The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
On Local Roundoff Errors in Floating-Point Arithmetic
Journal of the ACM (JACM)
On the distributions of significant digits and roundoff errors
Communications of the ACM
Improved Trailing Digits Estimates Applied to Optimal Computer Arithmetic
Journal of the ACM (JACM)
Loss of Significance in Floating Point Subtraction and Addition
IEEE Transactions on Computers
Analysis of Rounding Methods in Floating-Point Arithmetic
IEEE Transactions on Computers
Improved bound for stochastic formal correctness of numerical algorithms
Innovations in Systems and Software Engineering
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This paper analyzes the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base &bgr; and randomly chosen from a logarithmic distribution. The analysis shows that the nth digit for n ≥ 2 is actually approximately uniformly distributed. The approximation depends upon both n and the base&bgr;. It becomes better as n increases, and it is exact in the limit as n ⇒ ∞. A table of this distribution is presented for various &bgr; and n, along with a table of the maximum digit by digit deviation &Dgr; of the logarithmic distribution from the uniform distribution. Various asymptotic results for &Dgr; are included. These results have application in resolving open questions of Henrici, of Kaneko and Liu, and of Tsao.