; Golden ratio:
; phi/(1 + phi) = 1/phi
; => phi^2 = 1 + phi
; => phi^2 - phi - 1 = 0

; Let y be a fixed point of x -> 1 + 1/x
; Then y = 1 + 1/y => y^2 = y + 1 => y^2 - y - 1 = 0 => phi is a fixed point

(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? next guess)
          next
          (try next))))
  (try first-guess))

(define phi (fixed-point (lambda (x) (+ 1.0 (/ 1.0 x))) 1))

; 1 ]=> phi
; 
; ;Value: 1.6180327868852458
; 
; 1 ]=> (/ 1 phi)
; 
; ;Value: .6180344478216819
; 
; 1 ]=> (/ phi (+ 1 phi))
; 
; ;Value: .6180338134001252
; 
; 1 ]=> (- (* phi phi) phi 1)
; 
; ;Value: -2.687449610405679e-6