Two-loop five-point amplitudes in massless QCD with finite fields

I will discuss the analytic calculation of two-loop five-point helicity amplitudes in massless QCD. In our workflow, we perform the bulk of the computation using finite field arithmetic, avoiding the precision-loss problems of floating-point representation. The integrals are provided by the pentagon functions. We use numerical reconstruction techniques to bypass intermediate complexity and obtain compact forms for the rational coefficients. I will present results for NLO full-colour gluon-initiated diphoton-plus-jet production and NNLO leading-colour trijet production