Alpha max plus beta min algorithm

The alpha max plus beta min algorithm is a high-speed approximation of the square root of the sum of two squares. The square root of the sum of two squares, also known as Pythagorean addition, is a useful function, because it finds the hypotenuse of a right triangle given the two side lengths, the norm of a 2-D vector, or the magnitude of a complex number z=a+bi given the real and imaginary parts.

The algorithm avoids performing the square and square-root operations, instead using simple operations such as comparison, multiplication, and addition. Some choices of the α and β parameters of the algorithm allow the multiplication operation to be reduced to a simple shift of binary digits that is particularly well suited to implementation in high-speed digital circuitry.

The approximation is expressed as:

Where M a x is the maximum absolute value of a and b and M i n is the minimum absolute value of a and b.

For the closest approximation, the optimum values for α and β are α 0 = 2 cos ⁡ π 8 1 + cos ⁡ π 8 = 0.96043387... and β 0 = 2 sin ⁡ π 8 1 + cos ⁡ π 8 = 0.39782473... , giving a maximum error of 3.96%.