pseudo-identities which are not exact but the error is very small i would like to know more example of pseudo identities.. things that there are not equal but the error is about $ 0.01 $
for example $$ \pi ^{4} +\pi ^{5} =e^{6} $$
the error term is about $ 10^{-5} $
where can i see more of this amazing pseudo identities ? :D thanks

There are many examples given here at MSE. One of my favourites is that
$$
e^{\pi \sqrt{163}}=262 537 412 640 768 743.99999999999925
$$
is very close to an integer, see here. This has some serious number theoretical background, as is explained, and generalised, here.
