Editing Associative property (section)

===Nonassociativity of floating point calculation===

In mathematics, addition and multiplication of real numbers are associative.  By contrast, in computer science, addition and multiplication of [[floating point]] numbers are ''not'' associative, as different rounding errors may be introduced when dissimilar-sized values are joined in a different order.<ref>Knuth, Donald, [[The Art of Computer Programming]], Volume 3, section 4.2.2</ref>

To illustrate this, consider a floating point representation with a 4-bit [[significand]]:

{{block indent|1=(1.000<sub>2</sub>×2<sup>0</sup> + 1.000<sub>2</sub>×2<sup>0</sup>) +
1.000<sub>2</sub>×2<sup>4</sup> = 1.000<sub>2</sub>×2<sup>{{fontcolor|red|1}}</sup> + 1.000<sub>2</sub>×2<sup>4</sup> = 1.00{{fontcolor|red|1}}<sub>2</sub>×2<sup>4</sup>}}
{{block indent|1=1.000<sub>2</sub>×2<sup>0</sup> + (1.000<sub>2</sub>×2<sup>0</sup> +
1.000<sub>2</sub>×2<sup>4</sup>) = 1.000<sub>2</sub>×2<sup>{{fontcolor|red|0}}</sup> + 1.000<sub>2</sub>×2<sup>4</sup> = 1.00{{fontcolor|red|0}}<sub>2</sub>×2<sup>4</sup>}}

Even though most computers compute with 24 or 53 bits of significand,<ref>{{Cite book |title=IEEE Standard for Floating-Point Arithmetic |author=IEEE Computer Society |date=29 August 2008 |id=IEEE Std 754-2008|doi=10.1109/IEEESTD.2008.4610935 |ref=CITEREFIEEE_7542008 |isbn=978-0-7381-5753-5}}</ref> this is still an important source of rounding error, and approaches such as the [[Kahan summation algorithm]] are ways to minimise the errors. It can be especially problematic in parallel computing.<ref>{{Citation
| last1   = Villa
| first1   = Oreste
| last2   = Chavarría-mir
| first2   = Daniel
| last3   = Gurumoorthi
| first3   = Vidhya
| last4   = Márquez
| first4   = Andrés
| last5   = Krishnamoorthy
| first5   = Sriram
| title   = Effects of Floating-Point non-Associativity on Numerical Computations on Massively Multithreaded Systems
| url   = http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf
| access-date   = 8 April 2014
| archive-url   = https://web.archive.org/web/20130215171724/http://cass-mt.pnnl.gov/docs/pubs/pnnleffects_of_floating-pointpaper.pdf
| archive-date   = 15 February 2013
| url-status   = dead
}}</ref><ref name="Goldberg_1991">{{cite journal|last=Goldberg|first=David|author-link=David Goldberg (PARC)|date=March 1991|title=What Every Computer Scientist Should Know About Floating-Point Arithmetic|url=http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|journal=[[ACM Computing Surveys]]|volume=23|issue=1|pages=5–48|doi=10.1145/103162.103163|s2cid=222008826|access-date=20 January 2016|url-status=live|archive-url=https://web.archive.org/web/20220519083509/http://perso.ens-lyon.fr/jean-michel.muller/goldberg.pdf|archive-date=2022-05-19}}</ref>