; gaussian integers, part 1

(define (gs a b) (cons a b))
(define (re x) (car x))
(define (im x) (cdr x))

(define (gauss a b)
  (when (not (integer? a))
    (error 'gauss "must be integer"))
  (when (not (integer? b))
    (error 'gauss "must be integer"))
  (gs a b))

(define (gauss-from-complex x)
  (gauss (real-part x) (imag-part x)))

(define (gauss-to-complex x)
  (make-rectangular (re x) (im x)))

(define (gauss-zero? x)
  (and (zero? (re x)) (zero? (im x))))

(define (gauss-unit? x)
  (or (and (= (abs (re x)) 1) (zero? (im x)))
      (and (zero? (re x)) (= (abs (im x)) 1))))

(define (gauss-conjugate x)
  (gs (re x) (- (im x))))

(define (gauss-norm x)
  (define (square x) (* x x))
  (+ (square (re x)) (square (im x))))

(define (gauss-eql? x y)
  (and (= (re x) (re y))
       (= (im x) (im y))))

(define (gauss-add . xs)
  (define (add x y)
    (gs (+ (re x) (re y))
        (+ (im x) (im y))))
  (let loop ((xs xs) (zs (gs 0 0)))
    (if (null? xs) zs
      (loop (cdr xs) (add (car xs) zs))))

(define (gauss-negate x)
  (gs (- (re x)) (- (im x))))

(define (gauss-sub . xs)
  (define (sub x y)
    (gs (- (re x) (re y)) (- (im x) (im y))))
  (cond ((null? xs) (error 'gauss-sub "no operands"))
        ((null? (cdr xs)) (gauss-negate (car xs)))
        (else (let loop ((xs (cdr xs)) (zs (car xs)))
                (if (null? xs) zs
                  (loop (cdr xs) (sub zs (car xs))))))))

(define (gauss-mul . xs)
  (define (mul x y)
    (gs (- (* (re x) (re y))
           (* (im x) (im y)))
        (+ (* (re x) (im y))
           (* (im x) (re y)))))
  (let loop ((xs xs) (zs (gs 1 0)))
    (if (null? xs) zs
      (loop (cdr xs) (mul (car xs) zs)))))

(define (gauss-quotient num den)
  (let ((n (gauss-norm den))
        (r (+ (* (re num) (re den))
              (* (im num) (im den))))
        (i (- (* (re den) (im num))
              (* (re num) (im den)))))
    (gs (round (/ r n)) (round (/ i n)))))

(define (gauss-remainder num den quo)
  (gauss-sub num (gauss-mul den quo)))