Write a function that reduces compositions of linear operators

Question

You are given two functions g(x) and h(x), each of which takes an integer x and returns the number ax + b (where a and b are integers defined in the function).

Your task is to write a function f(g, h) that takes these two functions, and returns a function k(x) = g(h(x)), but where the procedure of k(x) is also simply of the form ax + b.

For example, given g(x) = 2x + 3 and h(x) = 3x + 4, you would need to return k(x) = 6x + 11, and not k(x) = 2(3x + 4) + 3.

Answer 1

GolfScript, 41 40 23 19 characters

{2,0+%/-{@*+}++}:k;

Thanks to Peter Taylor for pointing out this solution. Accepts two function bodies as arguments.

The following examples show how to use function k:

# As function
{2*3+} {3*4+} k         # => {11 6 @*+}

# Apply to value
5 {2*3+} {3*4+} k ~     # => 41

If one wants to retain the original output format, the following 22 character solution is possible:

{2,0+%/-{*}+\+{+}+}:k;

{2*3+} {3*4+} k         # => {6 * 11 +}

Explanation of the code:

{                       # on stack are functions g h
  2,                    # build array [0 1] -> g h [0 1]
  0+                    # append 0 -> g h [0 1 0]
  %                     # apply function h to each element -> g [h(0) h(1) h(0)]
  /                     # apply function g to each element
                        # and break up the array -> g(h(0)) g(h(1)) g(h(0))
  -                     # subtract last two -> g(h(0)) g(h(1))-g(h(0))
  {@*+}++               # append function body to these values
}:k;                    # save function to variable k

Answer 2

Mathematica 15

Define g, h as pure functions

g and h are defined as pure functions (awaiting a value or independent variable to be assigned)

g = 2 # + 3 &;
h = 3 # + 4 &;
g[x]
h[x]

3 + 2 x
4 + 3 x

---

Method #1: Compose g of h directly (15 chars)

g@h@x//Simplify

11 + 6 x

Or, alternatively: (17 chars)

g[h[x]]//Simplify

---

Method #2: Define f (25 chars= 13 + 10 chars for Simplify)

f=Composition

f[g, h][x]
f[g, h][x] // Simplify
Composition[g, h][x]//Simplify
f[g, h][1]
Table[f[g, h][x], {x, 1, 13}]

3 + 2 (4 + 3 x)
11 + 6 x
11 + 6 x
17
{17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89}

Answer 3

Haskell, 31

(f%g)x=f(g 0)+x*(f(g 1)-f(g 0))

Version in which the function body contains only variables (equivalent at run time, but closer to the question's form), 38 characters:

(f%g)x=a*x+b where h=f.g;b=h 0;a=h 1-b

Note that in both cases, since Haskell uses curried functions, there is no explicit construction of the composed function as it is implicit in % being defined with three arguments: function f, function g, value x.

Example use:

> ((\x -> 2*x + 3)%(\x -> 3*x + 4)) 0
11
> ((\x -> 2*x + 3)%(\x -> 3*x + 4)) 4
35

Ungolfed second form:

(f % g) x = a * x + b        -- equivalent: (f % g) = \x -> a * x + b
  where
    h = f . g                -- use built in function composition to bootstrap
    b = h 0                  -- constant term of composed function
    a = h 1 - h 0            -- linear term of composed function

Answer 4

vba, 89

Function f(g,h,x)
f=Evaluate(Replace(g,"x","*("&Replace(h,"x","*"&x)&")"))  
End Function

usage/result

?f("2x+3","3x+4",1)
17

Answer 5

Lua, 87

function f(g, h)local a,b=g(h(1))-g(h(0)),g(h(0))return function(x)return a*x+b end end

Answer 6

Ruby >=1.9, 55

f=->g,h{a=(g[1]-g[0])*(h[1]-h[0]);b=g[h[0]];->x{a*x+b}}

Less golfed:

def c(f,g)
  f_a=f[1]-f[0]
  g_a=g[1]-g[0]
  a=f_a*g_a
  b=f[g[0]]
  ->x{a*x+b}
end

Answer 7

SML (60 chars)

fun f g h=let val c=g(h 0)val m=g(h 1)-c in fn x=>m*x+c end;

Ungolfed:

fun f g h =
    let val c = g (h 0)
        val m = g (h 1) - c
     in fn x => m*x + c
    end;

Answer 8

PHP 72

Apparently I completely missed the point of this code-golf. I'm STILL not sure if I understand it correctly. Is no output required? Can I assume the methods of g and h? Here's a new version based off the old one:

<?=$f=function($g,$h,$x){return$k=($g[0]*$h[0]*$x)+($g[0]*$h[1])+$g[1];}

If this is still wrong, I give up.

---

PHP 176

The requirements are kind of confusing, but here's what I think you meant:

<?=f($argv);function f($a){$x=$a[3];$g=explode("x",$a[1]);$h=explode("x",$a[2]);return$k=($g[0]*$h[0])."*$x+".(($g[0]*$h[1])+$g[1])."=".(($g[0]*$h[0]*$x)+($g[0]*$h[1])+$g[1]);}

Usage: php f.php 2x+3 3x+4 5

Output: 6*5+11=41

Another Example

Usage: php f.php 4x-7 3x+4 2

Output: 12*2+9=33

Method

As you described, we can't simply output the answer with g(h(x)), we had to create function f(g,h) which determines what both the functions would be together, and return that as a new function: k(x).

Is this what OP meant?

Ungolfed

<?php

function f($a) {
    $x=$a[3];
    $g=explode("x",$a[1]);
    $h=explode("x",$a[2]);
    $k  = $g[0]*$h[0];
    $k .= "*";
    $k .= $x;
    $k .= "+";
    $k .= ($g[0]*$h[1])+$g[1];
    $k .= " = ";
    $k .= ($g[0]*$h[0]*$x) + ($g[0]*$h[1])+$g[1];
    return $k;
}

echo f($argv);

Answer 9

Erlang 54:

1> F = fun(G,H)->B=G(H(0)),A=G(H(1))-B,fun(X)->A*X+B end end.
2> (F(fun(X) -> 2*X+3 end, fun(X) -> 3*X + 4 end))(0).
11
3> (F(fun(X) -> 2*X+3 end, fun(X) -> 3*X + 4 end))(1) - v(2).
6

Answer 10

Common Lisp (108)

(defun f(g h)(let*((d(g 0))(c(-(g 1)d))(z(h 0))(e(-(h 1)z))(a(* c e))(b(+(* c z)d)))(lambda(x)(+(* a x)b))))

It could be shorter if I assumed expressions would have constants folded...but I didn't make that assumption to make sure I met the spec. Also, I have implemented Clojure's literal anonymous function syntax as a read macro. I decided against using that as well.

Answer 11

Postscript, 120

/f{1 index 0 get dup 
2 index 0 get mul exch
3 2 roll 2 get mul 
3 2 roll 2 get add[3 1 roll/mul cvx exch/add cvx]cvx}def

Ungolfed and commented with testing.

/g{2 mul 3 add}def
/h{3 mul 4 add}def

/f{
    1 index 0 get % g h g[0]
    dup 2 index 0 get mul % g h g[0] g[0]*h[0]
    exch 3 2 roll 2 get mul % g g[0]*h[0] g[0]*h[2]
    3 2 roll 2 get add % g[0]*h[0] g[0]*h[2]+g[2]
    [ 3 1 roll /mul cvx exch /add cvx ] cvx 
}def

//g //h f ==

Output:

GPL Ghostscript 9.10 (2013-08-30)
Copyright (C) 2013 Artifex Software, Inc.  All rights reserved.
This software comes with NO WARRANTY: see the file PUBLIC for details.
{6 mul 11 add}
GS>