[Numpy-discussion] Slightly off-topic - accuracy of C exp function?

Sat Apr 26 20:31:39 EDT 2014

On Sat, Apr 26, 2014 at 8:05 PM, <josef.pktd at gmail.com> wrote:

>
>
>
> On Sat, Apr 26, 2014 at 6:37 PM, Matthew Brett <matthew.brett at gmail.com>wrote:
>
>> Hi,
>>
>> On Wed, Apr 23, 2014 at 11:59 AM, Matthew Brett <matthew.brett at gmail.com>
>> wrote:
>> > Hi,
>> >
>> > On Wed, Apr 23, 2014 at 1:43 AM, Nathaniel Smith <njs at pobox.com> wrote:
>> >> On Wed, Apr 23, 2014 at 6:22 AM, Matthew Brett <
>> matthew.brett at gmail.com> wrote:
>> >>> Hi,
>> >>>
>> >>> I'm exploring Mingw-w64 for numpy building, and I've found it gives a
>> >>> slightly different answer for 'exp' than - say - gcc on OSX.
>> >>>
>> >>> The difference is of the order of the eps value for the output number
>> >>> (2 * eps for a result of ~2.0).
>> >>>
>> >>> Is accuracy somewhere specified for C functions like exp?  Or is
>> >>> accuracy left as an implementation detail for the C library author?
>> >>
>> >> C99 says (sec 5.2.4.2.2) that "The accuracy of the floating point
>> >> operations ... and of the library functions in <math.h> and
>> >> <complex.h> that return floating point results is implemenetation
>> >> defined. The implementation may state that the accuracy is unknown."
>> >> (This last sentence is basically saying that with regard to some
>> >> higher up clauses that required all conforming implementations to
>> >> document this stuff, saying "eh, who knows" counts as documenting it.
>> >> Hooray for standards!)
>> >>
>> >> Presumably the accuracy in this case is a function of the C library
>> >> anyway, not the compiler?
>> >
>> > Mingw-w64 implementation is in assembly:
>> >
>> >
>> http://sourceforge.net/p/mingw-w64/code/HEAD/tree/trunk/mingw-w64-crt/math/exp.def.h
>> >
>> >> Numpy has its own implementations for a
>> >> bunch of the math functions, and it's been unclear in the past whether
>> >> numpy or the libc implementations were better in any particular case.
>> >
>> > I only investigated this particular value, in which case it looked as
>> > though the OSX value was closer to the exact value (via sympy.mpmath)
>> > - by ~1 unit-at-the-last-place.  This was causing a divergence in the
>> > powell optimization path and therefore a single scipy test failure.  I
>> > haven't investigated further - was wondering what investigation I
>> > should do, more than running the numpy / scipy test suites.
>>
>> Investigating further, with this script:
>>
>> https://gist.github.com/matthew-brett/11301221
>>
>> The following are tests of np.exp accuracy for input values between 0
>> and 10, for numpy 1.8.1.
>>
>> If np.exp(x) performs perfectly, it will return the nearest floating
>> point value to the exact value of exp(x).  If it does, this scores a
>> zero for error in the tables below.  If 'proportion of zeros' is 1 -
>> then np.exp performs perfectly for all tested values of exp (as is the
>> case for linux here).
>>
>> OSX 10.9
>>
>> Proportion of zeros: 0.99789
>> Sum of error: 2.15021267458e-09
>> Sum of squared error: 2.47149370032e-14
>> Max / min error: 5.96046447754e-08 -2.98023223877e-08
>> Sum of squared relative error: 5.22456992025e-30
>> Max / min relative error: 2.19700100681e-16 -2.2098803255e-16
>> eps:  2.22044604925e-16
>> Proportion of relative err >= eps: 0.0
>>
>> Debian Jessie / Sid
>>
>> Proportion of zeros: 1.0
>> Sum of error: 0.0
>> Sum of squared error: 0.0
>> Max / min error: 0.0 0.0
>> Sum of squared relative error: 0.0
>> Max / min relative error: 0.0 0.0
>> eps:  2.22044604925e-16
>> Proportion of relative err >= eps: 0.0
>>
>> Mingw-w64 Windows 7
>>
>> Proportion of zeros: 0.82089
>> Sum of error: 8.08415331122e-07
>> Sum of squared error: 2.90045099615e-12
>> Max / min error: 5.96046447754e-08 -5.96046447754e-08
>> Sum of squared relative error: 4.18466468175e-28
>> Max / min relative error: 2.22041308226e-16 -2.22042100773e-16
>> eps:  2.22044604925e-16
>> Proportion of relative err >= eps: 0.0
>>
>> Take-home : exp implementation for mingw-w64 is exactly (floating
>> point) correct 82% of the time, and one unit-at-the-last-place off for
>> the rest [1].  OSX is off by 1 ULP only 0.2% of the time.
>>
>
>
> Windows 64 with MKL
>
> \WinPython-64bit-3.3.2.2\python-3.3.2.amd64>python
> "E:\Josef\eclipsegworkspace\statsmodels-git\local_scripts\local_scripts\try_exp_error.py"
> Proportion of zeros: 0.99793
> Sum of error: -2.10546855506e-07
> Sum of squared error: 3.33304327526e-14
> Max / min error: 5.96046447754e-08 -5.96046447754e-08
> Sum of squared relative error: 4.98420694339e-30
> Max / min relative error: 2.20881302691e-16 -2.18321571939e-16
> eps:  2.22044604925e-16
> Proportion of relative err >= eps: 0.0
>
>
> Windows 32 bit python with official MingW binaries
>
> Python 2.7.1 (r271:86832, Nov 27 2010, 18:30:46) [MSC v.1500 32 bit
> (Intel)] on win32
>
> Proportion of zeros: 0.99464
> Sum of error: -3.91621083118e-07
> Sum of squared error: 9.2239247812e-14
> Max / min error: 5.96046447754e-08 -5.96046447754e-08
> Sum of squared relative error: 1.3334972729e-29
> Max / min relative error: 2.21593462148e-16 -2.2098803255e-16
> eps:  2.22044604925e-16
> Proportion of relative err >= eps: 0.0
>
>
>
>>
>> Is mingw-w64 accurate enough?  Do we have any policy on this?
>>
>
> I wouldn't worry about a missing or an extra eps in our applications, but
> the competition is more accurate.
>

Just for comparison, I increased `until` to 300
the proportion of zeros and relative error stays about the same for both
MKL and your wheels

absolute error are huge, the following is MKL

Sum of error: -3.78802736366e+112
Sum of squared error: 1.51136754049e+225
Max / min error: 4.80981520952e+111 -3.84785216762e+112

(I looked a lot at the behavior of exp in the hundreds recently :(

As illustration why I don't care about one **relative** eps
>>> np.finfo(np.double).eps + 10 == 10
True

https://github.com/scipy/scipy/pull/3547 and many others

Josef

>
> Josef
>
>
>>
>> Cheers,
>>
>> Matthew
>>
>> [1] http://matthew-brett.github.io/pydagogue/floating_error.html
>> _______________________________________________
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>>
>
>
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