;; Design power (exponent) function that computes x to the nth power 
;; where n is a natural number

;; A Natural is one of: 
;; - 0
;; - (add1 Natural)

;-----------------------
;; Using Structural Recursion...

;; pow.v0 : Number Natural -> Number
;; x to the nth power (where n is a natural number)
(define (pow.v0 x n)
  (cond [(zero? n) 1]
        [else (* x (pow.v0 x (sub1 n)))]))

;-----------------------
;; Using Generative Recursion...

;; pow : Number Natural -> Number
;; x to the nth power (where n is a natural number)
;; HOW: if n is even: compute x^(n/2) using recursion, 
;;      then square that [idea: x^n = (x^(n/2))^2]
;;      if n is odd, compute (x^(n-1)) using recursion, 
;;      then multiply that by x [idea: x^n = (x * (x^(n-1)))]
(define (pow x n)
  (cond [(zero? n) 1]
        [(= n 1) x]
        ;; -- complex case
        [(even? n) 
         (local ((define y (pow x (/ n 2))))
           (* y y))]
        [(odd? n) (* x (pow x (sub1 n)))]))
  
(define (pow.v0 x n)
  (cond [(zero? n) 1]
        [else (* x (pow.v0 x (sub1 n)))]))

(check-expect (pow 4 0) 1)
(check-expect (pow 2 2) 4)
(check-expect (pow 4 2) 16)
(check-expect (pow 2 5) 32)
(check-expect (pow 10 4) 10000)