  Hyperuniform many-particle systems are characterized by a structure factor
$S({\mathbf{k}})$ that is precisely zero as $|\mathbf{k}|\rightarrow0$; and
stealthy hyperuniform systems have $S({\mathbf{k}})=0$ for the finite range $0
< |{\mathbf{k}}| \le K$, called the "exclusion region." Through a process of
collective-coordinate optimization, energy-minimizing disordered stealthy
hyperuniform systems of moderate size have been made to high accuracy, and
their novel physical properties have shown great promise. However, minimizing
$S(\mathbf{k})$ in the exclusion region is computationally intensive as the
system size becomes large. In this Letter, we present an improved methodology
to generate such states using double-double precision calculations on GPUs that
reduces the deviations from zero within the exclusion region by a factor of
approximately $10^{30}$ for systems sizes more than an order of magnitude
larger. We further show that this ultra-high accuracy is required to draw
conclusions about their corresponding characteristics, such as the nature of
the associated energy landscape and the presence or absence of Anderson
localization, which might be masked, even when deviations are relatively small.
