Multiple-precision Zero-finding Methods and the Complexity of Elementary Function Evaluation1

We consider methods for finding high-precision approximations to simple zeros of smooth functions. As an application, we give fast methods for evaluating the elementary functions log(x), exp(x), sin(x) etc. to high precision. For example, if x is a positive floating-point number with an n-bit fraction, then (under rather weak assumptions) an n-bit approximation to log(x) or exp(x) may be computed in time asymptotically equal to 13M(n) log2 n as n → ∞, where M(n) is the time required to multiply floating-point numbers with n-bit fractions. Similar results are given for the other elementary functions, and some analogies with operations on formal power series are mentioned.