Speed up comparisons coming from scalar products

[saraedum]

in libflatsurf we very commonly ask for a sign of the form sgn(a * b ± c * d). (E.g. when deciding collinearity, orthogonality, or relative length of two vectors.)

Currently, we cast everything to arb numbers, try there, and if this does not work, run the formula in e-antic. That's a bit inefficient since we then do the arb computation twice and this also creates a lot more allocations than necessary.

[videlec]

What do you suggest for this? a * b ± c * d is a very specific multivariate polynomial.

For the interface, I think it makes sense to have something for scalar products

int scalar_product_sign(renf_elem * x, renf_elem * y, slong len, renf_t nf)

Though, depending on the use case creating x and y might require allocation. Before doing the arb computation we can even implement a shortcut based on bit sizes of the entries of x and y.

Do you also want a dedicated 2-dimensional predicate for a * b - c * d

int determinant_2x2_sign(renf_elem_t a, renf_elem_t b, renf_elem_t c, renf_elem_t d, renf_t nf)