Unit in the last place

Builds on Floating-point arithmetic Machine epsilon

Definition

Consider a floating-point system with base $\beta$ , precision $p$ and minimal exponent $e_{\min}$ . Let $x \ne 0$ be a floating-point number written as $x = \pm (d_0 . d_1 \ldots d_{p-1})_\beta \times \beta^{e}$ , with $d_0 \ne 0$ if $e > e_{\min}$ . The unit in the last place of $x$ is $\mathrm{ulp}(x) = \beta^{\,e - p + 1},$ the value of the digit position $d_{p-1}$ .

Remarks

Example: with $\beta = 10$ and $p = 3$ , the number $x = 3.14 = 3.14 \times 10^{0}$ has $e = 0$ , so $\mathrm{ulp}(x) = 10^{-2}$ ; its neighbours are $3.13$ and $3.15$ . Non-example: in the same system, $0.001$ is not the ulp of $3.14$ , since $3.141$ is not a floating-point number of precision $3$ . The ulp of $1$ is $\beta^{1-p}$ , the machine epsilon. Several variants of the definition exist; some define $\mathrm{ulp}(x)$ for every real $x$ as the gap between the two floating-point numbers closest to $x$ . They agree away from powers of $\beta$ . Rounding to nearest gives an error of at most $\tfrac12 \mathrm{ulp}$ of the result.

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Tag	`NDS0H727BA`
URI	`m2://NDS0H727BA`
Kind	definition
Writing	W0 stub
MSC	65G50, 65Y04
Defines	Unit in the last place
Uses	Floating-point arithmetic, Machine epsilon

Unit in the last place

Definition

Remarks

Reading path

Further reading

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