(ns sicp.ex1_15
  (:use [sicp utils]
	[clojure.contrib trace test-is]))

;; exercise 1.15: sin (x) calculation
;;    a.  How many times is the procedure p applied when (sine 12.15)
;;        is evaluated?
;;    b.  What is the order of growth in space and number of steps (as
;;        a function of a) used by the process generated by the sine
;;        procedure when (sine a) is evaluated?
(defn p [x] (- (* 3 x) (* 4 (cube x))))

(defn sine [angle]
  (if (not (> (myabs angle) 0.1))
    angle
    (p (sine (/ angle 3.0)))))

;; solution to (a) => 5
;; (comment
;; user> (dotrace [p] (sine 12.15))
;; TRACE t2490: (p 0.049999999999999996)
;; TRACE t2490: => 0.1495
;; TRACE t2491: (p 0.1495)
;; TRACE t2491: => 0.4351345505
;; TRACE t2492: (p 0.4351345505)
;; TRACE t2492: => 0.9758465331678772
;; TRACE t2493: (p 0.9758465331678772)
;; TRACE t2493: => -0.7895631144708228
;; TRACE t2494: (p -0.7895631144708228)
;; TRACE t2494: => -0.39980345741334
;; -0.39980345741334
;; )

;; solution to b
;; both space and number of steps grows as log3(a) -> log a to the base 3.
;;
;; proof:
;;   a * (1/3)^n <= 0.1
;;   => take log to the base 3 on both the sides.

;; Note: Finding the order of space in a recursive process is sort of, equiv
;;       to finding the number of deferred operations. Which is in-turn the
;;       same as the depth of the evaluation tree.