I am playing around with Lisp (Scheme dialect in this case), so I decided to write some square root algorithms.

Here’s the first naive version I wrote. As you can see it only works with perfect squares (the first argument should always be 1, as it starts looking with it; the second argument is the number you want to find the square root of):

(define (squareRoot start x)
        (if (= (* start start) x)
                start
                (squareRoot (+ start 1) x)
        )               )

(display (squareRoot 1 9))
(display "\n")

The second algorithm I wrote uses Newton’s method. The idea is simple: take a guess for the square root, y, and then iteratively improve it by averaging y with x/y (every time you do this average you move guess toward the square root). You can see a similar algorithm on the Structure and Interpretation of Computer Programs (SICP) book:

(define (abso x) (if (< x 0) (* -1 x) x))

(define (squareRoot guess x)
        (if (< (abso(- (* guess guess) x)) 0.01)
        guess
        (squareRoot (/ (+ guess (/ x guess)) 2) x))
)

(display (squareRoot 1 10))
(display "\n")