为什么数字(不能)在双精度IEEE754中表示? [英] Why number are (not) representable in double precision IEEE754?

问题描述

我对 IEEE754 双精度感到困惑，我考虑了两个问题:
1. 为什么间隔-2 ⁵⁴ ，-2 ⁵⁴ + 2，-2 ⁵⁴ +4 ... 2 ⁵⁴ 是可代表的吗?

I am confused on IEEE754 double precision, I consider two questions:
1. Why each number from interval -2⁵⁴, -2⁵⁴+2, -2⁵⁴+4...2⁵⁴ is representable ?

2..为什么2 ⁵⁴ +2无法表示?

2. Why 2⁵⁴+2 is not representable ?

你能帮我吗?我了解 IEEE754 的工作方式-但是，看到存在问题.

Can you help me ? I understand way of working IEEE754 - however, I have a problem with seeing it.

推荐答案

IEEE 754 double的有效位(或尾数)中有53位. − 2 ⁵⁴ 可以精确表示为，

There are 53 bits in the significand (or mantissa) of an IEEE 754 double. −2⁵⁴ can be exactly represented, as

mantissa: 1.00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00 (bin)
exponent: 54
sign:     1

现在让我们暂时忘记符号位.与该解释无关.因此，假设我们有+2 ⁵⁴ .

Now let's forget the sign bit for a moment. It is irrelevant for this explanation. So assume we have +2⁵⁴.

有了这个指数，有效数字的 最低-最右- 位的值为2 ^-52 * 2 ⁵⁴ =4.因此2 ⁵⁴ + 4编码为:

With this exponent, the lowest -- rightmost -- bit of the significand has the value 2^-52 * 2⁵⁴ = 4. So 2⁵⁴ + 4 is encoded as:

mantissa: 1.00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 01 (bin)
exponent: 54                                                             ^
                                                                lowest bit

但两者之间没有任何价值.因此，您无法编码2 ⁵⁴ + 2 .

But there is no value inbetween. So you cannot encode 2⁵⁴ + 2.

为什么− 2 ⁵⁴ + 2 没问题?因为这与−(2 ⁵⁴ − 2)相同，所以表示为:

Why is this not a problem for −2⁵⁴ + 2? Because that is the same as −(2⁵⁴ − 2), and that is represented as:

mantissa: 1.11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11
exponent: 53 !!
sign:     1

指数53表示您的步长为2 ^-52 * 2 ⁵³ =2.然后，向0的下一个值是:

And the exponent 53 means you have steps of 2^-52 * 2⁵³ = 2. The next value toward 0 is then:

mantissa: 1.11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 10
exponent: 53
sign:     1

是− 2 ⁵⁴ + 4，或者实际上是−(2 ⁵⁴ − 4).然后您可以像这样继续下去，直到达到− 2 ⁵³ .

which is −2⁵⁴ + 4, or actually −(2⁵⁴ − 4). And you can go on like that until you reach −2⁵³.

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