; The fibonacci function uses double recursion, which 
; if written naively will run very slowly. Can we
; write it using tail recursion so that it is
; as efficient as an iterative loop?

; Here's the recursive version using named let

(define fibonacci1
  (lambda (n)
    (let fib ((i n))
      (cond
        ((= i 0) 0)
        ((= i 1) 1)
        (else (+ (fib (- i 1)) (fib (- i 2))))))))

; Now let's try to make it tail recursive

(define fibonacci2
  (lambda (n)
    (if (= n 0)
        0
        (let fib ((i n) (a1 1) (a2 0))
          (if (= i 1)
              a1
              (fib (- i 1) (+ a1 a2) a1))))))

; HOMEWORK: implement the QUICKSORT algorithm
; that sorts a list of integers into ascending order.
; Quicksort is doubly recursive. Is it possible to
; make it tail recursive?