### Chapter 2, Building Abstractions with Data

### Section - 2.5 - Systems with Generic Operations

#### Exercise 2.86

Well, it turned out I missed few bugs in 2.85(which I have not fixed as it is part of this exercise). This exercise is in a way a part of previous exercise. As to complete that exercise complex number can have rational parts too, which was required because of the problems encountered when adding `drop`

in that exercise.

Now, in this exercise this was asked explicitly to allow complex numbers with real and imaginary parts. I thought *bingo!*, I have already done it but the part that made me realised - I had left few issues - is that it explicitly asked for operation *sin*, *cos*. Soon I figured that I have not tested my last exercise with operations `div`

and `mul`

and used only the rectangular implementation of complex numbers. Thus in my test cases - sin and cos never came up!

Now, its just a matter of changing all operations used in rectangular and angular so that they can work fine.

Note that I have not defined `sine`

, `cosine`

and `arctan`

by apply-generic because it can be done without it - just by raising the argument passed to `scheme-number`

.

Also, I discovered one more bug in my code - again which should have been discovered earlier but because scheme is *not* static-typed - My implementation of `angle`

was wrong as I *forgot* to call div and just passed two arguments - which a static language should have caught as I was passing two arguments where only one is expected. I have left that bug in the previous exercise.

So here goes the complete code:

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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))
(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 (number? contents)
contents
(cons type-tag contents)
)
)
(define (has-type-tag? datum)
(cond
((number? datum) #t)
((pair? datum)
(contains (car datum) '(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 ((number? datum) datum)
((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 (accumulate op initial sequence)
(if (null? sequence)
initial
(op (car sequence)
(accumulate op initial (cdr sequence)))
)
)
(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) =)
;; 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)
'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)
(let ((g (gcd n d)))
(cons (/ n g) (/ d g))))
(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?)
; raise operation
(put-raise 'rational (lambda(r) (make-number (/ (numer (contents r)) (denom (contents r))))))
; set parent
(put-parent 'rational 'scheme-number)
; 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)
'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))
)
)
;; 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?)
; 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)
'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))
(install-rectangular-package)
(install-polar-package)
(install-scheme-number-package)
(install-rational-package)
(install-complex-package)

Examples/Output:

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> (display (mul (make-complex-from-real-imag 2 3) (make-complex-from-real-imag 3 4)))
(complex polar 18.027756377319946 . 1.9100889412489412)
> (display (mul (make-complex-from-real-imag 2.5 3) (make-complex-from-real-imag 3 4)))
(complex polar 19.525624189766635 . 1.8033532685998055)
> (display (mul (make-complex-from-real-imag (make-rational 5 2) 3) (make-complex-from-real-imag 3 4)))
(complex polar 19.525624189766635 . 1.8033532685998055)
> (display (add (make-complex-from-real-imag 2.5 -9) (make-complex-from-real-imag 1 9)))
(rational 7.0 . 2.0)
; add 0 converts to rectangular
> (display (add (make-number 0) (mul (make-complex-from-real-imag (make-rational 5 2) 3) (make-complex-from-real-imag 3 4))))
(complex rectangular -4.4999999999999964 rational 19.0 . 1.0)
> (display (add (make-rational 3 4) (mul (make-complex-from-real-imag (make-rational 5 2) 3) (make-complex-from-real-imag 3 4))))
(complex rectangular -3.7499999999999964 rational 19.0 . 1.0)