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

Fri Feb 9 17:17:44 EST 2018

Interesting...

```
static void
@NAME at _fill(@type@ *buffer, npy_intp length, void *NPY_UNUSED(ignored))
{
npy_intp i;
@type@ start = buffer[0];
@type@ delta = buffer[1];
delta -= start;
for (i = 2; i < length; ++i) {
buffer[i] = start + i*delta;
}
}
```

So, the second value is computed using the delta arange was given, but then
tries to get the delta back, which incurs errors:
```
>>> a = np.float32(-115)
>>> delta = np.float32(0.01)
>>> b = a + delta
>>> new_delta = b - a
>>> "%.16f" % delta
'0.0099999997764826'
>>> "%.16f" % new_delta
'0.0100021362304688'
```

Also, right there is a good example of where the use of fma() could be of
value.

Cheers!
Ben Root

On Fri, Feb 9, 2018 at 4:56 PM, Eric Wieser <wieser.eric+numpy at gmail.com>
wrote:

> 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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>>> NumPy-Discussion at python.org
>>> https://mail.python.org/mailman/listinfo/numpy-discussion
>>>
>>>
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