Redirect stdout and stderr, with timestamps on stderr only

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http://unix.stackexchange.com – I'm trying to hack in better logging for a test program that I didn't write. I want all lines of STDERR output to have timestamps, but if possible I want to leave STDOUT alone. I also want one file (all.txt) to log both stdout and stderr (either with or without STDERR timestamps), and a file dedicated to STDERR output (errors.txt, with timestamps) Also, I don't care whether anything is echoed to the terminal or not. This is a batch-style program. I did look at rsyslog and that might come in later, but this is a solution I should be able to hack in real quick...at least, if I knew more about (HowTos)