Is super-accuracy matters?

[johnb]

(10-20-2023 08:55 AM)EdS2 Wrote:  ... But you wouldn't want to have the limited precision of your calculator, or your value of pi, to be the limiting factor. You'd want headroom.

Exactly. Especially if your calculations are repetitive and build on each other, or are a very long chain of different manipulations. Having a few [extra] guard digits goes a long way towards avoiding an accumulation of roundoff errors.

Of course, big precision arithmetic won't save you from naïveté. The implementer is still responsible for couching their solutions in terms of operations that don't unnecessarily throw away precision or otherwise become degenerate. (For example, we should use the Kahan summation algorithm for repetitive simple operations, and we should rewrite expressions so we avoid the edge cases in things like trig functions --- something I've repeatedly nailed myself with over the years and thus been forced to wear the dunce cap.)