  The results of the computer investigation of the sign changes of the
difference between the number of twin primes $\pi_2(x)$ and the
Hardy--Littlewood conjecture $C_2\Li_2(x)$ are reported. It turns out that
$d_2(x)=\pi_2(x) - C_2\Li_2(x)$ changes the sign at unexpectedly low values of
$x$ and for $x<2^{48}=2.81\...\times10^{14}$ there are 477118 sign changes of
this difference. It is conjectured that the number of sign changes of $d_2(x)$
for $x\in (1, T)$ is given by $\sqrt T/\log(T)$. The running logarithmic
densities of the sets for which $d_2(x)>0$ and $d_2(x)<0$ are plotted for $x$
up to $2^{48}$.
