Sicp Exercise 1.39

Exercise 1.39.  A continued fraction representation of the tangent function was published in 1770 by the German mathematician J.H. Lambert:

where x is in radians. Define a procedure (tan-cf x k) that computes an approximation to the tangent function based on Lambert’s formula. K specifies the number of terms to compute, as in exercise 1.37.

Solution:

	(define (cont-frac
	n
	d
	k)
	(if
	(= k 1)
	(/
	(n k)
	(d k))
	(/
	(n k)
	(+
	(d k)
	(cont-frac
	n
	d
	(- k 1))))))
	(define (reverse-nd n k)
	(lambda (x)
	(n (- (+ k 1) x))))
	(define (cont-frac-rec n d k)
	(cont-frac
	(reverse-nd n k)
	(reverse-nd d k)
	k))
	(define (cont-frac-iter-helper
	result
	n
	d
	k)
	(if
	(= k 1)
	result
	(cont-frac-iter-helper
	(/
	(n (- k 1))
	(+
	(d (- k 1))
	result))
	n
	d
	(- k 1))))
	(define (cont-frac-iter n d k)
	(cont-frac-iter-helper
	(/ (n k) (d k))
	n
	d
	k))
	(define (dfun i)
	(- (* i 2) 1))
	(define d dfun)
	;; (cont-frac-rec n d 60)
	;; (cont-frac-iter n d 60)
	(define (tan-cf k x)
	(define (n k)
	(-
	0
	(* x x)))
	(*
	-1
	(/
	(cont-frac-iter n d k)
	x)))
	(tan-cf 60 (/ 3.1415926535 4))

view raw ContinuedFractionForTangentFunction.scm hosted with by GitHub