ex 2.56, 2.57 記号微分

2.58はまだ手を付けてない。
自分には時間がかかりそうなので、夜にじっくりとやる。

本文のコードから

;; 記号微分
(define (variable? e)
  (symbol? e))
(define (same-variable? v1 v2)
  (and (variable? v1) (variable? v2) (eq? v1 v2)))
(define (sum? e)
  (and (pair? e) (eq? (car e) '+)))
(define (addend e)
  (cadr e))
(define (augend e)
  (caddr e))
(define (=number? exp num)
  (and (number? exp) (= exp num)))
(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 (product? e)
  (and (pair? e) (eq? (car e) '*)))
(define (multiplier e)
  (cadr e))
(define (multiplicand e)
  (caddr e))
(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))))

;; 微分規則
(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)))
          (make-product (deriv (maltiplier exp) var)
                        (multipricand exp))))
        (else
         (error "unknown expression type --DERIV" exp))))

以下問題。

;; sicp ex 2.56
;; exponentiation は (** a b)の形で表される
(define (exponentiation? e)
  (and (pair? e) (eq? (car e) '**)))
(define (base e)
  (cadr e))
(define (exponent e)
  (caddr e))
(define (make-exponentiation a1 a2)
  (cond ((=number? a1 0) 0)
        ((=number? a2 0) 1)
        ((and (number? a1) (number a2)) (expr a1 a2))
        (else (list '** a1 a2))))
;; exponentiationの節を追加
(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 "unknown expression type --DERIV" exp))))

;; sicp ex 2.57
;; addend, multiplierは変更なし
(define (addend e)
  (cadr e))
(define (multiplier e)
  (cadr e))
;; 新たに (+ ...), (* ...)のリストを作る
(define (augend e)
  (let ((rest (cddr e)))
    (if (null? (cdr rest))
        (car rest)
        (cons '+ rest))))
(define (multiplicand e)
  (let ((rest (cddr e)))
    (if (null? (cdr rest))
        (car rest)
        (cons '* rest))))

注意すべきは、augend, multiplicand。
引数が2つ、つまり(+ 2 3)や(* 7 11)の形のときはこれまで通り、
それより引数が多い(+ 2 3 6)や(* 6 3 19 2)などのときは
(+ 3 6)、(* 3 19 2)と残りのものをこれまでと同じ形の式にする。
こうすれば、derivの実装は変えずにすむ。