$$\frac{44}{\left\lfloor\sqrt{\sqrt{(4+4)!}}\right\rfloor}$$
is equal to
the common approximation $22/7$
and scores
$$\frac{-\log_{10}\left(\frac{22}7-\pi\right)}6$$
which is $\approx0.483$.
Edit: Better yet is
$$\sqrt{\frac{44-4}4}$$
which scores
$$\frac{-\log_{10}\left(\sqrt{10}-\pi\right)}3$$
, or $\approx0.5614$.