I am a physicist and during some of my research I realized that $\log_2 \sec(\pi x)$ is *really well* approximated by $\frac{6x^2}{1-x}$ for small but positive $x$. Is there any reason this should be so? After all the former function looks quite unwieldy on first sight. In particular, I am wondering whether there is some context in mathematics where these functions would arise, with the hope of understanding why they should agree so well. (Note that if I instead simply Taylor expand the former function and keep the leading term, it performs much more poorly.)

EDIT: Here is a plot from WolframAlpha:

Note that at $x= \frac{1}{2}$, the former function has a singularity.

Edit: There were comments about the Taylor series and Padé approximants at $0$. The approximation is neither, and it is not exceptional at $0$, but it seems good on the interval $[0,1/3]$.