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

josef.pktd at gmail.com josef.pktd at gmail.com
Sat Apr 26 20:05:20 EDT 2014


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.

Josef


>
> Cheers,
>
> Matthew
>
> [1] http://matthew-brett.github.io/pydagogue/floating_error.html
> _______________________________________________
> NumPy-Discussion mailing list
> NumPy-Discussion at scipy.org
> http://mail.scipy.org/mailman/listinfo/numpy-discussion
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://mail.python.org/pipermail/numpy-discussion/attachments/20140426/a8c1553a/attachment.html>


More information about the NumPy-Discussion mailing list