Initial guess for babylonian method when calculating hypothenuse I have given side lengths $a$ and $b$ and wish to calculate hypothenuse $c$ using the Babylonian aka Newton-Raphson method for finding the square root of $c^2 = a^2 + b^2$. Since I know that $c^2$ is the result of adding $a^2$ and $b^2$ I figure there ought to be some clever way of using $a$ and $b$ to obtain a good initial guess for Babylonian. Thus far the best I've come up with for my initial guess is $|a| + |b|$. Is there something even better?

It is not entirely clear what operations are allowed. Without specifying it, some crazy things are possible even without taking the square root itself as an initial guess.

We'll assume we are allowed to use basic arithmetical operations. Without loss of generality, let $a>b>0$. Let $x=b/a$. Then $\sqrt{a^2+b^2}=a\sqrt{1+x^2}\approx a(1+x^2/2)$. So that is a good initial guess, especially if $x\ll1$. A more precise expansion can be obtained from Taylor series.

(Initially, this answer referred to another answer which is now gone, so I edited this one accordingly.)