  Multipoint vertex functions, and the four-point vertex in particular, are
crucial ingredients in many-body theory. Recent years have seen significant
algorithmic progress toward numerically computing their dependence on multiple
frequency arguments. However, such computations remain challenging and are
prone to suffer from numerical artifacts, especially in the real-frequency
domain. Here, we derive estimators for multipoint vertices that are numerically
more robust than those previously available. We show that the two central steps
for extracting vertices from correlators, namely the subtraction of
disconnected contributions and the amputation of external legs, can be achieved
accurately through repeated application of equations of motion, in a manner
that is symmetric with respect to all frequency arguments and involves only
fully renormalized objects. The symmetric estimators express the core part of
the vertex and all asymptotic contributions through separate expressions that
can be computed independently, without subtracting the large-frequency limits
of various terms with different asymptotic behaviors. Our strategy is general
and applies equally to the Matsubara formalism, the real-frequency
zero-temperature formalism, and the Keldysh formalism. We demonstrate the
advantages of the symmetric improved estimators by computing the Keldysh
four-point vertex of the single-impurity Anderson model using the numerical
renormalization group.
