  We describe a new method for the automated construction of one-loop
amplitudes based on the open-loop algorithm, where various operations are
performed on-the-fly while constructing the integrand. In particular, an
on-the-fly reduction interleaved with the construction steps of the amplitude
keeps the maximum tensor rank in the loop momentum at two throughout the
algorithm, thus drastically reducing the complexity of the calculation. The
full reduction to scalar integrals is unified with the amplitude construction
in a single recursion within the OpenLoops framework. This approach strongly
exploits the factorisation of one-loop integrands in a product of loop
segments. The on-the-fly approach, which is also applied to helicity summation
and the merging of different diagrams, increases the speed of the original
open-loop algorithm in a very significant way. A remarkably high level of
numerical stability is achieved by exploiting freedoms in reduction identities
and through simple expansions in rank-two Gram determinants. These features are
particularly attractive for NLO multi-leg and NNLO real-virtual calculations.
The new algorithm will be made public in a forthcoming release of the OpenLoops
program.
