  Building on the open-loop algorithm we introduce a new method for the
automated construction of one-loop amplitudes and their reduction to scalar
integrals. The key idea is that the factorisation of one-loop integrands in a
product of loop segments makes it possible to perform various operations
on-the-fly while constructing the integrand. Reducing the integrand on-the-fly,
after each segment multiplication, the construction of loop diagrams and their
reduction are unified in a single numerical recursion. In this way we entirely
avoid objects with high tensor rank, thereby reducing the complexity of the
calculations in a drastic way. Thanks to the on-the-fly approach, which is
applied also to helicity summation and for the merging of different diagrams,
the speed of the original open-loop algorithm can be further augmented in a
very significant way. Moreover, addressing spurious singularities of the
employed reduction identities by means of simple expansions in rank-two Gram
determinants, we achieve a remarkably high level of numerical stability. These
features of the new algorithm, which will be made publicly available in a
forthcoming release of the OpenLoops program, are particularly attractive for
NLO multi-leg and NNLO real-virtual calculations.

High Energy Physics - Phenomenology