(defun add-rat (x y)
  (make-rat (+ (* (numer x) (denom y))
	       (* (numer y) (denom x)))
	    (* (denom x) (denom y))))

(defun sub-rat (x y)
  (make-rat (- (* (numer x) (denom y))
	       (* (numer y) (denom x)))
	    (* (denom x) (denom y))))

(defun mul-rat (x y)
  (make-rat (* (numer x) (numer y))
	    (* (denom x) (denom y))))

(defun div-rat (x y)
  (make-rat (* (numer x) (denom y))
	    (* (numer y) (denom x))))

(defun equal-ratp (x y)
  (= (* (numer x) (denom y))
     (* (numer y) (denom x))))

(defun make-rat (n d)
  (let ((g (gcd n d)))
    (cons (/ n g) (/ d g))))

(defun numer (x) (car x))
(defun denom (x) (cdr x))

(defun print-rat (x)
  (format t "~a/" (numer x))
  (format t "~a~%" (denom x)))

;; Exercise 2.2
(defun make-segment (start end)
  (cons
   (make-point (x-point start) (y-point start))
   (make-point (x-point end) (y-point end))))

(defun start-segment (segment) (car segment))
(defun end-segment (segment) (cdr segment))

(defun average (x y)
  (/ (+ x y) 2))
(defun midpoint-segment (segment)
  (make-point (average (x-point (start-segment segment))
		       (x-point (end-segment segment)))
	      (average (y-point (start-segment segment))
		       (y-point (end-segment segment)))))

(defun make-point (x y)
  (cons x y))

(defun x-point (point) (car point))
(defun y-point (point) (cdr point))

(defun print-point (p)
  (format t "(~a,~a)~%" (x-point p) (y-point p)))

;; Exercise 2.7
(defun make-interval (a b)
  (cons a b))

(defun lower-bound (x)
  (car x))

(defun upper-bound (x)
  (cdr x))

(defun add-interval (a b)
  (make-interval (+ (lower-bound a) (lower-bound b))
		 (+ (upper-bound a) (upper-bound b))))

(defun mul-interval (a b)
  (let ((p1 (* (lower-bound a) (lower-bound b)))
	(p2 (* (lower-bound a) (upper-bound b)))
	(p3 (* (upper-bound a) (lower-bound b)))
	(p4 (* (upper-bound a) (upper-bound b))))
    (make-interval (min p1 p2 p3 p4)
		   (max p1 p2 p3 p4))))

(defun div-interval (a b)
  (mul-interval
   a
   (make-interval (/ 1.0 (upper-bound b))
		  (/ 1.0 (lower-bound b)))))

;; Exercise 2.8
(defun sub-interval (a b)
  (make-interval (- (lower-bound b) (lower-bound a))
		 (- (upper-bound b) (upper-bound a))))

;; Exercise 2.9
(defun width-interval (x)
  "Calculate width of an interval as half of the distance between lower
and upper bound."
  (/ (- (upper-bound x) (lower-bound x))
     2.0))

;; Exercise 2.10
(defun div-interval (a b)
  (if (zerop (width-interval b))
      (error "Cannot divide by 0 width interval")
   (mul-interval
    a
    (make-interval (/ 1.0 (upper-bound b))
		   (/ 1.0 (lower-bound b))))))

;; Exercise 2.12
(defun make-center-width (c w)
  (make-interval (- c w) (+ c w)))
(defun center (i)
  (/ (+ (lower-bound i) (upper-bound i) 2.0)))
(defun width (i)
  (/ (- (upper-bound i) (lower-bound i) 2.0)))

(defun make-center-percent (c p)
  (make-interval (* c (- 1 (/ p 100)))
		 (* c (+ 1 (/ p 100)))))

;; Exercise 2.17
(defun last-pair (l)
  (if (null (cdr l))
      l
      (last-pair (cdr l))))