How can I optimise the power series calculation of the exponential function? In an answer to the question Fastest way to calculate $e^x$ upto arbitrary number of decimals? there is a description of a method by which the number of terms needed to calcluate $e^x$ to a given precision can be halved:

Although the exponential function is not even (or odd), it satisfies a
  relation:      $$e^{-x} = 1/e^x$$    that implies it is a simple
  rational transformation of an even function:
$$e^x = (1 + x*f(x^2))/(1 - x*f(x^2))$$    which may be solved for a
  power series or continued fraction approximation of f(z).  This is the
  idea of symmetry as it applies to the exponential function.  Not
  only does it reduce the evaluaton of the exponential function on the
  whole real line to just the positive (or the negative) half, it also
  "contracts" the number of terms needed for a given accuracy by half
  (retaining only $x^2$ terms in the expansion).

Unfortunately, I am not able to understand how to 'solve for the power series approximation of f(z)'.
The context is that I am trying to improve the performance of code that calclulates $e^x$ to arbitrary precision.

What I've used in the past is the Pade acceleration. There are different levels you can use, i.e. how many terms to compute. For example:
$$
e^z \sim \frac{\left(1 + \frac{1}{2} z + \frac{1}{12}z^2\right)}{\left(1 - \frac{1}{2}z + \frac{1}{12} z^2\right)}
$$
is just one of many Pade approximations you can select. Another transformation is the Shanks. Information on both can be found online.
Paul Safier
