In Linux shell how to capture stderr to log file with datetime?

Question

In my case, cmd1 handle multi-line(seems like awk or sed).

But the reliability of cmd1 is not good (90% success rate maybe, that is to say, 10 lines are not dealed within 100 lines input in a execution), so I wonder if there is way to caputer error message into log file(instead of '/dev/null' below) with additional timestamp?

cmd1 2>>/dev/null | cmd2 | cmd3 > stdout.log

thanks a lot!