Ask Slashdot: How Reproducible Is Arithmetic In the Cloud?

goodminton writes "I'm research the long-term consistency and reproducibility of math results in the cloud and have questions about floating point calculations. For example, say I create a virtual OS instance on a cloud provider (doesn't matter which one) and install Mathematica to run a precise calculation. Mathematica generates the result based on the combination of software version, operating system, hypervisor, firmware and hardware that are running at that time. In the cloud, hardware, firmware and hypervisors are invisible to the users but could still impact the implementation/operation of floating point math. Say I archive the virutal instance and in 5 or 10 years I fire it up on another cloud provider and run the same calculation. What's the likelihood that the results would be the same? What can be done to adjust for this? Currently, I know people who 'archive' hardware just for the purpose of ensuring reproducibility and I'm wondering how this tranlates to the world of cloud and virtualization across multiple hardware types."

Comments

Fixed-point arithmetic

[mkremer]

Use Fixed-point arithmetic.
In Mathematica make sure to specify your precision.
Look at 'Arbitrary-Precision Numbers' and 'Machine-Precision Numbers' for more information on how Mathematica does this.

Re:Fixed-point arithmetic

[Giant Electronic Bra]

Trust me, its a subject I've studied. The problem here is that your system is unstable, tiny differences in inputs generate huge differences in output. You cannot simply take one set of inputs that produces what you think is the 'right answer' from that system and ignore all the rest! You have to explore the ensemble behavior of many different sets of inputs, and the overall set of responses of the system is your output, not any one specific run with specific inputs that would produce a totally different result if one was off by a tiny bit.

Of course Lorenz realized this. Simple experiments with an LDE will show you this kind of result. You simply cannot treat these systems the way you would ones which exhibit function-like behavior (at least within some bounds). Lorenz of course also realized THAT, but sadly not everyone has got the memo yet! lol.

Re:Fixed-point arithmetic

[tlhIngan]

Don't use floating point if you can avoid it.

If you can't, and the results are EXTREMELY important (remember, floating point is an APPROXIMATION of numbers), then you have to read What Every Computer Scientist Should Know About Floating Point Numbers. (Yes, it's an Oracle link, but if you google it, most of the links are PDFs while the Oracle one is HTML).

If you're worried about your cloud provider screwing with your results, then you're definitely doing it wrong (read that article).

And yes, lots of people, even scientists, do it wrong because the idealized notion of what a floating point type is and how it actually works in hardware is completely different. Floating point numbers are tricky - they're VERY easy to use, but they're also VERY easy to use wrongly, and it's only if you know how the actual hardware is doing the calculations can you structure your programs and algorithms to do it right.

And no actual hardware FPU or VPU (vector unit - some do floating point) implements the full IEEE spec. Many come close, but none implement it exactly - there's always an omission or two. Especially since a lot of FPUs provide extended precision that goes beyond IEEE spec.

Re:Fixed-point arithmetic

[goodminton]

Awesome link! I'm the OP and I really appreciate your response. The reason I'm looking into this is that I work with many scientists who use commercial software packages where they don't control the code or compiler and their results are archived and can be reanalyzed years later. I was recently helping someone revive an old server to perform just such a reanalysis and we had so much trouble getting the machine going again I started planning to clone/virtualize it. That got me thinking about where to put the virtual machine (dedicated hardware, cloud, etc) and it also got me curious about hypervisors. I found some papers indicating that commercial hypervisors can have variability in their floating point math performance and all of that culminated in my post. Thanks again.

Your chances are pretty darned good

[Red Jesus]

Mathematica in particular uses adaptive precision; if you ask it to compute some quantity to fifty decimal places, it will do so.

In general, if you want bit-for-bit reproducible calculations to arbitrary precision, the MPFR library may be right for you. It computes correctly-rounded special functions to arbitrary accuracy. If you write a program that calls MPFR routines, then even if your own approximations are not correctly-rounded, they will at least be reproducible.

If you want to do your calculations to machine precision, you can probably rely on C to behave reproducibly if you do two things: use a compiler flag like -mpc64 on GCC to force the elementary floating point operations (addition, subtraction, multiplication, division, and square root) to behave predictably, and use a correctly-rounded floating point library like crlibm (Sun also released a version of this at one point) to make the transcendental functions behave predictably.