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
- 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
<lang 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();</lang> 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.
<lang Ada>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;</lang>
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
<lang 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
}
</lang>
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.
<lang algol68>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</lang> Output:
+.500000000000000e +0 +.500000000000000e +0 +.500000000000000e +0
AppleScript
AppleScript does not have built-in functions like sine or cosine.
<lang applescript>-- 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}</lang>
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.
<lang AppleScript>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:</lang>
- Output:
<lang AppleScript>{0.5, 0.5, 0.5}</lang>
Arturo
<lang rebol>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]]
]</lang>
- Output:
sin/asin => 0.5 cos/acos => 0.4999999999999999 cube/croot => 0.5
AutoHotkey
By just me. Forum Post <lang AutoHotkey>#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</lang>
- Output:
Input: 0.500000000000000 Output: 0.500000000000000 0.500000000000000 0.500000000000001
Axiom
Using the interpreter: <lang Axiom>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]</lang> Using the Spad compiler: <lang Axiom>)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])</lang>
This would be called using: <lang Axiom>(fns * inv) 0.5</lang> Output: <lang Axiom>[0.5,0.5,0.5]</lang>
BBC BASIC
Strictly speaking you cannot return a function, but you can return a function pointer which allows the task to be implemented. <lang bbcbasic> 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%</lang>
Output:
1.234567 1.234567 2.345678 2.345678 3.456789 3.456789
Bori
<lang 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); }</lang> Output on Android phone: <lang bori>0.5 0.4999999999999999 0.5000000000000001</lang>
BQN
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 user defined function(funsB ← {𝕏⁼}¨ funsA).
<lang bqn>funsA ← ⟨⋆⟜2,-,•math.Sin,{𝕩×3}⟩ funsB ← ⟨√,-,•math.Asin,{𝕩÷3}⟩
comp ← funsA {𝕎∘𝕏}¨ funsB
•Show comp {𝕎𝕩}¨<0.5</lang> <lang bqn>⟨ 0.5000000000000001 0.5 0.5 0.5 ⟩</lang>
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.
<lang bracmat>( (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) )
)</lang> 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.
<lang c>#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;
}</lang>
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). <lang C>#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; }</lang>(Boring) output<lang>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</lang>
C#
<lang csharp>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)));
}
}
} </lang>
Output:
0.5 0.5 0.5
C++
<lang cpp>
- 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;
} </lang>
Ceylon
First, you need to import the numeric module in you module.ceylon file <lang ceylon>module rosetta "1.0.0" { import ceylon.numeric "1.2.1"; }</lang> And then you can use the math functions in your run.ceylon file <lang ceylon>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)); } }</lang>
Clojure
<lang 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)))
</lang> Output:
(0.5 0.4999999999999999 0.5000000000000001)
CoffeeScript
<lang 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]</lang> Output:
0.5 0.4999999999999999 0.5
Common Lisp
<lang 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)))</lang>
Output:
<lang lisp>(#<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</lang>
D
Using Standard Compose
<lang d>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));
}</lang>
- 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. <lang d>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));
}</lang>
Dart
<lang 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));
}
}</lang>
- Output:
0.49999999999999994 0.5 0.5000000000000001
Delphi
<lang 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.</lang>
- Output:
0.50 0.50 0.50
Dyalect
Create new functions from preexisting functions at run-time
Using partial application:
<lang Dyalect>func apply(fun, x) { y => fun(x, y) }
func sum(x, y) { x + y }
func sum2 = apply(sum, 2)</lang>
Store functions in collections
<lang Dyalect>func sum(x, y) { x + y } func doubleMe(x) { x + x }
var arr = [] arr.add(sum) arr.add(doubleMe) arr.add(arr.toString)</lang>
Use functions as arguments to other functions
<lang Dyalect>func Iterator.filter(pred) {
for x in this when pred(x) {
yield x
}
}
[1,2,3,4,5].iter().filter(x => x % 2 == 0)</lang>
Use functions as return values of other functions
<lang Dyalect>func flip(fun, x, y) {
(y, x) => fun(x, y)
}</lang>
Déjà Vu
<lang dejavu>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." </lang>
- 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.
<lang e>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]</lang>
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.
<lang e>def compose(f, g) {
return fn x { f(g(x)) }
}</lang>
<lang e>? 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</lang>
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.
EchoLisp
<lang scheme>
- 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
</lang>
Ela
Translation of Haskell:
<lang ela>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</lang>
Function (<<) is defined in standard prelude as:
<lang ela>(<<) f g x = f (g x)</lang>
Output (calculations are performed on 32-bit floats):
<lang ela>[0.5,0.5,0.499999989671302]</lang>
Elena
ELENA 5.0 : <lang elena>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
}</lang>
- Output:
0.5 0.5 0.5
Elixir
<lang 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)</lang>
- Output:
0.5 0.4999999999999999 0.5
Erlang
<lang 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 ). </lang>
- 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#
<lang fsharp>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()</lang>
Output:
0.500000 0.500000 0.500000
Factor
The constants A and B consist of arrays containing quotations (aka anonymous functions). <lang factor>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 . ;</lang>
- 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.
<lang fantom> 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)) }
}
} </lang>
Output:
0.5 0.4999999999999999 0.5
Forth
<lang 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</lang>
FreeBASIC
Like C, FreeBASIC doesn't have first class functions so I've contented myself by translating their code:
<lang freebasic>' 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</lang>
- 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
<lang 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 ]</lang>
Go
<lang 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))
}
}</lang> Output:
0.49999999999999994 0.5 0.5
Groovy
Solution: <lang groovy>def compose = { f, g -> { x -> f(g(x)) } }</lang>
Test program: <lang groovy>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) })</lang>
Output:
[0.5, 0.4999999999999999, 0.5000000000346574] [0.5, 0.4999999999999999, 0.5000000000346574]
Haskell
<lang 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 </lang>
- 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. <lang 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</lang> 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: <lang j> 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</lang>
<lang> A unqcol 0.5 0.479426 0.877583 0.125 0.25
BA unqcol 0.5
0.5 0.5 0.5 0.5</lang>
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:
<lang j> 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 </lang>
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.
<lang java5>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)); } }</lang> Output:
Compositions: 0.4999999999999999 0.49999999999999994 0.5000000000000001 Hard-coded compositions: 0.4999999999999999 0.49999999999999994 0.5000000000000001
<lang java5>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)); } }</lang>
JavaScript
ES5
<lang javascript>// 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));
}</lang>
ES6
<lang javascript>// 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));
}</lang>
Result is always:
0.5 0.4999999999999999 0.5
Julia
<lang 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</lang>
Output:
0.5 0.4999999999999999 0.5000000000000001
Kotlin
<lang scala>// version 1.0.6
fun compose(f: (Double) -> Double, g: (Double) -> Double ): (Double) -> Double = { f(g(it)) }
fun cube(d: Double) = d * d * d
fun main(args: Array<String>) {
val listA = listOf(Math::sin, Math::cos, ::cube) val listB = listOf(Math::asin, Math::acos, Math::cbrt) val x = 0.5 for (i in 0..2) println(compose(listA[i], listB[i])(x))
}</lang>
- Output:
0.5 0.4999999999999999 0.5000000000000001
Lambdatalk
Tested in [1] <lang Scheme> {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>
Lasso
<lang 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</lang>
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: <lang lingo>-- 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</lang>
- Output:
-- 1 -- 1 -- 1
User-defined arithmetic functions used in code above: <lang lingo>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)
end</lang>
Lua
<lang 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</lang>
Output: <lang>0.5 0.5 0.5</lang>
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) <lang M2000 Interpreter> 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 $("0.00")
For i=0 To 2 {
Print A(i)(B(i)(.5)), C(i)(.5)
}
} CheckFirst </lang>
Maple
The composition operator in Maple is denoted by "@". We use "zip" to produce the list of compositions. The cubing procedure and its inverse are each computed. <lang Maple> > A := [ sin, cos, x -> x^3 ]: > B := [ arcsin, arccos, rcurry( surd, 3 ) ]: > zip( `@`, A, B )( 2/3 );
[2/3, 2/3, 2/3]
> zip( `@`, B, A )( 2/3 );
[2/3, 2/3, 2/3]
</lang>
Mathematica / Wolfram Language
The built-in function Composition can do composition, a custom function that does the same would be compose[f_,g_]:=f[g[#]]&. However the latter only works with 2 arguments, Composition works with any number of arguments. <lang Mathematica>funcs = {Sin, Cos, #^3 &}; funcsi = {ArcSin, ArcCos, #^(1/3) &}; compositefuncs = Composition @@@ Transpose[{funcs, funcsi}]; Table[i[0.666], {i, compositefuncs}]</lang> gives back: <lang Mathematica>{0.666, 0.666, 0.666}</lang> Note that I implemented cube and cube-root as pure functions. This shows that Mathematica is fully able to handle functions as variables, functions can return functions, and functions can be given as an argument. Composition can be done in more than 1 way: <lang Mathematica>Composition[f,g,h][x] f@g@h@x x//h//g//f</lang> all give back: <lang Mathematica>f[g[h[x]]]</lang>
Maxima
<lang maxima>a: [sin, cos, lambda([x], x^3)]$ 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]</lang>
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
<lang Mercury>
- - 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).
</lang>
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.
<lang 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</lang>
- Output:
0.5 0.4999999999999999 0.5
Nemerle
<lang 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)]);
}
}</lang>
Use and Output
C:\Rosetta>ncc -o:FirstClassFunc FirstClassFunc.n C:Rosetta>FirstClassFunc [0.5, 0.5, 0.5]
newLISP
<lang 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) </lang>
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.
<lang nim>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)</lang>
Output:
0.5 0.4999999999999999 0.5
Objeck
<lang 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();
}
}</lang>
OCaml
<lang 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]</lang>
Octave
<lang 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</lang>
- Output:
0.50000 0.50000 0.50000
Oforth
<lang 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 </lang>
- Output:
1
Ol
<lang scheme>
- 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)) </lang>
Oz
This is now also compatible with Oz v 2.0 (To be executed in the Oz OPI, by typing ctl+. ctl+b)
<lang oz>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}}
end
</lang> This will output the following in the Emulator output window <lang oz> 0.5 0.5 0.5 </lang>
PARI/GP
<lang parigp>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)) )
};</lang> Usage note: In Pari/GP 2.4.3 the vectors can be written as <lang parigp> A=[sin, cos, x->x^2];
B=[asin, acos, x->sqrt(x)];</lang>
Output:
0.5000000000000000000000000000 0.5000000000000000000000000000 0.5000000000000000000000000000
Perl
<lang 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;</lang> 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.
<lang php>$compose = function ($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;
}</lang>
Output:
0.5 0.5 0.5
PicoLisp
<lang 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) )</lang>
Output:
0.500001 0.499999 0.500000
PostScript
<lang ps> % 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 ==
} for </lang>
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
<lang Prolog>:- 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). </lang> Output :
?- first_class. 0.5 0.4999999999999999 0.5000000000000001 true.
Python
<lang 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] >>></lang>
Or, equivalently:
<lang python>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()</lang>
- Output:
[0.49999999999999994, 0.5000000000000001, 0.5000000000000001]
R
<lang 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(f1i, f2i)(.5))
}</lang>
- Output:
[1] 0.5 [1] 0.5 [1] 0.5
Alternatively:
<lang R>sapply(mapply(compose,f1,f2),do.call,list(.5))</lang>
- Output:
[1] 0.5 0.5 0.5
Racket
<lang 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)))</lang>
- 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 a user-defined compose function, expressed as an infix operator, ∘. The .() construct invokes the function contained in the $_ (current topic) variable. <lang perl6>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</lang> Output:
0.5 0.5 0.5
Operators, both buildin and user-defined, are first class too.
<lang perl6>my @a = 1,2,3; my @op = &infix:<+>, &infix:<->, &infix:<*>; for flat @a Z @op -> $v, &op { say 42.&op($v) }</lang>
- Output:
43 40 126
REBOL
<lang 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. ]</lang>
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.
<lang rexx>/*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</lang>
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 2k − /2 (where k is an integer), finish it at 2k + /2, and then reverse themselves over 2k + /2 ───► 2k + 3/2.
Cosine and secant begin their period at 2k, finish it at 2k + , and then reverse themselves over 2k + ───► 2k + 2.
Tangent begins its period at 2k − /2, finishes it at 2k + /2, and then repeats it (forward) over 2k + /2 ───► 2k + 3/2.
Cotangent begins its period at 2k, finishes it at 2k + , and then repeats it (forward) over 2k + ───► 2k + 2.
The above text is from the Wikipedia webpage: http://en.wikipedia.org/wiki/Inverse_trigonometric_functions
Ruby
<lang 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]}</lang>
- 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.
<lang rust>#![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))
}</lang>
Scala
<lang 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) + " ")}</lang> Output:
0.5 0.4999999999999999 0.5000000000000001
Scheme
<lang 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)</lang> Output:
0.5 0.5 0.5
Sidef
<lang ruby>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)
}</lang>
- Output:
0.5 0.5 0.5
Slate
Compose is already defined in slate as (note the examples in the comment):
<lang slate>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}.</lang>
used as: <lang slate>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)].</lang>
Smalltalk
<lang 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 ].</lang>
Output:
0.4999999999999999 0.5 0.5000000000000001
Standard ML
<lang sml>- 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</lang>
Stata
In Mata it's not possible to get the address of a builtin function, so here we define user functions.
<lang stata>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)) }</lang>
SuperCollider
<lang 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 } </lang>
Swift
<lang swift>import Darwin func compose<A,B,C>(f: (B) -> C, g: (A) -> B) -> (A) -> C {
return { f(g($0)) }
} let funclist = [ { (x: Double) in sin(x) }, { (x: Double) in cos(x) }, { (x: Double) in pow(x, 3) } ] let funclisti = [ { (x: Double) in asin(x) }, { (x: Double) in acos(x) }, { (x: Double) in cbrt(x) } ] println(map(zip(funclist, funclisti)) { f, inversef in compose(f, inversef)(0.5) })</lang>
- Output:
[0.5, 0.5, 0.5]
Tcl
The following is a transcript of an interactive session:
<lang Tcl>% namespace path tcl::mathfunc ;# to import functions like abs() etc. % proc cube x {expr {$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</lang> 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.)
<lang ti89b>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</lang>
TXR
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.
<lang txrlisp>(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]))</lang>
- 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.
<lang Ursala>#import std
- import flo
functions = <sin,cos,times^/~& sqr> inverses = <asin,acos,math..cbrt>
- cast %eL
main = (gang (+)*p\functions inverses) 0.5</lang> In 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. <lang ecmascript>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))
}</lang>
- Output:
0.5 0.5 0.5
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. <lang zkl>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)</lang> 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.
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