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pleroyeggrobin
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Update documentation/Sin Cos.tex
Co-authored-by: Robin Leroy <egg.robin.leroy@gmail.com>
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documentation/Sin Cos.tex

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@@ -228,7 +228,7 @@ \subsection*{Argument Reduction Using the Two-Term Approximation}
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\abs n &\leq \iround{2^{\gk_1} \frac{\Pi}{2} \pa{1 + \gd_1} \frac{2}{\Pi} \pa{1 + \gd_2} \pa{1 + \gd_3}} \\
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&\leq \iround{2^{\gk_1} \pa{1 + \gg_3}}
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\end{align*}
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If $2^{\gk_1} \gg_3$ is small enough (less that $1/2$), the rounding cannot cause $n$ to exceed $2^{\gk_1}$. In practice we choose a relatively small value for $\gk_1$, so this condition is met.
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If $2^{\gk_1} \gg_3$ is small enough (less than $1/2$), the rounding cannot cause $n$ to exceed $2^{\gk_1}$. In practice we choose a relatively small value for $\gk_1$, so this condition is met.
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Now if $x$ is close to an odd multiple of $\frac{\Pi}{4}$ it is possible for misrounding to happen. In the following analysis we assume that $n > 0$. The results are symmetrical if $n < 0$. There are two possible kinds of misrounding, with different bounds.
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