Communications of the ACM
, Volume 10 (2) – Feb 1, 1967

/lp/association-for-computing-machinery/27-bits-are-not-enough-for-8-digit-accuracy-kxWPDdIfcS

- Publisher
- Association for Computing Machinery
- Copyright
- Copyright © 1967 by ACM Inc.
- ISSN
- 0001-0782
- DOI
- 10.1145/363067.363112
- Publisher site
- See Article on Publisher Site

i N E ugh for 2. The Simple Case 8-])iglit k c : racy [, I~ldNNI';T'J ('~(HA)I'IEI((} :~: We can represent each of the 90 decimal integers 10, 11, .-. , 98, 99 by the equivalent 7-bit integer, with no question regarding accuracy, as follows: (/+:/,ergd /(/cog:i:: (',rope*l,+/, (/~+w,b?'id(/~:,Masa. From the inequality 1 0 ~ < 2:, we are likely to conclude that we can represent 8-digit decimal floating-point numbers accurately by 27-bit floating~point numbers. However, we need 28 significant bits to represent some 8-digit numbers accurately. In general, we can show that if 10 ~' < 2 q ', then q significant bits are always enough for p-digit decimal accuracy. Finally, we can define a compact 27-blt floating-point representation that will give 28 significant bits, for numbers of practical importance. We can represent each of the 90 decimal fractions .10, .11, . . . , .98, .99 represented by the approximate binary fraction rounded to 7 [)Faces: .10 .11 .12 . 0 0 0 1 1 0 1 . 0 0 0 1 1 1 0 . 0 0 0 1 1 1 1 . 1 1 1 1 1 0 0 , 1 1 1 1

Communications of the ACM – Association for Computing Machinery

**Published: ** Feb 1, 1967

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