sub frac2bin { # convert a fraction to binary string (B32)
    my $frac = shift;
    my $bin ="";
    while (length($bin) < 32) {
      $bin = $bin . int($frac*2);
      $frac = $frac*2 - int($frac*2);
    }
    $bin;
  } # end sub frac2bin