Here is Gosper's algorithm for computing the square root of a rational number.

(define (gosper-sqrt a b c d) ;; Solve for ;; ax + b ;; ------ = x ;; cx + d (define (newtons-method f f-prime guess) (let ((dy (f guess))) (if (< (abs dy) 1) guess (let ((dy/dx (f-prime guess))) (newtons-method f f-prime (- guess (/ dy dy/dx))))))) (define (f x) (+ (* c x x) (* (- d a) x) (- b))) (define (f-prime x) (+ (* 2 c x) (- d a))) (let ((value (floor (newtons-method f f-prime b)))) (cons-stream value (gosper-sqrt (+ (* c value) d) c (+ (* (- a (* value c)) value) (- b (* value d))) (- a (* value c)))))) 1 ]=> (cf:render (gosper-sqrt 0 17 10 0)) 1.303840481040529742916594311485836883305618755782013091790079369... ;; base 10, 100 digits 1 ]=> (cf:render (gosper-sqrt 0 17 10 0) 10 100) 1.303840481040529742916594311485836883305618755782013091790079369896765385576397896545183528886788497...