Page 31 - ARM 64 Bit Assembly Language

14 Chapter 1

Sign Two’s
Binary Unsigned Magnitude Excess-127 Complement
00000000 0 0 −127 0
00000001 1 1 −126 1
. . . . .
. . . . .
. . . . .
01111110 126 126 −1 126
01111111 127 127 0 127
10000000 128 −0 1 −128
10000001 129 −1 2 −127
. . . . .
. . . . .
. . . . .
11111110 254 −126 127 −2
11111111 255 −127 128 −1

Figure 1.4: Four different representations for binary integers.

interpreting integers whose value can be both positive and negative. Programmers and hard-
ware designers have developed several standard schemes for encoding such numbers The
three main methods for storing and interpreting signed integer data are two’s complement,
sign-magnitude, and excess-N,Fig. 1.4 shows how the same binary pattern of bits can be in-
terpreted as a number in four different ways.

1.3.3.1 Sign-magnitude representation

The sign-magnitude representation simply reserves the most significant bit to represent the
sign of the number, and the remaining bits are used to store the magnitude of the number. This
method has the advantage that it is easy for humans to interpret, with a little practice. How-
ever, addition and subtraction are slightly complicated. The addition/subtraction logic must
compare the sign bits, complement one of the inputs if they are different, implement an end-
around carry, and complement the result if there was no carry from the most significant bit.
Complements are explained in Section 1.3.3.3. Because of the complexity, most integer CPUs
do not directly support addition and subtraction of integers in sign-magnitude form. However,
this method is commonly used for mantissa in floating-point numbers, as will be explained
in Chapter 8. Another drawback to sign-magnitude is that it has two representations for zero,
which can cause problems if the programmer is not careful.

1.3.3.2 Excess-(2 n−1 − 1) representation

Another method for representing both positive and negative numbers is by using an excess-N
representation. With this representation, the number that is stored is N greater than the actual
value. This representation is relatively easy for humans to interpret. Addition and subtraction