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I've met in a book the following algorithm that computes the exponential:

Input: $t, n$ ($n$ is the number of steps)

Output: $E_n$

$\begin{array}{l} \mbox{define $t_0 = 0$ ; $E_0 = 1$} \\ \mbox{build two sequences $t_i$ and $E_i$ as follows} \\ t_{n+1} = t_n + ln(1 + d_n 2^{-n}) \\ E_{n+1} = E_n(1 + d_n 2^{-n}) \\ d_n = \begin{cases} 1 & \mbox{if $t_n + ln(1 + 2^{-n}) \leq t$} \\ 0 & \mbox{otherwise} \end{cases} \end{array}$

It is taken from this book. My question is not theoretical (it resembles the restoring division, or square root, or reciprocal) but implementative. To me it looks like that such algorithm requires a LUT that stores the values of $ln(1 + 2^{-n})$ while the others algorithms i mentioned don't... Does the algorithm actually requires a LUT to be implemented? or could be somehow exploited the fact that $ln(1 + 2^{-n}) \approx 2^{-n}$?

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