I was pondering about Saruman's role as an opportunist ally of Sauron, when I realized that I had absolutely no idea how strong his army was compared to Sauron. We know Saruman fielded an impressive war economy in the relatively compact area of Isengard, and Sauron had behind him the might of the entire realm of Mordor, also implied to be heavily geared towards military production. Both have an obvious numerical superiority in the battles they fight.
However, I have no idea at all how large their armies were, overall. It is implied that Saruman's army was mostly gone after Helm's Deep, so I guess Sauron, who manages to keep fighting despite the crushing casualties at Pelennor fields, has by far more soldiers under his command. I'd like to see numbers, though. Is there anything in canon about the total amount of troops Saruman and Sauron could muster?