To use LLL to get good approximations to real numbers by algebraic numbers:
e.g.
lattice([[1,0,0,M], [0,1,0,M*x], [0,0,1,M*x^2]]);
in maple, where say  M:= 10^9 (I think) and x:=evalf(Pi) gives
[2453, -674, -34, 1468.], [-2195, -71, 245, 546.], ...
so e.g. 245 pi^2 -71 pi - 2195 ~~ 0  (=546*10^-9?).
Real root is  (71+sqrt(2156141))/490, correct to about 12 places.

Learned this from someone -- Richard Chen? -- but can't remember details.