  Fully relativistic first-principles electronic structure calculations based
on a noncollinear local spin density approximation (LSDA) are performed for
pyrochlore iridates Y$_2$Ir$_2$O$_7$ and Pr$_2$Ir$_2$O$_7$. The all-in, all-out
antiferromagnetic (AF) order is stablized by the on-site Coulomb repulsion
$U>U_c$ in the LSDA+$U$ scheme, with $U_c\sim1.1$~eV and 1.3~eV for
Y$_2$Ir$_2$O$_7$ and Pr$_2$Ir$_2$O$_7$, respectively. AF semimetals with and
without Weyl points and then a topologically trivial AF insulator successively
appear with further increasing $U$. For $U=1.3$~eV, Y$_2$Ir$_2$O$_7$ is a
topologically trivial narrow-gap AF insulator having an ordered local magnetic
moment $\sim0.5\mu_B$/Ir, while Pr$_2$Ir$_2$O$_7$ is barely a paramagnetic
semimetal with electron and hole concentrations of $0.016$/Ir, in overall
agreements with experiments. With decreasing oxygen position parameter $x$
describing the trigonal compression of IrO$_6$ octahedra, Pr$_2$Ir$_2$O$_7$ is
driven through a non-Fermi-liquid semimetal having only an isolated Fermi point
of $\Gamma_8^+$, showing a quadratic band touching, to a $Z_2$ topological
insulator.
