Digital Arithmetic

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CHAPTER 8

Floating-Point Representation, Algorithms, and Implementations

8.2.2

unbiased round to nearest

is

1

ifMd <2

1

ifMd >g

1

if Md -- g and

Mf --

even

1

if Md -- g and

Mf --

odd

Ms

Mf +r -f

Rnear(x)

= 8.36

Ms

Mf -{-r -f

Consequently, for this mode

RB[Rnear]

= 0 8.37

This roundoff mode is illustrated in Figure 8.5(a) for f = 2.

In summary, round to nearest (unbiased) produces the smallest possible

absolute error 1~ and has a zero bias. However, the implementation of this mode

requires an addition, so it is slow. We will see ways of reducing the delay for

specific operations.

Round Toward Zero (Truncation)

In ...