> This is not true. The mathematically correct result for sqrt might not be a representable value in floating point, so rounding may occur between the two steps. In that case, pow2(sqrt(x)) != x. > > --OwenI think what Christoph is saying is that x will always be at least as accurate as pow2(sqrt(x)), so it's only unsafe in so far as one's code is actually depending on an imprecise result. It's not an "as-if" optimization though, so it definitely would require "fast math" or something similar; however, it seems like it would be on the more benign side among optimizations that apply mathematical identities that are not necessarily identities on floats, as far as I can tell... -Stephen
On Apr 23, 2013, at 7:15 PM, Stephen Lin <swlin at post.harvard.edu> wrote:>> This is not true. The mathematically correct result for sqrt might not be a representable value in floating point, so rounding may occur between the two steps. In that case, pow2(sqrt(x)) != x. > > I think what Christoph is saying is that x will always be at least as > accurate as pow2(sqrt(x)), so it's only unsafe in so far as one's code > is actually depending on an imprecise result.Giving more-than-expected precision can be just as bad for the user as less. It tends to come up in situations where the optimization would break some symmetry, the same way that aggressively forming FMAs can break user code. Consider this example: float foo(float a, float b) { return pow2(a) - pow2(sqrt(b)); } float bar(float c) { return foo(sqrt(b), b); } The author *should* be able to assume that for any positive c, the only possible output values are Inf and zero. However, if we apply the pow2/sqrt peephole, suddenly non-zero finite outputs are possible. The equivalent example for FMA formation is x*x - x*x. If we convert that to "fma x, x, (-x*x)", you can get a non-zero finite result. It boils down to the fact that giving excess precision in some-places-but-not-others can lead to bad behavior. --Owen
On Wed, Apr 24, 2013 at 1:50 AM, Owen Anderson <resistor at mac.com> wrote:> > On Apr 23, 2013, at 7:15 PM, Stephen Lin <swlin at post.harvard.edu> wrote: > >>> This is not true. The mathematically correct result for sqrt might not be a representable value in floating point, so rounding may occur between the two steps. In that case, pow2(sqrt(x)) != x. >> >> I think what Christoph is saying is that x will always be at least as >> accurate as pow2(sqrt(x)), so it's only unsafe in so far as one's code >> is actually depending on an imprecise result. > > Giving more-than-expected precision can be just as bad for the user as less. It tends to come up in situations where the optimization would break some symmetry, the same way that aggressively forming FMAs can break user code. Consider this example: > > float foo(float a, float b) { > return pow2(a) - pow2(sqrt(b)); > } > > float bar(float c) { > return foo(sqrt(b), b); > } > > The author *should* be able to assume that for any positive c, the only possible output values are Inf and zero. However, if we apply the pow2/sqrt peephole, suddenly non-zero finite outputs are possible. The equivalent example for FMA formation is x*x - x*x. If we convert that to "fma x, x, (-x*x)", you can get a non-zero finite result. > > It boils down to the fact that giving excess precision in some-places-but-not-others can lead to bad behavior. > > --OwenOK, that makes sense--just clarifying what I thought Christoph meant. In any case, maybe it ought to be at least an option for "fast math" or in some other setting where mathematical identities that may gain or lose precision are taken advantage of. (I know GCC's version of "fast math" allows lots of things that are potentially more unsafe than this; I'm not familiar of the state of clang's equivalent). Stephen
On Apr 24, 2013, at 12:36 AM, Stephen Lin <swlin at post.harvard.edu> wrote:> By the way, I definitely believe you that this isn't a 100% safe > optimization, but I'm curious, is this really guaranteed here that c > will not be a non-zero finite value? I was under the impression that > FPU state could lead to slightly different results in these kinds of > cases, so one should almost never rely on the result of a computation > being exactly zero (outside of some simple primitive cases).That's a common misconception. Floating point computations are deterministic and a function only of their inputs (and a couple of modal switches like rounding mode). --Owen -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://lists.llvm.org/pipermail/llvm-dev/attachments/20130424/ffba8a29/attachment.html>
On 04/23/2013 10:50 PM, Owen Anderson wrote:> > On Apr 23, 2013, at 7:15 PM, Stephen Lin <swlin at post.harvard.edu> wrote: > >>> This is not true. The mathematically correct result for sqrt might not be a representable value in floating point, so rounding may occur between the two steps. In that case, pow2(sqrt(x)) != x. >> >> I think what Christoph is saying is that x will always be at least as >> accurate as pow2(sqrt(x)), so it's only unsafe in so far as one's code >> is actually depending on an imprecise result. > > Giving more-than-expected precision can be just as bad for the user as less. It tends to come up in situations where the optimization would break some symmetry, the same way that aggressively forming FMAs can break user code. Consider this example: > > float foo(float a, float b) { > return pow2(a) - pow2(sqrt(b)); > } > > float bar(float c) { > return foo(sqrt(b), b); > } > > The author *should* be able to assume that for any positive c, the only possible output values are Inf and zero. However, if we apply the pow2/sqrt peephole, suddenly non-zero finite outputs are possible. The equivalent example for FMA formation is x*x - x*x. If we convert that to "fma x, x, (-x*x)", you can get a non-zero finite result. > > It boils down to the fact that giving excess precision in some-places-but-not-others can lead to bad behavior. > > --Owen >Not sure exactly what the C/C++ standard says here but usually it's allowed to give more precision. It's also allowed for math to work as real math and not as limited precision math. If your code is requiring otherwise, then therein is the problem. Can't arcsin(sqrt(2)/2) == pi/4 ? I think so. When you are debugging a calculator that's what you want. People spend a lot of time trying to get the math library to give the expected results when known equations combining various operations are used. That is part of the sanity check of making sure that you have a good math library implementation. Scientists using the math library want the math to be as close to real math as possible. That was part of the point of the IEEE floating point standard. sin (n * pi) == 0 etc. Back in the day I knew W. Kahan (IEEE FP inventor) and I know that this was his belief.
Am Dienstag, 23. April 2013, 22:50:51 schrieben Sie:> [...] > Giving more-than-expected precision can be just as bad for the user as less. > It tends to come up in situations where the optimization would break some > symmetry, the same way that aggressively forming FMAs can break user code. > [...] > > It boils down to the fact that giving excess precision in > some-places-but-not-others can lead to bad behavior.Ok, I didn't think about excess precision resulting in problems. Now it's clear to me that fast-math is neccessary for this optimization. BTW: Is there a way to only get fast-math optimizations that don't change behaviour on NaNs and Infs? I think it would be an interesting possibility to allow arithmetic tranformations but only those preserving NaN and infinity bahavior to catch errors/corner cases. At least the pow(sqrt(x),2)=>x optimization falls into this category as I wrote before.> --OwenChristoph