  We construct and study the reduced, relative, genus one Gromov--Witten theory
of very ample pairs. These invariants form the principal component contribution
to relative Gromov--Witten theory in genus one and are relative versions of
Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute
theories by degeneration of the tangency conditions, and the resulting formulas
generalise a well-known recursive calculation scheme put forward by Gathmann in
genus zero. The geometric input is a desingularisation of the principal
component of the moduli space of genus one logarithmic stable maps to a very
ample pair, using the geometry of elliptic singularities. Our study passes
through general techniques for calculating integrals on logarithmic blowups of
moduli spaces of stable maps, which may be of independent interest.
