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
40037.48821639768 horner_fma
40037.49486325783 horner_compensated
40037.49486325783 horner_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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Python tracker <rep...@bugs.python.org>
<http://bugs.python.org/issue29282>
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