[llvm-dev] The most efficient way to implement an integer based power function pow in LLVM

[Steve (Numerics) Canon via llvm-dev]

> On Jan 12, 2017, at 2:21 PM, Mehdi Amini <mehdi.amini at apple.com> wrote:
> 
> 
>> On Jan 12, 2017, at 11:04 AM, Steve (Numerics) Canon <scanon at apple.com <mailto:scanon at apple.com>> wrote:
>> 
>>> 
>>> On Jan 12, 2017, at 12:58 PM, Friedman, Eli via llvm-dev <llvm-dev at lists.llvm.org <mailto:llvm-dev at lists.llvm.org>> wrote:
>>> 
>>> On 1/12/2017 9:33 AM, Mehdi Amini via llvm-dev wrote:
>>>>> On Jan 12, 2017, at 5:03 AM, Antoine Pitrou via llvm-dev <llvm-dev at lists.llvm.org <mailto:llvm-dev at lists.llvm.org>> wrote:
>>>>> 
>>>>> On Mon, 9 Jan 2017 11:43:17 -0600
>>>>> Wei Ding via llvm-dev <llvm-dev at lists.llvm.org <mailto:llvm-dev at lists.llvm.org>> wrote:
>>>>>> Hi,
>>>>>> 
>>>>>> I want an efficient way to implement function pow in LLVM instead of
>>>>>> invoking pow() math built-in. For algorithm part, I am clear for the logic.
>>>>>> But I am not quite sure for which parts of LLVM should I replace built-in
>>>>>> pow with another efficient pow implementation. Any comments and feedback
>>>>>> are appreciated. Thanks!
>>>>> In Numba, we have decided to optimize some usages of the power function
>>>>> in our own front-end, so that LLVM IR gets an already optimized form,
>>>>> as we have found that otherwise LLVM may miss some optimization
>>>>> opportunities. YMMV.
>>>> It seems to me that it would be more interesting to gather these misoptimization and fix LLVM to catch them.
>>>> 
>>>>> (e.g. we detect that the exponent is a compile-time constant and
>>>>> transform `x**3` into `x*x*x`)
>>>> This seems definitely in the scope of what LLVM could do, potentially with TTI.
>>> 
>>> LLVM already does this... but only if the pow() call is marked "fast".  IEEE 754 pow() is supposed to be correctly rounded, but (x*x)*x has an extra rounding step.
>> 
>> pow( ) is not supposed to be correctly rounded.
>> 
>> IEEE 754 recommends that a correctly rounded power function exist [9.2], but does not recommend or require that this be the default pow( ) function. [Note: it was not actually shown until quite late in the IEEE 754 revision process that it was even possible to have a reasonably efficient correctly-rounded pow( ) function, and such a function is at minimum an order of magnitude slower than a good sub-ulp-accurate-but-not-correctly-rounded implementation. Most 754 committee members would recommend that the default pow( ) function *not* be correctly rounded.]
> 
> So we should use x*x*x even without fast-math...
> 
>> In a good math library, however, pow( ) should absolutely be sub-ulp accurate, which means that it should *not* be implemented via log( ) and exp( ), and indeed, most math libraries don’t do that.
>> 
>> Just to provide some example data, for single-precision x in [1,2) on current OS X:
>> 
>> 	The worst-case error of x*x*x is 1.28736 ulp
>> 	The worst-case error of powf(x, 3) is 0.500013 ulp
>> 
>> 	The RMS error of x*x*x is 0.585066 ulp
>> 	The RMS error of powf(x,3) is 0.499984 ulp
> 
> … or maybe not! :)
> 
> I’m not sure what to think of these results. For instance what’s the perf impact of calling into powf(x, 3) on OSX vs using x*x*x?

Large! x*x*x has reciprocal throughput of 1 cycle, plus the opportunity for the compiler to autovectorize. powf(x, 3) is much harder to vectorize, and has reciprocal throughput of ~40 cycles on OS X (which is on the faster side; it’s more like 100 cycles on some platforms).

> At the source level, what could the user do to decide that 1.28ULP is acceptable for his use-case?

Write x*x*x. =)

I can see an argument for adding a flag that allows a few ulp of error in math functions (roughly what Intel’s LA versions in MKL or some of the YEPPP implementations provide), but there is a question of how many users would actually find it and use it.

– Steve

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