  We report high-precision measurements of the longitudinal double-spin
asymmetry, $A_{LL}$, for midrapidity inclusive jet and dijet production in
polarized $pp$ collisions at a center-of-mass energy of
$\sqrt{s}=200\,\mathrm{GeV}$. The new inclusive jet data are sensitive to the
gluon helicity distribution, $\Delta g(x,Q^2)$, for gluon momentum fractions in
the range from $x \simeq 0.05$ to $x \simeq 0.5$, while the new dijet data
provide further constraints on the $x$ dependence of $\Delta g(x,Q^2)$. The
results are in good agreement with previous measurements at
$\sqrt{s}=200\,\mathrm{GeV}$ and with recent theoretical evaluations of prior
world data. Our new results have better precision and thus strengthen the
evidence that $\Delta g(x,Q^2)$ is positive for $x > 0.05$.
