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Exercise 2.86.scm
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Exercise 2.86.scm
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; Exercise 2.85: This section mentioned a method for “simplifying” a data object by lowering it in the tower of types as far as possible. Design a procedure drop that accomplishes this for the tower described in Exercise 2.83. The key is to decide, in some general way, whether an object can be lowered. For example, the complex number 1.5+0i1.5+0i can be lowered as far as real, the complex number 1+0i1+0i can be lowered as far as integer, and the complex number 2+3i2+3i cannot be lowered at all. Here is a plan for determining whether an object can be lowered: Begin by defining a generic operation project that “pushes” an object down in the tower. For example, projecting a complex number would involve throwing away the imaginary part. Then a number can be dropped if, when we project it and raise the result back to the type we started with, we end up with something equal to what we started with. Show how to implement this idea in detail, by writing a drop procedure that drops an object as far as possible. You will need to design the various projection operations119 and install project as a generic operation in the system. You will also need to make use of a generic equality predicate, such as described in Exercise 2.79. Finally, use drop to rewrite apply-generic from Exercise 2.84 so that it “simplifies” its answers.
; Exercise 2.79: Define a generic equality predicate equ? that tests the equality of two numbers, and install it in the generic arithmetic package. This operation should work for ordinary numbers, rational numbers, and complex numbers.
(define (make-table)
(let ((local-table (list '*table*)))
(define (lookup key-1 key-2)
(let ((subtable (assoc key-1 (cdr local-table))))
(if subtable
(let ((record (assoc key-2 (cdr subtable))))
(if record
(cdr record)
false))
false)))
(define (insert! key-1 key-2 value)
(let ((subtable (assoc key-1 (cdr local-table))))
(if subtable
(let ((record (assoc key-2 (cdr subtable))))
(if record
(set-cdr! record value)
(set-cdr! subtable
(cons (cons key-2 value)
(cdr subtable)))))
(set-cdr! local-table
(cons (list key-1
(cons key-2 value))
(cdr local-table)))))
'ok)
(define (dispatch m)
(cond ((eq? m 'lookup-proc) lookup)
((eq? m 'insert-proc!) insert!)
(else (error "Unknown operation -- TABLE" m))))
dispatch))
(define operation-table (make-table))
(define get (operation-table 'lookup-proc))
(define put (operation-table 'insert-proc!))
(define coercion-table (make-table))
(define get-coercion (coercion-table 'lookup-proc))
(define put-coercion (coercion-table 'insert-proc!))
(define level-table (make-table))
(define get-level (level-table 'lookup-proc))
(define put-level (level-table 'insert-proc!))
(define (attach-tag type-tag contents)
(cons type-tag contents))
(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))
(else (error "Bad tagged datum: CONTENTS" datum))
)
)
(define (install-integer-package)
(define (tag x)
(cons 'integer x))
(put '=zero? '(integer)
(lambda (x) (tag (= x 0))))
(put 'value '(integer)
(lambda (x) x)
)
(put 'equ? '(integer integer)
(lambda (x y) (tag (round (= x y)))))
(put 'add '(integer integer)
(lambda (x y) (tag (round (+ x y)))))
(put 'sub '(integer integer)
(lambda (x y) (tag (round (- x y)))))
(put 'mul '(integer integer)
(lambda (x y) (tag (round (* x y)))))
(put 'div '(integer integer)
(lambda (x y) (tag (round (/ x y)))))
(put 'make 'integer
(lambda (x) (tag (round x))))
'integer-package-done)
(install-integer-package)
(define (install-scheme-number-package)
(define (tag x)
x)
(put '=zero? '(scheme-number)
(lambda (x) (tag (= x 0))))
(put 'equ? '(scheme-number scheme-number)
(lambda (x y) (tag (= x y))))
(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)))
'scheme-number-package-done)
(install-scheme-number-package)
(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))))
(define (equ? x y)
(if (and (= (numer x) (numer y)) (= (denom x) (denom y)))
true
false
)
)
(define (rational_zero? x)
(= 0 (numer x))
)
;; interface to rest of the system
(define (tag x) (attach-tag 'rational x))
(put '=zero? '(rational) rational_zero?)
(put 'equ? '(rational rational)
equ?
)
(put 'add '(rational rational)
(lambda (x y) (tag (add-rat x y))))
(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 'numer '(rational) numer)
(put 'denom '(rational) denom)
'rational-package-done)
(install-rational-package)
(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)
(atan (raiseto_to_scheme (imag-part z)) (raiseto_to_scheme (real-part z))))
(define (make-from-mag-ang r a)
(cons (raiseto_to_scheme (mul r (cos a))) (raiseto_to_scheme (mul r (sin 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))))
'rectangular-package-done)
(install-rectangular-package)
(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) (cos (angle z))))
(define (imag-part z)
(mul (magnitude z) (sin (angle z))))
(define (make-from-real-imag x y)
(cons (sqrt (add (square x) (square y)))
(atan (raiseto_to_scheme y) (raiseto_to_scheme 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))))
'polar-package-done)
(install-polar-package)
(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 (real-part z1) (real-part z2))
(add (imag-part z1) (imag-part z2))))
(define (sub-complex z1 z2)
(make-from-real-imag
(sub (real-part z1) (real-part z2))
(sub (imag-part z1) (imag-part z2))))
(define (mul-complex z1 z2)
(make-from-mag-ang
(mul (magnitude z1) (magnitude z2))
(add (angle z1) (angle z2))))
(define (div-complex z1 z2)
(make-from-mag-ang
(div (magnitude z1) (magnitude z2))
(sub (angle z1) (angle z2))))
(define (equ? z1 z2)
(if (and (equ? (real-part z1) (real-part z2)) (equ? (imag-part z1) (imag-part z2)))
true
false
)
)
;; interface to rest of the system
(define (tag z) (attach-tag 'complex z))
(put '=zero? '(complex)
(lambda (z1)
(and (= (real-part z1) 0) (= (imag-part z1) 0))))
(put 'equ? '(complex complex)
equ?)
(put 'real-part '(complex)
(lambda (x) (real-part x))
)
(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))))
'complex-package-done)
(install-complex-package)
(define (install-zero?-package)
(define (ordinary_zero? x)
(= x 0)
)
(define (complex_zero? x)
(= x 0)
)
(put 'zero '(scheme-number) ordinary_zero?)
(put 'zero '(complex) complex_zero?)
)
(define (apply-generic op . args)
(let ((type-tags (map type-tag args)))
(let ((proc (get op type-tags)))
(if proc
(apply proc (map contents args))
(error
"No method for these types:
APPLY-GENERIC"
(list op type-tags))))))
(define (add . args) (apply apply-generic (cons 'add args)))
(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 (equ? x y) (apply-generic 'equ? x y))
(define (make-integer n)
((get 'make 'integer) n))
(define (make-scheme-number n)
((get 'make 'scheme-number) n))
(define (make-rational n d)
((get 'make 'rational) n d))
(define (value n)
(apply-generic 'value n)
)
(define (numer z)
(apply-generic 'numer z))
(define (denom z)
(apply-generic 'denom z))
(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 (real-part z)
(apply-generic 'real-part z))
(define (imag-part z)
(apply-generic 'imag-part z))
(define (magnitude z)
(apply-generic 'magnitude z))
(define (angle z)
(apply-generic 'angle z))
(define (=zero? x)
(apply-generic '=zero? x))
(define (equ? x y) (apply-generic 'equ? x y))
(define (apply-generic op . args)
(let ((type-tags (map type-tag args)))
(let ((proc (get op type-tags)))
(if proc
(let ((result (apply proc (map contents args))))
(if (or
(eq? op 'mul)
(eq? op 'div)
(eq? op 'add)
(eq? op 'sub))
;for multi respresentation of complex
result
result)
)
(cond
((> (length args) 2)
(apply apply-generic
(cons op
(list (car args)
(apply apply-generic
(cons op
(cdr args)))) )))
((= (length args) 2)
(let ((type1 (car type-tags))
(type2 (cadr type-tags))
(a1 (car args))
(a2 (cadr args)))
(let ((level1 (get-level type1 'level))
(level2 (get-level type2 'level)))
(cond
((< level1 level2)
(apply-generic
op (raise a1) a2))
((> level1 level2)
(apply-generic
op a1 (raise a2)))
(else
(error
"No method for
these types"
(list
op
type-tags)))))))
(else (error
"No method for these types"
(list op type-tags))))))))
(define (install_transform_package)
; raise
(define integer->rational
(lambda (x) (make-rational (value x) 1))
)
(define rational->scheme-number
(lambda (x) (make-scheme-number (/ (numer x) (denom x))))
)
(define scheme-number->complex
(lambda (x) (make-complex-from-real-imag x 0))
)
; project
(define complex->scheme-number
(lambda (x) (make-scheme-number (real-part x)))
)
(define scheme-number->rational
(lambda (x) (make-rational (round x) 1))
)
(define rational->integer
(lambda (x) (make-integer (rational->scheme-number x)))
)
; raise
(put-coercion 'integer 'raise
integer->rational)
(put-coercion 'scheme-number 'raise
scheme-number->complex)
(put-coercion 'rational 'raise
rational->scheme-number)
; project
(put-coercion 'complex 'project complex->scheme-number)
(put-coercion 'scheme-number 'project scheme-number->rational)
(put-coercion 'rational 'project rational->integer)
(put-level 'integer 'level 1)
(put-level 'rational 'level 2)
(put-level 'scheme-number 'level 3)
(put-level 'complex 'level 4)
'install_transform_done)
(install_transform_package)
(define (project x)
(let ((lower (get-coercion (type-tag x) 'project)))
(if lower
(lower x)
false)))
(define (drop x)
(let ((low (project x)))
(if (and low (equ? (raise low) x))
(drop (project x))
x
)
)
)
(define (raise x)
(let ((raise (get-coercion (type-tag x) 'raise)))
(if raise
(raise x)
(error
"No raise for this types"
(type-tag x)))))
(define (raiseto_to_scheme x)
(if (eq? (type-tag x) 'scheme-number)
x
(raiseto_to_scheme (raise x))
)
)
; (define (drop x)
(define (square x) (mul x x))
(define (sqrt x)
(define (iter guess)
(let (
(good? (lambda (g) (< (abs (raiseto_to_scheme (- (square guess) g))) 0.001)))
(improve (lambda (g) (div (add g (div x g)) 2)))
)
(if (good? guess)
guess
(iter (improve guess))
)
)
)
(iter 1)
)
; )
(define c (make-rational 0 3))
(define d (make-rational 1 3))
(define e (make-integer 3.5))
(define f (make-integer 4))
(define a (make-complex-from-real-imag f e))
(define b (make-complex-from-real-imag c d))
; (raise a)
; (raise b)
(display (add a b))(newline)
(display (sub a b))(newline)
(display (mul a b))(newline)
(display (div a b))(newline)
(display (mul (add a b) (div a b)))(newline)
; Welcome to DrRacket, version 6.7 [3m].
; Language: SICP (PLaneT 1.18); memory limit: 128 MB.
; 'integer-package-done
; 'scheme-number-package-done
; 'rational-package-done
; 'rectangular-package-done
; 'polar-package-done
; 'complex-package-done
; 'install_transform_done
; (complex rectangular (rational 4 . 1) rational 13.0 . 3.0)
; (complex rectangular (rational 4 . 1) rational 11.0 . 3.0)
; (complex polar 1 . 2.356194490192345)
; (complex polar 1 . -0.7853981633974483)
; (complex polar 1 . 0.03997868712329)
; >