Chapter 2, Building Abstractions with Data

Section - 2.5 - Systems with Generic Operations

Exercise 2.90


Apparently I did not find it as difficult as it seemed from the hint. Perhaps I missed something :)

My understanding is problem requires to implement the polynomial system that can be used for both sparse and dense termlists and all operations can be performed between them. It has not asked to implement procedures that for the efficiency - It only asked to implement system which can enable us to later implement a polynomial system which is efficient for both dense and sparse. I believe so because as per my experience with the old problems - the problem generally gives an outline for what it seeks in the solution - here it only gave an outline to build a system which can enable operations on both term-list.

If the case is to implement the system where our polynomial also decides whether to use sparse or dense then it can also be implemented - we just need to think for the strategy - at how much packing is considered good for sparse and inside make-polynomial we can check with this strategy and do conversions(I have not made this change).

I made the following changes:

  • Implemented two terms-list packages - ‘dense-termlist, ‘sparse-termlist
  • make-term, coeff, and order are owned by the polynomial package(not by the term-list packages implemented) because they are terms not the list themselves and both packages - sparse-termlist and dense-termlist use them from the polynomial package.
  • There is an interesting way I worked around with the adjoin-term - this procedure have different implementation for each package and which should be the part of the respective term-list packages. But I can not apply adjoin-term in generic way because it takes two arguments - term, and termlist - where term does not have any type tag in my implementation because I made term to be owned by polynomial package instead of termlist package(which may be debatable). So, I made adjoin-term-proc - a procedure that returns a procedure! And the returned procedure takes an argument - term and adds it to the list. See the implementation for details.

Here is the complete code(Examples in the end):

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#lang sicp

(#%require (only racket/base error))
(#%require (only racket/base make-hash))
(#%require (only racket/base hash-set!))
(#%require (only racket/base hash-ref))

;Polynomial Package

(define (install-polynomial-package)
  ;; internal procedures
  ;; representation of poly
  (define (make-poly variable term-list)
    (cons variable term-list)
  )
  
  (define (variable p) (car p))
  (define (term-list p) (cdr p))
  (define (variable? x) (symbol? x))
  (define (same-variable? v1 v2) (and (variable? v1) (variable? v2) (eq? v1 v2)))
  ;; representation of terms and term lists
  
  (define (adjoin-term term list)
       ((apply-generic 'adjoin-term-proc list) term)
  )
  (define (first-term terms)
    (apply-generic 'first-term terms)
  )
  (define (rest-terms terms)
    (apply-generic 'rest-terms terms)
  )
  (define (empty-termlist? terms)
    (apply-generic 'empty-termlist? terms)
  )
  (define (make-term order coeff) (list order coeff))
  (define (order term) (car term))
  (define (coeff term) (cadr term))
  
  (define (add-poly p1 p2)
      (if (same-variable? (variable p1) (variable p2))
          (make-poly (variable p1)
                     (add-terms (term-list p1)
                            (term-list p2)
                     )
          )
          (error "Polys not in same var -- ADD-POLY" (list p1 p2))
      )
  )
  (define (mul-poly p1 p2)
      (if (same-variable? (variable p1) (variable p2))
          (make-poly (variable p1)
                     (mul-terms (term-list p1)
                                (term-list p2)
                     )
          )          
          (error "Polys not in same var -- MUL-POLY" (list p1 p2))
      )
  )
  (define (sub-poly p1 p2) (add-poly p1 (negate-poly p2)))
  (define (negate-poly p) (make-poly (variable p) (negate-termlist (term-list p))))

  (define (negate-termlist terms)
    (if (empty-termlist? terms)
        terms        
        (adjoin-term (make-term (order (first-term terms)) (apply-generic 'negate (coeff (first-term terms)))) (negate-termlist (rest-terms terms)))
    )
  )
  (define (add-terms L1 L2)
     (cond ((empty-termlist? L1) L2)
        ((empty-termlist? L2) L1)
        (else
           (let ((t1 (first-term L1)) (t2 (first-term L2)))
                (cond ((> (order t1) (order t2))
                           (adjoin-term
                                t1 (add-terms (rest-terms L1) L2))
                      )
                      ((< (order t1) (order t2))
                      (adjoin-term t2 (add-terms L1 (rest-terms L2))))
                      (else
                           (adjoin-term
                                (make-term (order t1)
                                           (add (coeff t1) (coeff t2))
                                )
                                (add-terms (rest-terms L1)
                                     (rest-terms L2)
                                )
                           )
                      )
                 )
             )
          )
        )
    )

    (define (mul-terms L1 L2)
        (if (empty-termlist? L1)
            L1
            (add-terms (mul-term-by-all-terms (first-term L1) L2)
                       (mul-terms (rest-terms L1) L2)
            )
        )
    )
    (define (mul-term-by-all-terms t1 L)
        (if (empty-termlist? L)
            L
            (let ((t2 (first-term L)))
                 (adjoin-term
                     (make-term (+ (order t1) (order t2))
                                (mul (coeff t1) (coeff t2))
                     )
                     (mul-term-by-all-terms t1 (rest-terms L))
                 )
            )
        )
    )

   ;; interface to rest of the system
   (define (tag p) (attach-tag 'polynomial p))
   (put 'add '(polynomial polynomial) 
        (lambda (p1 p2) (tag (add-poly p1 p2))))
   (put 'sub '(polynomial polynomial) 
        (lambda (p1 p2) (tag (add-poly p1 (negate-poly p2)))))
   (put 'mul '(polynomial polynomial) 
        (lambda (p1 p2) (tag (mul-poly p1 p2))))
   (put 'make 'polynomial
        (lambda (var terms) (tag (make-poly var terms))))
   (put 'make-term 'polynomial make-term)
   (put 'order 'polynomial order)
   (put 'coeff 'polynomial coeff)
   (put 'negate '(polynomial) (lambda (p) (tag (negate-poly p))))


   (define (=is-zero? poly)
       (define (iter terms)
         (if (apply-generic 'empty-termlist? terms)
             #t
             (and (=zero? (coeff (first-term terms))) (iter (rest-terms terms)))
         )
       )
       (iter (term-list poly))
   )
  
   (put '=zero? '(polynomial) =is-zero?)
  
  
  'done)

(define (make-polynomial var terms)
  ((get 'make 'polynomial) var terms))

(define (make-term order coeff)
  ((get 'make-term 'polynomial) order coeff))

(define (order term)
  ((get 'order 'polynomial) term))
    
(define (coeff term)
  ((get 'coeff 'polynomial) term))
        

;dense-termlist package
(define (install-dense-termlist-package)

    (define (adjoin-term-dense term term-list)
        (define (iter count terms)
            (if (= count 0)
                terms
                (iter (- count 1) (cons 0 terms))
            )
        )
        (let ((cof (coeff term))
              (count (- (order term) (length term-list)))
             )
             (cond
                ((=zero? cof) term-list)
                ((< count 0) (error "Can not add term - order of passed term is already present in the list"))
                (else (cons cof (iter count term-list)))
             )
        )
    )
    (define (first-term term-list) (make-term (- (length term-list) 1) (car term-list)))

       ;; interface to rest of the system
    (define (tag tl) (attach-tag 'dense-termlist tl))
  
    (put 'empty-dense-termlist 'dense-termlist (lambda () (tag '())))    
    (put 'first-term '(dense-termlist) first-term)
    (put 'rest-terms '(dense-termlist) (lambda (tl) (tag (cdr tl))))
    (put 'empty-termlist? '(dense-termlist) null?)
    (put 'adjoin-term-proc '(dense-termlist) (lambda(tl) (lambda(term) (tag (adjoin-term-dense term tl)))))
  
  'done)

(define (empty-dense-termlist)
  ((get 'make 'dense-termlist))
)

;sparse-termlist package
(define (install-sparse-termlist-package)
    (define (adjoin-term-sparse term term-list)
        (if (=zero? (coeff term))
            term-list
            (cons term term-list)
        )
    )
    
       ;; interface to rest of the system
    (define (tag tl) (attach-tag 'sparse-termlist tl))
  
    (put 'empty-sparse-termlist 'dense-termlist (lambda () (tag '())))    

    (put 'first-term '(sparse-termlist) (lambda (tl) (car tl)))
    (put 'rest-terms '(sparse-termlist) (lambda (tl) (tag (cdr tl))))
    (put 'empty-termlist? '(sparse-termlist) null?)
    (put 'adjoin-term-proc '(sparse-termlist) (lambda(tl) (lambda(term) (tag (adjoin-term-sparse term tl)))))
  
  'done)

(define (empty-sparse-termlist)
  ((get 'make 'sparse-termlist))
)

; Code below this is the arithmetic package we built in the last section
; Arithmetic Package


(define (sine x) (sin (contents (raise-to-type 'scheme-number x))))
(define (cosine x) (cos (contents (raise-to-type 'scheme-number x))))
(define (arctan x) (atan (contents (raise-to-type 'scheme-number x)))) 

(define (drop arg)
 (if (has-type-tag? arg)
     (let ((type (type-tag arg)))
       (let ((proc (get-projection type)))
          (if (not (null? proc))
             (let ((projected (proc arg)))
               (let ((raised (raise-to-type type projected)))
                 (if ((get 'equ? (list type type)) (contents raised) (contents arg))
                       (drop projected)
                       arg
                 )
               )
             )                
             arg
          )
       )
     )
     arg
  )
)

(define (apply-generic op . args)
  (let ((type-tags (map type-tag args)))
    (let ((proc (get op type-tags)))
      (if (not (null? proc))
          (drop (apply proc (map contents args)))
          (let ((has-same-type
                     (accumulate
                          (lambda (a b) (and a b))
                          #t
                          (map
                             (lambda(x) (equal? (car type-tags) x))
                             (cdr type-tags)
                          )
                     )
               ))
               (let ((highest-type (if has-same-type
                                       (get-parent (car type-tags))
                                       (find-highest-type type-tags)
                                   )
                    ))
                    (if (null? highest-type)
                       (error "Could not find a suitable ancestor" op args)
                       (let ((raised-args (map-until
                                                    '()
                                                    (lambda(arg)
                                                        (raise-to-type highest-type arg)
                                                    )
                                                    args
                                           )
                            ))
                           (if (not (null? raised-args))
                               (apply apply-generic op raised-args)
                               (error "Error. No suitable raise method found to raise all types into " highest-type " for raising types " type-tags)
                           )
                       )
                    )     
              )
          )
      )
   )
 )
)  

(define *type-table* (make-hash))

(define (put-parent type parent)
   (let ((already-set (hash-ref *type-table* (list 'parent type) '())))
     (if (null? already-set)
         (hash-set! *type-table*  (list 'parent type) parent)
         (error "Parent already defined for :" type " parent currently *present* is " already-set " the passed parent " parent)
     )
   )
)

(define (get-parent type)
  (hash-ref *type-table* (list 'parent type) '())
)

(define (put-child type child)
   (let ((already-set (hash-ref *type-table* (list 'child type) '())))
     (if (null? already-set)
         (hash-set! *type-table*  (list 'child type) child)
         (error "Child already defined for :" type " child currently *present* is " already-set " the passed child " child)
     )
   )
)

(define (get-child type)
  (hash-ref *type-table* (list 'child type) '())
)


(define *cast-table* (make-hash))

(define (put-raise type proc)
   (let ((already-set (hash-ref *cast-table* (list 'raise type) '())))
     (if (null? already-set)
         (hash-set! *cast-table*  (list 'raise type) proc)
         (error "Raise is already defined for :" type)
     )
   )
)

(define (get-raise type)
  (hash-ref *cast-table* (list 'raise type) '())
)

(define (put-projection type proc)
   (let ((already-set (hash-ref *cast-table* (list 'project type) '())))
     (if (null? already-set)
         (hash-set! *cast-table*  (list 'project type) proc)
         (error "Project is already defined for :" type)
     )
   )
)

(define (get-projection type)
  (hash-ref *cast-table* (list 'project type) '())
)

(define (get-ancestors type)
  (if (null? type)
      '()
       (cons type (get-ancestors (get-parent type)))
  )
)

(define (contains item item-list)
      (fold-left
         (lambda(result new)
             (or result (equal? item new))
         )
         #f
         item-list
      )
)
     

(define (find-highest-type type-tags)
  (let ((ancestors-of-each-type (map get-ancestors type-tags)))
     (let ((smallest-ancestor-set
               (accumulate (lambda(new remaining)
                               (if (< (size new) (size remaining))
                                   new
                                   remaining
                               )
                           )
                           (car ancestors-of-each-type)
                           ancestors-of-each-type
               )
          ))
          (define (find-type-present-in-each-ancestors type-list)
              (if (null? type-list)
                  '()
                  (let ((found (accumulate
                                      (lambda (a b) (and a b))
                                      #t
                                      (map
                                          (lambda(ancestors) (contains (car type-list) ancestors))
                                          ancestors-of-each-type)
                               )
                        ))
                        (if found
                            (car type-list)
                            (find-type-present-in-each-ancestors (cdr type-list))
                        )
                  )
              )
          )
          (find-type-present-in-each-ancestors smallest-ancestor-set)
       )
    )
)

(define (raise-to-type type arg)
     (cond ((null? arg) '())
           ((equal? type (type-tag arg)) arg)
           (else (let ((proc (get-raise (type-tag arg))))
                   (if (null? proc)
                       '()
                       (raise-to-type type (proc arg))
                   )
                 )
           )
    )       
)

(define (map-until until-val proc args)
   (if (null? args)
       '()
       (let ((arg (car args)))
           (let ((marg (proc arg)))
             (if (equal? marg until-val)
                 '()
                 (cons marg (map-until until-val proc (cdr args)))
             )
           )
      )
  )
)

(define (fold-left op initial sequence)
  (define (iter result rest)
    (if (null? rest)
        result
        (iter (op result (car rest))
              (cdr rest))))
  (iter initial sequence))


(define (raise n)
  ((get-raise (type-tag n)) n)
)

;use mul as x can be rational-number also
(define (square x) (mul x x))

(define *op-table* (make-hash))

(define (put op type proc)
  (hash-set! *op-table* (list op type) proc)
)

(define (get op type)
  (hash-ref *op-table* (list op type) '())
)

(define (attach-tag type-tag contents) 
   (if (equal? 'scheme-number type-tag)
       contents
       (cons type-tag contents)
   )
)

(define (has-type-tag? datum) 
  (cond 
        ((number? datum) #t)
        ((pair? datum)
                (contains (car datum) '(int complex rational))
        )
        (else #f)
  )
)

(define (type-tag datum) 
  (cond 
        ((number? datum) 'scheme-number)
        ((pair? datum) (car datum))
        (else (error "Bad tagged datum -- TYPE-TAG" datum))
  )
)
  
(define (contents datum)
  (cond 
        ((pair? datum) (cdr datum)) 
        (error "Bad tagged datum -- CONTENTS" datum)
        )
  )

(define (filter predicate sequence)
  (cond ((null? sequence) nil)
        ((predicate (car sequence))
         (cons (car sequence)
               (filter predicate (cdr sequence))))
        (else (filter predicate (cdr sequence)))))

(define (size list)
  (if (null? list) 0 (+ 1 (size (cdr list)))))

(define (list-and proc list)
    (define (iter list)
        (if (null? list)
            #t
            (let ((rs (proc (car list))))
              (if (boolean? rs)
                  (if rs (iter (cdr list)) #f)
                  (error "Proc should return boolean")
              )
            )
        )
    )
  
    (cond ((null? list) (error "Operation not defined for 0 items"))
          ((not (pair? list)) (error "Operation works only for lists"))
          (else (iter list))
    )
)                              
(define (accumulate op initial sequence)
  (if (null? sequence)
      initial
      (op (car sequence)
          (accumulate op initial (cdr sequence)))
  )
)


;;;;;;;;;;;;;;;;;;;;;;;;

(define (install-int-package)
  (define (tag x)
    (attach-tag 'int x))
  (put 'add '(int int)
       (lambda (x y) (tag (+ x y))))
  (put 'sub '(int int)
       (lambda (x y) (tag (- x y))))
  (put 'mul '(int int)
       (lambda (x y) (tag (* x y))))
  (put 'div '(int int)
       (lambda (x y) (tag (/ x y))))
  (put 'make 'int
       (lambda (x) (tag x)))  
  ;equ?
  (put 'equ? '(int int) =)
  (put 'negate '(int) -)
  
  ; raise operation
  (put-raise 'int (lambda(x) (make-rational (contents x) 1)))
  ; set parent
  (put-parent 'int 'rational)  
 
  (put '=zero? '(int) (lambda (x) (= x 0)))

  'done
)

(define (make-integer n)
  ((get 'make 'int) n)
)



;;;;;;;;;;;;;;;;;;;;;;;;
(define (install-scheme-number-package)
  (define (tag x)
    (attach-tag 'scheme-number x))
  (put 'add '(scheme-number scheme-number)
       (lambda (x y) (tag (+ x y))))
  (put 'sub '(scheme-number scheme-number)
       (lambda (x y) (tag (- x y))))
  (put 'mul '(scheme-number scheme-number)
       (lambda (x y) (tag (* x y))))
  (put 'div '(scheme-number scheme-number)
       (lambda (x y) (tag (/ x y))))
  (put 'make 'scheme-number
       (lambda (x) (tag x)))  
  ;equ?
  (put 'equ? '(scheme-number scheme-number) =)
  (put 'negate '(scheme-number) -)
  
  ;; following added to Scheme-number package
  (put 'exp '(scheme-number scheme-number)
       (lambda (x y) (tag (expt x y)))) ; using primitive expt

  ; raise operation
  (put-raise 'scheme-number (lambda(x) (make-complex-from-real-imag (contents x) 0)))
  ; set parent
  (put-parent 'scheme-number 'complex)  

  ;a simple way to convert from real to rational is by multiplying and dividing by a
  ; constant but it will only be correct to 4 places of decimal(4 zeroes in 10000)
  (define multiplier 10000)
  ; project operation
  (put-projection 'scheme-number (lambda(c) (make-rational (floor (* (contents c) multiplier)) multiplier)))
  ; set child
  (put-child 'scheme-number 'rational)
 
  (put '=zero? '(scheme-number) (lambda (x) (= x 0)))

  'done
)

(define (make-number n)
  ((get 'make 'scheme-number) n)
)  

(define (install-rational-package)
  ;; internal procedures
  (define (numer x) (car x))
  (define (denom x) (cdr x))
  (define (make-rat n d)
    (if (and (integer? n) (integer? d))
        (let ((nn (inexact->exact n))
              (dd (inexact->exact d))
             )
             (let ((g (gcd nn dd)))
                 (cons (/ nn g) (/ dd g))
             )
        )
        (error "Number and Denom shoud be integers" n d)
     )
  )
  (define (add-rat x y)
    (make-rat (+ (* (numer x) (denom y))
                 (* (numer y) (denom x)))
              (* (denom x) (denom y))))
  (define (sub-rat x y)
    (make-rat (- (* (numer x) (denom y))
                 (* (numer y) (denom x)))
              (* (denom x) (denom y))))
  (define (mul-rat x y)
    (make-rat (* (numer x) (numer y))
              (* (denom x) (denom y))))
  (define (div-rat x y)
    (make-rat (* (numer x) (denom y))
              (* (denom x) (numer y))))

  ;; equ?
  (define (equ? x y) 
    (and (= (numer x) (numer y)) (= (denom x) (denom y)))) 
     
  ;; interface to rest of the system
  (define (tag x) (attach-tag 'rational x))
  (put 'add '(rational rational)
       (lambda (x y) (tag (add-rat x y))))
  ; comment this to check the example suggested above for tree structure support
  (put 'sub '(rational rational)
       (lambda (x y) (tag (sub-rat x y))))
  (put 'mul '(rational rational)
       (lambda (x y) (tag (mul-rat x y))))
  (put 'div '(rational rational)
       (lambda (x y) (tag (div-rat x y))))
  (put 'make 'rational
       (lambda (n d) (tag (make-rat n d))))
  (put 'equ? '(rational rational) equ?)
  (put 'negate '(rational) (lambda (r) (make-rat (- (numer r)) (denom r))))

  ; raise operation
  (put-raise 'rational (lambda(r) (make-number (/ (numer (contents r)) (denom (contents r))))))
  ; set parent
  (put-parent 'rational 'scheme-number)

  ; project operation
  (put-projection 'rational (lambda(r) (make-integer (floor (/ (numer (contents r)) (denom (contents r)))))))
  ; set child
  (put-child 'rational 'int)
  
  ; uncomment this and comment above raise to make rational as child of complex instead of number
  ; (put-raise 'rational (lambda(r) (make-complex-from-real-imag (/ (numer (contents r)) (denom (contents r))) 0)))
  ; uncomment this and comment above raise to make rational as child of complex instead of number
  ; (put-parent 'rational 'complex)

  (put '=zero? '(rational) (lambda (x) (= (numer x) 0)))
  
  'done
  )

(define (make-rational n d)
  ((get 'make 'rational) n d)
  )

(define (install-complex-package)
  ;; imported procedures from rectangular and polar packages
  (define (make-from-real-imag x y)
    ((get 'make-from-real-imag 'rectangular) x y))
  (define (make-from-mag-ang r a)
    ((get 'make-from-mag-ang 'polar) r a))
  ;; internal procedures
  (define (add-complex z1 z2)
    (make-from-real-imag (add (apply-generic 'real-part z1) (apply-generic 'real-part z2))
                         (add (apply-generic 'imag-part z1) (apply-generic 'imag-part z2))))
  (define (sub-complex z1 z2)
    (make-from-real-imag (sub (apply-generic 'real-part z1) (apply-generic 'real-part z2))
                         (sub (apply-generic 'imag-part z1) (apply-generic 'imag-part z2))))
  (define (mul-complex z1 z2)
    (make-from-mag-ang (mul (apply-generic 'magnitude z1) (apply-generic 'magnitude z2))
                       (add (apply-generic 'angle z1) (apply-generic 'angle z2))))
  (define (div-complex z1 z2)
    (make-from-mag-ang (div (apply-generic 'magnitude z1) (apply-generic 'magnitude z2))
                       (sub (apply-generic 'angle z1) (apply-generic 'angle z2))))
  (define (equ? z1 z2) 
    (and
     (apply-generic 'equ? (apply-generic 'real-part z1) (apply-generic 'real-part z2))
     (apply-generic 'equ? (apply-generic 'imag-part z1) (apply-generic 'imag-part z2))
     )
  )

  (define (complex-negate z)
      (make-from-real-imag
           (apply-generic 'negate (apply-generic 'real-part z))
           (apply-generic 'negate (apply-generic 'real-part z))
      )
  )


  ;; interface to rest of the system
  (define (tag z) (attach-tag 'complex z))
  (put 'add '(complex complex)
       (lambda (z1 z2) (tag (add-complex z1 z2))))
  (put 'sub '(complex complex)
       (lambda (z1 z2) (tag (sub-complex z1 z2))))
  (put 'mul '(complex complex)
       (lambda (z1 z2) (tag (mul-complex z1 z2))))
  (put 'div '(complex complex)
       (lambda (z1 z2) (tag (div-complex z1 z2))))
  (put 'make-from-real-imag 'complex
       (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'complex
       (lambda (r a) (tag (make-from-mag-ang r a))))
  (put 'equ? '(complex complex) equ?)
  (put `nagate '(complex) complex-negate)

  ; raise operation is not defined for complex
  ; parent not defined for complex

  ; project operation
  (put-projection
          'complex
           (lambda(c)
               ;note that we can not call apply-generic here because this method is used by drop and drop is used in generic
               ;thus it can go into a loop
               ;so need to take care of type-tags here only
               ; c is complex number(includes tag complex) cc is rectangular/angular including the type-tag
               ; Note that, the repurcurssions of drop lead to the point that real and imag can be any number except number
               ; thus raising it to the real number type (because apart from complex numbers, my understanding is any number can be raised to real number
               (let ((cc (contents c)))
                     (raise-to-type 'scheme-number ((get 'real-part (list (type-tag cc))) (contents cc)))
               )
           )
  )
  ; set child
  (put-child 'complex 'scheme-number)

   ;; complex package:
   (define (=is-zero? z) 
       (and
           (= (apply-generic 'real-part z) 0)
           (= (apply-generic 'imag-part z) 0)
       )
   )
  (put '=zero? '(complex) =is-zero?)
  
  'done
)

(define (sqrt x)
  ;note the magic of apply-generic
  ;it will convert that rational into real because expt is only defined in
  ;scheme-number package and rational can be raised to scheme-number
  (exp x (make-rational 1 2))
)
        

(define (install-rectangular-package)
  ;; internal procedures
  (define (real-part z) (car z))
  (define (imag-part z) (cdr z))
  (define (make-from-real-imag x y) (cons x y))
  (define (magnitude z)
    (sqrt (add (square (real-part z))
             (square (imag-part z)))))
  (define (angle z)
    (arctan (div (imag-part z) (real-part z))))
  (define (make-from-mag-ang r a)
    (cons (mul r (cosine a)) (mul r (sine a))))
  ;; interface to the rest of the system
  (define (tag x) (attach-tag 'rectangular x))
  (put 'real-part '(rectangular) real-part)
  (put 'imag-part '(rectangular) imag-part)
  (put 'magnitude '(rectangular) magnitude)
  (put 'angle '(rectangular) angle)
  (put 'make-from-real-imag 'rectangular
       (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'rectangular
       (lambda (r a) (tag (make-from-mag-ang r a))))
  'done)
 
(define (install-polar-package)
  ;; internal procedures
  (define (magnitude z) (car z))
 
  (define (angle z) (cdr z))
  (define (make-from-mag-ang r a) (cons r a))
  (define (real-part z)
    (mul (magnitude z) (cosine (angle z))))
  (define (imag-part z)
    (mul (magnitude z) (sine (angle z))))
  (define (make-from-real-imag x y)
    (cons (sqrt (add (square x) (square y)))
          (arctan (div y x))))
  ;; interface to the rest of the system
  (define (tag x) (attach-tag 'polar x))
  (put 'real-part '(polar) real-part)
  (put 'imag-part '(polar) imag-part)
  (put 'magnitude '(polar) magnitude)
  (put 'angle '(polar) angle)
  (put 'make-from-real-imag 'polar
       (lambda (x y) (tag (make-from-real-imag x y))))
  (put 'make-from-mag-ang 'polar
       (lambda (r a) (tag (make-from-mag-ang r a))))
  'done)

(define (make-complex-from-real-imag x y)
  ((get 'make-from-real-imag 'complex) x y))
(define (make-complex-from-mag-ang r a)
  ((get 'make-from-mag-ang 'complex) r a))

(define (equ? x y)
  (apply-generic 'equ? x y)
)

(define (exp x y) (apply-generic 'exp x y))
(define (add x y) (apply-generic 'add x y))
(define (sub x y) (apply-generic 'sub x y))
(define (mul x y) (apply-generic 'mul x y))
(define (div x y) (apply-generic 'div x y))

(define (=zero? x) (apply-generic '=zero? x))
(define (negate x) (apply-generic 'negate x))


(install-rectangular-package)
(install-polar-package)
(install-scheme-number-package)
(install-rational-package)
(install-int-package)
(install-complex-package)
(install-sparse-termlist-package)
(install-dense-termlist-package)
(install-polynomial-package)

Test/Sample Output:

Note that - my integer representation adds tag - integer and drop is dropping to int when ‘add is performed. Thus (int 5) actually means 5.

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> (display (add (make-polynomial 'x (attach-tag 'sparse-termlist '((5 2) (3 2)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 3) (3 6))))))
(polynomial x sparse-termlist (5 (int . 5)) (3 (int . 8)))
> (display (add (make-polynomial 'x (attach-tag 'dense-termlist '(1 2 3))) (make-polynomial 'x (attach-tag 'dense-termlist '(4 5 6)))))
(polynomial x dense-termlist (int . 5) (int . 7) (int . 9))
> (display (add (make-polynomial 'x (attach-tag 'dense-termlist '(1 2 3))) (make-polynomial 'x (attach-tag 'sparse-termlist '((2 4) (1 5) (0 6))))))
(polynomial x sparse-termlist (2 (int . 5)) (1 (int . 7)) (0 (int . 9)))
> (display (sub (make-polynomial 'x (attach-tag 'dense-termlist '(1 2 3))) (make-polynomial 'x (attach-tag 'dense-termlist '(4 5 6)))))
(polynomial x dense-termlist (int . -3) (int . -3) (int . -3))
> (display (sub (make-polynomial 'x (attach-tag 'dense-termlist '(1 2 3))) (make-polynomial 'x (attach-tag 'sparse-termlist '((2 4) (1 5) (0 6))))))
(polynomial x sparse-termlist (2 (int . -3)) (1 (int . -3)) (0 (int . -3)))
> (display (add (make-polynomial 'x (attach-tag 'sparse-termlist '((5 2) (3 2)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((5 3) (3 6))))))
(polynomial x sparse-termlist (5 (int . 5)) (3 (int . 8)))
> (display (mul (make-polynomial 'x (attach-tag 'sparse-termlist '((1 1) (0 1)))) (make-polynomial 'x (attach-tag 'sparse-termlist '((1 1) (0 1))))))
(polynomial x sparse-termlist (2 (int . 1)) (1 (int . 2)) (0 (int . 1)))
> (display (mul (make-polynomial 'x (attach-tag 'dense-termlist '(1 1))) (make-polynomial 'x (attach-tag 'dense-termlist '(1 1)))))
(polynomial x dense-termlist (int . 1) (int . 2) (int . 1))
> (display (mul (make-polynomial 'x (attach-tag 'dense-termlist '(1 1))) (make-polynomial 'x (attach-tag 'sparse-termlist '((1 1) (0 1))))))
(polynomial x sparse-termlist (2 (int . 1)) (1 (int . 2)) (0 (int . 1)))