Bash output prefix and redirect to a file with timestamp

log_prefix.sh

#!/usr/bin/env bash

[ -f test.log ] && rm -f test.log

{
  {
    echo foo 1
    sleep 1
    echo bar 1 >&2
    echo foo 2
    sleep 1
    echo bar 2 >&2
  } \
    2> >(sed "s/^/$(date '+%Y-%m-%d %H:%M:%S') [ERR] /" >&2) \
    > >(sed "s/^/$(date '+%Y-%m-%d %H:%M:%S') [INF] /") \
    2>&1
} >>test.log

echo "-------------------"
cat test.log