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> One of the common interview questions I ask is computing the sum of an array of doubles. Barely 1% of applicants can even get the basics correctly such as hav

> One of the common interview questions I ask is computing the sum of an array of doubles. Barely 1% of applicants can even get the basics correctly such as having multiple partial sums. Ehhh... I've got multiple hats from my college classes. Yes, optimizations say that multiple partial sums is fastest. But... my floating-point analysis classes has my brain yelling "cancellation error". The only way to minimize cancellation error is to sort the numbers from biggest to smallest, and then separate them from positives-and-negatives. You add the numbers from smallest to largest in magnitude, only matching signs, and then finally combine them at the end. The only two partial sums you should be keeping are "positive numbers" and "negative numbers", and everything else needs to be folded together to minimize rounding and cancellation errors. -------- You know. For stability and minimization of errors. There's other, faster, methods to possibly be employed. But optimization of partial sums is probably the last thing I'm thinking if I see "float" or "double". If things are "int" or "long", I'll have a higher chance of actually thinking (and caring) about optimization, because stability/cancellation error isn't a thing with int or long.

mgaunard 3y ago
The reason you need multiple sums is because there is a loop dependency which causes every iteration to stall and prevents pipelining and saturating the multiple compute units of your superscalar processor. Yes, floating-point isn't associative, so the results aren't numerically equivalent, and that is why you need to do that optimization yourself since the compiler isn't allowed to do it for you. As per the exercise, there is no information about the distribution of the data, so there is no indication that any order would be any better than another for numerical stability. If you care about stability you'd implement a kahan sum, which isn't really affected by changing the order of the sum so that you can parallelize it.
- dragontamer 3y ago
  Kahan summing doesn't fix cancellation error. It just extends the precision by another 53-bits or so. The only way you fix cancellation error is summing the positives in one variable, and the negatives in another, and do a big cancellation at the end. Anyway, my overall point is not to criticize Kahan summing or other techniques. My point is that these techniques are incompatible with optimizations. The faster way is often times a less stable operation.

mgaunard 3y ago

The reason you need multiple sums is because there is a loop dependency which causes every iteration to stall and prevents pipelining and saturating the multiple compute units of your superscalar processor. Yes, floating-point isn't associative, so the results aren't numerically equivalent, and that is why you need to do that optimization yourself since the compiler isn't allowed to do it for you. As per the exercise, there is no information about the distribution of the data, so there is no indication that any order would be any better than another for numerical stability. If you care about stability you'd implement a kahan sum, which isn't really affected by changing the order of the sum so that you can parallelize it.

dragontamer 3y ago
Kahan summing doesn't fix cancellation error. It just extends the precision by another 53-bits or so. The only way you fix cancellation error is summing the positives in one variable, and the negatives in another, and do a big cancellation at the end. Anyway, my overall point is not to criticize Kahan summing or other techniques. My point is that these techniques are incompatible with optimizations. The faster way is often times a less stable operation.