  The CHY representation of scattering amplitudes is based on integrals over
the moduli space of a punctured sphere. We replace the punctured sphere by a
double-cover version. The resulting scattering equations depend on a parameter
$\Lambda$ controlling the opening of a branch cut. The new representation of
scattering amplitudes possesses an enhanced redundancy which can be used to
fix, modulo branches, the location of four punctures while promoting $\Lambda$
to a variable. Via residue theorems we show how CHY formulas break up into sums
of products of smaller (off-shell) ones times a propagator. This leads to a
powerful way of evaluating CHY integrals of generic rational functions, which
we call the $\Lambda$ algorithm.
