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

[Mehdi Amini via llvm-dev]

> On Jan 12, 2017, at 11:04 AM, Steve (Numerics) Canon <scanon at apple.com> wrote:
> 
>> 
>> On Jan 12, 2017, at 12:58 PM, Friedman, Eli via llvm-dev <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> wrote:
>>>> 
>>>> On Mon, 9 Jan 2017 11:43:17 -0600
>>>> Wei Ding via llvm-dev <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?
At the source level, what could the user do to decide that 1.28ULP is acceptable for his use-case?

— 
Mehdi
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