A fast algorithm for computing arctan x to n-bit precision for x in the interval (0, 1]...

A fast algorithm for computing arctan x to n-bit precision for x in the interval (0, 1] is as follows: Set a = 2^−n/2,

Then repeatedly update these variables by these formulas (in order from left to right and top to bottom):

After each sweep, print f = c ln[(1 + b)/(1 − b)]. Stop when 1 − a ?2⁻ⁿ. Write a double-precision routine to implement this algorithm and test it for various values of x. Compare the results to those obtained from the arctangent function on your computer system.

Note: This fast multiple-precision algorithm depends on the theory of elliptic integrals, using the arithmeticgeometric mean iteration and ascending Landen transformations. Other fast algorithms for trigonometric functions are discussed in Brent [1976].