Incorrect calculations by the computer: sin(x) near x=k*π

In this blog post, we explore incorrect calculations of the sine function sin(x) in MATLAB near numbers around $k\,\pi (k\neq0).$

Example 1. Given that $x_1=0.3769911184\textup{e}2\approx 12\,\pi$ and
$x_2=0.166504\textup{e}3\approx 53\,\pi\,,$ calculate $\sin(x_1)$ and $\sin(x_2)$ in MATLAB.

Let's just post the figure directly.

From the above screenshot, it can be seen that the two outputs from MATLAB are
$-3.077518\red{377555020}e-09$ and $4.106402475\red{102297}e-04.$

However, the correct values with 16 significant digits are -0.3077518861551721e-8 and 0.4106402475009074e-3, respectively (as provided by ISRealsoft).

Thus, the outputs of MATLAB contain 9 and 6 incorrect digits, with error rates of 9/16 = 56.25% and 6/16 = 37.5% respectively for the significant figures.