The number $$\frac{\ln(90)-\ln(83)}{\ln(87)-\ln(71)}$$ is an approximation of the number $$\int_0^1 \tan(x^2)dx$$
The approximation is the best for positive integers $\ a,b,c,d\ $ with $\ a<b\le 100\ $ and $c<d\le 100\ $ (See formula below). The error is $\ 2.2\cdot 10^{-8}$.
How can I find approximations of some real number of the form $$\frac{\ln(b)-\ln(a)}{\ln(d)-\ln(c)}$$ efficiently, where $a,b,c,d$ are positive integers with $a<b$ and $c<d$ ?
I do not see how I can apply continued fractions or linear dependency of real numbers to find quartupels $\ (a,b,c,d)\ $ giving good approximations. Any ideas ?