Graphics Reference
In-Depth Information
Table 11.2
The values assumed by IEEE-754 single-precision floating-point numbers based
on the values of the exponent, fraction, and sign fields.
Exponent
Fraction
Sign
Value
1)
S
2
E
−
127
0
<
E
<
255
V
=
(
−
∗
(1.
F
)
∗
=
=
=
=
E
0
F
0
S
0
V
0
=
=
=
=−
E
0
F
0
S
1
V
0
=
=
=
2
E
−
126
(denormalized numbers)
−
1)
S
∗
∗
E
0
F
0
V
(
(0.
F
)
E
=
255
F
=
0
S
=
0
V
=+
INF
E
=
255
F
=
0
S
=
1
V
=−
INF
E
=
255
F
=
0
V
=
NaN
(signaling if top fraction bit is 0,
else quiet
NaN
)
0
0
Figure 11.4
Denormalized (or subnormal) floating-point numbers fill in the gap immediately
surrounding zero.
An important detail about the introduction and widespread acceptance of the
IEEE standard is its strict accuracy requirements of the arithmetic operators. In short,
the standard states that the floating-point operations must produce results that are
correct in all bits. If the result is not
machine representable
(that is, directly representable
in the IEEE format), the result must be correctly rounded (with minimal change in
value) with respect to the currently set rounding rule to either of the two machine-
representable numbers surrounding the real number.
For rounding to be performed correctly, the standard requires that three extra bits
are used during floating-point calculations: two
guard bits
and one
sticky bit
. The two
guard bits act as two extra bits of precision, effectively making the mantissa 26 rather
than 24 bits. The sticky bit becomes set if one of the omitted bits beyond the guard
bits would become nonzero, and is used to help determine the correctly rounded
