Rationalise and check

Rationalising $\dfrac{1}{a + \sqrt{b}}$ gives $\dfrac{a - \sqrt{b}}{a^2 - b}$. You will write both forms and let the computer check that they agree.

The idea. Two decimals that should be equal can differ in the last digit, so compare them with a tiny allowance: abs(x - y) < 1e-12.

Your tasks.

1. In new_bottom, return $a^2 - b$.
2. In rationalised, return $\dfrac{a - \sqrt{b}}{a^2 - b}$.
3. Fill agree with True or False for each pair in pairs: do the two forms agree?