2.3.7 Dividing by Powers of Two

Dividing by a power of 2 can be performed using a right shift operation.

The two different shifts-logical and arithmetic-serve the shift operation for unsigned and two's-complement numbers, respectively.

Integer division always rounds toward zero.

1.Consider unsigned number for logical right shift by k. We claim this gives the same result as dividing by .

The examples show that the result of shifing consistently rounds toward zero, as is the convention for integer division.

 

Compared with figure 2.27 and figure 2.28, when k=0 and 1, , while when k = 4 or 8, .

U2B(12340):

                     0011000000110100

The negation of 12340 is:

                     1100111111001100

右移操作符：>>k

当 k = 0， 1，2 时，U2B(12340) + T2B(12340) = 0000000000000000 = 0

当 k >= 3 时，     U2B(12340) + T2B(12340) = 1111111111111111 =–1

 

Figure 2.28 show that the shifing is not consistent with dividing by , the following method correct this improper rounding by “biasing”

. The result show in figure 2.29.

 

 

 

Suppose that , where .

 

 

 

when =0, (x+y-1)/y = k; when r>0. (x+y-1)/y=k+1.