Binade

In software engineering and numerical analysis, a binade is a set of numbers in a binary floating-point format that all have the same sign and exponent. In other words, a binade is the interval $[2^{e},2^{e+1})$ or $(-2^{e+1},-2^{e}]$ for some integer value $e$ , that is, the set of real numbers or floating-point numbers $x$ of the same sign such that $2^{e}\leq |x|<2^{e+1}$ .^[1]^[2]^[3]

An axis with tick marks for each floating-point number between 0 and 1 with 3 bits of precision and minimum exponent −5, with an underbrace labelled +1.xy × 2^(−2) to highlight the ticks for numbers from 1/4, inclusive, to 1/2, exclusive. — The binade of exponent −2 in the floating-point numbers with 3 bits of precision and minimum exponent −5

Some authors use the convention of the closed interval $[2^{e},2^{e+1}]$ instead of a half-open interval,^[4] sometimes using both conventions in a single paper.^[5] Some authors additionally treat each of various special quantities such as NaN, infinities, and zeroes as its own binade,^[6] or similarly for the exceptional interval $(0,2^{\mathrm {emin} })$ of subnormal numbers.^[7]

References

^ Muller, Jean-Michel; Brunie, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Torres, Serge (2018). Handbook of Floating-Point Arithmetic (2nd ed.). Birkhäuser. pp. 418–419. doi:10.1007/978-3-319-76526-6. ISBN 978-3-319-76525-9.
^ Lefèvre, Vincent; Muller, Jean-Michel (2001). "Worst cases for correct rounding of the elementary functions in double precision" (PDF). 15th IEEE Symposium on Computer Arithmetic. ARITH 2001. IEEE. pp. 111–118. doi:10.1109/ARITH.2001.930110. ISSN 1063-6889.
^ Benet, Luis; Ferranti, Luca; Revol, Nathalie (2023). "A framework to test interval arithmetic libraries and their IEEE 1788-2015 compliance". Concurrency and Computation: Practice and Experience. 36: e7856. arXiv:2307.06953. doi:10.1002/cpe.7856. ISSN 1532-0626.
^ Coonen, Jerome T. (1981). "Underflow and the Denormalized Numbers". Computer. 14 (3). IEEE: 75–87. doi:10.1109/C-M.1981.220382. ISSN 0018-9162.
^ Hanrot, Guillaume; Lefèvre, Vincent; Stehlé, Damien; Zimmermann, Paul (2007). "Worst Cases of a Periodic Function for Large Arguments". 18th IEEE Symposium on Computer Arithmetic. ARITH 2007. pp. 133–140. doi:10.1109/ARITH.2007.37. ISSN 1063-6889.
^ Thomas, David B. (2015). "A general-purpose method for faithfully rounded floating-point function approximation in FPGAs". 22nd IEEE Symposium on Computer Arithmetic. ARITH 2015. pp. 42–49. doi:10.1109/ARITH.2015.27. ISSN 1063-6889.
^ Agrawal, Ankur; Mueller, Sylvia M.; Fleischer, Bruce M.; Choi, Jungwook; Wang, Naigang; Sun, Xiao; Gopalakrishnan, Kailash (2019). "DLFloat: A 16-b Floating Point format designed for Deep Learning Training and Inference". 26th IEEE Symposium on Computer Arithmetic. ARITH 2019. pp. 92–95. doi:10.1109/ARITH.2019.00023. ISSN 1063-6889.

[handbook-1] Muller, Jean-Michel; Brunie, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Torres, Serge (2018). Handbook of Floating-Point Arithmetic (2nd ed.). Birkhäuser. pp. 418–419. doi:10.1007/978-3-319-76526-6. ISBN 978-3-319-76525-9.

[2] Lefèvre, Vincent; Muller, Jean-Michel (2001). "Worst cases for correct rounding of the elementary functions in double precision" (PDF). 15th IEEE Symposium on Computer Arithmetic. ARITH 2001. IEEE. pp. 111–118. doi:10.1109/ARITH.2001.930110. ISSN 1063-6889.

[3] Benet, Luis; Ferranti, Luca; Revol, Nathalie (2023). "A framework to test interval arithmetic libraries and their IEEE 1788-2015 compliance". Concurrency and Computation: Practice and Experience. 36: e7856. arXiv:2307.06953. doi:10.1002/cpe.7856. ISSN 1532-0626.

[4] Coonen, Jerome T. (1981). "Underflow and the Denormalized Numbers". Computer. 14 (3). IEEE: 75–87. doi:10.1109/C-M.1981.220382. ISSN 0018-9162.

[5] Hanrot, Guillaume; Lefèvre, Vincent; Stehlé, Damien; Zimmermann, Paul (2007). "Worst Cases of a Periodic Function for Large Arguments". 18th IEEE Symposium on Computer Arithmetic. ARITH 2007. pp. 133–140. doi:10.1109/ARITH.2007.37. ISSN 1063-6889.

[6] Thomas, David B. (2015). "A general-purpose method for faithfully rounded floating-point function approximation in FPGAs". 22nd IEEE Symposium on Computer Arithmetic. ARITH 2015. pp. 42–49. doi:10.1109/ARITH.2015.27. ISSN 1063-6889.

[7] Agrawal, Ankur; Mueller, Sylvia M.; Fleischer, Bruce M.; Choi, Jungwook; Wang, Naigang; Sun, Xiao; Gopalakrishnan, Kailash (2019). "DLFloat: A 16-b Floating Point format designed for Deep Learning Training and Inference". 26th IEEE Symposium on Computer Arithmetic. ARITH 2019. pp. 92–95. doi:10.1109/ARITH.2019.00023. ISSN 1063-6889.

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Binade

See also

References