Phase issue when pulses == steps-1

[jbum]

I've produced a test suite of all the patterns from Toussaint's paper, which I'll include here.

let test_suite = [
    [[3,2], [1, 1, 0]],                
    [[4,3], [1, 1, 1, 0]],
    [[5,2], [1, 0, 1, 0, 0]],
    [[5,3], [1, 0, 1, 0, 1]],
    [[5,4], [1, 1, 1, 1, 0]],
    [[6,5], [1, 1, 1, 1, 1, 0]],
    [[7,2], [1, 0, 0, 1, 0, 0, 0]],
    [[7,3], [1, 0, 1, 0, 1, 0, 0]],
    [[7,4], [1, 0, 1, 0, 1, 0, 1]],
    [[7,5], [1, 0, 1, 1, 0, 1, 1]],
    [[7,6], [1, 1, 1, 1, 1, 1, 0]],
    [[8,3], [1, 0, 0, 1, 0, 0, 1, 0]],
    [[8,5], [1, 0, 1, 1, 0, 1, 1, 0]],
    [[8,7], [1, 1, 1, 1, 1, 1, 1, 0]],
    [[9,4], [1, 0, 1, 0, 1, 0, 1, 0, 0]],
    [[9,5], [1, 0, 1, 0, 1, 0, 1, 0, 1]],
    [[9,7], [1, 0, 1, 1, 1, 0, 1, 1, 1]],
    [[10,3], [1, 0, 0, 1, 0, 0, 1, 0, 0, 0]],
    [[10,7], [1, 0, 1, 1, 0, 1, 1, 0, 1, 1]],
    [[11,3], [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]],
    [[11,4], [1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0]],
    [[11,5], [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0]],
    [[12,5], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0]],
    [[12,7], [1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0]],
    [[12,11],[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0]],
    [[13,5], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0]],
    [[13,6], [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0]],
    [[13,9], [1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1]],
    [[14,3], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0]],
    [[14,9], [1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]],
    [[15,4], [1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0]],
    [[15,7], [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0]],
    [[16,5], [1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0]],
    [[16,7], [1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0]],
    [[16,9], [1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0]],
    [[17,7], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0]],
    [[17,8], [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0]],
    [[18,7], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0]],
    [[20,9], [1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0]],
    [[20,11],[1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0]],
    [[22,9], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0]],
    [[23,9], [1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0]],
    [[24,11],[1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]],
    [[24,13],[1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0]],
    [[30,19],[1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1]],
    [[34,15],[1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0 ]],
];

I've found that there is a discrepancy between Bjorklund's algorithm (which several implementations use) and Toussaint's examples, in that patterns like (4,5) in which the pulses are one less than the steps, are produced with a different rotation. So for example, for 4/5, Toussaint shows (1,1,1,1,0) and several implementations of Bjorklund's algorithm give (1,0,1,1,1). Your algorithm also outputs (1,0,1,1,1).

I would not call this a bug in your code, as it appears to be a property of Bjorklund's algorithm, but it is something to be aware of. I have a python implementation which corrects this issue, which matches Toussaint's slightly different result, which I can provide on request.