How to add timestamp to STDERR redirection

#!/bin/bash

DEBUG=1

LOG=$HOME/script_${0##*/}_$(date +%Y.%m.%d-%H.%M.%S-%N).log
ERROR=$HOME/error.log

exec 2> $ERROR
exec 1> >(tee -ai $LOG)

if [ $DEBUG = 0 ] ; then
    exec 4> >(xargs -i echo -e "[ debug ] {}")
else
    exec 4> /dev/null
fi

# test
echo " debug sth " >&4
echo " log sth normal "
type -f this_is_error
echo " errot sth ..." >&2
echo " finish ..." >&2>&4
# close descriptor 4
exec 4>&-

Redirections are taken in order. Try this:

Given a script -

$: cat tst
echo a
sleep 2
echo 1 >&2
echo b
sleep 2
echo 2 >&2
echo c
sleep 2
echo 3 >&2
echo d

I get the following

$: ./tst 2>&1 1>stdout | sed 's/^/echo $(date +%Y%m%dT%H%M%S) /; e'
20180925T084008 1
20180925T084010 2
20180925T084012 3

And as much as I dislike awk, it does avoid the redundant subcalls to date.

$: ./tst 2>&1 1>stdout | awk "{ print strftime(\"%Y%m%dT%H%M%S \") \$0; fflush() }"; >stderr

$: cat stderr
20180925T084414 1
20180925T084416 2
20180925T084418 3

$: cat stdout
a
b
c
d