[Python-ideas] Fused multiply-add (FMA)

[Stephan Houben]

Hi Juraj,

I think this would be a very useful addition to the `math' module.

The gating issue is probably C compiler support.
The most important non-C99 C compiler for Python is probably MS Visual
Studio.
And that one appears to support it:

https://msdn.microsoft.com/en-us/library/mt720715.aspx

So +1 on the proposal.

Stephan

2017-01-15 18:25 GMT+01:00 Juraj Sukop <juraj.sukop at gmail.com>:

> Hello!
>
> 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 `fma` function to `math.h` [6] 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.'''
>
> There is a simple module for Python 3 demonstrating the fused multiply-add
> operation which was build with simple `python3 setup.py build` under Linux
> [9].
>
> Any feedback is greatly appreciated!
>
> Juraj Sukop
>
> [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/
> [9] https://github.com/sukop/fma
>
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