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A detailed picture of the infrared factorisation properties of scattering-amplitude integrands can be exploited to derive finite-remainder representations which are amenable to direct numerical integration. This provides an alternative to the usual analytic approaches. In this talk, I will present a proof-of-concept result: integrand representations for generic two-loop QED amplitudes which exhibit local factorisation in all soft and collinear limits. I will also discuss a corresponding local IR subtraction formula and elaborate on obstacles and ideas for their generalisation to QCD.