(define eval-quadratic
  (lambda (a b c x)
    (+ (* a x x) (* b x) c)))

;; ax^2 + bx + c = 0
;; (-b +/- sqrt(b^2 - 4ac))/2a
(define quadratic1
  (lambda (a b c)
    (list (/ (+ (- b) (sqrt (- (* b b) (* 4 a c)))) 
	     (* 2 a))
	  (/ (- (- b) (sqrt (- (* b b) (* 4 a c)))) 
	     (* 2 a)))))

(define quadratic2
  (lambda (a b c)
    (let ((discriminant (sqrt (- (* b b) (* 4 a c))))
	  (denominator (* 2 a))
	  (-b (- b)))
      (list (/ (+ -b discriminant) denominator)
	    (/ (- -b discriminant) denominator)))))

(define quadratic3
  (lambda (a b c)
    ((lambda (discriminant denominator -b)
       (list (/ (+ -b discriminant) denominator)
	     (/ (- -b discriminant) denominator)))
     (sqrt (- (* b b) (* 4 a c)))
     (* 2 a)
     (- b))))

(define fact-helper
  (lambda (x acc)
    (if (= x 0)
	acc
	(fact-helper (- x 1) (* x acc)))))

(define fact
  (lambda (x)
    (fact-helper x 1)))

(define fact
  (lambda (x)
    (letrec
      ((loop
	(lambda (x acc)
	  (if (= x 0)
	      acc
	      (loop (- x 1) (* x acc))))))
      (loop x 1))))

(define append
  (lambda (ls1 ls2)
    (letrec
      ((loop
	(lambda (ls acc)
	  (if (null? ls)
	      acc
	      (loop (cdr ls) (cons (car ls) acc))))))
      (loop (loop ls1 ()) ls2))))


;; (let ((x 2) (y 3)) (+ x y)) =>
;; ((lambda (x y) (+ x y)) 2 3)

;; (let ((x 2) (y 3)) (+ x y)) => ((x 2) (y 3))
(define bindings cadr)

;; (let ((x 2) (y 3)) (+ x y)) => (+ x y)
(define body caddr)

;; (let ((x 2) (y 3)) (+ x y)) => (x y)
(define vars
  (lambda (expr)
    (vars-helper (cadr expr))))

;; ((x 2) (y 3)) => (x y)
(define vars-helper
  (lambda (bindings)
    (if (null? bindings)
	()
	(cons (caar bindings)
	      (vars-helper (cdr bindings))))))

;; (let ((x 2) (y 3)) (+ x y)) => (2 3)
(define vals
  (lambda (expr)
    (vals-helper (cadr expr))))

;; ((x 2) (y 3)) => (2 3)
(define vals-helper
  (lambda (bindings)
    (if (null? bindings)
	()
	(cons (cadar bindings)
	      (vals-helper (cdr bindings))))))

;; (let ((x 2) (y 3)) (+ x y)) => 
;; ((lambda (x y) (+ x y)) 2 3)
(define let->lambda
  (lambda (expr)
    (append (list (list 'lambda (vars expr) (body expr)))
	    (vals expr))))

;; (first-binding '(let* ((x 1) (y x)) (+ x y))) => (x 1)
(define first-binding
  (lambda (expr)
    (car (bindings expr))))

;; (rest-bindings '(let* ((x 1) (y x)) (+ x y))) => ((y x))
(define rest-bindings
  (lambda (expr)
    (cdr (bindings expr))))

;; (let*->let '(let* ((x 1) (y x)) (+ x y))) =>
;; (let ((x 1)) (let* ((y x)) (+ x y))) =>
;; (let ((x 1)) (let ((y x)) (let* () (+ x y)))) =>
;; (let ((x 1)) (let ((y x)) (+ x y)))

(define let*->let
  (lambda (expr)
    (if (null? (bindings expr))
	(body expr)
	(list 'let
	      (list (first-binding expr))
              (let*->let
                (list 'let*
                      (rest-bindings expr)
                      (body expr)))))))

;; (let*->lambda '(let* ((x 1) (y x)) (+ x y))) =>
;; ((lambda (x) ((lambda (y) (+ x y)) x)) 1)
(define let*->lambda
  (lambda (expr)
    (if (null? (bindings expr))
	(body expr)
	(let->lambda
	 (list 'let
	       (list (first-binding expr))
	             (let*->lambda
	               (list 'let*
	                     (rest-bindings expr)
	                     (body expr))))))))

;; for experts...
(define let->lambda
  (lambda (expr)
    `((lambda ,(vars expr) ,(body expr))
      ,@(vals expr))))

(define let*->let
  (lambda (expr)
    (if (null? (bindings expr))
	(body expr)
	`(let (,(first-binding expr))
	   ,(let*->let
	     `(let* ,(rest-bindings expr)
		,(body expr)))))))

;; (tel ((x 2)) (+ x x)) => 4
(define-macro tel
  (lambda (bindings body)
    `((lambda ,(vars-helper bindings) ,body)
      ,@(vals-helper bindings))))

;; (tel* ((x 1) (y x)) (+ x x)) => 2
(define-macro tel*
  (lambda (bindings body)
    (if (null? bindings)
        body
	`(tel (,(car bindings))
	   ,(tel* ,(cdr bindings)
	      ,body)))))