See attached chart.
The first column is for the simple L=Y+X, R=Y-X with normalization. The second column is the same as the first except it uses clipping instead of normalization. The third column is a slightly (very slightly) more complicated algorithm which I believe gives the same results as what you get from the WPI library (and may indeed even be mathematically identical... I haven't checked).
The first 8 rows of the chart are the values around the perimeter of the large square in
this diagram, starting at the top center and going around the square clockwise. Notice that all three algorithms give the same values for these points.
The remaining rows show the outputs for intermediate joystick values. The three algorithms differ somewhat for these inputs.
None of these algorithms uses any trigonometry.