#lang sicp

(define tolerance 0.00001)

(define (fixed-point f first-guess)
    (define (close-enough? v1 v2)
        (< (abs (- v1 v2)) tolerance))
    (define (try guess)
        (let ((next (f guess)))
            (if (close-enough? guess next)
                next
                (try next))))
    (try first-guess))

; below is phi
;
; proof: f(x) = 1 + 1/x
; f(phi) = 1 + 1/((1+sqrt(5))/2) = 1 + 2/(1+sqrt(5)) =
; (1+sqrt(5))/(1+sqrt(5)) + 2/(1+sqrt(5)) =
; ( 3+sqrt(5) ) / (1+sqrt(5)) (multiply and divide by (1-sqrt(5)))
; you get: (1+sqrt(5))/2 qed
;
(define phi (fixed-point (lambda (x) (+ 1 (/ 1 x))) 2.0))
(define solution (lambda (x) (+ 1 (/ 1 x))))