4 ms·

Here is a fascinating post by Stefan Karpinski, one of the creators of Julia: https://discourse.julialang.org/t/array-ordering-and-naive-summation/1929 https://

by celrod 8y ago

Here is a fascinating post by Stefan Karpinski, one of the creators of Julia: https://discourse.julialang.org/t/array-ordering-and-naive-summation/1929 https://discourse.julialang.org/t/array-ordering-and-naive-s... He shows off a function named "sumsto" which takes a single positive double precision floating point number "x" as an argument. "sumsto" always returns the same vector of 2046 floating point numbers -- just in an order so that naive left-to-right summation returns "x". Just changing the order of that vector lets you sum to almost any positive double precision number. If you want to run the code, I didn't see where "realmax" was defined, but this works: realmax() = prevfloat(typemax(Float64))

joe_the_user 8y ago
You think that's bad? Theoretical math has "conditionally convergent series'" [1] Sum in order and it can convergence to a given value. Rearrange the terms and any real number is possible. I have a hunch there could be a bit of relation between these things. [1] https://en.wikipedia.org/wiki/Conditional_convergence https://en.wikipedia.org/wiki/Conditional_convergence
- StefanKarpinski 8y ago
  It's an interesting connection. Any finite collection of real values (represented in floating-point or otherwise) has a definite correct sum. If you're constrained to the floating-point to represent that value, you should ideally round the true value to the closest float and return that. Computing this true sum of a sequence of floating-point values is surprisingly hard though. The state of the art seems to be Radford Neal's paper on superaccumulators from 2015: https://arxiv.org/abs/1505.05571 https://arxiv.org/abs/1505.05571 Fortunately pairwise summation (Julia's default) is fast and fairly hard to trip up, although it certainly is possible to trick it.
StefanKarpinski 8y ago
`realmax` got renamed to `floatmax` in 1.0; similarly, `realmin` got renamed to `floatmin` (they're both very much specific to floating point types).

by celrod 8y ago

joe_the_user 8y ago

You think that's bad? Theoretical math has "conditionally convergent series'" [1] Sum in order and it can convergence to a given value. Rearrange the terms and any real number is possible. I have a hunch there could be a bit of relation between these things. [1] https://en.wikipedia.org/wiki/Conditional_convergence https://en.wikipedia.org/wiki/Conditional_convergence

StefanKarpinski 8y ago
It's an interesting connection. Any finite collection of real values (represented in floating-point or otherwise) has a definite correct sum. If you're constrained to the floating-point to represent that value, you should ideally round the true value to the closest float and return that. Computing this true sum of a sequence of floating-point values is surprisingly hard though. The state of the art seems to be Radford Neal's paper on superaccumulators from 2015: https://arxiv.org/abs/1505.05571 https://arxiv.org/abs/1505.05571 Fortunately pairwise summation (Julia's default) is fast and fairly hard to trip up, although it certainly is possible to trick it.

StefanKarpinski 8y ago

`realmax` got renamed to `floatmax` in 1.0; similarly, `realmin` got renamed to `floatmin` (they're both very much specific to floating point types).