如果某个系统上的浮点数存储具有符号位，3位指数和4位有效数字:

问题描述

(假设没有隐含位，没有偏差，指数使用2的补码表示法，并且允许全零和全1的指数.)我正在尝试查找如果系统标准化则可以表示的最大和最小数字.我认为最大的数字是:

(Assume no bits are implied, there is no biasing, exponents use two’s complement notation, and exponents of all zeros and all ones are allowed.)I am trying to find the largest and smallest number that can be represented if the system is normalized. I thought that the largest number would be:

.1111 x 2^4 = 0 100 1111 = 15

，最小的:

1.0 x 2^-4 = 0 000 0001 = 0.0625

但是我看到的答案是:

Largest: .1111 x 2^3 = 111.1 = 7.5

Smallest: 0.1 x 2^-4 = .00001 = 0.03125

我不明白以4位有效位数怎么可能回答这两个问题.

I do not understand how either answer would even be possible in a 4-bit significand.

推荐答案

这里有一些要解密的东西.

There're a few things to decipher here.

• 指数:3位，2的补码，无偏差.这意味着指数可以表示范围在 -4 (对应于 100 )到 3 (对应于 011 )之间的值>).

• Exponent: 3-bits, 2's complement, no bias. This means the exponent can represent values in the range -4 (corresponding to 100) to 3 (corresponding to 011).

没有隐含位的规范化:这意味着有效数始终以 1 开头.

Normalized without implied bit: This means significand always starts with a 1.

将它们放在一起时，可以写的最大数量为:

When you put these together, the maximum number you can write is:

 0 011 1111 = 2^3 * (2^-1 + 2^-2 + 2^-3 + 2^-4) = 7.5

由于浮点值在 0 附近是对称的，因此通过翻转上方的符号位，可以写入的最小值为 -7.5 .但是我想您的老师要求的是最低限度的正值(即非零).在这种情况下，我们选择指数越小越好，并仅保留有效位数的第一位以满足标准化要求.我们得到:

Since floating point values are symmetic around 0, the minumum value you can write is -7.5 by flipping the sign bit above. But I guess your teacher is going for minumum strictly positive (i.e., non-zero) value. In that case, we pick the exponent to be as small as possible, and just keep the first bit of the significand to satisfy the normalized requirement. We get:

 0 100 1000 = 2^-4 * 2^-1 = 2^-5 = 0.03125

希望如此！

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