SICP exercise 1.39

From Drewiki

Problem

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

Here's one implementation of Lambert's continued fraction representation of the tangent function, using the cont-frac function from exercise 1.37:

We can test the procedure by comparing the results to the tan primitive provided by Chicken Scheme 3.1 on a MacBook Pro running Mac OS X 10.5:

Output:

0.0

Output:

0

Here's an approximation of π:

Output:

0.999999998205103

Output:

0.999999998205103

Output:

-1.00000000538469

Output:

-1.00000000539582

Looks good. Values near a multiple of π (i.e., those for which tan x is nearly or exactly zero) need more iterations:

Output:

-3.58979302983757e-09

Output:

-5.48300699971479e-09

Output:

-3.58979298522367e-09