feat: added support for comparison operators

[kashbrti]

Description

added support for comparison operators between bignums

Problem*

Resolves

Summary*

Additional Context

PR Checklist*

• I have tested the changes locally.
• I have formatted the changes with Prettier and/or cargo fmt on default settings.

[kashbrti]

We still need to add way more testing to this. also need to address the changes from the PR with the edge case bugs.

[kashbrti]

I was first thinking about skipping the if statement by casting the bool to Field and then computing intended_lhs = (1- comp_result as Field)* lhs + comp_result * rhs but not sure whether looping over the limbs and doing field multiplications would be any better than a conditional.

[kashbrti]

@TomAFrench

[TomAFrench]

If you just want to swap the lhs and rhs then you can do

let (lhs, rhs) = if comp_result {
  (rhs, lhs)
} else {
  (lhs, rhs)
}

This would end up with needing to merge two [Field;N] values but that's more efficient than duplicating the logic currently in the if-statement. (This could be made better by just returning the element-wise difference between the two arrays rather than directly reordering them)

[TomAFrench]

The best thing to do in this situation however would be to integrate the benchmarking system I added to https://github.com/noir-lang/noir-library-starter.

[kashbrti]

thanks I was wondering if there's a way to do a swap in a single line :D

[kashbrti]

wonder how bad this is in brillig @TomAFrench. should I do what I did in the constrained version and just duplicate the code?

[TomAFrench]

Could you not refactor this function to have a private implementation which accepts two U60Reprs?

pub(crate) unconstrained fn __cmp_remainder<let N: u32>(
    lhs: [Field; N],
    rhs: [Field; N],
) -> (bool, [Field; N], [bool; N], [bool; N]) {
    let mut a_u60: U60Repr<N, 2> = U60Repr::from(lhs);
    let mut b_u60: U60Repr<N, 2> = U60Repr::from(rhs);
    let mut temp_u60: U60Repr<N, 2> = U60Repr::one();
    let underflow = b_u60.gte(a_u60);

    if underflow {
        __cmp_remainder_priv(b_u60, a_u60);
    } else {
        __cmp_remainder_priv(a_u60, b_u60);
    }

Again, benchmarking would be way to make these decisions.

[kashbrti]

yep I can make a private function that does the second part (computing carries etc). will do that now.

[kashbrti]

did the refactoring. I think we can refactor it a lot here. these operations are done over and over in all the with_flags functions.