Here's my solution. You'll be needing File::Tee and IPC::System::Simple. I took your script and put it on my desktop. Then I wrote another script to call your script. If you have any further questions, post them.

#!/usr/bin/perl

use strict;
no warnings;
use File::Tee qw(tee);
use IPC::System::Simple qw(capture system);

my $test = '/root/Desktop/your_dup.pl';

tee(STDOUT, '>', 'stdout.txt');
tee(STDERR, '>',  'error.log');

my $file = system("perl $test");
print STDOUT $file, "\n";
print STDERR $file, "\n";

close STDERR;
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And your_dup.pl

use Test::More qw(no_plan);
$\="\n";
# here will be file name which will store the log o/p
print "hello world";
pass("hello world");
fail("do nothing");
diag("hi friend");
use_ok("fake_module");
print "bye";
[download]