## Same Programs + Different Computers = Different Weather Forecasts 240

knorthern knight writes

*"Most major weather services (US NWS, Britain's Met Office, etc) have their own supercomputers, and their own weather models. But there are some models which are used globally. A new paper has been published, comparing outputs from one such program on different machines around the world. Apparently, the same code, running on different machines, can produce different outputs due to accumulation of differing round-off errors. The handling of floating-point numbers in computing is a field in its own right. The paper apparently deals with 10-day weather forecasts. Weather forecasts are generally done in steps of 1 hour. I.e. the output from hour 1 is used as the starting condition for the hour 2 forecast. The output from hour 2 is used as the starting condition for hour 3, etc. The paper is paywalled, but the abstract says: 'The global model program (GMP) of the Global/Regional Integrated Model system (GRIMs) is tested on 10 different computer systems having different central processing unit (CPU) architectures or compilers. There exist differences in the results for different compilers, parallel libraries, and optimization levels, primarily due to the treatment of rounding errors by the different software systems. The system dependency, which is the standard deviation of the 500-hPa geopotential height averaged over the globe, increases with time. However, its fractional tendency, which is the change of the standard deviation relative to the value itself, remains nearly zero with time. In a seasonal prediction framework, the ensemble spread due to the differences in software system is comparable to the ensemble spread due to the differences in initial conditions that is used for the traditional ensemble forecasting.'"*
## Re:Have these people never heard of IEEE754???? (Score:5, Insightful)

differentrounding perturbations. Perturb weather or some other systems, and you will get a completely different trajectory.That said, many applied fields, including meteorology, could benefit from more well-disciplined computational science approaches. But don't expect all that much of a difference.

## Re:Have these people never heard of IEEE754???? (Score:2, Insightful)

I was in particular thinking about the section on rounding in IEEE754. You are also overlooking that badly conditioned != behaves in a random fashion. My guess is they did not involve the numerics people in the optimization process, which is a complete fail when you know your problem is not well conditioned.

## Re:Have these people never heard of IEEE754???? (Score:5, Insightful)

When floating point roundoff errors grow big enough to affect the outcome of the simulation, you have long since reached the point where you are not predicting anything useful any longer. It is not exactly a problem if the results differ at that point.

## Re:Have these people never heard of IEEE754???? (Score:4, Insightful)

When floating point roundoff errors grow big enough to affect the outcome of the simulation, you have long since reached the point where you are not predicting anything useful any longer.

This is not true. If the model predicts rain at 2 pm two days out and different rounding moves it to 3 pm, that is still a useful forecast in a lot of cases.

## Re:Damn you people (Score:5, Insightful)

Precision is the point. Mathematical chaos diverges exponentially. This means that if you have a value of 9.3440281 in one calculation and it returns 3.5 and a value of 9.344028147 in another, that you can get completely different results (where the second case returns 8.1). Now you say: well, let's just make it more precise then! So you put in the value of 9.34402814672 and get a completely different result (1.7), and so on*. If you weren't dealing with mathematical chaos, you would continually refine the values down (e.g. 3.5, 3.45, 3.467, etc.).

* Note: I should be careful with this layman's description to point out that more precise values technically shrink the window down. But since it is exponentially divergent in the first place, this might not ever do you any good in a realistic setting. Ref Lyapunov exponents [wikipedia.org] and mathematical chaos [wikipedia.org]

## Re:Have these people never heard of IEEE754???? (Score:5, Insightful)

Almost nothing you do with IEEE754 floating point numbers is correct in the strict mathematical sense. You can't even represent 0.1 (1/10) as an IEEE754 floating point number. There are entire series of lectures on the topic of scientific computing with floating point numbers. The errors are usually small enough that a few simple rules keep you safe (e.g., never compare floating point numbers for equality), but when you do many iterations, the errors can accumulate and mess with your results, and if in that case you do the calculations in a different order, the accumulated error will mess with your results in a different way. That's what's happening here.

## Utterly Unsurprising (Score:2, Insightful)

Floating Point arithmetic is not associative.

Everyone who reads Stack Overflow knows this, because every who doesn't know this posts to Stack Overflow asking why they get weird results.

Everyone who does numerical simulation or scientific programming work knows this because they've torn their hair out at least once wondering if they have a subtle bug or if it's just round-off error.

Everyone who does cross-platform work knows this because different platforms implement compilers (and IEEE-754) in slightly different ways.

Everyone who does parallel programming knows this because holy shit will you see round-off differences when you through many minutes of TFlops at a problem and it sequences difference every time.

Everyone who looks at the standards knows this because for Chrissakes, Fused-Multiply-Add is standards compliant but _optional_.

Why is this remotely news?

## Re:I've seen this before (Score:5, Insightful)

This often happens when the simulation results are influenced by variations in the accuracy of the built-in functions. Every floating point unit (FPU) returns an approximation of the correct result to an arbitrary level of accuracy, and the accuracy level of these results varies considerably when built-in functions like sqrt(), sin(), cos(), ln(), and exp() are considered. Normally, the accuracy of these results is pretty high. However, the initial 8087 FPU hardware from Intel was pretty old, and it necessarily made approximations.

At one point, Cyrix released an 80287 clone FPU that was faster and more accurate than Intel's 80287 equivalent. This broke many programs. Since then, Intel and AMD have been developing FPUs that are compatible with the 8087, ideally at least as accurate, and much faster. The GPU vendors have been doing something similar, however in video games, speed is more important than accuracy. For compatibility reasons (CPUs) and speed reasons (GPUs), vendors have focused on returning fast, compatible and reasonably accurate results.

In terms of accuracy, the results of the key transcendental functions, exponential functions, logarithmic functions, and the sqrt function should be viewed with suspicion. At high-accuracy levels, the least-significant bits of the results may vary considerably between processor generations, and CPU/GPU vendors. Additionally, slight differences in the results of double-precision floating point to 64-bit integer conversion functions can be detected, especially when 80-bit intermediate values are considered. Given these approximations, getting repeatable results for accuracy-sensitive simulations is tough.

It is likely that the articles weather simulations and the parent poster's simulations have differing results due to the approximations in built-in functions. Inaccuracies in the built-in functions are often much more significant that the differences due to round-off errors.

## Re:I've seen this before (Score:4, Insightful)

In theory both should have been the same, if they stuck rigidly to the IEEE specifications. There may be other explanations though.

Sometimes compilers create multiple code paths optimized for different CPU architectures. One might use SSE4 and be optimized for Intel CPUs, another might use AMD extensions and be tuned for performance on their hardware. There was actually some controversy when it was discovered that Intel's compiler disabled code paths that would execute quickly on AMD CPUs just because they were not Intel CPUs. Anyway, the point is that perhaps one machine was using different code and different super-scalar instructions, which operate at different word lengths. Compilers sometimes extend a 64 bit double to 80 bit super-scalar registers, for example.

Or one machine was a Pentium. Intel will never live that one down.

## Re:Have these people never heard of IEEE754???? (Score:5, Insightful)

"Remember, the desired result here is not a set of identical numbers everywhere. It is an accurate simulation."

*An* accurate simulation is not the desired result either, an accurate model is. Without reproducibility you don't have a model.

Reproducibility is important always.

## Re: Have these people never heard of IEEE754???? (Score:4, Insightful)

another one says the earth will absorb the heat Which one do you trust?

I think I'd have to go with the one that doesn't redefine "absorb" to mean "magically disappear".