SICP_2.56-2.57

#lang racket

;;;;;;;;;;;;;;;只需先写出按愿望和公式而构造的函数
;;;;;;;;;;;;;;;再加上一些特定的选择函数和构造函数就可以实现这么强大的功能

;;;;;;;;;;;;;;;quote 和 list 的区别
(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) (- (exponent exp) 1)))
                       (deriv (base exp) var)))
        (else
         (error "unknow expression type --DERIV" exp))))

(define (variable? x)
  (symbol? x))

(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))

(define (=number? exp num)
  (and (number? exp) (= exp num)))

(define (** a b)
  (define (iter n result)
      (cond ((= 0 b) 1)
          ((or (< n b) (= n b)) (iter (+ n 1) (* a result)))
          ((> n b) result)))
  (iter 1 1))

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

;(define (addend s) (cadr s))

;(define (augend s) (caddr s))

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

;(define (multiplier p) (cadr p))

;(define (multiplicand p) (caddr p))

(define (exponentiation? x)
  (and (pair? x) (eq? (car x) '**)))

(define (base e) (cadr e))

(define (exponent e) (caddr e))

(define (make-exponentiation b1 b2)
  (cond ((=number? b2 0) 1)
        ((=number? b2 1) b1)
        ((and (number? b1) (number? b2)) (** b1 b2))
        (else (list '** b1 b2))))

(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 (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))))

;;;;;;;;;;;;test
;(deriv '(* (* x y) (+ x 3)) 'x)
;(deriv '(** (+ x y) 4) 'x)
;(deriv '(+ (* x y) x) 'x)

;;;;;;;;;;;;;2.58
(define (addend s)
  (cadr s))

(define (augend s)
  (if (null? (cdddr s))
      (caddr s)
      (cons '+ (cddr s))))

(define (multiplier p)
  (cadr p))

(define (multiplicand p)
  (if (null? (cdddr p))
      (caddr p)
      (cons '* (cddr p))))

;;;;;;;;;;;;test
(deriv '(+ (* x 4) (+ x x) (* x x)) 'x)
(deriv '(* (+ x 4) (* x 4) y (** x 3)) 'y)

1.这几题进一步强化了按愿望实现函数的方法

2.quoto 和 list 的区别  参考

（1）' 是quoto 的缩写

（2）' 后的内容会原样返回，因为 ' 就是个省略了的单引号 

（= '3 3）    ->#t

（car  '（list a b））   ->list    (和sicp_2.55同理)

（3）list 是个函数

（list （+ 1 3） （+ 2 5）） -> (4 7)