6 ms·

Computation errors that will be discovered only with great difficulties, when someone is careful enough to use some additional verification methods for the nume

Computation errors that will be discovered only with great difficulties, when someone is careful enough to use some additional verification methods for the numeric results. The magnitude of the error caused by an underflow is completely unpredictable, which is why in any serious program it is unacceptable to ignore the underflows. Even when the errors happen to be discovered soon enough, people will be puzzled about their origin and they may waste a lot of time analyzing their algorithms, because there are a lot of things that can cause numeric errors. Because at their origin the underflows will neither signal any exception nor generate any unusual value, the errors will be normally caught much later, perhaps after thousands of other operations, when their cause will not be obvious. This fight between the people who want to get only correct results from computers and the people who do not care whether the results are erroneous as long as the results are obtained after a delay shorter by a few percents than for obtaining correct results has continued for decades, almost since electronic computers have been invented. Errors are acceptable in games or for some other graphics or audio applications that generate ephemeral images or sounds, but they are not acceptable for any engineering purposes. The option "-Ofast" is like smoking. There is no doubt that it is a bad habit, but whoever wants to smoke should be free to do it. On the other hand, exactly like a smoker should not be permitted to smoke in a closed room with non-smokers, the option "-Ofast" and the related options should not be permitted to alter the behavior of any other programs that are linked with an object file compiled with it. The behavior of "-Ofast" where it affects globally the content of MXCSR is unacceptable. The right behavior would have been for any function compiled with "-Ofast" to save MXCSR, put in it any desired value for the duration of the function execution and restore it at function exit. Moreover, when invoking any other function it should restore the original MXCSR before the call and put again in it the desired local value after the function returns back.

bjourne 2y ago
It's not so easy because -ffast-math can actually be more correct than running without.
- adrian_b 2y ago
  In most cases it is unpredictable when such an improvement happens and what is unpredictable about correctness is useless. It is much more frequent for -ffast-math to generate bigger errors than to benefit from an accidental cancellation of the errors. The only case when the result of underflows is predictable is when the values whose computation may result in an underflow are immediately added to other values that are known to be big enough, and they are not used for anything else. It is not frequent to have so much information about which expressions will underflow and which will not, and about the range of possible values of the values that would be added. When there really exists so much information about what must be computed, then there are chances that the computation can be rearranged in a way that would avoid the underflows, in which case there would be no need to change MXCSR to ignore the underflows, because they would not happen.
  - bjourne 2y ago
    > It is much more frequent for -ffast-math to generate bigger errors than to benefit from an accidental cancellation of the errors. In my experience underflows are very uncommon and indicate that you're doing something wrong, whereas fp contractions are extremely common. So I disagree with you.
    - mbauman 2y ago
      fp contract isn't a uniform "decrease in error" either, though. As a simple example, it introduces error in the straightforward: a*b - a*b
    - adrian_b 2y ago
      I agree that using fused multiply-add in almost all cases improves the accuracy of the results. Nevertheless, I consider that a compiler option that gives you good FMA, but only together with bad behavior at underflows, is something exceedingly stupid. FMA is a standard operation, while ignoring underflows is rightly prohibited by the floating-point standard. There should never have been a compiler option that mixes standard operations with non-standard operations. In C or C++, you can use the "fma" function to get FMA where desired. This is more awkward, but it is safe. Any better solution requires a change in the programming language. For instance, it could be required that adjacent * and + must always be contracted into an FMA, unless separated by parentheses. This would allow the programmer to select between FMA and FMUL+FADD, as desired. FMA has appeared in 1990 (in IBM POWER) and it has been the greatest advance in floating-point computation after the introduction of Intel 8087 in 1980 (which has resulted in the IEEE standard a few years later). It is weird that 34 years later, most, if not all, programming languages do not handle well the generation of FMA operations.
jcranmer 2y ago
> The behavior of "-Ofast" where it affects globally the content of MXCSR is unacceptable. > The right behavior would have been for any function compiled with "-Ofast" to save MXCSR, put in it any desired value for the duration of the function execution and restore it at function exit. Moreover, when invoking any other function it should restore the original MXCSR before the call and put again in it the desired local value after the function returns back. I completely agree here. The problem is that introducing floating-point environment into the programming model requires disabling some of the transformations you want to do with fast-math in the first place, so doing it this way kind of doesn't work to enable what you wanted to do. Furthermore, one of the SPEC benchmarks gets like a 30% speedup if you turn on denormal flushing, and that's the kind of change that gives compiler engineer management heartburn if you want to tell them to forgo that speedup because the optimization is dumb. I am grateful for blog posts like this one because this does help build a case for getting the compiler to stop doing stupid stuff like globally enabling DAZ/FTZ bits.
- adrian_b 2y ago
  Not all CPUs behave badly when handling underflows, denormals or other exceptional cases. On some CPUs the penalties for exceptional cases are negligible (e.g. on AMD Zen), so it is impossible for any SPEC benchmark to get a 30% speedup when ignoring underflows. There are also bad CPU models, which have a microprogrammed handling of the exceptional cases, which can add e.g. a penalty of somewhat more than one hundred CPU cycles for each underflow, which is very similar to the penalty for a load from the main memory (which misses the caches). If one of the SPEC benchmarks has so many underflows that on a bad CPU model it can be sped up by 30% when ignoring the underflows, then it is pretty much guaranteed that when the underflows are ignored the results computed by that benchmark are erroneous. Cheating at the SPEC benchmark by ignoring the underflows could be easily avoided by changing the benchmarking rules, by providing a file with the valid results and by not accepting any benchmark claims where the results do not match those known to be good. You can cheat at any SPEC benchmark with bogus compiler options, if there is no constraint on the compiled program to be correct. For instance you could make a compilation option that deletes 90% of the generated machine code, resulting in a 10 times faster benchmark program. For some reason, the compiler writers that handle the generation of code for floating-point operations feel that they are licensed to generate incorrect code, even when similarly incorrect code would be rejected in other contexts. The fact that the floating-point operations are inexact, so they are accompanied by some inherent errors, does not give the right to a compiler writer or library writer to introduce arbitrary errors in the computations. The reason why a standard exists is precisely for guaranteeing that the errors are bounded and you can predict their effects.
HappMacDonald 2y ago
This sounds quite a lot like my allergy to using floating point for doing anything critical at all. It's more difficult to force deterministic results on all platforms for float than it is for known-bitwidth-integers.

bjourne 2y ago

It's not so easy because -ffast-math can actually be more correct than running without.

adrian_b 2y ago
In most cases it is unpredictable when such an improvement happens and what is unpredictable about correctness is useless. It is much more frequent for -ffast-math to generate bigger errors than to benefit from an accidental cancellation of the errors. The only case when the result of underflows is predictable is when the values whose computation may result in an underflow are immediately added to other values that are known to be big enough, and they are not used for anything else. It is not frequent to have so much information about which expressions will underflow and which will not, and about the range of possible values of the values that would be added. When there really exists so much information about what must be computed, then there are chances that the computation can be rearranged in a way that would avoid the underflows, in which case there would be no need to change MXCSR to ignore the underflows, because they would not happen.
- bjourne 2y ago
  > It is much more frequent for -ffast-math to generate bigger errors than to benefit from an accidental cancellation of the errors. In my experience underflows are very uncommon and indicate that you're doing something wrong, whereas fp contractions are extremely common. So I disagree with you.
  - mbauman 2y ago
    fp contract isn't a uniform "decrease in error" either, though. As a simple example, it introduces error in the straightforward: a*b - a*b
  - adrian_b 2y ago
    I agree that using fused multiply-add in almost all cases improves the accuracy of the results. Nevertheless, I consider that a compiler option that gives you good FMA, but only together with bad behavior at underflows, is something exceedingly stupid. FMA is a standard operation, while ignoring underflows is rightly prohibited by the floating-point standard. There should never have been a compiler option that mixes standard operations with non-standard operations. In C or C++, you can use the "fma" function to get FMA where desired. This is more awkward, but it is safe. Any better solution requires a change in the programming language. For instance, it could be required that adjacent * and + must always be contracted into an FMA, unless separated by parentheses. This would allow the programmer to select between FMA and FMUL+FADD, as desired. FMA has appeared in 1990 (in IBM POWER) and it has been the greatest advance in floating-point computation after the introduction of Intel 8087 in 1980 (which has resulted in the IEEE standard a few years later). It is weird that 34 years later, most, if not all, programming languages do not handle well the generation of FMA operations.

jcranmer 2y ago

> The behavior of "-Ofast" where it affects globally the content of MXCSR is unacceptable. > The right behavior would have been for any function compiled with "-Ofast" to save MXCSR, put in it any desired value for the duration of the function execution and restore it at function exit. Moreover, when invoking any other function it should restore the original MXCSR before the call and put again in it the desired local value after the function returns back. I completely agree here. The problem is that introducing floating-point environment into the programming model requires disabling some of the transformations you want to do with fast-math in the first place, so doing it this way kind of doesn't work to enable what you wanted to do. Furthermore, one of the SPEC benchmarks gets like a 30% speedup if you turn on denormal flushing, and that's the kind of change that gives compiler engineer management heartburn if you want to tell them to forgo that speedup because the optimization is dumb. I am grateful for blog posts like this one because this does help build a case for getting the compiler to stop doing stupid stuff like globally enabling DAZ/FTZ bits.

adrian_b 2y ago
Not all CPUs behave badly when handling underflows, denormals or other exceptional cases. On some CPUs the penalties for exceptional cases are negligible (e.g. on AMD Zen), so it is impossible for any SPEC benchmark to get a 30% speedup when ignoring underflows. There are also bad CPU models, which have a microprogrammed handling of the exceptional cases, which can add e.g. a penalty of somewhat more than one hundred CPU cycles for each underflow, which is very similar to the penalty for a load from the main memory (which misses the caches). If one of the SPEC benchmarks has so many underflows that on a bad CPU model it can be sped up by 30% when ignoring the underflows, then it is pretty much guaranteed that when the underflows are ignored the results computed by that benchmark are erroneous. Cheating at the SPEC benchmark by ignoring the underflows could be easily avoided by changing the benchmarking rules, by providing a file with the valid results and by not accepting any benchmark claims where the results do not match those known to be good. You can cheat at any SPEC benchmark with bogus compiler options, if there is no constraint on the compiled program to be correct. For instance you could make a compilation option that deletes 90% of the generated machine code, resulting in a 10 times faster benchmark program. For some reason, the compiler writers that handle the generation of code for floating-point operations feel that they are licensed to generate incorrect code, even when similarly incorrect code would be rejected in other contexts. The fact that the floating-point operations are inexact, so they are accompanied by some inherent errors, does not give the right to a compiler writer or library writer to introduce arbitrary errors in the computations. The reason why a standard exists is precisely for guaranteeing that the errors are bounded and you can predict their effects.

HappMacDonald 2y ago

This sounds quite a lot like my allergy to using floating point for doing anything critical at all. It's more difficult to force deterministic results on all platforms for float than it is for known-bitwidth-integers.