[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Change by Mark Dickinson :


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[issue29282] Fused multiply-add: proposal to add math.fma()

[Juraj Sukop]

Juraj Sukop  added the comment:

FWIW, there is a new implementation of FMA [1] which is licensed very 
permissively [2]. Perhaps it could be used here as well..?

[1] https://github.com/smasher164/fma
[2] 
https://github.com/smasher164/fma/commit/4e85d2388c7d4d850be12df918f9431ca687f57a

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[issue29282] Fused multiply-add: proposal to add math.fma()

[STINNER Victor]

STINNER Victor  added the comment:

> For Windows, you need to do much more than this: it's not just about handling 
> NaNs and infinities, it's about reimplementing the entire function from 
> scratch to give correctly rounded results. Without correctly-rounded results, 
> there's very little point in having fma.

Would it make sense to only make the function available on non-Windows 
platforms? As we do for other Unix-only functions in the os module. Maybe even 
skip more platforms if they provide a broken implementation.

We could implement a test suite in configure to decide if fma() fits our 
requirements or not.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson  added the comment:

Okay, looks like Windows is happy in the PR's continuous integration. If the 
buildbots are also happy, then I'm content to have this pushed through.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson  added the comment:

> If tests continue to fail on some platforms, I plan to manually handle NaN 
> and INF in the C code, before calling libc fma().

For Windows, you need to do much more than this: it's not just about handling 
NaNs and infinities, it's about reimplementing the entire function from scratch 
to give correctly rounded results. Without correctly-rounded results, there's 
very little point in having fma.

If it were a couple of niche platforms that gave bad results, then we could 
push this through. But it's Windows. :-(

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[issue29282] Fused multiply-add: proposal to add math.fma()

[STINNER Victor]

STINNER Victor  added the comment:

I converted https://hg.python.org/cpython/rev/b33012ef1417 written by Mark 
Dickinson into a GitHub PR: PR 17987. I still expect tests failures. I plan to 
use the PR as a starting point to implement math.fma(). If tests continue to 
fail on some platforms, I plan to manually handle NaN and INF in the C code, 
before calling libc fma().

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[issue29282] Fused multiply-add: proposal to add math.fma()

[STINNER Victor]

Change by STINNER Victor :


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pull_requests: +17391
stage: commit review -> patch review
pull_request: https://github.com/python/cpython/pull/17987

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Steven D'Aprano]

Change by Steven D'Aprano :


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[issue29282] Fused multiply-add: proposal to add math.fma()

[Caleb Donovick]

Change by Caleb Donovick :


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[issue29282] Fused multiply-add: proposal to add math.fma()

[Nico Schlömer]

Nico Schlömer  added the comment:

Okay, thanks for the info.

As a stop-gap measure, I've created pyfma [1, 2]. Install with
```
pip install pyfma
```
and use with
```
pyfma.fma(3.0, 2.0, 1.0)
```
Only works on Unix reliable then, but that's all I care about. :)

Cheers,
Nico

[1] https://github.com/nschloe/pyfma
[2] https://pypi.python.org/pypi/pyfma

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson  added the comment:

> Is this because of the inf/NaN discrimination hiccups [...]

No, more than that. If it were just that, we could work around it by adding the 
appropriate special case handling before calling the libm fma (as has been 
done, reluctantly, with some of the other math module functions; see the source 
for math.pow, for example).

But the fma implementation on Windows is fundamentally broken. For finite 
numbers, it simply doesn't give what it's supposed to (a * b + c, computed with 
a _single_ rounding). Since that single rounding is most of the point of fma, 
that makes the libm fma not fit for purpose on Windows.

It _is_ possible, with care, to code up a not-too-inefficient fma emulation 
using clever tricks like Veltkamp splitting and Dekker multiplication. I have 
half such an implementation sitting on my home computer, but so far have not 
had the cycles to finish it (and it's not high on the priority list right now).

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Nico Schlömer]

Nico Schlömer  added the comment:

> Existing libm implementations don't work,

Okay. Is this because of the inf/NaN discrimination hiccups mentioned above or 
are there any other pitfalls?

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson  added the comment:

> Do I read this thread correctly assuming that this hasn't been implemented 
> yet?

Yes. Existing libm implementations don't work, so simply wrapping the libm 
function isn't enough. And writing an implementation from scratch is 
non-trivial.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Nico Schlömer]

Nico Schlömer  added the comment:

Do I read this thread correctly assuming that this hasn't been implemented yet? 
If not, I would probably make my own little library for this -- I really need 
the feature for the precision.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Eric Fahlgren]

Changes by Eric Fahlgren :


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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

The patch needs tests for the case where a*b overflows and c is infinite 
(either of the same sign as a*b or not). This combination should never return 
NaN, but a poor emulation of fma might do so.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

Also failures on Gentoo: here b is positive (possibly +inf), and c is finite, 
so we expect an infinite result. Instead, we're apparently getting a NaN. I 
don't have a good guess about what's causing this: the rest of the tests are 
passing, so it's unlikely that we're using a bad FMA emulation. Maybe an 
optimization bug?

==
ERROR: test_fma_infinities (test.test_math.FMATests)
--
Traceback (most recent call last):
  File 
"/buildbot/buildarea/3.x.ware-gentoo-x86.installed/build/target/lib/python3.7/test/test_math.py",
 line 1482, in test_fma_infinities
self.assertEqual(math.fma(math.inf, b, c), math.inf)
ValueError: invalid operation in fma

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Roundup Robot]

Roundup Robot added the comment:

New changeset b5a5f13500b9 by Mark Dickinson in branch 'default':
Issue #29282: Backed out changeset b33012ef1417
https://hg.python.org/cpython/rev/b5a5f13500b9

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

Failures on the Windows buildbot 
(http://buildbot.python.org/all/builders/AMD64%20Windows8.1%20Non-Debug%203.x/builds/238/steps/test/logs/stdio)
 shown below.

It looks as though Windows is emulating the FMA operation on this machine (and 
not doing a particularly good job of it). That means that if we want to support 
Windows (and we do), we may have to emulate ourselves, preferably using 
something a bit more efficient than the fractions.Fraction module.

I'll let the buildbots complete, to see what else fails, and then roll back the 
commit. The patch clearly isn't good enough in its current state.


==
ERROR: test_fma_overflow (test.test_math.FMATests)
--
Traceback (most recent call last):
  File "D:\buildarea\3.x.ware-win81-release\build\lib\test\test_math.py", line 
1565, in test_fma_overflow
self.assertEqual(math.fma(a, b, -c),
OverflowError: overflow in fma

==
FAIL: test_fma_zero_result (test.test_math.FMATests)
--
Traceback (most recent call last):
  File "D:\buildarea\3.x.ware-win81-release\build\lib\test\test_math.py", line 
1524, in test_fma_zero_result
self.assertIsNegativeZero(math.fma(tiny, -tiny, 0.0))
  File "D:\buildarea\3.x.ware-win81-release\build\lib\test\test_math.py", line 
1642, in assertIsNegativeZero
msg="Expected a negative zero, got {!r}".format(value)
AssertionError: False is not true : Expected a negative zero, got 0.0

==
FAIL: test_random (test.test_math.FMATests)
--
Traceback (most recent call last):
  File "D:\buildarea\3.x.ware-win81-release\build\lib\test\test_math.py", line 
1623, in test_random
self.assertEqual(math.fma(a, b, c), expected)
AssertionError: 0.5506672157701096 != 0.5506672157701097

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Serhiy Storchaka]

Serhiy Storchaka added the comment:

Cross fingers...

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Roundup Robot]

Roundup Robot added the comment:

New changeset b33012ef1417 by Mark Dickinson in branch 'default':
Issue #29282: add fused multiply-add function, math.fma.
https://hg.python.org/cpython/rev/b33012ef1417

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

Thanks, Serhiy. I'm going to hold off committing this for 24 hours or so, 
because I want to follow the buildbots when I do (and I don't have time for 
that right now). I wouldn't be at all surprised to see platform-specific test 
failures.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Serhiy Storchaka]

Serhiy Storchaka added the comment:

Then LGTM unconditionally.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

Ah, the dev guide says 80 characters. 
(https://docs.python.org/devguide/documenting.html)

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

> lines in What's New are too long.

Thanks. Fixed (I think). I'm not sure what the limit is, but the lines are now 
all <= 79 characters long.

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Added file: http://bugs.python.org/file46346/math_fma3.patch

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Serhiy Storchaka]

Serhiy Storchaka added the comment:

LGTM except that lines in What's New are too long.

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stage: patch review -> commit review

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Serhiy Storchaka]

Changes by Serhiy Storchaka :


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Removed message: http://bugs.python.org/msg285838

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Serhiy Storchaka]

Serhiy Storchaka added the comment:

Have you missed the patch?

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

Whoops; looks like I failed to attach the updated patch. Here it is.

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Added file: http://bugs.python.org/file46345/math_fma2.patch

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

Serhiy, Victor: thanks for the reviews. Here's a new patch. Differences w.r.t. 
the old one:

- Converted to argument clinic.
- Updated docstring to talk about special cases.
- More tests, as suggested by Serhiy.
- whatsnew entry and ..versionadded in docs.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Serhiy Storchaka]

Serhiy Storchaka added the comment:

LGTM except that needed the versionadded directive and What's New entry. And 
maybe few additional tests.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Changes by Mark Dickinson :


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stage:  -> patch review

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

And here's a patch.

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Added file: http://bugs.python.org/file46321/math_fma.patch

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

Here's a pure Python reference implementation, with tests.

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Added file: http://bugs.python.org/file46304/fma_reference.py

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

> Isn't the behaviour of quiet NaNs kindof implementation-dependent already?

Not as far as IEEE 754-2008 is concerned, and not as far as Python's math 
module is concerned, either: handling of special cases is, as far as I know, 
both consistent across platforms and compliant with IEEE 754. That's not true 
across Python as a whole, but it should be true for the math module.

If you find an exception to the above statement, please do open a bug report!

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Antoine Pitrou]

Antoine Pitrou added the comment:

Isn't the behaviour of quiet NaNs kindof implementation-dependent already?

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

An implementation note: IEEE 754-2008 leaves it to the implementation to decide 
whether FMA operations like:

0 * inf + nan

and

inf * 0 + nan

(where nan represents a quiet NaN and the inf and 0 can have arbitrary signs) 
signal the invalid operation FPE or not. (Ref: 7.2(c) in IEEE 754-2008.) 

I'd suggest that in this case we follow what Intel does in its x64 chips with 
FMA3 support.(according to ). If I'm reading the table in section 2.3 of the 
Intel Advanced Vector Extensions Programming Reference correctly, Intel does 
*not* signal the invalid operation FPE in this case. That is, we're following 
the usual rule of quiet NaN in => quiet NaN out with no exception. This does 
unfortunately conflict with the IBM decimal specification and Python's decimal 
module, where these operations *do* set the "invalid" flag (see the spec, and 
test fmax0809 in the decimal test set).

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Mark Dickinson]

Mark Dickinson added the comment:

> The performance argument unlikely is applicable in this case.

Agreed. This is mainly about accuracy, not speed: the FMA operation is a 
fundamental building block for floating-point arithmetic, is useful in some 
numerical algorithms, and essential in others (especially when doing things 
like double-double arithmetic). It would be valuable to have when prototyping 
numerical algorithms in pure Python.

Given that it's supported in C99 and on current Windows, I'm +1 on including it 
in the math module.

Note that implementing this it not quite as straightforward as simply wrapping 
the libm version, since we'll also want the correct exceptional behaviour, for 
consistency with the rest of the math module: i.e., we should be raising 
ValueError where the fma operation would signal the invalid FPE, and 
OverflowError where the fma operation would signal the overflow FPE.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Serhiy Storchaka]

Serhiy Storchaka added the comment:

The performance argument unlikely is applicable in this case. I suppose that an 
overhead of function call in Python make two operators faster than one function 
call.

Alternatives to fma() for exact computations are integer arithmetic (if all 
values can be scaled to integers), fractions and decimal numbers.

But since fma() is a part of C (C99), C++ (C++11) and POSIX (POSIX.1-2001) 
standards for long time, I don't have objections against including it in the 
math module.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Antoine Pitrou]

Antoine Pitrou added the comment:

I don't know. If I want to compute a dot product, the first thing I'll do is 
import Numpy and then use the `@` operator on Numpy arrays.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Juraj Sukop]

Juraj Sukop added the comment:

I would say because it has wide applicability, especially considering the 
amount of code it adds. It is similar in spirit to `copysign` or `fsum` which 
are already there and C99 includes it anyway. For simpler things like dot 
product or polynomial evaluation importing Numpy might be too much of an 
dependency.

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Antoine Pitrou]

Antoine Pitrou added the comment:

What's the point of adding this in the math module rather than a more 
specialized library like Numpy?

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[issue29282] Fused multiply-add: proposal to add math.fma()

[STINNER Victor]

STINNER Victor added the comment:

Thread on python-ideas:
https://mail.python.org/pipermail/python-ideas/2017-January/044300.html

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[issue29282] Fused multiply-add: proposal to add math.fma()

[STINNER Victor]

Changes by STINNER Victor :


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[issue29282] Fused multiply-add: proposal to add math.fma()

[STINNER Victor]

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Juraj Sukop]

Changes by Juraj Sukop :


Added file: http://bugs.python.org/file46300/setup.py

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Juraj Sukop]

Changes by Juraj Sukop :


Added file: http://bugs.python.org/file46298/xmathmodule.c

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Juraj Sukop]

Changes by Juraj Sukop :


Added file: http://bugs.python.org/file46299/example.py

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[issue29282] Fused multiply-add: proposal to add math.fma()

[Juraj Sukop]

New submission from Juraj Sukop:

Fused multiply-add (henceforth FMA) is an operation which calculates the 
product of two numbers and then the sum of the product and a third number with 
just one floating-point rounding. More concretely:

r = x*y + z

The value of `r` is the same as if the RHS was calculated with infinite 
precision and the rounded to a 32-bit single-precision or 64-bit 
double-precision floating-point number [1].

Even though one FMA CPU instruction might be calculated faster than the two 
separate instructions for multiply and add, its main advantage comes from the 
increased precision of numerical computations that involve the accumulation of 
products. Examples which benefit from using FMA are: dot product [2], 
compensated arithmetic [3], polynomial evaluation [4], matrix multiplication, 
Newton's method and many more [5].

C99 includes [6] `fma` function to `math.h` and emulates the calculation if the 
FMA instruction is not present on the host CPU [7]. PEP 7 states that "Python 
versions greater than or equal to 3.6 use C89 with several select C99 features" 
and that "Future C99 features may be added to this list in the future depending 
on compiler support" [8].

This proposal is then about adding new `fma` function with the following 
signature to `math` module:

math.fma(x, y, z)

'''Return a float representing the result of the operation `x*y + z` with 
single rounding error, as defined by the platform C library. The result is the 
same as if the operation was carried with infinite precision and rounded to a 
floating-point number.'''

Attached is a simple module for Python 3 demonstrating the fused multiply-add 
operation. On my machine, `example.py` prints:

40037.524591982365   horner_double
40037.48821639768horner_fma
40037.49486325783horner_compensated
40037.49486325783horner_decimal

[1] https://en.wikipedia.org/wiki/Multiply%E2%80%93accumulate_operation
[2] S. Graillat, P. Langlois, N. Louvet. Accurate dot products with FMA. 2006
[3] S. Graillat, Accurate Floating Point Product and Exponentiation. 2007.
[4] S. Graillat, P. Langlois, N. Louvet. Improving the compensated Horner 
scheme with a Fused Multiply and Add. 2006
[5] J.-M. Muller, N. Brisebarre, F. de Dinechin, C.-P. Jeannerod, V. Lefèvre, 
G. Melquiond, N. Revol, D. Stehlé, S. Torres. Handbook of Floating-Point 
Arithmetic. 2010. Chapter 5
[6] ISO/IEC 9899:TC3, "7.12.13.1 The fma functions", Committee Draft - 
Septermber 7, 2007
[7] https://git.musl-libc.org/cgit/musl/tree/src/math/fma.c
[8] https://www.python.org/dev/peps/pep-0007/

--
components: Library (Lib)
messages: 285547
nosy: juraj.sukop
priority: normal
severity: normal
status: open
title: Fused multiply-add: proposal to add math.fma()
type: enhancement
versions: Python 3.7

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