591
Floating-point Math Instructions Section 3-15
It is not necessary for the user to be aware of the IEEE754 data format when
reading and writing floating-point data. It is only necessary to remember that
floating point values occupy two words each.
Numbers Expressed as Floating-point Values
The following types of floating-point numbers can be used.
Note A non-normalized number is one whose absolute value is too small to be
expressed as a normalized number. Non-normalized numbers have fewer sig-
nificant digits. If the result of calculations is a non-normalized number (includ-
ing intermediate results), the number of significant digits will be reduced.
Normalized Numbers Normalized numbers express real numbers. The sign bit will be 0 for a positive
number and 1 for a negative number.
The exponent (e) will be expressed from 1 to 254, and the real exponent will
be 127 less, i.e., –126 to 127.
The mantissa (f) will be expressed from 0 to 2
33
– 1, and it is assume that, in
the real mantissa, bit 2
33
is 1 and the binary point follows immediately after it.
Normalized numbers are expressed as follows:
(–1)
(sign s)
x 2
(exponent e)–127
x (1 + mantissa x 2
–23
)
Example
Sign: –
Exponent: 128 – 127 = 1
Mantissa: 1 + (2
22
+ 2
21
) x 2
–23
= 1 + (2
–1
+ 2
–2
) = 1 + 0.75 = 1.75
Value: –1.75 x 2
1
= –3.5
Non-normalized Numbers Non-normalized numbers express real numbers with very small absolute val-
ues. The sign bit will be 0 for a positive number and 1 for a negative number.
The exponent (e) will be 0, and the real exponent will be –126.
The mantissa (f) will be expressed from 1 to 2
33
– 1, and it is assume that, in
the real mantissa, bit 2
33
is 0 and the binary point follows immediately after it.
Non-normalized numbers are expressed as follows:
(–1)
(sign s)
x 2
–126
x (mantissa x 2
–23
)
Example
Sign: –
Exponent: –126
Mantissa: 0 + (2
22
+ 2
21
) x 2
–23
= 0 + (2
–1
+ 2
–2
) = 0 + 0.75 = 0.75
Value: –0.75 x 2
–126
15
n+1
n
70
f
se
6
Mantissa (f) Exponent (e)
0 Not 0 and
not all 1’s
All 1’s (255)
0 0 Normalized number Infinity
Not 0 Non-normalized
number
NaN
1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
31 30 23 22 0
0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
31 30 23 22 0
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