lua – 使用浮点数或双精度数而不是整数

对于64位双精度,最大的可表示整数是253(9007199254740992),对于32位浮点数,最大的可表示整数是224(16777216).请参阅 the Wikipedia page for IEEE floating point numbers的基准数字.

在Lua中验证这一点非常简单：

local maxdouble = 2^53

-- one less than the maximum can be represented precisely
print (string.format("%.0f",maxdouble-1)) --> 9007199254740991
-- the maximum itself can be represented precisely
print (string.format("%.0f",maxdouble))   --> 9007199254740992
-- one more than the maximum gets rounded down
print (string.format("%.0f",maxdouble+1)) --> 9007199254740992 again

如果我们没有方便的IEEE定义的字段大小,知道我们对浮点数设计的了解,我们可以使用可能值的简单循环来确定这些值：

#include <stddef.h>
#include <stdint.h>
#include <stdio.h>
#define min(a,b) (a < b ? a : b)
#define bits(type) (sizeof(type) * 8)
#define testimax(test_t) { 
  uintmax_t in = 1,out = 2; 
  size_t pow = 0,limit = min(bits(test_t),bits(uintmax_t)); 
  while (pow < limit && out == in + 1) { 
    in = in << 1; 
    out = (test_t) in + 1; 
    ++pow; 
  } 
  if (pow == limit) 
    puts(#test_t " is as precise as longest integer type"); 
  else printf(#test_t " conversion imprecise for 2^%d+1:n" 
    "   in: %llun  out: %llunn",pow,in + 1,out); 
}

int main(void)
{
    testimax(float);
    testimax(double);
    return 0;
}

The output of the above code：

float conversion imprecise for 2^24+1:
   in: 16777217
  out: 16777216

double conversion imprecise for 2^53+1:
   in: 9007199254740993
  out: 9007199254740992

当然,由于浮点精度的工作方式,64位双精度可以表示远大于264的数字,因为浮动指数增长为正. The Wikipedia page on double-precision floating-point描述：

Between 2⁵²=4,503,599,627,370,496 and 2⁵³=9,007,199,254,740,992 the representable numbers are exactly the integers. For the next range,from 2⁵³ to 2⁵⁴,everything is multiplied by 2,so the representable numbers are the even ones,etc. Conversely,for the previous range from 2⁵¹ to 2⁵²,the spacing is 0.5,etc.

double可以容纳的绝对最大值列在该页面的下方：0x7fefffffffffffff,其计算为(1(1 – 2-52))* 21023,或大致为1.7976931348623157e308.