Iterated square root expansions for the inverse cosine and inverse hyperbolic cosine

Abstract:

Let ${R_1} = \sqrt {2 + 2x}$, ${R_{k + 1}} = \sqrt {|{2 + {R_k}}|}$. Then ${2^k}\sqrt {|{2 - {R_k}}|}$ and ${2^k}{\{ |{6 - 2\sqrt {3 + 3{R_k}} }|\} ^{1/2}}$ both converge to $\arccos x$ if $|x| \leqq 1$ and to $\operatorname {arccosh} x$ if $1 \leqq x < \infty$. Truncation errors for the two expressions are of the order of 2$^{-2k}$ and 2$^{-4k}$, respectively.