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Function composition (computer science)

Programming mechanism

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Programmers frequently apply functions to results of other functions, and almost all programming languages allow it. In some cases, the composition of functions is interesting as a function in its own right, to be used later. Such a function can always be defined but languages with first-class functions make it easier.

The ability to easily compose functions encourages factoring (breaking apart) functions for maintainability and code reuse. More generally, big systems might be built by composing whole programs.

Narrowly speaking, function composition applies to functions that operate on a finite amount of data, each step sequentially processing it before handing it to the next. Functions that operate on potentially infinite data (a stream or other codata) are known as filters, and are instead connected in a pipeline, which is analogous to function composition and can execute concurrently.

Composing function calls

For example, suppose we have two functions f and g, as in z = f(y) and y = g(x). Composing them means we first compute y = g(x), and then use y to compute z = f(y). Here is the example in the C language:

<span id="mwJQ"></span><span class="kt" id="mwJg">float</span><span class="w" id="mwJw"> </span><span class="n" id="mwKA">x</span><span class="p" id="mwKQ">,</span><span class="w" id="mwKg"> </span><span class="n" id="mwKw">y</span><span class="p" id="mwLA">,</span><span class="w" id="mwLQ"> </span><span class="n" id="mwLg">z</span><span class="p" id="mwLw">;</span> <span class="c1" id="mwMA">// ...</span> <span class="n" id="mwMQ">y</span><span class="w" id="mwMg"> </span><span class="o" id="mwMw">=</span><span class="w" id="mwNA"> </span><span class="n" id="mwNQ">g</span><span class="p" id="mwNg">(</span><span class="n" id="mwNw">x</span><span class="p" id="mwOA">);</span> <span class="n" id="mwOQ">z</span><span class="w" id="mwOg"> </span><span class="o" id="mwOw">=</span><span class="w" id="mwPA"> </span><span class="n" id="mwPQ">f</span><span class="p" id="mwPg">(</span><span class="n" id="mwPw">y</span><span class="p" id="mwQA">);</span>

The steps can be combined if we don't give a name to the intermediate result:

<span id="mwRA"></span><span class="n" id="mwRQ">z</span><span class="w" id="mwRg"> </span><span class="o" id="mwRw">=</span><span class="w" id="mwSA"> </span><span class="n" id="mwSQ">f</span><span class="p" id="mwSg">(</span><span class="n" id="mwSw">g</span><span class="p" id="mwTA">(</span><span class="n" id="mwTQ">x</span><span class="p" id="mwTg">));</span>

Despite differences in length, these two implementations compute the same result. The second implementation requires only one line of code and is colloquially referred to as a "highly composed" form. Readability and hence maintainability is one advantage of highly composed forms, since they require fewer lines of code, minimizing a program's "surface area". DeMarco and Lister empirically verify an inverse relationship between surface area and maintainability. On the other hand, it may be possible to overuse highly composed forms. A nesting of too many functions may have the opposite effect, making the code less maintainable.

In a stack-based language, functional composition is even more natural: it is performed by concatenation, and is usually the primary method of program design. The above example in Forth:

g f

Which will take whatever was on the stack before, apply g, then f, and leave the result on the stack. See postfix composition notation for the corresponding mathematical notation.

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