### Compound Functions

We have identified in Lisp some of the elements that must appear in any powerful programming language:

• Numbers and arithmetic operations are primitive data and functions.
• Nesting of combinations provides a means of combining operations.
• Definitions that associate names with values provide a limited means of abstraction.

Now we will learn about function definitions, a much more powerful abstraction technique by which a compound operation can be given a name and then referred to as a unit.

We begin by examining how to express the idea of "squaring." We might say, "To square something, multiply it by itself." This is expressed in our language as

``````(defun square (x) (* x x))
``````

We can understand this in the following way:

``````(defun   square (x)        (*       x      x))
^        ^     ^          ^       ^      ^
|        |     |          |       |      |
To      square something, multiply it by itself.
``````

We have here a compound function, which has been given the name `square`. The function represents the operation of multiplying something by itself. The thing to be multiplied is given a local name, `x`, which plays the same role that a pronoun plays in natural language. Evaluating the definition creates this compound function and associates it with the name `square`.1

The general form of a function definition is

``````(defun <name> (<formal parameters>) <body>)
``````

The `<name>` is a symbol to be associated with the function definition in the environment.2 The `<formal parameters>` are the names used within the body of the function to refer to the corresponding arguments of the function. The `<body>` is an expression that will yield the value of the function application when the formal parameters are replaced by the actual arguments to which the function is applied.3 The `<name>` and the` <formal parameters>` are grouped within parentheses, just as they would be in an actual call to the function being defined.

Having defined square, we can now use it:

``````> (square 21)
441
> (square (+ 2 5))
49
> (square (square 3))
81
``````

We can also use square as a building block in defining other functions. For example, \$\$x^2\$\$ + \$\$y^2\$\$ can be expressed as

``````(+ (square x) (square y))
``````

We can easily define a function `sum-of-squares` that, given any two numbers as arguments, produces the sum of their squares:

``````> (defun sum-of-squares (x y)
(+ (square x) (square y)))
sum-of-squares
> (sum-of-squares 3 4)
25
``````

Now we can use `sum-of-squares` as a building block in constructing further functions:

``````> (defun f (a)
(sum-of-squares (+ a 1) (* a 2)))
f
> (f 5)
136
``````

Compound functions are used in exactly the same way as primitive functions. Indeed, one could not tell by looking at the definition of `sum-of-squares` given above whether `square` was built into the interpreter, like `+` and `*`, or defined as a compound function.

1

Observe that there are two different operations being combined here: we are creating the function, and we are giving it the name `square`. It is possible, indeed important, to be able to separate these two notions -- to create functions without naming them, and to give names to functions that have already been created. We will see how to do this in the section Constructing Functions Using Lambda.

2

Throughout this book, we will describe the general syntax of expressions by using italic symbols delimited by angle brackets -- e.g., `<name>` -- to denote the "slots" in the expression to be filled in when such an expression is actually used.

3

More generally, the body of the function can be a sequence of expressions. In this case, the interpreter evaluates each expression in the sequence in turn and returns the value of the final expression as the value of the function application.