Add a user defined function and its inverse
>>> cube = lambda x: x * x * x
>>> croot = lambda x: x ** (1/3.0)
>>> # First class functions allow run-time creation of functions from functions
>>> # return function compose(f,g)(x) == f(g(x))
>>> compose = lambda f1, f2: ( lambda x: f1(f2(x)) )
>>> # first class functions should be able to be members of collection types
>>> funclist = [sin, cos, cube]
>>> funclisti = [asin, acos, croot]
>>> # Apply functions from lists as easily as integers
>>> [compose(inversef, f)(.5) for f, inversef in zip(funclist, funclisti)]
[0.5, 0.4999999999999999, 0.5]
>>>

[edit] R

cube <- function(x) x^3
croot <- function(x) x^(1/3)
compose <- function(f, g) function(x){f(g(x))}
 
f1 <- c(sin, cos, cube)
f2 <- c(asin, acos, croot)
 
for(i in 1:3) {
print(compose(f1[[i]], f2[[i]])(.5))
}
Output:
[1] 0.5
[1] 0.5
[1] 0.5

Alternatively:

sapply(mapply(compose,f1,f2),do.call,list(.5))
Output:
[1] 0.5 0.5 0.5

[edit] Racket

 
#lang racket
 
(define (compose f g) (λ (x) (f (g x))))
(define (cube x) (expt x 3))
(define (cube-root x) (expt x (/ 1 3)))
(define funlist (list sin cos cube))
(define ifunlist (list asin acos cube-root))
 
(for ([f funlist] [i ifunlist])
(displayln ((compose i f) 0.5)))
 

The output:

 
0.5
0.4999999999999999
0.5
 

[edit] REBOL

This example is incomplete. Fails to demonstrate that the result of applying the composition of each function in A and its inverse in B to a value, is the original value Please ensure that it meets all task requirements and remove this message.
rebol [
Title: "First Class Functions"
Author: oofoe
Date: 2009-12-05
URL: http://rosettacode.org/wiki/First-class_functions
]

 
; Functions "foo" and "bar" are used to prove that composition
; actually took place by attaching their signatures to the result.
 
foo: func [x][reform ["foo:" x]]
bar: func [x][reform ["bar:" x]]
 
cube: func [x][x * x * x]
croot: func [x][power x 1 / 3]
 
; "compose" means something else in REBOL, so I "fashion" an alternative.
 
fashion: func [f1 f2][
do compose/deep [func [x][(:f1) (:f2) x]]]
 
A: [foo sine cosine cube]
B: [bar arcsine arccosine croot]
 
while [not tail? A][
fn: fashion get A/1 get B/1
source fn ; Prove that functions actually got composed.
print [fn 0.5 crlf]
 
A: next A B: next B ; Advance to next pair.
]

[edit] REXX

The REXX language doesn't have any trig functions built-in, nor the square root function.
The only REXX functions that have an inverse are:

  •   d2x ◄──► x2d
  •   d2c ◄──► c2d
  •   c2x ◄──► x2c.

These six functions (generally) only support non-negative intergers, so a special test in the program below only
supplies appropriate integers when testing out the first function listed in the   A   collection.

/*REXX pgm demonstrating first-class functions (as a list of functions).*/
A = 'd2x square sin cos' /*list of functions to test. */
B = 'x2d sqrt Asin Acos' /*inverse functions of the above.*/
w=digits() /*W=width of numbers to be shown.*/
 
do j=1 for words(A); say /*do functions: A,B collections.*/
say center("number",w) center('function',3*w+1) center("inverse",4*w)
say copies("─",w) copies("─",3*w+1) copies("─",4*w)
if j<2 then call test j, 20 60 500 /*x2d,d2x; integers only.*/
else call test j, 0 0.5 1 2 /*all else, floating pt. */
end /*j*/
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────INVOKE subroutine───────────────────*/
invoke: parse arg fn,v; q='"'; if datatype(v,'N') then q=
_=fn||'('q||v||q')'; interpret 'func='_; return func
/*──────────────────────────────────TEST subroutine─────────────────────*/
test: procedure expose A B w; parse arg fu,xList /*xList=bunch of #s*/
do k=1 for words(xList); x=word(xList,k)
numeric digits digits()+5 /*higher precision.*/
fun=word(A,fu); funV=invoke(fun,x)  ; funInvoke=_
inv=word(B,fu); invV=invoke(inv,funV); invInvoke=_
numeric digits digits()-5 /*restore precision*/
if datatype(funV,'N') then funV=funV/1 /*round to digits()*/
if datatype(invV,'N') then invV=invV/1 /*round to digits()*/
say center(x,w) right(funInvoke,2*w)'='left(funV,w),
right(invInvoke,3*w)'='left(invV,w)
end /*k*/
return
/*──────────────────────────────────subroutines (functions)─────────────*/
Asin: procedure; arg x; if x<-1 | x>1 then call AsinErr; s=x*x
if abs(x)>=.7 then return sign(x)*Acos(sqrt(1-s)); z=x; o=x; p=z
do j=2 by 2; o=o*s*(j-1)/j; z=z+o/(j+1); if z=p then leave; p=z; end
return z
 
cos: procedure; arg x; x=r2r(x); a=abs(x); numeric fuzz min(9,digits()-9)
if a=pi() then return -1; if a=pi()/2 | a=2*pi() then return 0
if a=pi()/3 then return .5; if a=2*pi()/3 then return -.5
return .sinCos(1,1,-1)
 
sin: procedure; arg x; x=r2r(x); numeric fuzz min(5,digits()-3)
if abs(x)=pi() then return 0; return .sinCos(x,x,1)
 
.sinCos: parse arg z 1 p,_,i; x=x*x
do k=2 by 2; _=-_*x/(k*(k+i));z=z+_;if z=p then leave;p=z;end; return z
 
sqrt: procedure; parse arg x; if x=0 then return 0; m.=9; p=digits(); i=
numeric digits 9; if x<0 then do; x=-x; i='i'; end; numeric form; m.0=p
parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'E'_%2; m.1=p
do j=2 while p>9; m.j=p; p=p%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=.5*(g+x/g); end /*k*/
numeric digits m.0; return (g/1)i
 
pi: return, /*a bit of overkill, but hey !! */
3.1415926535897932384626433832795028841971693993751058209749445923078164062862
 
Acos: procedure; arg x; if x<-1|x>1 then call AcosErr; return .5*pi()-Asin(x)
r2r: return arg(1)//(2*pi()) /*normalize radians►1 unit circle*/
square: return arg(1)**2
tellErr: say; say '*** error! ***'; say; say arg(1); say; exit 13
tanErr: call tellErr 'tan('||x") causes division by zero, X="||x
AsinErr: call tellErr 'Asin(x), X must be in the range of -1 ──► +1, X='||x
AcosErr: call tellErr 'Acos(x), X must be in the range of -1 ──► +1, X='||x

output

 number             function                         inverse
───────── ──────────────────────────── ────────────────────────────────────
   20                d2x(20)=14                            x2d(14)=20
   60                d2x(60)=3C                          x2d("3C")=60
   500              d2x(500)=1F4                        x2d("1F4")=500

 number             function                         inverse
───────── ──────────────────────────── ────────────────────────────────────
    0              square(0)=0                             sqrt(0)=0
   0.5           square(0.5)=0.25                       sqrt(0.25)=0.5
    1              square(1)=1                             sqrt(1)=1

 number             function                         inverse
───────── ──────────────────────────── ────────────────────────────────────
    0                 sin(0)=0                             Asin(0)=0
   0.5              sin(0.5)=0.4794255      Asin(0.47942553860419)=0.5
    1                 sin(1)=0.8414709      Asin(0.84147098480862)=1
    2                 sin(2)=0.9092974      Asin(0.90929742682567)=1.1415926

 number             function                         inverse
───────── ──────────────────────────── ────────────────────────────────────
    0                 cos(0)=1                             Acos(1)=0
   0.5              cos(0.5)=0.8775825      Acos(0.87758256189038)=0.5
    1                 cos(1)=0.5403023      Acos(0.54030230586810)=1
    2                 cos(2)=-0.416146     Acos(-0.41614683654753)=2

The reason why   Asin[sin(n)]   may not equal   n:

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2π.

Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2, and then reverse themselves over 2πk + π/2 to 2πk + 3π/2.

Cosine and secant begin their period at 2πk, finish it at 2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π.

Tangent begins its period at 2πk − π/2, finishes it at 2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2.

Cotangent begins its period at 2πk, finishes it at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.


The above text is from the wikipedia webpage:   http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

[edit] Ruby

cube = proc{|x| x ** 3}
croot = proc{|x| x ** (1.quo 3)}
compose = proc {|f,g| proc {|x| f[g[x]]}}
funclist = [Math.method(:sin), Math.method(:cos), cube]
invlist = [Math.method(:asin), Math.method(:acos), croot]
 
puts funclist.zip(invlist).map {|f, invf| compose[invf, f][0.5]}
Output:
0.5
0.4999999999999999
0.5

[edit] Scala

import math._
 
// functions as values
val cube = (x: Double) => x * x * x
val cuberoot = (x: Double) => pow(x, 1 / 3d)
 
// higher order function, as a method
def compose[A,B,C](f: B => C, g: A => B) = (x: A) => f(g(x))
 
// partially applied functions in Lists
val fun = List(sin _, cos _, cube)
val inv = List(asin _, acos _, cuberoot)
 
// composing functions from the above Lists
val comp = (fun, inv).zipped map (_ compose _)
 
// output results of applying the functions
comp foreach {f => print(f(0.5) + " ")}

Output:

0.5   0.4999999999999999   0.5000000000000001

[edit] Scheme

(define (compose f g) (lambda (x) (f (g x))))
(define (cube x) (expt x 3))
(define (cube-root x) (expt x (/ 1 3)))
 
(define function (list sin cos cube))
(define inverse (list asin acos cube-root))
 
(define x 0.5)
(define (go f g)
(if (not (or (null? f)
(null? g)))
(begin (display ((compose (car f) (car g)) x))
(newline)
(go (cdr f) (cdr g)))))
 
(go function inverse)

Output:

0.5
0.5
0.5

[edit] Slate

This example is incomplete. Fails to demonstrate that the result of applying the composition of each function in A and its inverse in B to a value, is the original value Please ensure that it meets all task requirements and remove this message.

Compose is already defined in slate as (note the examples in the comment):

m@(Method traits) ** n@(Method traits)
"Answers a new Method whose effect is that of calling the first method
on the results of the second method applied to whatever arguments are passed.
This composition is associative, i.e. (a ** b) ** c = a ** (b ** c).
When the second method, n, does not take a *rest option or the first takes
more than one input, then the output is chunked into groups for its
consumption. E.g.:
#; `er ** #; `er applyTo: {'a'. 'b'. 'c'. 'd'} => 'abcd'
#; `er ** #name `er applyTo: {#a. #/}. => 'a/'"
[
n acceptsAdditionalArguments \/ [m arity = 1]
ifTrue:
[[| *args | m applyTo: {n applyTo: args}]]
ifFalse:
[[| *args |
m applyTo:
([| :stream |
args do: [| *each | stream nextPut: (n applyTo: each)]
inGroupsOf: n arity] writingAs: {})]]
].
 
#**`er asMethod: #compose: on: {Method traits. Method traits}.

used as:

n@(Number traits) cubed [n raisedTo: 3].
n@(Number traits) cubeRoot [n raisedTo: 1 / 3].
define: #forward -> {#cos `er. #sin `er. #cube `er}.
define: #reverse -> {#arcCos `er. #arcSin `er. #cubeRoot `er}.
 
define: #composedMethods -> (forward with: reverse collect: #compose: `er).
composedMethods do: [| :m | inform: (m applyWith: 0.5)].

[edit] Smalltalk

Works with: GNU Smalltalk
|forward reverse composer compounds|
"commodities"
Number extend [
cube [ ^self raisedTo: 3 ]
].
Number extend [
cubeRoot [ ^self raisedTo: (1 / 3) ]
].
 
forward := #( #cos #sin #cube ).
reverse := #( #arcCos #arcSin #cubeRoot ).
 
composer := [ :f :g | [ :x | f value: (g value: x) ] ].
 
"let us create composed funcs"
compounds := OrderedCollection new.
 
1 to: 3 do: [ :i |
compounds add: ([ :j | composer value: [ :x | x perform: (forward at: j) ]
value: [ :x | x perform: (reverse at: j) ] ] value: i)
].
 
compounds do: [ :r | (r value: 0.5) displayNl ].

Output:

0.4999999999999999
0.5
0.5000000000000001

[edit] Standard ML

- fun cube x = Math.pow(x, 3.0);
val cube = fn : real -> real
- fun croot x = Math.pow(x, 1.0 / 3.0);
val croot = fn : real -> real
- fun compose (f, g) = fn x => f (g x); (* this is already implemented in Standard ML as the "o" operator
= we could have written "fun compose (f, g) x = f (g x)" but we show this for clarity *)
val compose = fn : ('a -> 'b) * ('c -> 'a) -> 'c -> 'b
- val funclist = [Math.sin, Math.cos, cube];
val funclist = [fn,fn,fn] : (real -> real) list
- val funclisti = [Math.asin, Math.acos, croot];
val funclisti = [fn,fn,fn] : (real -> real) list
- ListPair.map (fn (f, inversef) => (compose (inversef, f)) 0.5) (funclist, funclisti);
val it = [0.5,0.5,0.500000000001] : real list

[edit] Swift

import Darwin
func compose<A,B,C>(f: (B) -> C, g: (A) -> B) -> (A) -> C {
return { f(g($0)) }
}
let funclist = [ { (x: Double) in sin(x) }, { (x: Double) in cos(x) }, { (x: Double) in pow(x, 3) } ]
let funclisti = [ { (x: Double) in asin(x) }, { (x: Double) in acos(x) }, { (x: Double) in cbrt(x) } ]
println(map(Zip2(funclist, funclisti)) { f, inversef in compose(f, inversef)(0.5) })
Output:
[0.5, 0.5, 0.5]

[edit] Tcl

The following is a transcript of an interactive session:

Works with: tclsh version 8.5
% namespace path tcl::mathfunc ;# to import functions like abs() etc.
% proc cube x {expr {$x**3}}
% proc croot x {expr {$x**(1/3.)}}
% proc compose {f g} {list apply {{f g x} {{*}$f [{*}$g $x]}} $f $g}
 
% compose abs cube ;# returns a partial command, without argument
apply {{f g x} {{*}$f [{*}$g $x]}} abs cube
 
% {*}[compose abs cube] -3 ;# applies the partial command to argument -3
27
 
% set forward [compose [compose sin cos] cube] ;# omitting to print result
% set backward [compose croot [compose acos asin]]
% {*}$forward 0.5
0.8372297964617733
% {*}$backward [{*}$forward 0.5]
0.5000000000000017

Obviously, the (C) library implementation of some of the trigonometric functions (on which Tcl depends for its implementation) on the platform used for testing is losing a little bit of accuracy somewhere.

[edit] TI-89 BASIC

See the comments at Function as an Argument#TI-89 BASIC for more information on first-class functions or the lack thereof in TI-89 BASIC. In particular, it is not possible to do proper function composition, because functions cannot be passed as values nor be closures.

Therefore, this example does everything but the composition.

(Note: The names of the inverse functions may not display as intended unless you have the “TI Uni” font.)

Prgm
Local funs,invs,composed,x,i
 
Define rc_cube(x) = x^3 © Cannot be local variables
Define rc_curt(x) = x^(1/3)
 
Define funs = {"sin","cos","rc_cube"}
Define invs = {"sin","cos","rc_curt"}
 
Define x = 0.5
Disp "x = " & string(x)
For i,1,3
Disp "f=" & invs[i] & " g=" & funs[i] & " f(g(x))=" & string(#(invs[i])(#(funs[i])(x)))
EndFor
 
DelVar rc_cube,rc_curt © Clean up our globals
EndPrgm

[edit] TXR

Translation of: Racket

Translation notes: we use op to create cube and inverse cube anonymously and succinctly. chain composes a variable number of functions, but unlike compose, from left to right, not right to left.

@(do
(defvar funlist [list sin
cos
(op expt @1 3)])
 
(defvar invlist [list asin
acos
(op expt @1 (/ 1 3))])
 
(each ((f funlist) (i invlist))
(prinl [(chain f i) 0.5])))
Output:
0.5
0.5
0.5
0.5


[edit] Ursala

The algorithm is to zip two lists of functions into a list of pairs of functions, make that a list of functions by composing each pair, "gang" the list of functions into a single function returning a list, and apply it to the argument 0.5.

#import std
#import flo
 
functions = <sin,cos,times^/~& sqr>
inverses = <asin,acos,math..cbrt>
 
#cast %eL
 
main = (gang (+)*p\functions inverses) 0.5

In more detail,

  • (+)*p\functions inverses evaluates to (+)*p(inverses,functions) by definition of the reverse binary to unary combinator (\)
  • This expression evaluates to (+)*p(<asin,acos,math..cbrt>,<sin,cos,times^/~& sqr>) by substitution.
  • The zipping is indicated by the p suffix on the map operator, (*) so that (+)*p evaluates to (+)* <(asin,sin),(acos,cos),(cbrt,times^/~& sqr)>.
  • The composition ((+)) operator is then mapped over the resulting list of pairs of functions, to obtain the list of functions <asin+sin,acos+cos,cbrt+ times^/~& sqr>.
  • gang<aisn+sin,acos+cos,cbrt+ times^/~& sqr> expresses a function returning a list in terms of a list of functions.

output:

<5.000000e-01,5.000000e-01,5.000000e-01>

[edit] zkl

In zkl, methods bind their instance so something like x.sin is the sine method bound to x (whatever real number x is). eg var a=(30.0).toRad().sin; is a method and a() will always return 0.5 (ie basically a const in this case). Which means you can't just use the word "sin", it has to be used in conjunction with an instance.

var a=T(fcn(x){x.toRad().sin()}, fcn(x){x.toRad().cos()}, fcn(x){x*x*x});
var b=T(fcn(x){x.asin().toDeg()}, fcn(x){x.acos().toDeg()}, fcn(x){x.pow(1.0/3)});
 
var H=Utils.Helpers;
var ab=H.zipWith(H.fcomp,b,a);
ab.run(True,5.0); //-->L(5.0,5.0,5.0)
 
a.run(True,5.0) //-->L(0.0871557,0.996195,125)

fcomp is the function composition function, fcomp(b,a) returns the function (x)-->b(a(x)). List.run(True,x) is inverse of List.apply/map, it returns a list of list[i](x)

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