Precompute grids and coeffs

[oxinabox]

grids and coeffs are always the same for standard grids for given p and q.

Precomputing them helps decrease the cost assoicated with adapt in the gradients.
This also fixes up some of the type instability with the central gradient, during adapt.

using BenchmarkTools
using FiniteDifferences

const _fdm = central_fdm(2,1);
const _fdm5 = central_fdm(5,1);

const xs = collect(1:0.1:200);
f(x) = sum(sin, x);
@btime grad($_fdm, $f, $xs);
@btime grad($_fdm5, $f, $xs);

Current

julia> @btime grad($_fdm, $f, $xs);
  202.552 ms (154789 allocations: 4.26 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  413.311 ms (162753 allocations: 5.45 MiB)

---

New with Precomputed coeffs

julia> @btime grad($_fdm, $f, $xs);
  195.610 ms (53759 allocations: 1.96 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  386.412 ms (53759 allocations: 1.87 MiB)

[oxinabox]

From stacking this ontop of #60:

julia> @btime grad($_fdm, $f, $xs);
  193.883 ms (51768 allocations: 1.93 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  400.359 ms (51768 allocations: 1.84 MiB)

With #60 alone:

julia> @btime grad($_fdm, $f, $xs);
  200.205 ms (49777 allocations: 2.66 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  398.832 ms (59732 allocations: 3.87 MiB)

interesting that it seems to make the central_fdm(2) case slightly worse.
Not sure whats up with that.

[oxinabox]

From stacking this ontop of #60:

julia> @btime grad($_fdm, $f, $xs);
  193.883 ms (51768 allocations: 1.93 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  400.359 ms (51768 allocations: 1.84 MiB)

With #60 alone:

julia> @btime grad($_fdm, $f, $xs);
  200.205 ms (49777 allocations: 2.66 MiB)

julia> @btime grad($_fdm5, $f, $xs);
  398.832 ms (59732 allocations: 3.87 MiB)

interesting that it seems to make the central_fdm(2) case slightly worse.
Not sure whats up with that.

[KristofferC]

Instead of precomputing you could memoize. Avoids having to determine upfront what exact combinations should be precomputed. Taking the cost once per session is probably ok.

[oxinabox]

Should this be defined in a constant?
Or should it be declared outside the eval, and then inlined in as a local literal
and thus should constant fold

[oxinabox]

Instead of precomputing you could memoize. Avoids having to determine upfront what exact combinations should be precomputed. Taking the cost once per session is probably ok.

Yeah, good idea.

I think I might have looked into it, and struggled to workout how to do it without getting allocations.
The precomputes use tuples which are very compiler friendly.

But I can't recall if I did, so I should check again

[oxinabox]

No point with #73