The only thing that you did not consider is that x*x < x for x < 1. That's why your code doesn't work for input < 1.

Apart from that a couple of small comments concerning your code.

• Don't use prototypes for your subroutines unless you know exactly what you are doing.
• Don't use the variables $a and $b, they are special and used for sort.
• Instead of writing my ($x, $y, $z) = (shift, shift, shift); simply write my ($x, $y, $z) = @_;.

FWIW, here is my take without having looked any other solutions to include monsieur_champs'.

print sqrt($_), "\t", mysqrt($_, .0000001), "\n" for qw/.25 .50 .75 1 
+10 50 100 1000 31415926/;
sub mysqrt {
    my ($tgt, $err, $try, $len) = @_;
    return 1 if $tgt == 1;
    if (! defined $try) {
        ($len, $try) = $tgt < 1
            ? ((1 - $tgt) / 2, $tgt + (1 - $tgt) / 2)
            : ($tgt / 2, $tgt / 2);
        return mysqrt($tgt, $err, $try, $len);
    }
    my $dif = $tgt - ($try * $try);
    return $try if abs($dif) < $err;
    $len /= 2;
    $try = $dif > 0 ? $try + $len : $try - $len;
    return mysqrt($tgt, $err, $try, $len);
}
[download]

Update: The adjustment for values < 1 can be mathematically simplified. Additionally, a much faster converging algorithm would be $try = (($tgt / $try) + $try) / 2.