Incorrect calculations: sec(x) near x=k*π+π/2

In this blog post, we explore incorrect calculations of the secant function sec(x) in MATLAB near numbers around $k\,\pi+\frac{\pi}{2}.$

Example 1. Given that $x_1=0.1099557429\textup{e}2 \approx 3\,\pi+\frac{\pi}{2}$ and
$x_2=0.8011061266654\textup{e}2 \approx 25\,\pi+\frac{\pi}{2}\,,$ calculate $\sec(x_1)$ and $\sec(x_2)$ in MATLAB.

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

From the above screenshot, it can be seen that the outputs from MATLAB are
$4.10555\red{5334642597}\textup{e}+08$ and $3.\red{580150224915013}\textup{e}+12$ , respectively.

However, the correct values with 16 significant digits are 0.4105556037464873e9 and 0.3670813182326778e13, respectively (as provided by ISRealsoft).

Thus, the output of MATLAB contains only 6 and 1 correct digits, with error rates of (16-6)/16 = 62.5% and (16-1)/16 = 93.75% respectively for the significant figures.

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