Suppose I have the function $$f(x) = \sqrt{1+x^4} - \sqrt{1-x^3}$$
I want to compute it when $x$ is very small, but this introduces lots of big errors because both terms in the function are of similar magnitude.
So I was thinking I could rearrange it so that this subtraction of terms of similar magnitude for small $x$ goes away... but how could it be changed?
My only thinking was that I could use the Taylor Series expansion of $f$ at $0$.... but this is obviously a lot worse than if I just had another form for the general function that was easily computable for small values of $x$.
I tried doing log/exponent tricks and refactoring so that I get an addition or something, but have not been successful.
You could write (rationalising the denominator) $$f(x)=\frac{x^4+x^3}{\sqrt{1+x^4}+\sqrt{1-x^3}}.$$