什么类型的数字可以用二进制浮点表示？ [英] What types of numbers are representable in binary floating-point?

问题描述

我已经读了很多关于花车的内容，但都是不必要的。我知道我已经知道了，但是我只想知道一件事：

我知道， 1 / pow（2，n），其中 n 是一个整数的小数部分可以精确地表示浮点数字。这意味着如果我给自己加上3200万次 1/32 ，我会得到 1,000,000 。

那么像 1 /（32 + 16）？这是两个两个权力的总和，这个工作吗？或者它是 1/32 + 1/16 的工作？这是我困惑的地方，所以如果有人可以澄清，我会感激。

总结为：

• 如果分母的素因子分解只包含2，则可以用二进制表示一个数字。（即分母是一种权力的二）
  
  $ b $所以 1 /（32 + 16）不能用二进制表示，因为分母的因子是3。但是 1/32 + 1/16 = 3/32 就是。
  
  

  也就是说，可以用浮点型表示。例如，在IEEE double 所以 1/2 + 1/2 ^ 500 中只有53位尾数，是不可表示的。

  
  
  

  因此，只要指数的范围不超过53次幂，就可以做两次幂的和。

  
  
  

  ---

  
  
  

  将其推广到其他基础：
  
  

  
  

  • 如果分母的素因式分解仅由2和5组成，那么数字可以用10进行精确表示。
    

  • 有理数<如果 X 的分母的素因式分解，code> X code>只包含 N 因式分解中的素数。

  
  
  
  

  I've read a lot about floats, but it's all unnecessarily involved. I think I've got it pretty much understood, but there's just one thing I'd like to know for sure:

  I know that, fractions of the form 1/pow(2,n), with n an integer, can be represented exactly in floating point numbers. This means that if I add 1/32 to itself 32 million times, I would get exactly 1,000,000.

  What about something like 1/(32+16)? It's one over the sum of two powers of two, does this work? Or is it 1/32+1/16 that works? This is where I'm confused, so if anyone could clarify that for me I would appreciate it.

  解决方案

  The rule can be summed up as this:

  • A number can be represented exactly in binary if the prime factorization of the denominator contains only 2. (i.e. the denominator is a power-of-two)

  So 1/(32 + 16) is not representable in binary because it has a factor of 3 in the denominator. But 1/32 + 1/16 = 3/32 is.

  That said, there are more restrictions to be representable in a floating-point type. For example, you only have 53 bits of mantissa in an IEEE double so 1/2 + 1/2^500 is not representable.

  So you can do sum of powers-of-two as long as the range of the exponents doesn't span more than 53 powers.

  ---

  To generalize this to other bases:

  • A number can be exactly represented in base 10 if the prime factorization of the denominator consists of only 2's and 5's.

  • A rational number X can be exactly represented in base N if the prime factorization of the denominator of X contains only primes found in the factorization of N.

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