(Undoubtedly) stupid question: would it be any faster to project (x, y) to the unit circle (x', y'), then compute acos(x') or asin(y'), and then correct the res

(Undoubtedly) stupid question: would it be any faster to project (x, y) to the unit circle (x', y'), then compute acos(x') or asin(y'), and then correct the result based on the signs of x & y? When converting Cartesian coordinates to polar, the value of r=HYPOT(x, y) is needed anyway, so the projection to the unit circle would be a single division by r.