In the process of investigating this problem I encountered the following $\log_2$-plots of bitwise errors of the Taylor expressions of growing number of significant terms:

enter image description here By these plots it's quite evident that the locality of the Taylor expressions impact the precision for large $x$ a lot. However I have not (yet) managed to prove how to solve the problem of saving calculations with angular halfing as proposed in the previous question, but what if we switch focus to another direction: How to design polynomials so that we get a more uniform distribution of error.

Own ideas:

Some obvious candidates could be $$P(x) = \frac 1 n \sum_{k=0}^n T_{k/n}(x), T_{k/n}\text{ being Taylor polynomial expanded at } x=\frac{k+1/2}{n+1}\cdot \frac{\pi}{2}$$

Or maybe trying to estimate $${\bf c_o} = \min_{\bf c} \left\{\int_0^{\frac{\pi}{2}}\left|\sum_{k=0}^N c_kx^k-\sin(x)\right|dx\right\}$$

Do you think these would be feasible ideas or could some other approach be better?


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