Obviously the machinery of QFT allows us to calculate processes, such as QED diagrams, to great precision, and whilst it is effective, it seems there are many processes that make calculations (say by hand) significantly slow.
Are there any recent developments in our machinery to compute Feynman diagrams that makes it faster to analytically compute matrix elements, widths and cross sections?
Nowadays, almost nobody uses Feynman diagrams for precision calculations (that is, anything beyond a tree level 4 or 5 point amplitude). There is a whole field dedicated to finding better methods of calculating scattering amplitudes (using recursion and unitarity for example). In fact this is how most calculations for the LHC are done in practice.
A nice review of some of the techniques is https://arxiv.org/abs/1308.1697. Most of the work is motivated by the observation that scattering amplitudes are much simpler than one could have thought by just staring at the Feynman rules. An example of this is the famous Parke-Taylor amplitude https://en.wikipedia.org/wiki/MHV_amplitudes. Take a look and compare to the QCD Feynman rules.
There are a number of computer algebra systems for evaluating Feynman diagrams and doing other particle physics calculations, such as FeynCalc, FORM, GiNAC, Package-X, etc.
