Binade Explained

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} ) or ( -2 ^{e} , -2 ^{e]} for some integer value e , that is, the set of real numbers or floating-point numbers x of the same sign such that2 ^{e} \leq |x| < 2 ^{e} .^{[1] [2] [3]}

Some authors use the convention of the closed interval

[ 2 ^{e,} 2 ^{e} ] 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 ^{emin} ) of subnormal numbers.^[7]

See also

• Floating-point arithmetic
• IEEE 754
• Significand

Notes and References

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