You could use IPC::Open3, here is a basic example of collecting stdout separate from stderr. To test it's functioning, try entering "2 + 2" then try "2 / 0".

#!/usr/bin/perl
use warnings;
use strict;
use IPC::Open3;
use IO::Select;

my $pid = open3(\*WRITE, \*READ,\*ERROR,"bc");

my $sel = new IO::Select();

$sel->add(\*READ);
$sel->add(\*ERROR);

my($error,$answer)=('','');

while(1){

 print "Enter command\n";
 chomp(my $query = <STDIN>);
 
 #send query to bash
 print WRITE "$query\n";

 foreach my $h ($sel->can_read)
 {
   my $buf = '';
   if ($h eq \*ERROR)
   {
     sysread(ERROR,$buf,4096);
     if($buf){print "ERROR-> $buf\n"}
   }
   else
   {
     sysread(READ,$buf,4096);
     if($buf){print "$query = $buf\n"}
   }
 }
}

waitpid($pid, 1);
# It is important to waitpid on your child process,  
# otherwise zombies could be created.
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