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

[Eric Wieser]

Can’t arange and linspace operations with floats be done internally

Yes, and they probably should be - they’re done this way as a hack because
the api exposed for custom dtypes is here
<https://github.com/numpy/numpy/blob/81e15e812574d956fcc304c3982e2b59aa18aafb/numpy/core/include/numpy/ndarraytypes.h#L507-L511>,
(example implementation here
<https://github.com/numpy/numpy/blob/81e15e812574d956fcc304c3982e2b59aa18aafb/numpy/core/src/multiarray/arraytypes.c.src#L3711-L3721>)
- essentially, you give it the first two elements of the array, and ask it
to fill in the rest.
​

On Fri, 9 Feb 2018 at 13:17 Matthew Harrigan <harrigan.matthew at gmail.com>
wrote:

> I apologize if I'm missing something basic, but why are floats being
> accumulated in the first place?  Can't arange and linspace operations with
> floats be done internally similar to `start + np.arange(num_steps) *
> step_size`?  I.e. always accumulate (really increment) integers to limit
> errors.
>
> On Fri, Feb 9, 2018 at 3:43 PM, Benjamin Root <ben.v.root at gmail.com>
> wrote:
>
>>
>>
>> On Fri, Feb 9, 2018 at 12:19 PM, Chris Barker <chris.barker at noaa.gov>
>> wrote:
>>
>>> On Wed, Feb 7, 2018 at 12:09 AM, Ralf Gommers <ralf.gommers at gmail.com>
>>> wrote:
>>>>
>>>>  It is partly a plea for some development of numerically accurate
>>>>> functions for computing lat/lon grids from a combination of inputs: bounds,
>>>>> counts, and resolutions.
>>>>>
>>>>
>>> Can you be more specific about what problems you've run into -- I work
>>> with lat-lon grids all the time, and have never had a problem.
>>>
>>> float32 degrees gives you about 1 meter accuracy or better, so I can see
>>> how losing a few digits might be an issue, though I would argue that you
>>> maybe shouldn't use float32 if you are worried about anything close to 1m
>>> accuracy... -- or shift to a relative coordinate system of some sort.
>>>
>>
>> The issue isn't so much the accuracy of the coordinates themselves. I am
>> only worried about 1km resolution (which is approximately 0.01 degrees at
>> mid-latitudes). My concern is with consistent *construction* of a
>> coordinate grid with even spacing. As it stands right now. If I provide a
>> corner coordinate, a resolution, and the number of pixels, the result is
>> not terrible (indeed, this is the approach used by gdal/rasterio). If I
>> have start/end coordinates and the number of pixels, the result is not bad,
>> either (use linspace). But, if I have start/end coordinates and a
>> resolution, then determining the number of pixels from that is actually
>> tricky to get right in the general case, especially with float32 and large
>> grids, and especially if the bounding box specified isn't exactly divisible
>> by the resolution.
>>
>>
>>>
>>> I have been playing around with the decimal package a bit lately,
>>>>>
>>>>
>>> sigh. decimal is so often looked at a solution to a problem it isn't
>>> designed for. lat-lon is natively Sexagesimal -- maybe we need that dtype
>>> :-)
>>>
>>> what you get from decimal is variable precision -- maybe a binary
>>> variable precision lib is a better answer -- that would be a good thing to
>>> have easy access to in numpy, but in this case, if you want better accuracy
>>> in a computation that will end up in float32, just use float64.
>>>
>>
>> I am not concerned about computing distances or anything like that, I am
>> trying to properly construct my grid. I need consistent results regardless
>> of which way the grid is specified (start/end/count, start/res/count,
>> start/end/res). I have found that loading up the grid specs (using in a
>> config file or command-line) using the Decimal class allows me to exactly
>> and consistently represent the grid specification, and gets me most of the
>> way there. But the problems with arange() is frustrating, and I have to
>> have extra logic to go around that and over to linspace() instead.
>>
>>
>>>
>>> and I discovered the concept of "fused multiply-add" operations for
>>>>> improved accuracy. I have come to realize that fma operations could be used
>>>>> to greatly improve the accuracy of linspace() and arange().
>>>>>
>>>>
>>> arange() is problematic for non-integer use anyway, by its very
>>> definition (getting the "end point" correct requires the right step, even
>>> without FP error).
>>>
>>> and would it really help with linspace? it's computing a delta with one
>>> division in fp, then multiplying it by an integer (represented in fp --
>>> why? why not keep that an integer till the multiply?).
>>>
>>
>> Sorry, that was a left-over from a previous draft of my email after I
>> discovered that linspace's accuracy was on par with fma(). And while
>> arange() has inherent problems, it can still be made better than it is now.
>> In fact, I haven't investigated this, but I did recently discover some unit
>> tests of mine started to fail after a numpy upgrade, and traced it back to
>> a reduction in the accuracy of a usage of arange() with float32s. So,
>> something got worse at some point, which means we could still get accuracy
>> back if we can figure out what changed.
>>
>>
>>>
>>> In particular, I have been needing improved results for computing
>>>>> latitude/longitude grids, which tend to be done in float32's to save memory
>>>>> (at least, this is true in data I come across).
>>>>>
>>>>
>>>> If you care about saving memory *and* accuracy, wouldn't it make more
>>>> sense to do your computations in float64, and convert to float32 at the
>>>> end?
>>>>
>>>
>>> that does seem to be the easy option :-)
>>>
>>
>> Kinda missing the point, isn't it? Isn't that like saying "convert all
>> your data to float64s prior to calling np.mean()"? That's ridiculous.
>> Instead, we made np.mean() upcast the inner-loop operation, and even allow
>> an option to specify what the dtype that should be used for the aggregator.
>>
>>
>>>
>>>
>>>> Now, to the crux of my problem. It is next to impossible to generate a
>>>>> non-trivial numpy array of coordinates, even in double precision, without
>>>>> hitting significant numerical errors.
>>>>>
>>>>
>>> I'm confused, the example you posted doesn't have significant errors...
>>>
>>
>> Hmm, "errors" was the wrong word. "Differences between methods" might be
>> more along the lines of what I was thinking. Remember, I am looking for
>> consistency.
>>
>>
>>>
>>>
>>>> Which has lead me down the path of using the decimal package (which
>>>>> doesn't play very nicely with numpy because of the lack of casting rules
>>>>> for it). Consider the following:
>>>>> ```
>>>>> $ cat test_fma.py
>>>>> from __future__ import print_function
>>>>> import numpy as np
>>>>> res = np.float32(0.01)
>>>>> cnt = 7001
>>>>> x0 = np.float32(-115.0)
>>>>> x1 = res * cnt + x0
>>>>> print("res * cnt + x0 = %.16f" % x1)
>>>>> x = np.arange(-115.0, -44.99 + (res / 2), 0.01, dtype='float32')
>>>>> print("len(arange()): %d  arange()[-1]: %16f" % (len(x), x[-1]))
>>>>> x = np.linspace(-115.0, -44.99, cnt, dtype='float32')
>>>>> print("linspace()[-1]: %.16f" % x[-1])
>>>>>
>>>>> $ python test_fma.py
>>>>> res * cnt + x0 = -44.9900015648454428
>>>>> len(arange()): 7002  arange()[-1]:       -44.975044
>>>>> linspace()[-1]: -44.9900016784667969
>>>>> ```
>>>>> arange just produces silly results (puts out an extra element...
>>>>> adding half of the resolution is typically mentioned as a solution on
>>>>> mailing lists to get around arange()'s limitations -- I personally don't do
>>>>> this).
>>>>>
>>>>
>>> The real solution is "don't do that" arange is not the right tool for
>>> the job.
>>>
>>
>> Well, it isn't the right tool because as far as I am concerned, it is
>> useless for anything but integers. Why not fix it to be more suitable for
>> floating point?
>>
>>
>>>
>>> Then there is this:
>>>
>>> res * cnt + x0 = -44.9900015648454428
>>> linspace()[-1]: -44.9900016784667969
>>>
>>> that's as good as you are ever going to get with 32 bit floats...
>>>
>>
>> Consistency is the key thing. I am fine with one of those values, so long
>> as that value is what happens no matter which way I specify my grid.
>>
>>
>>>
>>> Though I just noticed something about your numbers -- there should be a
>>> nice even base ten delta if you have 7001 gaps -- but linspace produces N
>>> points, not N gaps -- so maybe you want:
>>>
>>>
>>> In [*17*]: l = np.linspace(-115.0, -44.99, 7002)
>>>
>>>
>>> In [*18*]: l[:5]
>>>
>>> Out[*18*]: array([-115.  , -114.99, -114.98, -114.97, -114.96])
>>>
>>>
>>> In [*19*]: l[-5:]
>>>
>>> Out[*19*]: array([-45.03, -45.02, -45.01, -45.  , -44.99])
>>>
>>>
>>> or, in float32 -- not as pretty:
>>>
>>>
>>> In [*20*]: l = np.linspace(-115.0, -44.99, 7002, dtype=np.float32)
>>>
>>>
>>> In [*21*]: l[:5]
>>>
>>> Out[*21*]:
>>>
>>> array([-115.        , -114.98999786, -114.98000336, -114.97000122,
>>>
>>>        -114.95999908], dtype=float32)
>>>
>>>
>>> In [*22*]: l[-5:]
>>>
>>> Out[*22*]: array([-45.02999878, -45.02000046, -45.00999832, -45.        ,
>>> -44.99000168], dtype=float32)
>>>
>>>
>>> but still as good as you get with float32, and exactly the same result
>>> as computing in float64 and converting:
>>>
>>>
>>>
>>> In [*25*]: l = np.linspace(-115.0, -44.99, 7002).astype(np.float32)
>>>
>>>
>>> In [*26*]: l[:5]
>>>
>>> Out[*26*]:
>>>
>>> array([-115.        , -114.98999786, -114.98000336, -114.97000122,
>>>
>>>        -114.95999908], dtype=float32)
>>>
>>>
>>> In [*27*]: l[-5:]
>>>
>>> Out[*27*]: array([-45.02999878, -45.02000046, -45.00999832, -45.        ,
>>> -44.99000168], dtype=float32)
>>>
>>
>> Argh! I got myself mixed up between specifying pixel corners versus pixel
>> centers. rasterio has been messing me up on this.
>>
>>
>>>
>>>
>>>>> So, does it make any sense to improve arange by utilizing fma() under
>>>>> the hood?
>>>>>
>>>>
>>> no -- this is simply not the right use-case for arange() anyway.
>>>
>>
>> arange() has accuracy problems, so why not fix it?
>>
>> >>> l4 = np.arange(-115, -44.99, 0.01, dtype=np.float32)
>> >>> np.median(np.diff(l4))
>> 0.0099945068
>> >>> np.float32(0.01)
>> 0.0099999998
>>
>> There is something significantly wrong here if arange(), which takes a
>> resolution parameter, can't seem to produce a sequence with the proper
>> delta.
>>
>>
>>
>>>
>>>
>>>> Also, any plans for making fma() available as a ufunc?
>>>>>
>>>>
>>> could be nice -- especially if used internally.
>>>
>>>
>>>> Notice that most of my examples required knowing the number of grid
>>>>> points ahead of time. But what if I didn't know that? What if I just have
>>>>> the bounds and the resolution? Then arange() is the natural fit, but as I
>>>>> showed, its accuracy is lacking, and you have to do some sort of hack to do
>>>>> a closed interval.
>>>>>
>>>>
>>> no -- it's not -- if you have the bounds and the resolution, you have an
>>> over-specified problem. That is:
>>>
>>> x_min + (n * delta_x) == x_max
>>>
>>> If there is ANY error in either delta_x or x_max (or x_min), then you'll
>>> get a missmatch. which is why arange is not the answer (you can make the
>>> algorithm a bit more accurate, I suppose but there is still fp limited
>>> precision -- if you can't exactly represent either delta_x or x_max, then
>>> you CAN'T use the arange() definition and expect to work consistently.
>>>
>>> The "right" way to do it is to compute N with: round((x_max - x_min) /
>>> delta), and then use linspace:
>>>
>>> linspace(x_min, x_max, N+1)
>>>
>>> (note that it's too bad you need to do N+1 -- if I had to do it over
>>> again, I'd use N as the number of "gaps" rather than the number of points
>>> -- that's more commonly what people want, if they care at all)
>>>
>>> This way, you get a grid with the endpoints as exact as they can be, and
>>> the deltas as close to each-other as they can be as well.
>>>
>>> maybe you can do a better algorithm in linspace to save an ULP, but it's
>>> hard to imagine when that would matter.
>>>
>>
>> Yes, it is overspecified. My problem is that different tools require
>> different specs (ahem... rasterio/gdal), and I have gird specs coming from
>> other sources. And I need to produce data onto the same grid so that tools
>> like xarray won't yell at me when I am trying to do an operation between
>> gridded data that should have the same coordinates, but are off slightly
>> because they were computed differently for whatever reason.
>>
>> I guess I am crying out for some sort of tool that will help the
>> community stop making the same mistakes. A one-stop shop that'll allow us
>> to specify a grid in a few different ways and still produce the right
>> thing, and even do the inverse... provide a coordinate array and get grids
>> specs in whatever form we want. Maybe even have options for dealing with
>> pixel corner vs. pixel centers, too? There are additional fun problems such
>> as padding out coordinate arrays, which np.pad doesn't really do a great
>> job with.
>>
>> Cheers!
>> Ben Root
>>
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>>
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