[Numpy-discussion] improving arange()? introducing fma()?

Thu Feb 22 14:33:28 EST 2018

Sorry, I have been distracted with xarray improvements the past couple of
weeks.

Some thoughts on what has been discussed:

First, you are right...Decimal is not the right module for this. I think
instead I should use the 'fractions' module for loading grid spec
information from strings (command-line, configs, etc). The tricky part is
getting the yaml reader to use it instead of converting to a float under
the hood.

Second, what has been pointed out about the implementation of arange
actually helps to explain some oddities I have encountered. In some
situations, I have found that it was better for me to produce the reversed
sequence, and then reverse that array back and use it.

Third, it would be nice to do what we can to improve arange()'s results.
Would we be ok with a PR that uses fma() if it is available, but then falls
back on a regular multiply and add if it isn't available, or are we going
to need to implement it ourselves for consistency?

Lastly, there definitely needs to be a better tool for grid making. The
problem appears easy at first, but it is fraught with many pitfalls and
subtle issues. It is easy to say, "always use linspace()", but if the user
doesn't have the number of pixels, they will need to calculate that using
--- gasp! -- floating point numbers, which could result in the wrong
answer. Or maybe their first/last positions were determined by some other
calculation, and so the resulting grid does not have the expected spacing.
Another problem that I run into is starting from two different sized grids
and padding them both to be the same spec -- and getting that to match what
would come about if I had generated the grid from scratch.

Getting these things right is hard. I am not even certain that my existing
code for doing this even right. But, what I do know is that until we build
such a tool, users will continue to incorrectly use arange() and
linspace(), and waste time trying to re-invent the wheel badly, assuming
they even notice their mistakes in the first place! So, should such a tool
go into numpy, given how fundamental it is to generate a sequence of
floating point numbers, or should we try to put it into a package like
rasterio or xarray?

Cheers!
Ben Root

On Thu, Feb 22, 2018 at 2:02 PM, Chris Barker <chris.barker at noaa.gov> wrote:

> @Ben: Have you found a solution to your problem? Are there thinks we could
> do in numpy to make it better?
>
> -CHB
>
>
> On Mon, Feb 12, 2018 at 9:33 AM, Chris Barker <chris.barker at noaa.gov>
> wrote:
>
>> I think it's all been said, but a few comments:
>>
>> On Sun, Feb 11, 2018 at 2:19 PM, Nils Becker <nilsc.becker at gmail.com>
>> wrote:
>>
>>> Generating equidistantly spaced grids is simply not always possible.
>>>
>>
>> exactly -- and linspace gives pretty much teh best possible result,
>> guaranteeing tha tthe start an end points are exact, and the spacing is
>> within an ULP or two (maybe we could make that within 1 ULP always, but not
>> sure that's worth it).
>>
>>
>>> The reason is that the absolute spacing of the possible floating point
>>> numbers depends on their magnitude [1].
>>>
>>
>> Also that the exact spacing may not be exactly representable in FP -- so
>> you have to have at least one space that's a bit off to get the end points
>> right (or have the endpoints not exact).
>>
>>
>>> If you - for some reason - want the same grid spacing everywhere you may
>>> choose an appropriate new spacing.
>>>
>>
>> well, yeah, but usually you are trying to fit to some other constraint.
>> I'm still confused as to where these couple of ULPs actually cause
>> problems, unless you are doing in appropriate FP comparisons elsewhere.
>>
>> Curiously, either by design or accident, arange() seems to do something
>>> similar as was mentioned by Eric. It creates a new grid spacing by adding
>>> and subtracting the starting point of the grid. This often has similar
>>> effect as adding and subtracting N*dx (e.g. if the grid is symmetric around
>>> 0.0). Consequently, arange() seems to trade keeping the grid spacing
>>> constant for a larger error in the grid size and consequently in the end
>>> point.
>>>
>>
>> interesting -- but it actually makes sense -- that is the definition of
>> arange(), borrowed from range(), which was designed for integers, and, in
>> fact, pretty much mirroered the classic C index for loop:
>>
>>
>> for (int i=0; i<N; i++) {
>> ...
>>
>>
>> or in python:
>>
>> i = start
>> while i < stop:
>>     i += step
>>
>> The problem here is that termination criteria -- i < stop -- that is the
>> definition of the function, and works just fine for integers (where it came
>> from), but with FP, even with no error accumulation, stop may not be
>> exactly representable, so you could end up with a value for your last item
>> that is about (stop-step), or you could end up with a value that is a
>> couple ULPs less than step -- essentially including the end point when you
>> weren't supposed to.
>>
>> The truth is, making a floating point range() was simply a bad idea to
>> begin with -- it's not the way to define a range of numbers in floating
>> point. Whiuch is why the docs now say "When using a non-integer step, such
>> as 0.1, the results will often not
>> be consistent.  It is better to use ``linspace`` for these cases."
>>
>> Ben wants a way to define grids in a consistent way -- make sense. And
>> yes, sometimes, the original source you are trying to match (like GDAL)
>> provides a starting point and step. But with FP, that is simply
>> problematic. If:
>>
>> start + step*num_steps != stop
>>
>> exactly in floating point, then you'll need to do the math one way or
>> another to get what you want -- and I'm not sure anyone but the user knows
>> what they want -- do you want step to be as exact as possible, or do you
>> want stop to be as exact as possible?
>>
>> All that being said -- if arange() could be made a tiny bit more
>> accurate with fma or other numerical technique, why not? it won't solve
>> the problem, but if someone writes and tests the code (and it does not
>> require compiler or hardware features that aren't supported everywhere
>> numpy compiles), then sure. (Same for linspace, though I'm not sure it's
>> possible)
>>
>> There is one other option: a new function (or option) that makes a grid
>> from a specification of: start, step, num_points. If that is really a
>> common use case (that is, you don't care exactly what the end-point is),
>> then it might be handy to have it as a utility.
>>
>> We could also have an arange-like function that, rather than < stop,
>> would do "close to" stop. Someone that understands FP better than I might
>> be able to compute what the expected error might be, and find the closest
>> end point within that error. But I think that's a bad specification --
>> (stop - start) / step may be nowhere near an integer -- then what is the
>> function supposed to do??
>>
>>
>> BTW: I kind of wish that linspace specified the number of steps, rather
>> than the number of points, that is (num+points - 1) that would save a
>> little bit of logical thinking. So if something new is made, let's keep
>> that in mind.
>>
>>
>>
>>> 1. Comparison to calculations with decimal can be difficult as not all
>>> simple decimal step sizes are exactly representable as
>>>
>> finite floating point numbers.
>>>
>>
>> yeah, this is what I mean by inappropriate use of Decimal -- decimal is
>> not inherently "more accurate" than fp -- is just can represent _decimal_
>> numbers exactly, which we are all use to -- we want  1 / 10 to be exact,
>> but dont mind that 1 / 3 isn't.
>>
>> Decimal also provided variable precision -- so it can be handy for that.
>> I kinda wish Python had an arbitrary precision binary floating point built
>> in...
>>
>> -CHB
>>
>> --
>>
>> Christopher Barker, Ph.D.
>> Oceanographer
>>
>> Emergency Response Division
>> NOAA/NOS/OR&R            (206) 526-6959   voice
>> 7600 Sand Point Way NE   (206) 526-6329   fax
>> Seattle, WA  98115       (206) 526-6317   main reception
>>
>> Chris.Barker at noaa.gov
>>
>
>
>
> --
>
> Christopher Barker, Ph.D.
> Oceanographer
>
> Emergency Response Division
> NOAA/NOS/OR&R            (206) 526-6959   voice
> 7600 Sand Point Way NE   (206) 526-6329   fax
> Seattle, WA  98115       (206) 526-6317   main reception
>
> Chris.Barker at noaa.gov
>
> _______________________________________________
> NumPy-Discussion mailing list
> NumPy-Discussion at python.org
> https://mail.python.org/mailman/listinfo/numpy-discussion
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://mail.python.org/pipermail/numpy-discussion/attachments/20180222/7cdf3601/attachment-0001.html>