Olver (SIAM J. Numer. Anal., v. 15, 1978, pp. 368-393) suggested relative precision as an attractive substitute for relative error in round-off error analysis. He remarked that in certain respects the error measure
d
(
x
¯
,
x
)
=
min
{
α
|
1
−
α
⩽
x
/
x
¯
⩽
1
/
(
1
−
α
)
}
d(\bar x,x) = \min \{ \alpha |1 - \alpha \leqslant x/\bar x \leqslant 1/(1 - \alpha )\}
,
x
¯
≠
0
\bar x \ne 0
,
x
/
x
¯
>
0
x/\bar x > 0
is even more favorable, through it seems to be inferior because of two drawbacks which are not shared by relative precision: (i) the inequality
d
(
x
¯
k
,
x
k
)
⩽
|
k
|
d
(
x
¯
,
x
)
d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)
is not true for
0
>
|
k
|
>
1
0 > |k| > 1
. (ii)
d
(
x
¯
,
x
)
d(\bar x,x)
is not defined for complex
x
¯
,
x
\bar x,x
. In this paper the definition of
d
(
⋅
,
⋅
)
d( \cdot , \cdot )
is replaced by
d
(
x
¯
,
x
)
=
|
x
¯
−
x
|
/
max
{
|
x
¯
|
,
|
x
|
}
d(\bar x,x) = |\bar x - x|/\max \{ |\bar x|,|x|\}
. This definition is equivalent to the first in case
x
¯
≠
0
\bar x \ne 0
,
x
/
x
¯
>
0
x/\bar x > 0
, and is free of (ii). The inequality
d
(
x
¯
k
,
x
k
)
⩽
|
k
|
d
(
x
¯
,
x
)
d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)
is replaced by the more universally valid inequality
d
(
x
¯
k
,
x
k
)
⩽
|
k
|
d
(
x
¯
,
x
)
/
(
1
−
δ
)
,
δ
=
max
{
d
(
x
¯
,
x
)
,
|
k
|
d
(
x
¯
,
x
)
}
d({\bar x^k},{x^k}) \leqslant |k|d(\bar x,x)/(1 - \delta ),\delta = \max \{ d(\bar x,x),|k|d(\bar x,x)\}
. The favorable properties of
d
(
⋅
,
⋅
)
d( \cdot , \cdot )
are preserved in the complex case. Moreover, its definition may be generalized to linear normed spaces by
d
(
x
¯
,
x
)
=
‖
x
¯
−
x
‖
/
max
{
‖
x
¯
‖
,
‖
x
‖
}
d(\bar x,x) = \left \| {\bar x - x} \right \|/\max \{ \left \| {\bar x} \right \|,\left \| x \right \|\}
. Its properties in such spaces raise the possibility that with further investigation it might become the basis for error analysis in some vector, matrix, and function spaces.