Is super-accuracy matters?

[EdS2]

I think this is related: we might ask ourselves, where in the real world might we need unexpectedly great accuracy?

In physics, the Fine-structure constant seems to have been experimentally confirmed to 11 digits, and that is held to be the most extreme example. Notably, some other apparently very accurate physical constants are now exact numbers by way of re-definition. (For example the speed of light is exactly 299792458 ms⁻¹, and the Avogadro number is exactly 6.02214076E23)

One of the examples often seen is in finance: calculations involving some small rate of interest applied over very many terms ultimately involve transcendental functions and scaled integers. A friend of mine, at the time working in finance, was sorely challenged to reconcile the value of some financial instrument to some precise number of Japanese Yen. Of course, the calculation had to agree with some official way of doing the maths. In the US, at least, I gather that the HP-12C has now become enshrined as the official way to get the one true exactly correct answer. But getting a million dollar sum correct to one cent is still only ten digits or so. Even a trillion dollars of value expressed in yen is just 15 digits. It might be that 30 digits of accuracy will suffice in realistic TVM calculations even in this case.

For myself, I'm very attracted by the 40-odd digits in Free42 and DM42, and very much appreciate efforts to get errors down to as few ULPs as is practical. And I'm thrilled at the as-many-digits-as-you-have-time-to-scroll accuracy of the Android calculator.