I'm really surprised that computing binomial numbers for floating point real/complex numbers so slow in Pari-Gp. For example: binomial(1140000.78,420000) takes about 6 seconds and more than 22 MB Ram on my computer.
A much faster way: F(n,k)=gamma(n+1)/gamma(k+1)/gamma(n-k+1) gives binomial(n,k)