Alpha max plus beta min algorithm

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The locus of points that give the same value in the algorithm, for different values of alpha and beta.

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 is the maximum absolute value of a and b and is the minimum absolute value of a and b.

For the closest approximation, the optimum values for and are and , giving a maximum error of 3.96%.

Largest error (%) Mean error (%)
1/1 1/2 11.80 8.68
1/1 1/4 11.61 3.20
1/1 3/8 6.80 4.25
7/8 7/16 12.50 4.91
15/16 15/32 6.25 3.08
3.96 2.41
Alpha Max Beta Min approximation.png

See also[edit]

  • Hypot, a precise function or algorithm that is also safe against overflow and underflow


  • Lyons, Richard G. Understanding Digital Signal Processing, section 13.2. Prentice Hall, 2004 ISBN 0-13-108989-7.
  • Griffin, Grant. DSP Trick: Magnitude Estimator.