(defn expt [b n]
  (if (= n 0)
    1
    (* b (expt b (- n 1)))))

(defn expt2 [b n]
  (defn expt-iter [b n s]
    (if (= n 0)
      s
      (expt-iter b (- n 1) (* s b))))
  (expt-iter b n 1))

(defn fast-expt [b n]
  (defn even? [n]
    (= (mod n 2) 0))
  (defn square [n]
    (* n n))
  (defn expt-iter [b n s]
    (cond
      (= n 0) s
      (even? n) (square (expt-iter b (/ n 2) s))
      (* b (expt-iter b (- n 1) s))))
  (expt-iter b n 1))

(defn fast-expt2 [b n]
  (defn even? [n]
    (= (mod n 2) 0))
  (defn square [n]
    (* n n))
  (defn expt-iter [b n s]
    (cond
      (= n 0) s
      (even? n) (expt-iter (square b) (/ n 2) s)
      (expt-iter b (- n 1) (* b s))))
  (expt-iter b n 1))

(defn count-fast-expt2 [b n]
  (var counter 0)
  (defn even? [n]
    (= (mod n 2) 0))
  (defn square [n]
    (* n n))
  (defn expt-iter [b n s]
    (set counter (+ counter 1))
    (cond
      (= n 0) s
      (even? n) (expt-iter (square b) (/ n 2) s)
      (expt-iter b (- n 1) (* b s))))
  (expt-iter b n 1)
  counter)

# case 1 (n mod 2 = 0):
#   f(b,n) = f(b,n/2)^2
# case 2 (n mod 2 = 1):
#   f(b,n) = f(b,n-1)*b
# iterative approach:
# case 1 (n mod 2 = 0):
#   f(b,n) = f(b^2, n/2)
# case 2 (n mod 2 = 1):
#   f(b,n) = f(b, n-1)*b

# examples:
# n = 1
#   f(b,1) = f(b,0)*b = 1 * b = b
# n = 2
#   f(b,2) = f(b^2,1) = 1 * b^2 = b^2
# n = 3
#   f(b,3) = f(b,2)*b = f(b^2, 1) * b = b * b^2 = b^3
# n = 4
#   f(b,4) = f(b^2,2) = f(b^4, 1) = 1 * b^4 = b^4
# n = 5
#   f(b,5) = f(b,4)*b = f(b^4, 1) = b * b^5 = b^5

(def k 100000)
(defn lo [n]
  (cond
    (= n k) (printf "%d,%d" n (count-fast-expt2 0.99 n))
    (do
      (printf "%d,%d" n (count-fast-expt2 0.99 n))
      (lo (+ n 1)))))

(lo 0)