Some code to go with an exploration of SICP exercise 1.13, a proof of closed form of Fibonacci function.

sicp_ex_1.13.scm

;; expt is three characters too long on the REPL
;; I use on my iPhone
(define (^ x n) (expt x n))

;; Golden Ratio, phi and the conjugate, psi
(define psi (/ (- 1 (sqrt 5)) 2))
(define phi (/ (+ 1 (sqrt 5)) 2))

;; Linear recursive Fib
(define (fib n) 
  (cond ( (= n 0) 0)
        ( (= n 1) 1)
	( else ( + (fib (- n 1) )
                   (fib (- n 2) )))))

;; Approximation of Fib via Golden Ratio,
;; (phi^n)/sqrt( 5 )
(define (golden-fib n)
  (/ (^ phi n) (sqrt 5)))

;; Fib by the closed form equation we just proved
(define (closed-fib n)
 (/ (- (^ phi n) (^ psi n)) (sqrt 5)))

Supporting scheme code to compute Fibonacci in various ways, from a discussion of the solution to exercise 1.13 in SICP at http://www.enigmaticape.com/blog/sicp-exercise-1-13/