  We perform molecular dynamics simulations to compress binary hard spheres
into jammed packings as a function of the compression rate $R$, size ratio
$\alpha$, and number fraction $x_S$ of small particles to determine the
connection between the glass-forming ability (GFA) and packing efficiency in
bulk metallic glasses (BMGs). We define the GFA by measuring the critical
compression rate $R_c$, below which jammed hard-sphere packings begin to form
"random crystal" structures with defects. We find that for systems with $\alpha
\gtrsim 0.8$ that do not de-mix, $R_c$ decreases strongly with $\Delta \phi_J$,
as $R_c \sim \exp(-1/\Delta \phi_J^2)$, where $\Delta \phi_J$ is the difference
between the average packing fraction of the amorphous packings and random
crystal structures at $R_c$. Systems with $\alpha \lesssim 0.8$ partially
de-mix, which promotes crystallization, but we still find a strong correlation
between $R_c$ and $\Delta \phi_J$. We show that known metal-metal BMGs occur in
the regions of the $\alpha$ and $x_S$ parameter space with the lowest values of
$R_c$ for binary hard spheres. Our results emphasize that maximizing GFA in
binary systems involves two competing effects: minimizing $\alpha$ to increase
packing efficiency, while maximizing $\alpha$ to prevent de-mixing.
