Incorrect calculations: csc(x) near x=k*π

In this blog post, we explore incorrect calculations of the cosecant function csc(x) in MATLAB near numbers around $k\,\pi.$

Example 1. Given that $x_1=0.942477796123\textup{e}1 \approx 3\,\pi$ and
$x_2=0.7853981633974\textup{e}2 \approx 25\,\pi\,,$ calculate $\csc(x_1)$ and $\csc(x_2)$ in MATLAB.

Using MATLAB, we computed the values of $\csc(x_1)$ and $\csc(x_2).$ The result is shown in the following screenshot.

From the above screenshot, it can be seen that the outputs from MATLAB are
$-2.17098\red{8725424315}\textup{e}+09$ and $2.069\red{879330499836}\textup{e}+11$ , respectively.

However, the correct values with 16 significant digits are -0.217098558923038e10 and 0.2069981278717736e12, respectively (as provided by ISRealsoft).

Thus, the output of MATLAB contains only 6 and 4 correct digits, with error rates of (16-6)/16 = 62.5% and (16-4)/16 = 75% respectively for the significant figures.

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