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 | z | = a 2 + b 2 {\displaystyle |z|={\sqrt {a^{2}+b^{2}}}} 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 | z | = α M a x + β M i n , {\displaystyle |z|=\alpha \,\mathbf {Max} +\beta \,\mathbf {Min} ,} where M a x {\displaystyle \mathbf {Max} } is the maximum absolute value of a and b, and M i n {\displaystyle \mathbf {Min} } is the minimum absolute value of a and b.
For the closest approximation, the optimum values for α {\displaystyle \alpha } and β {\displaystyle \beta } are α 0 = 2 cos π 8 1 + cos π 8 = 0.960433870103... {\displaystyle \alpha _{0}={\frac {2\cos {\frac {\pi }{8}}}{1+\cos {\frac {\pi }{8}}}}=0.960433870103...} and β 0 = 2 sin π 8 1 + cos π 8 = 0.397824734759... {\displaystyle \beta _{0}={\frac {2\sin {\frac {\pi }{8}}}{1+\cos {\frac {\pi }{8}}}}=0.397824734759...} , giving a maximum error of 3.96%.
| α {\displaystyle \alpha \,\!} | β {\displaystyle \beta \,\!} | 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 |
| α 0 {\displaystyle \alpha _{0}} | β 0 {\displaystyle \beta _{0}} | 3.96 | 2.41 |
Improvements
When α < 1 {\displaystyle \alpha <1} , | z | {\displaystyle |z|} becomes smaller than M a x {\displaystyle \mathbf {Max} } (which is geometrically impossible) near the axes where M i n {\displaystyle \mathbf {Min} } is near 0. This can be remedied by replacing the result with M a x {\displaystyle \mathbf {Max} } whenever that is greater, essentially splitting the line into two different segments.
| z | = max ( M a x , α M a x + β M i n ) . {\displaystyle |z|=\max(\mathbf {Max} ,\alpha \,\mathbf {Max} +\beta \,\mathbf {Min} ).}Depending on the hardware, this improvement can be almost free.
Using this improvement changes which parameter values are optimal, because they no longer need a close match for the entire interval. A lower α {\displaystyle \alpha } and higher β {\displaystyle \beta } can therefore increase precision further.
Increasing precision: When splitting the line in two like this one could improve precision even more by replacing the first segment by a better estimate than M a x {\displaystyle \mathbf {Max} } , and adjust α {\displaystyle \alpha } and β {\displaystyle \beta } accordingly.
| z | = max ( | z 0 | , | z 1 | ) , {\displaystyle |z|=\max {\big (}|z_{0}|,|z_{1}|{\big )},} | z 0 | = α 0 M a x + β 0 M i n , {\displaystyle |z_{0}|=\alpha _{0}\,\mathbf {Max} +\beta _{0}\,\mathbf {Min} ,} | z 1 | = α 1 M a x + β 1 M i n . {\displaystyle |z_{1}|=\alpha _{1}\,\mathbf {Max} +\beta _{1}\,\mathbf {Min} .}| α 0 {\displaystyle \alpha _{0}} | β 0 {\displaystyle \beta _{0}} | α 1 {\displaystyle \alpha _{1}} | β 1 {\displaystyle \beta _{1}} | Largest error (%) |
|---|---|---|---|---|
| 1 | 0 | 7/8 | 17/32 | −2.65% |
| 1 | 0 | 29/32 | 61/128 | +2.4% |
| 1 | 0 | 0.898204193266868 | 0.485968200201465 | ±2.12% |
| 1 | 1/8 | 7/8 | 33/64 | −1.7% |
| 1 | 5/32 | 27/32 | 71/128 | 1.22% |
| 127/128 | 3/16 | 27/32 | 71/128 | −1.13% |
Beware however, that a non-zero β 0 {\displaystyle \beta _{0}} would require at least one extra addition and some bit-shifts (or a multiplication), probably nearly doubling the cost and, depending on the hardware, possibly defeat the purpose of using an approximation in the first place.
See also
- 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.
External links
- "Extension to three dimensions". Stack Exchange. May 14, 2015.
References
Assim, Ara Abdulsatar Assim (2021). "ASIC implementation of high-speed vector magnitude & arctangent approximator". Computing, Telecommunication and Control. 71 (4): 7–14. doi:10.18721/JCSTCS.14401. https://infocom.spbstu.ru/en/article/2021.71.01/ ↩