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#!/usr/bin/perl
use warnings;
...
printf( "\$ub = %.23f\n", $ub );
print "\$lb == \$ub ", $lb == $ub ? 'true' : 'false', "\n";
print "\$changed = $changed\n";

P:\>fround
$mode = 0
...
$ub = 0.33333333333333337000000
$lb == $ub false
$changed = 0