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The linked NASA article points out that with 40 digits of pi you could compute the circumference of the visible universe to an accuracy equal to the diameter of a hydrogen atom. I'm gonna say there's no practical application that would require even 40 digits, never mind a few hundred

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You need more digits than that to accurately compute double-precision trigonometric functions (if the input is close to pi, you need enough accurate digits left after performing range reduction).

This paper claims you need 2/pi accurate to 1144 bits which is about 345 decimal digits: https://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.p...

As a counterpoint, no real computation I've performed on a computer needed to compute the cos of 2^1023 radians. I can't imagine such a scenario either.

You can either implement the functions accurately or inaccurately. Implementing them inaccurately is a slippery slope. Intel botched the hardware implementations in their processors not only for large inputs but also for inputs nearish to multiples of pi:

http://notabs.org/fpuaccuracy/index.htm

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