```
(define (square x) (* x x))
(define (sum-of-squares x y) (+ (square x) (square y)))
(define (f x) (sum-of-squares x (+ x 1)))
```

In Applicative Order Evaluation arguments to functions get evaluated (reduced as much as possible)
before being passed to a function. Here’s a trace of what would happen do a function call `f`

defined above with arguments of 3 and (+ 3 1).

```
(f 3)
(sum-of-squares 3 (+ 3 1))
(sum-of-squares 3 4)
(+ (square 3) (square 4))
(+ (* 3 3) (* 4 4))
(+ 9 16)
25
```

In Normal Order Evaluation a full expansion of all the function application happens first and then the arguments are evaluated.

```
(f 3)
(sum-of-squares 3 (+ 3 1))
(+ (square 3) (square (+ 3 1))
(+ (* 3 3) (* (+ 3 1) (+ 3 1)))
(+ 9 (* 4 4))
(+ 9 16)
25
```

The best case and simplest case I can think of illustrating the difference between the two is the following:
Suppose there is a defined function `rand`

which will return a random integer, `(rand 100)`

returns an random int
between 0 and 100. In this case I pretend that `(rand 100)`

returned 13.

## Applicative Evaluation

```
(f (rand 100))
(sum-of-squares 13)
(sum-of-squares 13 (+ 13 1))
(sum-of-squares 13 14)
(+ (square 13) (square 14))
(+ (* 13 13) (* 14 14))
(+ 169 196)
365
```

The chances of running `(f (rand 100))`

under applicative order again and getting the same answer is ^{1}⁄_{100}.

## Normal Order Evaluation

```
(f (rand 100))
(sum-of-squares (rand 100) (+ (rand 100) 1))
(+ (square (rand 100)) (square (+ (rand 100) 1))
(+ (* (rand 100) (rand 100)) (* (+ (rand 100) 1) (+ (rand 100) 1)))
```

I want to stop here to note that all 4 calls to (rand 100) should, if the `rand`

function is any
good, evaluate to 4 random integers between 0 and 100. Only if there is a lot of luck involved
(the chances of the this Normal Order `f`

being the same its Applicative Order cousin is about
1 in one hundred million) will the rest of this statement evaluate to the same result as the
the Applicative Evaluation earlier.