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~~I am looking for a way to *numericaly* calculate $\sin x$ for large $x$.~~
Background is from software programming (java, if that should matter).
I have seen some realy horrible code (don't ask for details unless you want to wake up screaming at night from dreaming about it).
What remains after refactoring and clean-up is the question in the first line.
I have seen a paper approaching this in two steps:
* First, considering only values in $[-\frac\pi 4,\frac\pi 4]$ so the Taylor series converges quickly.
* Second, a software algorithm *specialized* on data representation with bits and bytes.
I would like to see a *mathematical* solution to that problem, preferably one that gives lower and upper bounds as rational numbers.

I am looking for a way to *numerically* calculate $\sin x$ for large $x$.
Background is from software programming (java, if that should matter).
I have seen some realy horrible code (don't ask for details unless you want to wake up screaming at night from dreaming about it).
What remains after refactoring and clean-up is the question in the first line.
I have seen a paper approaching this in two steps:
* First, considering only values in $[-\frac\pi 4,\frac\pi 4]$ so the Taylor series converges quickly.
* Second, a software algorithm *specialized* on data representation with bits and bytes.
I would like to see a *mathematical* solution to that problem, preferably one that gives lower and upper bounds as rational numbers.

Numerical taylor-expansion