;;; Ray Strode

;;; Ex. 1.31 a. ===============================================
(define (product f a next b)
  (define (iter a result) 
    (if (> a b)
        result
        (iter (next a) (* (f a) result))))
  (iter a 1))

;;; n * n - 1 * n - 2 * ... * 1
(define (factorial n)
    (product (lambda (n) n) 1 (lambda (n) (+ n 1)) n))

;;; pi/4 = (2 * 4 * 4 * 6 * 6 * 8 * ...)/(3 * 3 * 5 * 5 * 7 * 7)
;;; pi/4 ~= (2 * 4 * 4 * ... * f(n))/(3 * 3 * 5 * 5 * ... * g(n))
;;;      ~= 2/f(n) * ((2 * 4 * ... * f(n))/(3 * 5 * 7 * ... * g(n)))^2 
(define (pi/4 n)
  (define (next x) (+ x 2))
  (define (f x) x)
  (define (even x) (+ (* 2 x) 2))
  (define (odd x)  (+ (* 2 x) 1))
  (define (square x) (* x x))
  (define (numerator)
    (/ (* 2 (square (product f 4.0 next (even n))))
       (even n)))
  (define (denominator)
    (square (product f 3.0 next (odd n))))
  (/ (numerator) (denominator)))

(define (pi n) (* 4 (pi/4 n)))

;;; Ex. 1.32 a. ===============================================
(define (acummulate combiner null-value f a next b)
  (define (iter a result)
    (if (> a b)
        result
        (iter (next a) (combiner (f a) result))))
  (iter a null-value))

;;; Exercise 1.43 =============================================
(define (repeated f n)
  (define (compose f g) (lambda (x) (f (g x))))
  (if (< n 1) (lambda(x) x)
      (compose f (repeated f (- n 1)))))

;;; Exercise 2.56 ============================================
(define (deriv exp var)
  (define (variable? x) (symbol? x))

  (define (same-variable? v1 v2)
    (and (variable? v1) (variable? v2) (eq? v1 v2)))

  (define (=number? exp num)
    (and (number? exp) (= exp num)))

  (define (make-sum a1 a2)
    (cond ((=number? a1 0) a2)
          ((=number? a2 0) a1)
          ((and (number? a1) (number? a2)) (+ a1 a2))
          (else (list '+ a1 a2))))

  (define (make-product m1 m2)
    (cond ((or (=number? m1 0) (=number? m2 0)) 0)
          ((=number? m1 1) m2)
          ((=number? m2 1) m1)
          ((and (number? m1) (number? m2)) (* m1 m2))
          (else (list '* m1 m2))))

  (define (make-exponentiation b e)
    (cond ((=number? e 0) 1)   
          ((=number? e 1) b)
          ((=number? b 0) 0)
          ((=number? b 1) 1)
          ((and (number? b) (number? e)) 
            (product (lambda (n) b) 1 (lambda (n) (+ n 1)) e))
          (else (list '** b e))))

  (define (sum? x)
    (and (pair? x) (eq? (car x) '+)))

  (define (addend s) (cadr s))

  (define (augend s) (caddr s))

  (define (product? x)
    (and (pair? x) (eq? (car x) '*)))

  (define (multiplier p) (cadr p))

  (define (multiplicand p) (caddr p))

  (define (exponentiation? x)
    (and (pair? x) (eq? (car x) '**))) 

  (define (base b) (cadr b))

  (define (exponent e) (caddr e))

  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
           (deriv (augend exp) var)))
        ((product? exp)
         (make-sum
           (make-product (multiplier exp)
             (deriv (multiplicand exp) var))
           (make-product (deriv (multiplier exp) var)
             (multiplicand exp))))
        ((exponentiation? exp)
         (if (same-variable? (base exp) var) 
           (make-product 
             (make-product 
               (exponent exp)
               (make-exponentiation (base exp) (make-sum (exponent exp) -1)))
             (deriv (base exp) var))
           0))
        (else
          (error "unknown expression type -- DERIV" exp))))

;;; Exercise 2.57 ============================================
(define (deriv exp var)
  (define (variable? x) (symbol? x))

  (define (same-variable? v1 v2)
    (and (variable? v1) (variable? v2) (eq? v1 v2)))

  (define (=number? exp num)
    (and (number? exp) (= exp num)))

  (define (make-sum a1 a2)
    (cond ((=number? a1 0) a2)
          ((=number? a2 0) a1)
          ((and (number? a1) (number? a2)) (+ a1 a2))
          (else (list '+ a1 a2))))

  (define (make-product m1 m2)
    (cond ((or (=number? m1 0) (=number? m2 0)) 0)
          ((=number? m1 1) m2)
          ((=number? m2 1) m1)
          ((and (number? m1) (number? m2)) (* m1 m2))
          (else (list '* m1 m2))))

  (define (make-exponentiation b e)
    (cond ((=number? e 0) 1)   
          ((=number? e 1) b)
          ((=number? b 0) 0)
          ((=number? b 1) 1)
          ((and (number? b) (number? e)) 
            (product (lambda (n) b) 1 (lambda (n) (+ n 1)) e))
          (else (list '** b e))))

  (define (sum? x)
    (and (pair? x) (eq? (car x) '+)))

  (define (addend s) (cadr s))

  (define (augend s) 
    (if (null? (cdddr s)) (caddr s)
        (cons '+ (cddr s))))

  (define (product? x)
    (and (pair? x) (eq? (car x) '*)))

  (define (multiplier p) (cadr p))

  (define (multiplicand p) (caddr p))

  (define (exponentiation? x)
    (and (pair? x) (eq? (car x) '**))) 

  (define (base b) (cadr b))

  (define (exponent e) (caddr e))

  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
           (deriv (augend exp) var)))
        ((product? exp)
         (make-sum
           (make-product (multiplier exp)
             (deriv (multiplicand exp) var))
           (make-product (deriv (multiplier exp) var)
             (multiplicand exp))))
        ((exponentiation? exp)
         (if (same-variable? (base exp) var) 
           (make-product 
             (make-product 
               (exponent exp)
               (make-exponentiation (base exp) (make-sum (exponent exp) -1)))
             (deriv (base exp) var))
           0))
        (else
          (error "unknown expression type -- DERIV" exp))))