kouddy

(define (square x)
  (* x x))

(define (sum-of-squares x y)
  (+ (square x) (square y)))

(define (sum-of-highest-squares x y z)
  (cond ((< x y z) (sum-of-squares y z))
        ((< y z x) (sum-of-squares z x))
        ((< z x y) (sum-of-squares x y))))

kouddy

;;;Re-written version of the code
(define (a-plus-abs-b a b)
  (if (> b 0) 
     (+ a b) 
     (- a b)))

kouddy

(test 0 p)

;;;Applicative-order
(test 0 p) ;;; Substitute p with p
(test 0 p) ;;; Substitute p with p
(test 0 p) ;;; Substitute p with p

;;;Normal order
(test 0 p)
(if (= 0 0) 0 p) ;;;p's value not needed

kouddy

(define (square x) (* x x))

(define (average x y)
  (/ (+ x y) 2)) 

(define (improve guess x)
  (average guess (/ x guess)))

;;; Newly implemented guess
;;; Good enough when |guess - improved-guess|/guess < 0.001 (0.1%)

kouddy

(define (square x) (* x x))

(define (improve guess x)
  (/ (+ (/ x (square guess)) (* 2 guess)) 3))

(define (good-enough? guess improved-guess x)
  (< (/ (abs (- guess improved-guess)) guess) 0.001))

(define (cubic-root-iter guess x)
  (if (good-enough? guess (improve guess x) x)

kouddy

;;; First process
(+ 4 5)
(inc (+ 3 5))
(inc (inc (+ 2 5)))
(inc (inc (inc (+ 1 5))))
(inc (inc (inc (inc (+ 0 5)))))
(inc (inc (inc (inc 5))))
(inc (inc (inc 6)))
(inc (inc 7))
(inc 8)

kouddy

;;;(A 1 10)
(A 0 (A 1 9)
(* 2 (A 0 (A 1 8)))
(* 2 (* 2 (A 0 (A 1 7))))
(* 2 (* 2 (* 2 (A 0 (A 1 6)))))
(* 2 (* 2 (* 2 (* 2 (A 0 (A 1 5))))))
(* 2 (* 2 (* 2 (* 2 (* 2 (A 0 (A 1 4)))))))
(* 2 (* 2 (* 2 (* 2 (* 2 (* 2 (A 0 (A 1 3))))))))
(* 2 (* 2 (* 2 (* 2 (* 2 (* 2 (* 2 (A 0 (A 1 2)))))))))
(* 2 (* 2 (* 2 (* 2 (* 2 (* 2 (* 2 (* 2 (A 0 (A 1 1))))))))))

kouddy

;;;(A 2 4)
;;; We know that (A 1 n) = 2^n, so steps to evaluate (A 1 n) are omitted here.
(A 1 (A 2 3))
(A 1 (A 1 (A 2 2)))
(A 1 (A 1 (A 1 (A 2 1))))
(A 1 (A 1 (A 1 2)))
...
(A 1 (A 1 4))
...
(A 1 16)

kouddy

;;;(A 3 3)
;;; We know that (A 1 n) = 2^n, so steps to evaluate (A 1 n) are omitted here.
;;; We also know that (A 2 n) = 2^2^...n times, so steps to evaluate (A 2 n) are omitted here.
(A 3 3)
(A 2 (A 3 2))
(A 2 (A 2 (A 3 1)))
(A 2 (A 2 2))
...
(A 2 4)
...

kouddy

(define (f n)
  (if (< n 3)
      n
      (+ (f (- n 1))
         (* 2 (f (- n 2)))
         (* 3 (f (- n 3))))))