Incorrect calculations by the computer: cos(x) near x=k*π+π/2

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

Example 1. Given that $x_1=0.392699082\textup{e}2\approx 12\,\pi+\frac{\pi}{2}$ and
$x_2=0.1115265392024\textup{e}3\approx 35\,\pi+\frac{\pi}{2}\,,$ calculate $\cos(x_1)$ and $\cos(x_2)$ in MATLAB.

Let's just post the figure directly.

From the above screenshot, it can be seen that the two outputs from MATLAB are
$-3.012758\red{778139255}e-08$ and $-3.765\red{778342090408}e-11.$

However, the correct values with 16 significant digits are -0.3012758451921695e-7 and -0.3765996542384011e-10, respectively (as provided by ISRealsoft).

Thus, the outputs of MATLAB contain 9 and 12 incorrect digits, with error rates of 9/16 = 56.25% and 12/16 = 75% respectively for the significant figures.