Alpha max plus beta min algorithm

The alpha max plus beta min algorithm¹ 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 ${\displaystyle |z|={\sqrt {a^{2}+b^{2}}}}$ 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 $${\displaystyle |z|=\alpha \,\mathbf {Max} +\beta \,\mathbf {Min} ,}$$ where ${\displaystyle \mathbf {Max} }$ is the maximum absolute value of a and b, and ${\displaystyle \mathbf {Min} }$ is the minimum absolute value of a and b.

For the closest approximation, the optimum values for ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are ${\displaystyle \alpha _{0}={\frac {2\cos {\frac {\pi }{8}}}{1+\cos {\frac {\pi }{8}}}}=0.960433870103...}$ and ${\displaystyle \beta _{0}={\frac {2\sin {\frac {\pi }{8}}}{1+\cos {\frac {\pi }{8}}}}=0.397824734759...}$, giving a maximum error of 3.96%.

${\displaystyle \alpha \,\!}$	${\displaystyle \beta \,\!}$	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
${\displaystyle \alpha _{0}}$	${\displaystyle \beta _{0}}$	3.96	2.41

When ${\displaystyle \alpha <1}$, ${\displaystyle |z|}$ becomes smaller than ${\displaystyle \mathbf {Max} }$ (which is geometrically impossible) near the axes where ${\displaystyle \mathbf {Min} }$ is near 0. This can be remedied by replacing the result with ${\displaystyle \mathbf {Max} }$ whenever that is greater, essentially splitting the line into two different segments.

${\displaystyle |z|=\max(\mathbf {Max} ,\alpha \,\mathbf {Max} +\beta \,\mathbf {Min} ).}$

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 ${\displaystyle \alpha }$ and higher ${\displaystyle \beta }$ can therefore increase precision further.

This can be improved even further by, instead of using ${\displaystyle |z_{0}|=1\cdot \mathbf {Max} +0\cdot \mathbf {Min} }$ as the second estimate, use a second pair of parameters ${\displaystyle \alpha _{0}}$ and ${\displaystyle \beta _{0}}$, with ${\displaystyle \alpha _{1}}$ and ${\displaystyle \beta _{1}}$ adjusted accordingly.

${\displaystyle |z|=\max \left(|z_{0}|,|z_{1}|\right),}$

${\displaystyle |z_{0}|=\alpha _{0}\,\mathbf {Max} +\beta _{0}\,\mathbf {Min} ,}$

${\displaystyle |z_{1}|=\alpha _{1}\,\mathbf {Max} +\beta _{1}\,\mathbf {Min} .}$

${\displaystyle \alpha _{0}}$	${\displaystyle \beta _{0}}$	${\displaystyle \alpha _{1}}$	${\displaystyle \beta _{1}}$	Largest error (%)
1	0	7/8	17/32	−2.66%
1	0	29/32	61/128	+2.40%
1	0	0.898204193266868	0.485968200201465	±2.12%
1	1/8	7/8	33/64	−1.67%
1	5/32	27/32	71/128	+1.21%
127/128	3/16	27/32	71/128	−1.12%

Of course, a non-zero ${\displaystyle \beta _{0}}$ requires at least one extra addition and some bit-shifts (or a multiplication), nearly doubling the cost and, depending on the hardware, possibly defeating the purpose of using an approximation in the first place.