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    Store $x= log($X)
    Store $y= log($Y)
    Want  $z= log( $X + $Y )
    But we don't know $X and $Y, just $x and $y so
    Want  $z= log( exp($x) + exp($y) )

    $z= log( exp($x) + exp($y) )
      = log( exp($x)*( 1 + exp($y)/exp($x) ) )
      = log( exp($x) * ( 1 + exp($y-$x) ) )
      = log(exp($x)) + log( 1 + exp($y-$x) )
      = $x + log( 1 + exp($y-$x) )