Comment by TeMPOraL

> Is our tooling too bad for this?

Floating points are fundamentally too bad for this. We use them because they're fast, which usually more than compensates for inaccuracies FP math introduces.

(One, dealing with FP errors is mostly a fixed cost - there's a branch of CS/mathematics specializing in it, producing formally proven recipes for computing specific things in way that minimize or at least give specific bounds on errors. That's work that can be done once, and reused forever. Two, most programmers are oblivious to those issues anyway, and we've learned to live with the bugs :).)

When your parallel map-reduce is just doing matrix additions and multiplications, guaranteeing order of execution comes with serious overhead. For one, you need to have all partial results available together before reducing, so either the reduction step needs to have enough memory to store a copy of all the inputs, or it needs to block the units computing those inputs until all of them finish. Meanwhile, if you drop the order guarantee, then the reduction step just needs one fixed-size accumulator, and every parallel unit computing the inputs is free to go and do something else as soon as it's done.

So the price you pay for deterministic order is either a reduction of throughput or increase in on-chip memory, both of which end up translating to slower and more expensive hardware. The incentives strongly point towards not giving such guarantees if it can be avoided - keep in mind that GPUs have been designed for videogames (and graphics in general), and for this, floating point inaccuracies only matter when they become noticeable to the user.