2.3.7 Dividing by Powers of Two

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原文链接：http://www.cnblogs.com/baiweiguo/archive/2013/01/17/2864715.html

本文深入探讨了如何利用逻辑右移操作来完成整数除以2的运算，并详细解释了不同类型的移位操作（逻辑与算术移位）在无符号数和补码数上的区别。文章还介绍了整数除法的基本规则及常见陷阱，通过实例展示了移位操作在实际编程中的应用。

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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 $2^{k}$ .

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, $U2B_{\omega}^{k}(x)+T2B_{\omega}^{k}=0$ , while when k = 4 or 8, $U2B_{\omega}^{k}+T2B_{\omega}^{k}=-1$ .

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 $2^{k}$ , the following method correct this improper rounding by “biasing”

$y=2^{k}-1$ . The result show in figure 2.29.

Suppose that $x=ky+r$ , where $0\le r<y$ .

$\frac{x+y-1}{y}=\frac{ky+r+y-1}{y}=k+\frac{r+y-1}{y}$

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

转载于:https://www.cnblogs.com/baiweiguo/archive/2013/01/17/2864715.html