# 27 bits are not enough for 8-digit accuracy

27 bits are not enough for 8-digit accuracy 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications of the ACM Association for Computing Machinery

# 27 bits are not enough for 8-digit accuracy

, Volume 10 (2) – Feb 1, 1967
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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

### Abstract

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

### Journal

Communications of the ACMAssociation for Computing Machinery

Published: Feb 1, 1967