在16,32和64位IEEE-754系统中可以表示的范围是什么？ [英] What range of numbers can be represented in a 16-, 32- and 64-bit IEEE-754 systems?

问题描述

我知道一点关于如何表示浮点数，但恐怕还不够。

总的问题是：

对于一个给定的精度（对于我来说，基数为10的精确小数位数），可以表示16,32 - 和64位IEEE-754系统？

具体来说，我只对16位和32位系统感兴趣，位数精确到+/- 0.5（一个地方）或+/- 0.0005（千分之一位置）。

解决方案

从MATLAB文档中为函数绘制答案 EPS a>，但它应该普遍适用于 IEEE-754 浮点数。

对于给定的浮点数 X ，如果

  2 ^ E <= abs（X） 2 ^（E + 1）
  

然后从 X 到下一个最大的可表示的浮点数（ epsilon ）是：

  epsilon = 2 ^（E-52 ）％对于64位浮点数（双精度）
 epsilon = 2 ^（E-23）％对于32位浮点数（单精度）
 epsilon = 2 ^（E-10）对于16位浮点数（半精度）
  

上面的等式允许我们计算如下： / p>

• 对于半精度 ...

  
  
  

  如果您需要+/- 0.5（或2 ^ -1）的准确度，最大的数量可以是2 ^ 10。任何大于这个的值和浮点数之间的距离大于0.5。
  
  

  如果你想得到+/- 0.0005（大约2 ^ -11）的精度，该数字的最大大小是1.任何大于这个值的浮点数和大于0.0005之间的距离。




• 对于单精度 ...

  
  
  

  如果您想要+/- 0.5（或2 ^ -1）的精确度，则数字的最大大小为2 ^ 23。任何大于这个的值和浮点数之间的距离大于0.5。
  
  

  如果你想得到+/- 0.0005（大约2 ^ -11）的精度，最大的数量可以是2 ^ 13。任何大于这个的值和浮点数之间的距离大于0.0005。

.org / wiki / Double_precisionrel =noreferrer> double precision ...




如果您想要+/- 0.5（或2 ^ -1），最大可以是2 ^ 52。任何大于这个的值和浮点数之间的距离大于0.5。



如果你想得到+/- 0.0005（大约2 ^ -11）的精度，最大的数量可以是2 ^ 42。任何大于这个和浮点数之间的距离大于0.0005。

I know a little bit about how floating-point numbers are represented, but not enough, I'm afraid.

The general question is:

For a given precision (for my purposes, the number of accurate decimal places in base 10), what range of numbers can be represented for 16-, 32- and 64-bit IEEE-754 systems?

Specifically, I'm only interested in the range of 16-bit and 32-bit numbers accurate to +/-0.5 (the ones place) or +/- 0.0005 (the thousandths place).

解决方案

I'm drawing this answer from the MATLAB documentation for the function EPS, but it should apply universally to IEEE-754 floating point numbers.

For a given floating point number X, if

2^E <= abs(X) < 2^(E+1)

then the distance from X to the next largest representable floating point number (epsilon) is:

epsilon = 2^(E-52)    % For a 64-bit float (double precision)
epsilon = 2^(E-23)    % For a 32-bit float (single precision)
epsilon = 2^(E-10)    % For a 16-bit float (half precision)

The above equations allow us to compute the following:

• For half precision...

  If you want an accuracy of +/-0.5 (or 2^-1), the maximum size that the number can be is 2^10. Any larger than this and the distance between floating point numbers is greater than 0.5.

  If you want an accuracy of +/-0.0005 (about 2^-11), the maximum size that the number can be is 1. Any larger than this and the distance between floating point numbers is greater than 0.0005.

• For single precision...

  If you want an accuracy of +/-0.5 (or 2^-1), the maximum size that the number can be is 2^23. Any larger than this and the distance between floating point numbers is greater than 0.5.

  If you want an accuracy of +/-0.0005 (about 2^-11), the maximum size that the number can be is 2^13. Any larger than this and the distance between floating point numbers is greater than 0.0005.

• For double precision...

  If you want an accuracy of +/-0.5 (or 2^-1), the maximum size that the number can be is 2^52. Any larger than this and the distance between floating point numbers is greater than 0.5.

  If you want an accuracy of +/-0.0005 (about 2^-11), the maximum size that the number can be is 2^42. Any larger than this and the distance between floating point numbers is greater than 0.0005.

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