Replacing Square Roots by Pythagorean Sums

An algorithm is presented for computing a “Pythagorean sum” a ⊕ b = √(a
²
+ b
²
) directly from a and b without computing their squares or taking a square root. No destructive floating point overflows or underflows are possible. The algorithm can be extended to compute the Euclidean norm of a vector. The resulting subroutine is short, portable, robust, and accurate, but not as efficient as some other possibilities. The algorithm is particularly attractive for computers where space and reliability are more important than speed.