while (<>) {
    s!http://www.theirsite.com/\W!http://our.server1/theirsite/\n!i    ||
    s!http://www.theirsite.com/!http://our.server1/theirsite/|i ||
    s!http://www.dummy.com/\W!http://our.server2/dummy/\n!i ||
    ...
    logme ($&)
}