In this paper, we consider a fast and high-precision algorithm of the basic elementary functions. Firstly, we discuss the power series expansion of functions which are related to the Bernoulli number B2n or Euler number E2n, such as tan x, sec x, tanh x and so on, and study the corresponding fast computation. For the basic elementary functions, hyperbolic functions and inverse hyperbolic functions, we derive a fast and arbitrarily accurate algorithm based on the power series expansion in the complex domain. The algorithm proposed in this paper is suitable for all elementary functions due to that the exponential and logarithm functions can be expressed by the power series. The feature of this algorithm is that it can be easily coded and it is self-contained for the computation of the elementary functions.
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