If the concern is first ensuring errors don't accumulate and second an emphasis on minimum variation in period then the following code should help:

use strict;
use warnings;
use Time::HiRes qw/time sleep/;

my $nextTick = int (time());

while (1) {
    $nextTick += 1.0; # We want to stuff again in 1 second

    my $wait = 0.99 - ($nextTick - time()); # allow 10ms slop in sleep
+ period

    sleep $wait if $wait > 0.0;

    1 while time() < $nextTick; # spin until time is up

    RunStuff();
}


sub RunStuff {
    # kick off required sub tasks here
    print time(), "\n";
}
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The key is to calculate when the next interesting epoch is, then sleep for most of the time until then. The 1 while ... loop is a busy wait until the next epoch time has arrived.