Thursday, 2 August 2018

Implementing Euclidean division

While implementing various kinds of division in haroldbot, I had to look up/work out how to implement different kinds of signed division in terms of unsigned division. The common truncated division (written as /s in this post and in haroldbot, /t in some other places) is natural result of using your intuition from ℝ and writing the definition based on signs and absolute values, ensuring that the division only happens between non-negative numbers (making its meaning unambiguous) and that the result is an integer: $$\DeclareMathOperator{\sign}{sign} D /_s d = \sign(d)\cdot\sign(D)\cdot\left\lfloor\cfrac{\left|D\right|}{\left|d\right|}\right\rfloor$$ That definition leads to a plot like this, showing division by 3 as an example:

Of course the absolute values and sign functions create symmetry around the origin, and that seems like a reasonable symmetry to have. But that little plateau around the origin often makes the mirror at the origin a kind of barrier that you can run into, leading to the well-documented downsides of truncated division.

The alternative floored division and Euclidean division have a different symmetry, which does not lead to that little plateau, instead the staircase pattern simply continues:

The point of symmetry, marked by the red cross, is at (-0.5, -0.5). Flipping around -0.5 may remind you of bitwise complement, especially if you have read my earlier post visualizing addition, subtraction and bitwise complement, and mirroring around -0.5 is no more than a conditional complement. So Euclidean division may be implemented with positive division as: $$\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\xor}{\bigoplus} D /_e d = \sign(d)\cdot(\sgn(D)\xor\left\lfloor\cfrac{D\xor\sgn(D)}{\left|d\right|}\right\rfloor)$$ Where the sgn function is -1 for negative numbers and 0 otherwise, and the circled plus is XOR. XORing with the sgn is a conditional complement, with the inner XOR being responsible for the horizontal component of the symmetry and the outer XOR being responsible for the vertical component.

It would have been even nicer if the symmetry of the divisor also worked that way, but unfortunately that doesn't quite work out. For the divisor, the offset introduced by mirroring around -0.5 would affect the size of the steps of the staircase instead of just their position.

The /e and %e operators should be live in haroldbot soon, though like all forms of division the general case is really too hard, even for the circuit-SAT based truth checker (the BDD engine stands no chance at all).

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