計算機程序的構造和解釋 練習題2.58

(a)小題

#lang racket

(define (deriv exp var )
  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
                   (deriv (augend exp) var)))
        ((product? exp)
         (make-sum
          (make-product (multiplier exp)
                        (deriv (multiplicand exp) var))
          (make-product (deriv (multiplier exp) var)
                        (multiplicand exp))))
        ((exponentiation? exp)
         (make-product
          (make-product (exponent exp)
                        (make-exponentiation (base exp)
                                             (make-sum (exponent exp) -1)))
          (deriv (base exp) var)))
        (else
         (error "unknown expression type -- DERIV" exp))))
               
(define (variable? x) (symbol? x))
(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))
(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list '+ a1 a2))))
(define (=number? exp num)
  (and (number? exp) (= exp num)))

(define (make-product m1 m2)
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list '* m1 m2))))

(define (sum? x)
  (and (pair? x) (eq? (cadr x) '+)))
(define (addend s) (car s))
(define (augend s) (caddr s))
(define (product? x)
  (and (pair? x) (eq? (cadr x) '*)))
(define (multiplier p) (car p))
(define (multiplicand p) (caddr p))
(define (exp x n)
  (if (= n 0) 1
      (* x (exp x (- n 1)))))
(define (make-exponentiation e1 e2)
  (cond ((=number? e2 0) 1)
        ((=number? e2 1) e1)
        ((and (number? e1) (number? e2)) (exp e1 e2))
        (else (list '** e1 e2))))
(define (exponentiation? x)
  (and (pair? x) (eq? (cadr x) '**)))
(define (base s) (car s))
(define (exponent s) (caddr s))

(deriv '(x + (3 * (x + (y + 2)))) 'x)
(deriv '(x + (3 * ((x ** y) + (y + 2)))) 'x)

運行結果

4
'(+ 1 (* 3 (* y (** x (+ y -1)))))

(b)小題，這道題比較難的一點是優先級的部分，首先我們可以考慮按照優先級高低順序來劃分表達式，比如表達式xa+xb，就可以先拿加號劃分成xa和xb，再各自用乘法進行求導計算，這樣就能保證優先級。這樣就需要修改判斷sum？的方法，和兩個選擇過程，只要表達式裏面有加號，就要先進行加法求導，然後再進行乘法的求導運算，最後再進行求冪的算。

#lang racket

(define (deriv exp var )
  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
                   (deriv (augend exp) var)))
        ((product? exp)
         (make-sum
          (make-product (multiplier exp)
                        (deriv (multiplicand exp) var))
          (make-product (deriv (multiplier exp) var)
                        (multiplicand exp))))
        ((exponentiation? exp)
         (make-product
          (make-product (exponent exp)
                        (make-exponentiation (base exp)
                                             (make-sum (exponent exp) -1)))
          (deriv (base exp) var)))
        (else
         (error "unknown expression type -- DERIV" exp))))

 (define (operator o)
  (cond ((memq '+ o) '+)
        ((memq '* o) '*)
        ((memq '** o) '**)
        (else 'unknown)))

(define (variable? x) (symbol? x))
(define (same-variable? v1 v2) 
  (and (variable? v1) (variable? v2) (eq? v1 v2)))
(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list '+ a1 a2))))
(define (=number? exp num)
  (and (number? exp) (= exp num)))

(define (make-product m1 m2)
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list '* m1 m2))))

(define (sum? x)
  (and (pair? x) (eq? (operator x) '+)))

(define (addend s)
  (define (iter x)
    (cond ((null? x) null)
          ((eq? (car x) '+) null)
          (else (append (list (car x)) (iter (cdr x))))))
  (let ((result (iter s)))
    (if (= 1 (length result)) (car result) result)))

(define (augend s)
  (let ((result (cdr (memq '+ s))))
   (if (= 1 (length result)) (car result) result)))

(define (product? x)
  (and (pair? x) (eq? (operator x) '*)))

(define (multiplier p)
  (define (iter x)
    (cond ((null? x) null)
          ((eq? (car x) '*) null)
          (else (append (list (car x)) (iter (cdr x))))))
  (let ((result (iter p)))
    (if (= 1 (length result)) (car result) result)))
(define (multiplicand p)
  (let ((result (cdr (memq '* p))))
   (if (= 1 (length result)) (car result) result)))

(define (exp x n)
  (if (= n 0) 1
      (* x (exp x (- n 1)))))
(define (make-exponentiation e1 e2)
  (cond ((=number? e2 0) 1)
        ((=number? e2 1) e1)
        ((and (number? e1) (number? e2)) (exp e1 e2))
        (else (list '** e1 e2))))
(define (exponentiation? x)
  (and (pair? x) (eq? (operator x) '**)))

(define (base p)
  (define (iter x)
    (cond ((null? x) null)
          ((eq? (car x) '**) null)
          (else (append (list (car x)) (iter (cdr x))))))
  (let ((result (iter p)))
    (if (= 1 (length result)) (car result) result)))

(define (exponent p)
    (let ((result (cdr (memq '** p))))
   (if (= 1 (length result)) (car result) result)))
   
(deriv '(x + (3 * (x + (y + 2)))) 'x)
(deriv '(x + 3 * (x + y + 2)) 'x)
(deriv '(x + x ** 4 * 5 + 3 * (x + y + 2)) 'x)

運行結果（運行結果暫時沒寫一個簡化函數，可以簡化顯示結果）

4
4
'(+ 1 (+ (* (* 4 (** x 3)) 5) 3))