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Just dropping this on here for documentation/searchability. I do not think we should merge this functionality. In fact, I do not think the paper that introduces this solver strategy should have been published at all.
The TL;DR is that the scalars used to determine step sizes are literally unrepresentable even for fairly loose tolerances.
Pasting in my commentary from the implementation as detailed rationale for trashing the whole solver strategy.
I also took a look at using Decimal or BigRational types to circumvent this limitation, since it's only necessary for a small number of scalar calculations. Unfortunately, because the resulting rational is still unrepresentable as a float, it makes no meaningful difference.