Compute e^x given starting approximation I have been looking for an algorithm to calculate $e^x$ to arbitrary precision (millions of digits).  The best way I know is to use Taylor polynomials.  However, the Taylor polynomial algorithm seems to have no way to enhance the precision of already existing estimates of $e^x$.

For example:

I already have an exact approximation for $e^{0.7}$ calculated to 1000 decimal places, but now I want an exact approximation of $e^{0.700000001}$ calculated to 100000 decimal places.

Because $e^{0.7}$ is very close to $e^{0.700000001}$, I should be able to use $e^{0.7}$ as a starting estimate for $e^{0.700000001}$.  However, using Taylor polynomials, I know of no way to input a starting approximation.

Does anyone know of an algorithm that efficiently converges to $e^x$ given an initial estimate?

To clarify, I'm looking for an algorithm that can actually surpass the accuracy of the given estimation.  For example, if I give it an estimate to 1000 digits, it should still be able to converge accurately beyond 10000000 digits.  This rules out the optimization $e^{0.70001} = e^{0.7} \cdot e^{0.00001}$.
The obvious answer to "I have an approximate solution to an equation and I want to improve it" is Newton-Raphson. If you have an approximation $y \approx e^x$ then you can improve it to $$y' = y - \frac{(\ln y) - x}{y^{-1}} = y(1 + x - \ln y)$$

Now you need an efficient way of calculating $\ln y$, but here I note that Wikipedia, commenting on high precision logarithms, says (my emphasis):


  Another alternative for extremely high precision calculation is the formula
  $$\ln x \approx {\frac {\pi }{2M(1,4/s)}}-m\ln 2,$$
  where $M$ denotes the arithmetic-geometric mean of $1$ and $4/s$, and
  $$\displaystyle s=x2^{m}&gt;2^{p/2},$$
  with $m$ chosen so that $p$ bits of precision is attained. (For most purposes, the value of $8$ for $m$ is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently.


The references given are Practically fast multiple-precision evaluation of log(x), Sasaki and Kanada (1982), Journal of Information Processing 5(4) pp247–250; and Fast Computations of the Exponential Function, Ahrendt (1999), Stacs Lecture Notes in Computer Science 1564 pp302–312.