Computer Hardware User's Guide
Floating-Point Addition and Subtraction
5-32
5.6 Floating-Point Addition and Subtraction
In floating-point addition and subtraction, two floating-point numbers α and
b
can be defined as:
α = α(
man
) × 2
α(
exp
)
b
=
b
(
man
) × 2
b
(
exp
)
The sum (or difference) of α and
b
can be defined as:
c
= α ±
b
=(α(
man
) ± (
b
(
man
) × 2
–(α(
exp
)–
b
(
exp
))
) × 2
α(
exp
)
, if α(
exp
) ≥
b
(
exp
)
=(α(
man
) × 2
–(
b
(
exp
)–α(
exp
))
) ±
b
(
man
)) × 2
b
(
exp
)
, if α(
exp
) <
b
(
exp
)
Figure 5–17 shows the flowchart for floating-point addition. Because this flow-
chart assumes signed data, it is also appropriate for floating-point subtraction.
In this figure, it is assumed that α(
exp
) ≤
b
(
exp
).
In step 1, the source exponents, α(
exp
) and
b
(
exp
), are compared, and
c
(
exp
) is set equal to the largest of the two source exponents.
In step 2,
d
is set to the difference of the two exponents.
In step 3, the mantissa with the smallest exponent, in this case α(
man
),
is right-shifted
d
bits to align the mantissas.
In step 4, after the mantissas have been aligned, they are added.
In steps 5 through 7, a check for a special case of
c
(
man
). If
c
(
man
) is 0
(step 5), then
c
(
exp
) is set to its most negative value (step 8) to yield the
correct representation of 0. If
c
(
man
) has overflowed
c
(step 6), then in
step 9
c
(
man
) is right-shifted one bit and 1 is added to
c
(
exp
). In step 10,
the result is normalized.
Steps 11 through 13 check for special cases of
c
(
exp
). If
c
(
exp
) has over-
flowed (step 11) in the positive direction, then step 14 sets
c
(
exp
) to the
most positive extended-precision format value. If
c
(
exp
) has overflowed
(step 11) in the negative direction, then step 14 sets
c
(
exp
) to the most
negative extended-precision format value. If
c
(
exp
) has underflowed (step
12), then step 15 sets
c
to 0; that is,
c
(
man
) = 0 and
c
(
exp
) = –128. If no
overflow or underflow occurred, then
c
is not modified.