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A statistical study of the accuracy of floating point number systems

A statistical study of the accuracy of floating point number systems,10.1145/362003.362013,Communications of The ACM,H. Kuki,W. J. Cody

A statistical study of the accuracy of floating point number systems   (Citations: 21)
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This paper presents the statistical results of tests of the accuracy of certain arithmetic systems in evaluating sums, products and inner products, and analytic error estimates for some of the computations. The arithmetic systems studied are 6-digit hexadecimal and 22-digit binary floating point number representations combined with the usual chop and round modes of arithmetic with various numbers of guard digits, and with a modified round mode with guard digits. In a certain sense, arithmetic systems differing only in their use of binary or hexadecimal number representations are shown to be approximately statistically equivalent in accuracy. Further, the usual round mode with guard digits is shown to be statistically superior in accuracy to the usual chop mode in all cases save one. The modified round mode is found to be superior to the chop mode in all cases.
Journal: Communications of The ACM - CACM , vol. 16, no. 4, pp. 223-230, 1973
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    • ...The reduction in precision caused by using only a small number of guard digits is discussed by Kuki and Cody [18]...
    • ...Thus we prefer rms to other probabilistic measures of precision such as the expected value of | | (McKeeman [21]), the expected value of log2 | | (Kuki and Codi [18]), and the expected error in “units in the last place” (Kahan [15])...
    • ...The rounding rule for systems S1 to S5 is the “R*-mode” of Kuki and Cody [18]: fl(x) is defined to be the floating-point number closest to x, and ties are broken by choosing fl(x) so that its least significant fraction bit is one...
    • ...A number z was drawn from a uniform distribution on [0,1], then numbers x1,...,xn were drawn independently from a uniform distribution on [ Z,Z], where Z = 256z is a scale factor used to avoid a bias in favour of any of the number systems (see Kuki and Cody [18])...

    Richard P. Brent. On the precision attainable with various floating-point number systems

    • ...corresponding binary system, but this is not necessarily the case (see [22])...

    William J. Cody. Basic concepts for computational software

    • ...One result that might have broad interest concerns the behavior of the mean value of multiplicative error as a function of the base fl. What had been observed empirically for even base by Cody [2] and by Kukl and Cody [10], and had been proven theoretically in [6], is that two guard flits are better than one when postmultlphcatlve normalization occurs before symmetric rounding...
    • ...Of course adding a second guard flit (or even just a fifth binary guard bit) would further reduce the mean error, by an order of magmtude in u. This was observed by Cody [2] and by Kuki and Cody [10]...

    Joaquín Bustozet al. Improved Trailing Digits Estimates Applied to Optimal Computer Arithme...

    • ...We use R*-rounding [15, 33] after postnormalization, with four guard digits...

    Richard P. Brent. A Fortran Multiple-Precision Arithmetic Package

    • ...We use R*-rounding [15, 33] after postnormalization, with four guard digits...

    RICHARD P. BRENT. A Fortran Multiple-Precision Arithmetic

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