> 1 is not equal to 1 - e^(-e^1000). But for Richardson and Fitch's algorithm to detect that, it would require more steps than there are atoms in the universe.
> They needed something faster.
I'm disappointed after this paragraph I expected a better algorithm and instead they decided to give up. Fredrik Johansson in his paper "Calcium: computing in exact real and complex fields" gives a partial algorithm for the problem and writes "Algorithm 2 is inspired by Richardson’s algorithm, but incomplete: it will
find logarithmic and exponential relations, but only if the extension tower is flattened (in
other words, we must avoid extensions such as e^log(z) or √z^2), and it does not handle all
algebraic functions.
Much like the Risch algorithm, Richardson’s algorithm has apparently never been implemented fully. We presume that Mathematica and Maple use similar heuristics to ours,
but the details are not documented [6], and we do not know to what extent True/False
answers are backed up by a rigorous certification in those system".
> 1 is not equal to 1 - e^(-e^1000). But for Richardson and Fitch's algorithm to detect that, it would require more steps than there are atoms in the universe.
> They needed something faster.
I'm disappointed after this paragraph I expected a better algorithm and instead they decided to give up. Fredrik Johansson in his paper "Calcium: computing in exact real and complex fields" gives a partial algorithm for the problem and writes "Algorithm 2 is inspired by Richardson’s algorithm, but incomplete: it will find logarithmic and exponential relations, but only if the extension tower is flattened (in other words, we must avoid extensions such as e^log(z) or √z^2), and it does not handle all algebraic functions. Much like the Risch algorithm, Richardson’s algorithm has apparently never been implemented fully. We presume that Mathematica and Maple use similar heuristics to ours, but the details are not documented [6], and we do not know to what extent True/False answers are backed up by a rigorous certification in those system".