A simple estimation/approximation of square root

Question

Someone works in a kitchen cabinetry workshop and his calculator/phone doesn't have a square root function. He intends to use Pythagorean theorem to calculate right triangle sides for cabinet doors etc. like:

a^2 + b^2 = c^2

He asked me to provide him a formula consisting of -, +, / and * operations. Note that his math knowledge is not sophisticated, therefor his is not able to use iterative approaches. I researched for a rational estimation, but I couldn't find a proper one. I wonder if anybody can help.

Answer 1

Since we can assume that $a$ and $b$ are measured in $m$ and barely won't exceed $10$ , the following approximation should be good enough for $a\ge b$ :

$$\sqrt{a^2+b^2}\approx \frac{a(5b^4+20a^2b^2+16a^4)}{b^4+12a^2b^2+16a^4}$$

Answer 2

Thanks to @Peter comment and this Taylor expansion post, the square root of a number x can be approximated by such a formula:

$\sqrt{x}\approx 1+\dfrac{(x-1)}{2}-\dfrac{(x-1)^2}{8}+\dfrac{(x-1)^3}{16}-\dfrac{5(x-1)^4}{128}+\dfrac{7(x-1)^5}{256}-\dfrac{21(x-1)^6}{1024}+\dfrac{231(x-1)^7}{14336}$

I'm going to introduce this equation to the wood-working guy. Hopefully it is simple enough.