  We present three dimensional smoothed particle hydrodynamics (SPH)
calculations of irradiation-driven warping of accretion discs. Initially
unwarped planar discs are unstable to the radiation reaction when the disc is
illuminated by a central radiation source. The disc warps and tilts and
precesses slowly in a retrograde direction; its shape continuously flexes in
response to the changing orientation of the Roche potential. We simulate ten
systems: eight X-ray binaries, one cataclysmic variable (CV), and a `generic'
low mass X-ray binary (LMXB). We adopt system parameters from observations and
tune a single parameter: our model X-ray luminosity ($L_{*}$) to reproduce the
observed or inferred super-orbital periods. Without exception, across a wide
range of parameter space, we find an astonishingly good match between the
observed $L_{X}$ and the model $L_{*}$. We conclude irradiation-driven warping
is the mechanism underlying the long periods in X-ray binaries. Our Her X-1
simulation simultaneously reproduces the observed $L_{X}$, the "main-" and
"short-high" X-ray states and the orbital inclination. Our simulations of SS
433 give a maximum warp angle of $18.6^{\circ}$, a good match to the cone
traced by the jets, but this angle is reached only in the outer disc. In all
cases, the overall disc tilt is less than $\degrees{13}$ and the maximum disc
warp is less than and or equal to $\degrees{21}$.
