On Fri, Feb 9, 2018 at 12:19 PM, Chris Barker wrote:

On Wed, Feb 7, 2018 at 12:09 AM, Ralf Gommers 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

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..., (example implementation here https://github.com/numpy/numpy/blob/81e15e812574d956fcc304c3982e2b59aa18aafb...) - 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 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 wrote:

...

On Fri, Feb 9, 2018 at 12:19 PM, Chris Barker wrote:

...

On Wed, Feb 7, 2018 at 12:09 AM, Ralf Gommers 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

_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion

_______________________________________________ NumPy-Discussion mailing list NumPy-Discussion@python.org https://mail.python.org/mailman/listinfo/numpy-discussion

I just want to add a few links related to the topic: https://mail.python.org/pipermail/numpy-discussion/2007-September/029129.htm... https://quantumwise.com/forum/index.php?topic=110.0#.VIVgyIctjRZ https://mail.python.org/pipermail/numpy-discussion/2012-February/060238.html http://nbviewer.jupyter.org/github/mgeier/python-audio/blob/master/misc/aran... cheers, Matthias On Fri, Feb 9, 2018 at 11:17 PM, Benjamin Root wrote:

Interesting...

``` static void @NAME@_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@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@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@gmail.com>
>> wrote:
>>
>>>
>>>
>>> On Fri, Feb 9, 2018 at 12:19 PM, Chris Barker <chris.barker@noaa.gov>
>>> wrote:
>>>
>>>> On Wed, Feb 7, 2018 at 12:09 AM, Ralf Gommers <ralf.gommers@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
>>>
>>> _______________________________________________
>>> NumPy-Discussion mailing list
>>> NumPy-Discussion@python.org
>>> https://mail.python.org/mailman/listinfo/numpy-discussion
>>>
>>>
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Hey,

Anyone that understands FP better than I do:

In the above code, you are multiplying the step by an integer -- is there any precision loss when you do that??

Elementary operations (add, sub, mul, div) are demanded to be correctly rounded (cr) by IEEE, i.e. accurate within +/- 0.5 ulp. Consequently, a cr multiplication followed by a cr addition will be accurate within +/-1 ulp. This is also true if the first multiplicand is an integer. Using FMA will reduce this to +/- 0.5 ulp. This increase in accuracy of the grid calculation should not be relevant - but it also does not hurt. Still I would suggest adding the FMA operation to numpy, e.g. np.fma(a, b, c). There are several places in numpy that could benefit from the increased accuracy, e.g. evaluation of polynomials using Horner's method. In cases like this due to iteration and consequent error propagation the accuracy benefit of using FMA can be far larger. There may also be a performance benefit on platforms that implement FMA in hardware (although I am not sure about that). Cheers Nils

@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 wrote:

I think it's all been said, but a few comments:

On Sun, Feb 11, 2018 at 2:19 PM, Nils Becker 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

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@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@noaa.gov

On Thu, 2018-02-22 at 14:33 -0500, Benjamin Root wrote:

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?

I am not sure I like the idea: 1. It sounds like it might break code 2. It sounds *not* like a fix, but rather a "make it slightly less bad, but it is still awful" Using fma inside linspace might make linspace a bit more exact possible, and would be a good thing, though I am not sure we have a policy yet for something that is only used sometimes, nor am I sure it actually helps. It also would be nice to add stable summation to numpy in general (in whatever way), which maybe is half related but on nobody's specific todo list.

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.

Maybe you are right, but right now I have no clue what that tool would do :). If we should add it to numpy likely depends on what exactly it does and how complex it is. I once wanted to add a "step" argument to linspace, but didn't in the end, largely because it basically enforced in a very convoluted way that the step fit exactly to a number of steps (up to floating point precision) and body was quite sure it was a good idea, since it would just be useful for a little convenience when you do not want to calculate the steps. Best, Sebastian

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 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 wrote:

...

I think it's all been said, but a few comments:

On Sun, Feb 11, 2018 at 2:19 PM, Nils Becker 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

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@noaa.gov

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