The effectiveness of compiling the tree-recursive Fibonacci procedure

Carry out an analysis like the one in exercise 5.45 to determine the effectiveness of compiling the tree-recursive Fibonacci procedure

(define (fib n)
  (if (< n 2)
      n
      (+ (fib (- n 1)) (fib (- n 2)))))

compared to the effectiveness of using the special-purpose Fibonacci machine of figure 5.12. (For measurement of the interpreted performance, see exercise 5.29 .) For Fibonacci, the time resource used is not linear in n; hence the ratios of stack operations will not approach a limiting value that is independent of n.

(controller
   (assign continue (label fib-done))
 fib-loop
   (test (op <) (reg n) (const 2))
   (branch (label immediate-answer))
   ;; set up to compute Fib(n - 1)
   (save continue)
   (assign continue (label afterfib-n-1))
   (save n)                           ; save old value of n
   (assign n (op -) (reg n) (const 1)); clobber n to n - 1
   (goto (label fib-loop))            ; perform recursive call
 afterfib-n-1                         ; upon return, val contains Fib(n - 1)
   (restore n)
   (restore continue)
   ;; set up to compute Fib(n - 2)
   (assign n (op -) (reg n) (const 2))
   (save continue)
   (assign continue (label afterfib-n-2))
   (save val)                         ; save Fib(n - 1)
   (goto (label fib-loop))
 afterfib-n-2                         ; upon return, val contains Fib(n - 2)
   (assign n (reg val))               ; n now contains Fib(n - 2)
   (restore val)                      ; val now contains Fib(n - 1)
   (restore continue)
   (assign val                        ;  Fib(n - 1) +  Fib(n - 2)
           (op +) (reg val) (reg n)) 
   (goto (reg continue))              ; return to caller, answer is in val
 immediate-answer
   (assign val (reg n))               ; base case:  Fib(n) = n
   (goto (reg continue))
 fib-done)
Figure 5.12: Controller for a machine to compute Fibonacci numbers.


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