First-class functions
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
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
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.)
- Related task
ActionScript
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
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.
Aikido
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
}ALGOL 68
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))
ODOutput:
+.500000000000000e +0 +.500000000000000e +0 +.500000000000000e +0
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}
Putting math libraries aside for the moment (we can always shell out to bash functions like bc), a deeper issue is that the architectural position of functions in the AppleScript type system is simply a little too incoherent and second class to facilitate really frictionless work with first-class functions. (This is clearly not what AppleScript was originally designed for).
Incoherent, in the sense that built-in functions and operators do not have the same place in the type system as user functions. The former are described as 'commands' in parser errors, and have to be wrapped in user handlers if they are to be used interchangeably with other functions.
Second class, in the sense that user functions (or 'handlers' in the terminology of Apple's documentation), are properties of scripts. The scripts are autonomous first class objects, but the handlers are not. Functions which accept other functions as arguments will internally need to use an mReturn or mInject function which 'lifts' handlers into script object types. Functions which return functions will similarly have to return them embedded in such script objects.
Once we have a function like mReturn, however, we can readily write higher order functions like map, zipWith and mCompose below.
on run
set fs to {sin_, cos_, cube_}
set afs to {asin_, acos_, croot_}
-- Form a list of three composed function objects,
-- and map testWithHalf() across the list to produce the results of
-- application of each composed function (base function composed with inverse) to 0.5
script testWithHalf
on |λ|(f)
mReturn(f)'s |λ|(0.5)
end |λ|
end script
map(testWithHalf, zipWith(mCompose, fs, afs))
--> {0.5, 0.5, 0.5}
end run
-- Simple composition of two unadorned handlers into
-- a method of a script object
on mCompose(f, g)
script
on |λ|(x)
mReturn(f)'s |λ|(mReturn(g)'s |λ|(x))
end |λ|
end script
end mCompose
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
on zipWith(f, xs, ys)
set lng to min(length of xs, length of ys)
set lst to {}
tell mReturn(f)
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, item i of ys)
end repeat
return lst
end tell
end zipWith
-- min :: Ord a => a -> a -> a
on min(x, y)
if y < x then
y
else
x
end if
end min
on sin:r
(do shell script "echo 's(" & r & ")' | bc -l") as real
end sin:
on cos:r
(do shell script "echo 'c(" & r & ")' | bc -l") as real
end cos:
on cube:x
x ^ 3
end cube:
on croot:x
x ^ (1 / 3)
end croot:
on asin:r
(do shell script "echo 'a(" & r & "/sqrt(1-" & r & "^2))' | bc -l") as real
end asin:
on acos:r
(do shell script "echo 'a(sqrt(1-" & r & "^2)/" & r & ")' | bc -l") as real
end acos:
- Output:
{0.5, 0.5, 0.5}
Arturo
cube: function [x] -> x^3
croot: function [x] -> x^(1//3)
names: ["sin/asin", "cos/acos", "cube/croot"]
funclist: @[var 'sin, var 'cos, var 'cube]
invlist: @[var 'asin, var 'acos, var 'croot]
num: 0.5
loop 0..2 'f [
result: call funclist\[f] @[num]
print [names\[f] "=>" call invlist\[f] @[result]]
]
- Output:
sin/asin => 0.5 cos/acos => 0.4999999999999999 cube/croot => 0.5
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
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) 0.5Output:
[0.5,0.5,0.5]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
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
0.5000000000000001BQN
BQN has full support for first class functions in the context of this wiki page. A more detailed article on BQN's implementation of functional programming is discussed here.
∘ (Atop) is used here to compose the two lists of functions given. Higher order functions are then used to apply the functions on a value. There is a slight change in value due to floating point inaccuracy.
BQN has provisions to invert builtin functions using ⁼ (Undo), and that can also be used for this demonstration, if we remove the block function (funsB ← {𝕏⁼}¨ funsA).
funsA ← ⟨⋆⟜2,-,•math.Sin,{𝕩×3}⟩
funsB ← ⟨√,-,•math.Asin,{𝕩÷3}⟩
comp ← funsA {𝕎∘𝕏}¨ funsB
•Show comp {𝕎𝕩}¨<0.5
⟨ 0.5000000000000001 0.5 0.5 0.5 ⟩
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
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;
}
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
C#
using System;
class Program
{
static void Main(string[] args)
{
var cube = new Func<double, double>(x => Math.Pow(x, 3.0));
var croot = new Func<double, double>(x => Math.Pow(x, 1 / 3.0));
var functionTuples = new[]
{
(forward: Math.Sin, backward: Math.Asin),
(forward: Math.Cos, backward: Math.Acos),
(forward: cube, backward: croot)
};
foreach (var ft in functionTuples)
{
Console.WriteLine(ft.backward(ft.forward(0.5)));
}
}
}
Output:
0.5 0.5 0.5
C++
#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;
}
Ceylon
First, you need to import the numeric module in you module.ceylon file
module rosetta "1.0.0" {
import ceylon.numeric "1.2.1";
}
And then you can use the math functions in your run.ceylon file
import ceylon.numeric.float {
sin, exp, asin, log
}
shared void run() {
function cube(Float x) => x ^ 3;
function cubeRoot(Float x) => x ^ (1.0 / 3.0);
value functions = {sin, exp, cube};
value inverses = {asin, log, cubeRoot};
for([func, inv] in zipPairs(functions, inverses)) {
print(compose(func, inv)(0.5));
}
}
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)
CoffeeScript
# 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
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 func in (list #'sin #'cos #'cube )
for inverse in (list #'asin #'acos #'cube-root)
for composed = (compose inverse func)
do (format t "~&(~A ∘ ~A)(~A) = ~A~%"
inverse
func
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
D
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
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));
}
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
Delphi
program First_class_functions;
{$APPTYPE CONSOLE}
uses
System.SysUtils,
System.Math;
type
TFunctionTuple = record
forward, backward: TFunc<Double, Double>;
procedure Assign(forward, backward: TFunc<Double, Double>);
end;
TFunctionTuples = array of TFunctionTuple;
var
cube, croot, fsin, fcos, faSin, faCos: TFunc<Double, Double>;
FunctionTuples: TFunctionTuples;
ft: TFunctionTuple;
{ TFunctionTuple }
procedure TFunctionTuple.Assign(forward, backward: TFunc<Double, Double>);
begin
self.forward := forward;
self.backward := backward;
end;
begin
cube :=
function(x: Double): Double
begin
result := x * x * x;
end;
croot :=
function(x: Double): Double
begin
result := Power(x, 1 / 3.0);
end;
fsin :=
function(x: Double): Double
begin
result := Sin(x);
end;
fcos :=
function(x: Double): Double
begin
result := Cos(x);
end;
faSin :=
function(x: Double): Double
begin
result := ArcSin(x);
end;
faCos :=
function(x: Double): Double
begin
result := ArcCos(x);
end;
SetLength(FunctionTuples, 3);
FunctionTuples[0].Assign(fsin, faSin);
FunctionTuples[1].Assign(fcos, faCos);
FunctionTuples[2].Assign(cube, croot);
for ft in FunctionTuples do
Writeln(ft.backward(ft.forward(0.5)):2:2);
readln;
end.
- Output:
0.50 0.50 0.50
Dyalect
Create new functions from preexisting functions at run-time
Using partial application:
func apply(fun, x) { y => fun(x, y) }
func sum(x, y) { x + y }
let sum2 = apply(sum, 2)Store functions in collections
func sum(x, y) { x + y }
func doubleMe(x) { x + x }
var arr = []
arr.Add(sum)
arr.Add(doubleMe)
arr.Add(arr.ToString)Use functions as arguments to other functions
func Iterator.Filter(pred) {
for x in this when pred(x) {
yield x
}
}
[1,2,3,4,5].Iterate().Filter(x => x % 2 == 0)Use functions as return values of other functions
func flip(fun, x, y) {
(y, x) => fun(x, y)
}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
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) = 0.5000000000000001Note: 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.
EchoLisp
;; adapted from Racket
;; (compose f g h ... ) is a built-in defined as :
;; (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])
(writeln ((compose i f) 0.5)))
→
0.5
0.4999999999999999
0.5
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 funclistiFunction (<<) 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]Elena
ELENA 6.x :
import system'routines;
import system'math;
import extensions'routines;
import extensions'math;
extension op
{
compose(f,g)
= f(g(self));
}
public program()
{
var fs := new object[]{ mssgconst sin<mathOp>[1], mssgconst cos<mathOp>[1], (x => power(x, 3.0r)) };
var gs := new object[]{ mssgconst arcsin<mathOp>[1], mssgconst arccos<mathOp>[1], (x => power(x, 1.0r / 3)) };
fs.zipBy(gs, (f,g => 0.5r.compose(f,g)))
.forEach(printingLn)
}- Output:
0.5 0.5 0.5
Elixir
defmodule First_class_functions do
def task(val) do
as = [&:math.sin/1, &:math.cos/1, fn x -> x * x * x end]
bs = [&:math.asin/1, &:math.acos/1, fn x -> :math.pow(x, 1/3) end]
Enum.zip(as, bs)
|> Enum.each(fn {a,b} -> IO.puts compose([a,b], val) end)
end
defp compose(funs, x) do
Enum.reduce(funs, x, fn f,acc -> f.(acc) end)
end
end
First_class_functions.task(0.5)
- Output:
0.5 0.4999999999999999 0.5
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
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
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 }
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
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
FreeBASIC
Like C, FreeBASIC doesn't have first class functions so I've contented myself by translating their code:
' FB 1.05.0 Win64
#Include "crt/math.bi" '' include math functions in C runtime library
' Declare function pointer type
' This implicitly assumes default StdCall calling convention on Windows
Type Class2Func As Function(As Double) As Double
' A couple of functions with the above prototype
Function functionA(v As Double) As Double
Return v*v*v '' cube of v
End Function
Function functionB(v As Double) As Double
Return Exp(Log(v)/3) '' same as cube root of v which would normally be v ^ (1.0/3.0) in FB
End Function
' A function taking a function as an argument
Function function1(f2 As Class2Func, val_ As Double) As Double
Return f2(val_)
End Function
' A function returning a function
Function whichFunc(idx As Long) As Class2Func
Return IIf(idx < 4, @functionA, @functionB)
End Function
' Additional function needed to treat CDecl function pointer as StdCall
' Get compiler warning otherwise
Function cl2(f As Function CDecl(As Double) As Double) As Class2Func
Return CPtr(Class2Func, f)
End Function
' A list of functions
' Using C Runtime library versions of trig functions as it doesn't appear
' to be possible to apply address operator (@) to FB's built-in versions
Dim funcListA(0 To 3) As Class2Func = {@functionA, cl2(@sin_), cl2(@cos_), cl2(@tan_)}
Dim funcListB(0 To 3) As Class2Func = {@functionB, cl2(@asin_), cl2(@acos_), cl2(@atan_)}
' Composing Functions
Function invokeComposed(f1 As Class2Func, f2 As Class2Func, val_ As double) As Double
Return f1(f2(val_))
End Function
Type Composition
As Class2Func f1, f2
End Type
Function compose(f1 As Class2Func, f2 As Class2Func) As Composition Ptr
Dim comp As Composition Ptr = Allocate(SizeOf(Composition))
comp->f1 = f1
comp->f2 = f2
Return comp
End Function
Function callComposed(comp As Composition Ptr, val_ As Double ) As Double
Return comp->f1(comp->f2(val_))
End Function
Dim ix As Integer
Dim c As Composition Ptr
Print "function1(functionA, 3.0) = "; CSng(function1(whichFunc(0), 3.0))
Print
For ix = 0 To 3
c = compose(funcListA(ix), funcListB(ix))
Print "Composition"; ix; "(0.9) = "; CSng(callComposed(c, 0.9))
Next
Deallocate(c)
Print
Print "Press any key to quit"
Sleep
- Output:
function1(functionA, 3.0) = 27 Composition 0(0.9) = 0.9 Composition 1(0.9) = 0.9 Composition 2(0.9) = 0.9 Composition 3(0.9) = 0.9
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 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 ]
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
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]
Haskell
cube :: Floating a => a -> a
cube x = x ** 3.0
croot :: Floating a => a -> a
croot x = x ** (1/3)
-- compose already exists in Haskell as the `.` operator
-- compose :: (a -> b) -> (b -> c) -> a -> c
-- compose f g = \x -> g (f x)
funclist :: Floating a => [a -> a]
funclist = [sin, cos, cube ]
invlist :: Floating a => [a -> a]
invlist = [asin, acos, croot]
main :: IO ()
main = print $ zipWith (\f i -> f . i $ 0.5) funclist invlist
- Output:
[0.5,0.4999999999999999,0.5000000000000001]
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
J
Explicit version
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
Tacit (unorthodox) version
In J only adverbs and conjunctions (functionals) can produce verbs (functions)... Unless they are forced to cloak as verbs (functions). (Note that this takes advantage of a bug/feature of the interpreter ; see unorthodox tacit .) The resulting functions (which correspond to functionals) can take and produce functions:
train =. (<'`:')(0:`)(,^:)&6 NB. Producing the function train corresponding to the functional `:6
inverse=. (<'^:')(0:`)(,^:)&_1 NB. Producing the function inverse corresponding to the functional ^:_1
compose=. (<'@:')(0:`)(,^:) NB. Producing the function compose corresponding to the functional @:
an =. <@:((,'0') ; ]) NB. Producing the atomic representation of a noun
of =. train@:([ ; an) NB. Evaluating a function for an argument
box =. < @: train"0 NB. Producing a boxed list of the trains of the components
]A =. box (1&o.)`(2&o.)`(^&3) NB. Producing a boxed list containing the Sin, Cos and Cubic functions
┌────┬────┬───┐
│1&o.│2&o.│^&3│
└────┴────┴───┘
]B =. inverse &.> A NB. Producing their inverses
┌────────┬────────┬───────┐
│1&o.^:_1│2&o.^:_1│^&3^:_1│
└────────┴────────┴───────┘
]BA=. B compose &.> A NB. Producing the compositions of the functions and their inverses
┌────────────────┬────────────────┬──────────────┐
│1&o.^:_1@:(1&o.)│2&o.^:_1@:(2&o.)│^&3^:_1@:(^&3)│
└────────────────┴────────────────┴──────────────┘
BA of &> 0.5 NB. Evaluating the compositions at 0.5
0.5 0.5 0.5
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.
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
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));
}
}
JavaScript
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));
}
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
jq
Also works with gojq, the Go implementation of jq
Also works with fq, a Go implementation of a large subset of jq
jq does not support functions as "first class objects" in the sense specified in the task description but it does give near-first-class status to functions and functional expressions, which can for example, be passed as arguments to functions. In fact, it is quite straightforward, though possibly misleading, to transcribe the Wren entry to jq, as follows:
# Apply g first
def compose(f; g): g | f;
def A: [sin, cos, .*.*.];
def B: [asin, acos, pow(.; 1/3) ];
0.5
| range(0;3) as $i
| compose( A[$i]; B[$i] )However this transcription is inefficient because at each iteration (i.e. for each $i), all three components of A and of B are computed. To avoid this, one would have to ignore A and B, and instead write:
0.5 | compose(sin; asin), compose(cos; acos), compose(pow(.;3); pow(.; 1/3))
- Output:
Using gojq:
0.5 0.5000000000000001 0.5000000000000001
Julia
#!/usr/bin/julia
function compose(f::Function, g::Function)
return x -> f(g(x))
end
value = 0.5
for pair in [(sin, asin), (cos, acos), (x -> x^3, x -> x^(1/3))]
func, inverse = pair
println(compose(func, inverse)(value))
end
Output:
0.5 0.4999999999999999 0.5000000000000001
Kotlin
import kotlin.math.*
fun compose(f: (Double) -> Double, g: (Double) -> Double ): (Double) -> Double = { f(g(it)) }
fun cube(d: Double) = d * d * d
fun main() {
val listA = listOf(::sin, ::cos, ::cube)
val listB = listOf(::asin, ::acos, ::cbrt)
val x = 0.5
for (i in 0..2) println(compose(listA[i], listB[i])(x))
}
- Output:
0.5 0.4999999999999999 0.5000000000000001
Lambdatalk
Tested in [1]
{def cube {lambda {:x} {pow :x 3}}}
{def cuberoot {lambda {:x} {pow :x {/ 1 3}}}}
{def compose {lambda {:f :g :x} {:f {:g :x}}}}
{def fun sin cos cube}
{def inv asin acos cuberoot}
{def display {lambda {:i}
{br}{compose {nth :i {fun}}
{nth :i {inv}} 0.5}}}
{map display {serie 0 2}}
Output:
0.5
0.49999999999999994
0.5000000000000001
Lang
fp.cube = ($x) -> return parser.op($x ** 3)
fp.cuberoot = ($x) -> return parser.op($x ** (1/3))
# fn.concat can be used as compose
&funcs $= [fn.sin, fn.cos, fp.cube]
&invFuncs $= [fn.asin, fn.acos, fp.cuberoot]
$pair
foreach($[pair], fn.arrayZip(&funcs, &invFuncs)) {
parser.op(fn.println(($pair[0] ||| $pair[1])(.5)))
}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
Lingo
Lingo does not support functions as first-class objects. But with the limitations described under Function composition the task can be solved:
-- sin, cos and sqrt are built-in, square, asin and acos are user-defined
A = [#sin, #cos, #square]
B = [#asin, #acos, #sqrt]
testValue = 0.5
repeat with i = 1 to 3
-- for implementation details of compose() see https://www.rosettacode.org/wiki/Function_composition#Lingo
f = compose(A[i], B[i])
res = call(f, _movie, testValue)
put res = testValue
end repeat- Output:
-- 1 -- 1 -- 1
User-defined arithmetic functions used in code above:
on square (x)
return x*x
end
on asin (x)
res = atan(sqrt(x*x/(1-x*x)))
if x<0 then res = -res
return res
end
on acos (x)
return PI/2 - asin(x)
endLua
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
M2000 Interpreter
Cos, Sin works with degrees, Number pop number from stack of values, so we didn't use a variable like this POW3INV =Lambda (x)->x**(1/3)
Module CheckFirst {
RAD = lambda -> number/180*pi
ASIN = lambda RAD -> {
Read x : x=Round(x,10)
If x>=0 and X<1 Then {
=RAD(abs(2*Round(ATN(x/(1+SQRT(1-x**2))))))
} Else.if x==1 Then {
=RAD(90)
} Else error "asin exit limit"
}
ACOS=lambda ASIN (x) -> PI/2 - ASIN(x)
POW3 = Lambda ->number**3
POW3INV =Lambda ->number**(1/3)
COSRAD =lambda ->Cos(number*180/pi)
SINRAD=lambda ->Sin(number*180/pi)
Composed=lambda (f1, f2) -> {
=lambda f1, f2 (x)->{
=f1(f2(x))
}
}
Dim Base 0, A(3), B(3), C(3)
A(0)=ACOS, ASIN, POW3INV
B(0)=COSRAD, SINRAD, POW3
C(0)=Composed(ACOS, COSRAD), Composed(ASIN, SINRAD), Composed(POW3INV, POW3)
Print $("0.00")
For i=0 To 2 {
Print A(i)(B(i)(.5)), C(i)(.5)
}
}
CheckFirstMaple
The composition operator in Maple is denoted by "@". We use "zip" to produce the list of compositions. The cubing procedure and its inverse are each computed.
> A := [ sin, cos, x -> x^3 ]:
> B := [ arcsin, arccos, rcurry( surd, 3 ) ]:
> zip( `@`, A, B )( 2/3 );
[2/3, 2/3, 2/3]
> zip( `@`, B, A )( 2/3 );
[2/3, 2/3, 2/3]Mathematica / Wolfram Language
The built-in function Composition can do composition, a custom function that does the same would be compose[f_,g_]:=f[g[#]]&. However the latter only works with 2 arguments, Composition works with any number of arguments.
funcs = {Sin, Cos, #^3 &};
funcsi = {ArcSin, ArcCos, #^(1/3) &};
compositefuncs = Composition @@@ Transpose[{funcs, funcsi}];
Table[i[0.666], {i, compositefuncs}]
gives back:
{0.666, 0.666, 0.666}
Note that I implemented cube and cube-root as pure functions. This shows that Mathematica is fully able to handle functions as variables, functions can return functions, and functions can be given as an argument. Composition can be done in more than 1 way:
Composition[f,g,h][x]
f@g@h@x
x//h//g//f
all give back:
f[g[h[x]]]
Maxima
a: [sin, cos, lambda([x], x^3)]$
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]
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:
- A list of "forward" functions is provided (sin, cosine and a lambda that calls ln).
- A list of "reverse" functions is provided (asin, acosine and a lambda that calls exp).
- 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.
- The results are returned and printed when all function pairs have been processed.
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).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.)
min
Note concat is what performs the function composition, as functions are lists in min.
('sin 'cos (3 pow)) =A
('asin 'acos (1 3 / pow)) =B
(A bool) (
0.5 A first B first concat -> puts!
A rest #A
B rest #B
) while- Output:
0.5 0.4999999999999999 0.5
Nemerle
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)]);
}
}
Use and Output
C:\Rosetta>ncc -o:FirstClassFunc FirstClassFunc.n C:Rosetta>FirstClassFunc [0.5, 0.5, 0.5]
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)
Nim
Note that when defining a sequence @[a, b, c], “a” defines the type of the elements. So, if there is an ambiguity, we need to precise the type.
For instance @[math.sin, cos] wouldn’t compile as there exists two “sin” functions, one for “float32” and one for “float64”.
So, we have to write either @[(proc(x: float64): float64]) math.sin, cos] to avoid the ambiguity or make sure there is no ambiguity as regards the first element.
Here, in first sequence, we put in first position our function “sin” defined only for “float64” and in second position the standard one “math.cos”. And in second sequence, we used the type MF64 = proc(x: float64): float64 to suppress the ambiguity.
from math import nil # Require qualifier to access functions.
type MF64 = proc(x: float64): float64
proc cube(x: float64) : float64 =
math.pow(x, 3)
proc cuberoot(x: float64) : float64 =
math.pow(x, 1/3)
proc compose[A](f: proc(x: A): A, g: proc(x: A): A) : (proc(x: A): A) =
proc c(x: A): A =
f(g(x))
return c
proc sin(x: float64) : float64 =
math.sin(x)
proc acos(x: float64) : float64 =
math.arccos(x)
var fun = @[sin, math.cos, cube]
var inv = @[MF64 math.arcsin, acos, cuberoot]
for i in 0..2:
echo compose(inv[i], fun[i])(0.5)
Output:
0.5 0.4999999999999999 0.5
Objeck
use Collection.Generic;
lambdas Func {
Double : (FloatHolder) ~ FloatHolder
}
class FirstClass {
function : Main(args : String[]) ~ Nil {
vector := Vector->New()<Func2Holder <FloatHolder, FloatHolder> >;
# store functions in collections
vector->AddBack(Func2Holder->New(\Func->Double : (v)
=> v * v)<FloatHolder, FloatHolder>);
# new function from preexisting function at run-time
vector->AddBack(Func2Holder->New(\Func->Double : (v)
=> Float->SquareRoot(v->Get()))<FloatHolder, FloatHolder>);
# process collection
each(i : vector) {
# return value of other functions and pass argument to other function
Show(vector->Get(i)<Func2Holder>->Get()<FloatHolder, FloatHolder>);
};
}
function : Show(func : (FloatHolder) ~ FloatHolder) ~ Nil {
func(13.5)->Get()->PrintLine();
}
}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]
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
Oforth
: compose(f, g) #[ g perform f perform ] ;
[ #cos, #sin, #[ 3 pow ] ] [ #acos, #asin, #[ 3 inv powf ] ] zipWith(#compose)
map(#[ 0.5 swap perform ]) conform(#[ 0.5 == ]) println- Output:
1
Ol
; creation of new function from preexisting functions at run-time
(define (compose f g) (lambda (x) (f (g x))))
; storing functions in collection
(define (quad x) (* x x x x))
(define (quad-root x) (sqrt (sqrt x)))
(define collection (tuple quad quad-root))
; use functions as arguments to other functions
; and use functions as return values of other functions
(define identity (compose (ref collection 2) (ref collection 1)))
(print (identity 11211776))
Oz
This is now also compatible with Oz v 2.0 (To be executed in the Oz OPI, by typing ctl+. ctl+b)
declare
fun {Compose F G}
fun {$ X}
{F {G X}}
end
end
fun {Cube X} {Number.pow X 3.0} 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}}
endThis will output the following in the Emulator output window
0.5
0.5
0.5PARI/GP
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
PascalABC.NET
function Cube(x: real) := x**3;
function CubeRoot(x: real) := x**(1/3);
function Composition<T>(f1,f2: T -> T): T -> T := x -> f1(f2(x));
begin
var A := Arr(Sin,Cos,Cube);
var B := Arr(ArcSin,ArcCos,CubeRoot);
for var i:=0 to A.Length-1 do
Println(Composition(A[i],B[i])(0.5));
// Build-in composition f1 * f2
foreach var (f1,f2) in A.Zip(B) do
Println((f1 * f2)(0.5));
end.
- Output:
0.5 0.5 0.5 0.5 0.5 0.5
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
Phix
There is not really any direct support for this sort of thing in Phix, but it is all pretty trivial to manage explicitly.
In the following, as it stands, constant m cannot be used the same way as a routine_id, and a standard routine_id cannot be passed to the first argument of call_composite, but tagging ctable entries so that you know exactly what to do with them does not sound difficult to me.
sequence ctable = {} function compose(integer f, g) ctable = append(ctable,{f,g}) integer cdx = length(ctable) return cdx end function function call_composite(integer cdx, atom x) integer {f,g} = ctable[cdx] return f(g(x)) end function function plus1(atom x) return x+1 end function function halve(atom x) return x/2 end function constant m = compose(halve,plus1) ?call_composite(m,1) -- displays 1 ?call_composite(m,4) -- displays 2.5
PHP
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
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
PostScript
% PostScript has 'sin' and 'cos', but not these
/asin { dup dup 1. add exch 1. exch sub mul sqrt atan } def
/acos { dup dup 1. add exch 1. exch sub mul sqrt exch atan } def
/cube { 3 exp } def
/cuberoot { 1. 3. div exp } def
/compose { % f g -> { g f }
[ 3 1 roll exch
% procedures are not executed when encountered directly
% insert an 'exec' after procedures, but not after operators
1 index type /operatortype ne { /exec cvx exch } if
dup type /operatortype ne { /exec cvx } if
] cvx
} def
/funcs [ /sin load /cos load /cube load ] def
/ifuncs [ /asin load /acos load /cuberoot load ] def
0 1 funcs length 1 sub { /i exch def
ifuncs i get funcs i get compose
.5 exch exec ==
} forProlog
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.
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]
>>>
Or, equivalently:
'''First-class functions'''
from math import (acos, cos, asin, sin)
from inspect import signature
# main :: IO ()
def main():
'''Composition of several functions.'''
pwr = flip(curry(pow))
fs = [sin, cos, pwr(3.0)]
ifs = [asin, acos, pwr(1 / 3.0)]
print([
f(0.5) for f in
zipWith(compose)(fs)(ifs)
])
# GENERIC FUNCTIONS ------------------------------
# compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
def compose(g):
'''Right to left function composition.'''
return lambda f: lambda x: g(f(x))
# curry :: ((a, b) -> c) -> a -> b -> c
def curry(f):
'''A curried function derived
from an uncurried function.'''
return lambda a: lambda b: f(a, b)
# flip :: (a -> b -> c) -> b -> a -> c
def flip(f):
'''The (curried or uncurried) function f with its
two arguments reversed.'''
if 1 < len(signature(f).parameters):
return lambda a, b: f(b, a)
else:
return lambda a: lambda b: f(b)(a)
# zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]
def zipWith(f):
'''A list constructed by zipping with a
custom function, rather than with the
default tuple constructor.'''
return lambda xs: lambda ys: [
f(a)(b) for (a, b) in zip(xs, ys)
]
if __name__ == '__main__':
main()
- Output:
[0.49999999999999994, 0.5000000000000001, 0.5000000000000001]
Quackery
Preamble. Quackery has first class functions, but it doesn't have floating point numbers.
However, as it is implemented in Python, we can drop down into Python for that functionality, and represent floating point numbers as strings in Quackery. This code implements that, with a light sprinkle of syntactic sugar for the compiler so we can use f 0.5 rather than $ "0.5" to represent the floating point number 0.5
[ $ \
try:
float(string_from_stack())
except:
to_stack(False)
else:
to_stack(True)
\ python ] is isfloat ( $ --> b )
[ nextword
dup isfloat not if
[ $ '"f" needs to be followed by a number.'
message put bail ]
' [ ' ] swap nested join
nested swap dip join ] builds f ( [ $ --> [ $ )
[ $ \
import math
a = string_from_stack()
a = str(math.sin(float(a)))
string_to_stack(a) \ python ] is sin ( $ --> $ )
[ $ \
import math
a = string_from_stack()
a = str(math.asin(float(a)))
string_to_stack(a) \ python ] is asin ( $ --> $ )
[ $ \
import math
a = string_from_stack()
a = str(math.cos(float(a)))
string_to_stack(a) \ python ] is cos ( $ --> $ )
[ $ \
import math
a = string_from_stack()
a = str(math.acos(float(a)))
string_to_stack(a) \ python ] is acos ( $ --> $ )
[ $ \
a = string_from_stack()
b = string_from_stack()
c = str(float(b) * float(a))
string_to_stack(c) \ python ] is f* ( $ $ --> $ )
[ $ \
a = string_from_stack()
b = string_from_stack()
c = str(float(b) / float(a))
string_to_stack(c) \ python ] is f/ ( $ $ --> $ )
[ $ \
a = string_from_stack()
b = string_from_stack()
c = str(float(b) ** float(a))
string_to_stack(c) \ python ] is f** ( $ $ --> $ )…and now the task…
[ dup dup f* f* ] is cubed ( $ --> $ )
[ f 1 f 3 f/ f** ] is cuberoot ( $ --> $ )
[ table sin cos cubed ] is A ( n --> [ )
[ table asin acos cuberoot ] is B ( n --> [ )
[ dip nested nested join ] is compose ( x x --> [ )
[ dup dip A B compose do ] is ->A->B-> ( f n --> f )
' [ f 0.5 f 1.234567 ]
witheach
[ do 3 times
[ dup echo$
say " -> "
i^ A echo
say " -> "
i^ B echo
say " -> "
dup i^ ->A->B-> echo$
cr ]
drop cr ]- Output:
0.5 -> sin -> asin -> 0.5 0.5 -> cos -> acos -> 0.4999999999999999 0.5 -> cubed -> cuberoot -> 0.5 1.234567 -> sin -> asin -> 1.234567 1.234567 -> cos -> acos -> 1.234567 1.234567 -> cubed -> cuberoot -> 1.234567
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
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)))
- Output:
0.5 0.4999999999999999 0.5
Raku
(formerly Perl 6) Here we use the Z ("zipwith") metaoperator to zip the 𝐴 and 𝐵 lists with builtin composition operator, ∘ (or just o). The .() construct invokes the function contained in the $_ (current topic) variable.
my \𝐴 = &sin, &cos, { $_ ** <3/1> }
my \𝐵 = &asin, &acos, { $_ ** <1/3> }
say .(.5) for 𝐴 Z∘ 𝐵
- Output:
0.5 0.4999999999999999 0.5000000000000001
It is not clear why we don't get exactly 0.5, here.
Operators, both builtin and user-defined, are first class too.
my @a = 1,2,3;
my @op = &infix:<+>, &infix:<->, &infix:<*>;
for flat @a Z @op -> $v, &op { say 42.&op($v) }
- Output:
43 40 126
REBOL
REBOL [
Title: "First Class Functions"
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.
]
REXX
The REXX language doesn't have any trigonometric functions built-in, nor the square root function, so several higher-math functions are included herein as RYO functions.
The only REXX functions that have an inverse are:
- d2x ◄──► x2d
- d2c ◄──► c2d
- c2x ◄──► x2c
These six functions (generally) only support non-negative integers, so a special test in the program below only
supplies appropriate integers when testing the first function listed in the A collection.
/*REXX program demonstrates first─class functions (as a list of the names of functions).*/
A = 'd2x square sin cos' /*a list of functions to demonstrate.*/
B = 'x2d sqrt Asin Acos' /*the inverse functions of above list. */
w=digits() /*W: width of numbers to be displayed.*/
/* [↓] collection of A & B functions*/
do j=1 for words(A); say; say /*step through the list; 2 blank lines*/
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 /*functions X2D, D2X: integers only. */
else call test j, 0 0.5 1 2 /*all other functions: floating point.*/
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Acos: procedure; parse arg x; if x<-1|x>1 then call AcosErr; return .5*pi()-Asin(x)
r2r: return arg(1) // (pi()*2) /*normalize radians ──► 1 unit circle*/
square: return arg(1) ** 2
pi: pi=3.14159265358979323846264338327950288419716939937510582097494459230; return pi
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
/*──────────────────────────────────────────────────────────────────────────────────────*/
Asin: procedure; parse 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; parse arg x; x=r2r(x); a=abs(x); Hpi=pi*.5
numeric fuzz min(6,digits()-3); if a=pi() then return -1
if a=Hpi | a=Hpi*3 then return 0 ; if a=pi()/3 then return .5
if a=pi()*2/3 then return -.5; return .sinCos(1,1,-1)
/*──────────────────────────────────────────────────────────────────────────────────────*/
sin: procedure; parse 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
/*──────────────────────────────────────────────────────────────────────────────────────*/
invoke: parse arg fn,v; q='"'; if datatype(v,"N") then q=
_=fn || '('q || v || q")"; interpret 'func='_; return func
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); m.=9; numeric form
numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_%2
h=d+6; do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/
numeric digits d; return g/1
/*──────────────────────────────────────────────────────────────────────────────────────*/
test: procedure expose A B w; parse arg fu,xList; d=digits() /*xList: numbers. */
do k=1 for words(xList); x=word(xList, k)
numeric digits d+5 /*higher precision.*/
fun=word(A, fu); funV=invoke(fun, x) ; fun@=_
inv=word(B, fu); invV=invoke(inv, funV); inv@=_
numeric digits d /*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(fun@, 2*w)'='left(left('', funV>=0)funV, w),
right(inv@, 3*w)'='left(left('', invV>=0)invV, w)
end /*k*/
return
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
2 square(2)= 4 sqrt(4)= 2
number function inverse
───────── ──────────────────────────── ────────────────────────────────────
0 sin(0)= 0 Asin(0)= 0
0.5 sin(0.5)= 0.479425 Asin(0.47942553860419)= 0.5
1 sin(1)= 0.841470 Asin(0.84147098480862)= 1
2 sin(2)= 0.909297 Asin(0.90929742682567)= 1.141592
number function inverse
───────── ──────────────────────────── ────────────────────────────────────
0 cos(0)= 1 Acos(1)= 0
0.5 cos(0.5)= 0.877582 Acos(0.87758256188987)= 0.5
1 cos(1)= 0.540302 Acos(0.54030230586810)= 1
2 cos(2)=-0.416146 Acos(-0.41614683650659)= 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 2 .
Sine and cosecant begin their period at 2 k − /2 (where k is an integer), finish it at 2 k + /2, and then reverse themselves over 2 k + /2 ───► 2 k + 3 /2.
Cosine and secant begin their period at 2 k, finish it at 2 k + , and then reverse themselves over 2 k + ───► 2 k + 2 .
Tangent begins its period at 2 k − /2, finishes it at 2 k + /2, and then repeats it (forward) over 2 k + /2 ───► 2 k + 3 /2.
Cotangent begins its period at 2 k, finishes it at 2 k + , and then repeats it (forward) over 2 k + ───► 2 k + 2 .
The above text is from the Wikipedia webpage: http://en.wikipedia.org/wiki/Inverse_trigonometric_functions
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
Rust
This solution uses a feature of Nightly Rust that allows us to return a closure from a function without using the extra indirection of a pointer. Stable Rust can also accomplish this challenge -- the only difference being that compose would return a Box<Fn(T) -> V> which would result in an extra heap allocation.
#![feature(conservative_impl_trait)]
fn main() {
let cube = |x: f64| x.powi(3);
let cube_root = |x: f64| x.powf(1.0 / 3.0);
let flist : [&Fn(f64) -> f64; 3] = [&cube , &f64::sin , &f64::cos ];
let invlist: [&Fn(f64) -> f64; 3] = [&cube_root, &f64::asin, &f64::acos];
let result = flist.iter()
.zip(&invlist)
.map(|(f,i)| compose(f,i)(0.5))
.collect::<Vec<_>>();
println!("{:?}", result);
}
fn compose<'a, F, G, T, U, V>(f: F, g: G) -> impl 'a + Fn(T) -> V
where F: 'a + Fn(T) -> U,
G: 'a + Fn(U) -> V,
{
move |x| g(f(x))
}
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
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
Sidef
func compose(f,g) {
func (*args) {
f(g(args...))
}
}
var cube = func(a) { a.pow(3) }
var croot = func(a) { a.root(3) }
var flist1 = [Num.method(:sin), Num.method(:cos), cube]
var flist2 = [Num.method(:asin), Num.method(:acos), croot]
for a,b (flist1 ~Z flist2) {
say compose(a, b)(0.5)
}
- Output:
0.5 0.5 0.5
Slate
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)].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
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
Stata
In Mata it's not possible to get the address of a builtin function, so here we define user functions.
function _sin(x) {
return(sin(x))
}
function _asin(x) {
return(asin(x))
}
function _cos(x) {
return(cos(x))
}
function _acos(x) {
return(acos(x))
}
function cube(x) {
return(x*x*x)
}
function cuberoot(x) {
return(sign(x)*abs(x)^(1/3))
}
function compose(f,g,x) {
return((*f)((*g)(x)))
}
a=&_sin(),&_cos(),&cube()
b=&_asin(),&_acos(),&cuberoot()
for(i=1;i<=length(a);i++) {
printf("%10.5f\n",compose(a[i],b[i],0.5))
}
SuperCollider
a = [sin(_), cos(_), { |x| x ** 3 }];
b = [asin(_), acos(_), { |x| x ** (1/3) }];
c = a.collect { |x, i| x <> b[i] };
c.every { |x| x.(0.5) - 0.5 < 0.00001 }
Swift
import Darwin
func compose<A,B,C>(f: (B) -> C, g: (A) -> B) -> (A) -> C {
return { f(g($0)) }
}
let funclist = [ { (x: Double) in sin(x) }, { (x: Double) in cos(x) }, { (x: Double) in pow(x, 3) } ]
let funclisti = [ { (x: Double) in asin(x) }, { (x: Double) in acos(x) }, { (x: Double) in cbrt(x) } ]
println(map(zip(funclist, funclisti)) { f, inversef in compose(f, inversef)(0.5) })
- Output:
[0.5, 0.5, 0.5]
Tcl
The following is a transcript of an interactive session:
% namespace path tcl::mathfunc ;# to import functions like abs() etc.
% proc cube x {expr {$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.
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
EndPrgmTXR
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.
(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
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) 0.5In more detail,
(+)*p\functions inversesevaluates 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
psuffix on the map operator, (*) so that(+)*pevaluates 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>
Wren
In Wren, there is a distinction between functions and methods. Essentially, the former are independent objects which can do all the things required of 'first class functions' and the latter are subroutines which are tied to a particular class. As sin, cos etc. are instance methods of the Num class, we need to wrap them in functions to complete this task.
var compose = Fn.new { |f, g| Fn.new { |x| f.call(g.call(x)) } }
var A = [
Fn.new { |x| x.sin },
Fn.new { |x| x.cos },
Fn.new { |x| x * x * x }
]
var B = [
Fn.new { |x| x.asin },
Fn.new { |x| x.acos },
Fn.new { |x| x.pow(1/3) }
]
var x = 0.5
for (i in 0..2) {
System.print(compose.call(A[i], B[i]).call(x))
}
- Output:
0.5 0.5 0.5
XBS
func cube(x:number):number{
send x^3;
}
func cuberoot(x:number):number{
send x^(1/3);
}
func compose(f:function,g:function):function{
send func(n:number){
send f(g(n));
}
}
const a:[function]=[math.sin,math.cos,cube];
const b:[function]=[math.asin,math.acos,cuberoot];
each a as k,v{
log(compose(v,b[k])(0.5))
}- Output:
0.5 0.4999999999999999 0.5000000000000001
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=b.zipWith(H.fcomp,a); //-->list of deferred calculations
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). The True is to return the result, False is just do it for the side effects.