Quantics grid rounding problem

[thkloss]

grid::Quantics may return the wrong bitvector for a given coordinate.
In the given test example, we sample over all 2**nBit gridpoints and compare the corresponding bitvector from the grid.coord_to_id(x)
method and a reference implementation. Both differ from time to time, and one can check by looking at the actual output that the grid.coord_to_id(x) method does not always increment the bits properly.
This behaviour is probably due to some rounding issues, due to the intermediate conversion to a floating point number.

[thkloss]

run the test with: xfac_test "Test quantics rounding"

[elv3rs]

Ive tried taking a look at this to determine if it was causing convergence problems i had (it wasnt), but, heres what i found:

bitvec = grid.coord_to_id(grid.id_to_coord(bitvec)) occasionally does not return the same bitvec.
Since the bitvec losslessly maps to an integer, we can write the operation as follows:

x = a + deltaX * i
i = int( (x-a) / deltaX )

which algebraically works just fine, stuff cancels and we get:
i = int(i)

Where casting to int means truncation, i.e.
i = floor(i)

In reality though, we are using floats, which incurr some error at each step. Thus:
i = floor(i+eps)

Now, if eps is positive, we are fine due to truncation, for negative values though, we run into a problem (e.g. 1.99999999->1).

One simple solution therefore is to round before converting back to an integer.
i = floor(round(i+eps))

Another one would be adding a small bias (e.g. 2.4999999->2).
i = floor(i+eps+0.5)

Apparently, there is already a fix in #20, but fixing the mapping might cause some tci regression tests to fail, needing further investigation.