为什么代表1.0和2.0的位串如此不同？ [英] Why are the bit strings representing 1.0 and 2.0 so different?

问题描述

我最近开始使用Julia，并且找到了位函数，它返回数字参数的位串表示。例如：

  julia> （1.0）
0011111111110000000000000000000000000000000000000000000000000000
  

然而，在玩这个功能的时候，惊讶地发现 bits 为 1.0 和 2.0


  julia>位（1.0）
0011111111110000000000000000000000000000000000000000000000000000
 
 julia>位（2.0）
0100000000000000000000000000000000000000000000000000000000000000
  

我预计这两个值是相似的...



这些位的含义是什么？我隐约记得一些关于编码指数的位（从我的数值分析类） ，但我真的不记得它，我没有设法找到一个很好的描述在线...

解决方案

为什么位（1.0）和位（2.0）的 ASCIIString $ c>是如此不同，您需要了解一下（！）关于 IEEE-754（二进制）浮点数字。每个这样的双精度数字存储为一个64位的字，细分分为三部分：



• 符号位（0表示非负数，1表示非正数），接下来是
  
• 偏向指数（11位），然后是
  
• 有效位（52位）。




  normalized number （例如 1.0 和<$ c
$ b $ pre $（$ 1 $ ^ $ $ $ $ $ $ $） sign_bit x 1.significand x 2 ^（biased_exponent - bias）


  （对于双精度浮点数字，这个偏差的值是2 ^ 10 - 1 = 1023）
  
  现在，
  
  

  
  

  • 1.0 = +1.000 .. （1023-bias）
    
    

    和1023对应于（2）基11中的（0）1111111111，所以相应的位串是
    
    

      0 01111111111 0000000000000000000000000000000000000000000000000000 
      

  
  

  • 2.0 = +1.000 ... x 2 ^（1024 - 偏差）
    
    <1024>和1024对应于基数为2的10000000000，所以相应的位串是

    
    $ b $

      0 10000000000 0000000000000000000000000000000000000000000000000000 
      

  
  

  • 3.0 = +1.100 ... x 2 ^（1024 - bias）
    
    

    所以对应的位串是
    
    

      0 10000000000 1000000000000000000000000000000000000000000000000000 
      

  
  
  
  
  $ p
  $ b $ p总之，你可以得到 2.0 通过增加 1.0 这个2的幂乘-1的位串中的偏指数部分。增加这样一个数字会导致它的所有位二进制表示改变，以同样的方式递增数字9999（以十进制表示）导致所有数字改变。
  

  I recently started using Julia and I came upon the bits function, which returns the bit-string representation of its numeric argument. For example:

  julia> bits(1.0)
  "0011111111110000000000000000000000000000000000000000000000000000"
  

  However, while playing with this function, I was surprised to discover that bits returns very different bit strings for 1.0 and 2.0:

  julia> bits(1.0)
  "0011111111110000000000000000000000000000000000000000000000000000"
  
  julia> bits(2.0)
  "0100000000000000000000000000000000000000000000000000000000000000"
  

  I would have expected those two values to be similar...

  What is the meaning of those bits? I vaguely recall something about bits encoding the exponent (from my numerical-analysis class), but I really do not remember it well and I did not manage to find a good description online...

  解决方案

  To understand why the ASCIIString values of bits(1.0) and bits(2.0) are "so different", you need to know a bit (!) about IEEE-754 (binary) floating-point numbers. Each such double-precision number is stored as a 64-bit word, broken down into three parts:

  • the sign bit (0 for nonnegative numbers, 1 for nonpositive numbers), followed by
  • the biased exponent (11 bits), followed by
  • the significand (52 bits).

  The value of a normalized number (such as 1.0 and 2.0) can be obtained by using the following formula:

  (-1)^sign_bit x 1.significand x 2^(biased_exponent - bias)
  

  (For double-precision floating-point numbers, the bias has a value of 2^10 - 1 = 1023)

  Now,

  • 1.0 = +1.000... x 2^(1023 - bias)

    and 1023 corresponds to (0)1111111111 in base 2, so the corresponding bit string is

    0 01111111111 0000000000000000000000000000000000000000000000000000
    

  • 2.0 = +1.000... x 2^(1024 - bias)

    and 1024 corresponds to 10000000000 in base 2, so the corresponding bit string is

    0 10000000000 0000000000000000000000000000000000000000000000000000
    

  • 3.0 = +1.100... x 2^(1024 - bias)

    so the corresponding bit string is

    0 10000000000 1000000000000000000000000000000000000000000000000000
    

  etc.

  In summary, you can obtain the bit string of 2.0 by incrementing the biased-exponent part in the bit string of 1.0, which is a power of 2, minus 1. Incrementing such a number causes all the bits of its binary representation to change, in the same way that incrementing the number 9999 (in decimal representation) causes all the digits to change.

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