View source for Binade

{{Short description|Interval of binary floating-point numbers with a common sign and exponent}}
[[File:Binade.svg|thumb|alt=An axis with tick marks for each floating-point number between 0 and 1 with 3 bits of precision and minimum exponent &minus;5, with an underbrace labelled +1.xy &times; 2^(&minus;2) to highlight the ticks for numbers from 1/4, inclusive, to 1/2, exclusive.|The binade of exponent &minus;2 in the floating-point numbers with 3 bits of precision and minimum exponent &minus;5]]

In [[software engineering]] and [[numerical analysis]], a '''binade''' is a set of numbers in a binary [[Floating-point arithmetic|floating-point format]] that all have the same sign and exponent.
In other words, a binade is the [[Interval (mathematics)|interval]]
<math>[2^e, 2^{e + 1})</math> or <math>(-2^{e + 1}, -2^e]</math>
for some integer value&nbsp;<math>e</math>, that is, the set of real numbers or floating-point numbers <math>x</math> of the same sign such that
<math>2^e \leq |x| < 2^{e + 1}</math>.<ref name="handbook">{{cite book
|last1=Muller
|first1=Jean-Michel
|last2=Brunie
|first2=Nicolas
|last3=de Dinechin
|first3=Florent
|last4=Jeannerod
|first4=Claude-Pierre
|last5=Joldes
|first5=Mioara
|last6=Lefèvre
|first6=Vincent
|last7=Melquiond
|first7=Guillaume
|last8=Revol
|first8=Nathalie
|author8-link=Nathalie Revol
|last9=Torres
|first9=Serge
|title=Handbook of Floating-Point Arithmetic
|date=2018
|publisher=Birkhäuser
|isbn=978-3-319-76525-9
|at=pp. 418–419
|doi=10.1007/978-3-319-76526-6
|edition=2nd
|url=https://doi.org/10.1007/978-3-319-76526-6
}}</ref><ref>{{cite conference
|author-last1=Lefèvre
|author-first1=Vincent
|author-last2=Muller
|author-first2=Jean-Michel
|title=Worst cases for correct rounding of the elementary functions in double precision
|book-title=15th IEEE Symposium on Computer Arithmetic
|conference=ARITH 2001
|year=2001
|publisher=IEEE
|pages=111–118
|doi=10.1109/ARITH.2001.930110
|url=http://www.acsel-lab.com/arithmetic/arith15/papers/ARITH15_Lefevre.pdf
|issn=1063-6889
}}</ref><ref>{{cite journal
|author-last1=Benet
|author-first1=Luis
|author-last2=Ferranti
|author-first2=Luca
|author-last3=Revol
|author-first3=Nathalie
|author-link3=Nathalie Revol
|title=A framework to test interval arithmetic libraries and their IEEE 1788-2015 compliance
|journal=Concurrency and Computation: Practice and Experience
|year=2023
|volume=36
|issn=1532-0626
|doi=10.1002/cpe.7856
|pages=e7856
|doi-access=free
|arxiv=2307.06953
}}</ref>

Some authors use the convention of the closed interval <math>[2^e, 2^{e + 1}]</math> instead of a half-open interval,<ref>{{cite journal
|author-last=Coonen
|author-first=Jerome T.
|title=Underflow and the Denormalized Numbers
|journal=Computer
|year=1981
|volume=14
|number=3
|pages=75–87
|publisher=IEEE
|url=https://ieeexplore.ieee.org/document/1667289
|doi=10.1109/C-M.1981.220382
|issn=0018-9162
}}</ref>
sometimes using both conventions in a single paper.<ref>{{cite conference
|author-last1=Hanrot
|author-first1=Guillaume
|author-last2=Lefèvre
|author-first2=Vincent
|author-last3=Stehlé
|author-first3=Damien
|author-last4=Zimmermann
|author-first4=Paul
|author-link4=Paul Zimmermann (mathematician)
|title=Worst Cases of a Periodic Function for Large Arguments
|book-title=18th IEEE Symposium on Computer Arithmetic
|conference=ARITH 2007
|year=2007
|pages=133–140
|doi=10.1109/ARITH.2007.37
|url=https://ieeexplore.ieee.org/document/4272859
|issn=1063-6889
}}</ref>
Some authors additionally treat each of various special quantities such as [[NaN]], infinities, and zeroes as its own binade,<ref>{{cite conference
|author-last=Thomas
|author-first=David B.
|title=A general-purpose method for faithfully rounded floating-point function approximation in FPGAs
|book-title=22nd IEEE Symposium on Computer Arithmetic
|conference=ARITH 2015
|year=2015
|pages=42–49
|doi=10.1109/ARITH.2015.27
|url=https://ieeexplore.ieee.org/document/7203794
|issn=1063-6889
}}</ref>
or similarly for the exceptional interval <math>(0, 2^{\mathrm{emin}})</math> of [[Subnormal number|subnormal numbers]].<ref>{{cite conference
|author-last1=Agrawal
|author-first1=Ankur
|author-last2=Mueller
|author-first2=Sylvia M.
|author-last3=Fleischer
|author-first3=Bruce M.
|author-last4=Choi
|author-first4=Jungwook
|author-last5=Wang
|author-first5=Naigang
|author-last6=Sun
|author-first6=Xiao
|author-last7=Gopalakrishnan
|author-first7=Kailash
|title=DLFloat: A 16-b Floating Point format designed for Deep Learning Training and Inference
|book-title=26th IEEE Symposium on Computer Arithmetic
|conference=ARITH 2019
|year=2019
|pages=92–95
|doi=10.1109/ARITH.2019.00023
|url=https://ieeexplore.ieee.org/document/8877411
|issn=1063-6889
}}</ref>

==See also==
* [[Floating-point arithmetic]]
* [[IEEE 754]]
* [[Significand]]

==References==
{{reflist}}

[[Category:Computer arithmetic]]
[[Category:Floating point]]