approximate $\sqrt{x+y}$

Question

How can one approximate the expression

$$\sqrt{x+y}$$

I think this can achieved by a Taylor expansion but I don't know how.

Answer 1

If $x\gg |y|$, the binomial theorem gives a Taylor series $\sqrt{x}\left(1+\tfrac{y}{x}\right)^{1/2}$.

Answer 2

A very simple approximation, often used in early computers, is, if $x \ge y >0$, $$ \sqrt{x^2+y^2} = x \cdot \sqrt{\frac{1+y^2}{x^2}} \approx x \cdot \frac{1+y^2}{2x^2} = \frac{x+y^2}{2x}. $$

Answer 3

If $x\gg y \ge 0$, then $\sqrt{x+y} = \sqrt{x}\sqrt{1+y/x} \approx \sqrt{x}(1+y/(2x)...)$