  We prove an explicit inverse Chevalley formula in the equivariant $K$-theory
of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley
formula, we mean a formula for the product of an equivariant scalar with a
Schubert class, expressed as a $\mathbb{Z}[q^{\pm 1}]$-linear combination of
Schubert classes twisted by equivariant line bundles. Our formula applies to
arbitrary Schubert classes in semi-infinite flag manifolds of simply-laced type
and equivariant scalars $e^{\lambda}$, where $\lambda$ is an arbitrary
minuscule weight. By a result of Stembridge, our formula completely determines
the inverse Chevalley formula for arbitrary weights in simply-laced type,
except for type $E_8$. The combinatorics of our formula is governed by the
quantum Bruhat graph, and the proof is based on a limit from the double affine
Hecke algebra. As such, our formula also provides an explicit determination of
all nonsymmetric $q$-Toda operators for minuscule weights in ADE type.
