Relative distance—an error measure in round-off error analysis

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( x,x) = min \{ α |1 - α x/ x 1/(1 - α )\} , x ¯ ≠ 0 x ≠ 0 , x / x ¯ > 0 x/ 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({ x^k},{x^k}) |k|d( x,x) is not true fo