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java - 关于乘法的奇怪的 java.lang.Double 簇

转载 作者：行者123 更新时间：2023-11-30 11:15:02 25 4
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关于这些 java.lang.Double 的东西真的很奇怪......这是 [Clojure] 代码 - 与 Java 相同，用于所有 [与执行相关的] 意图和目的:

(doseq [x (range 100)
        :let [x33 (str (+ x 0.33))
              x99 (str (+ x 0.99))
              tx33 (* 100 (Double/parseDouble x33))
              tx99 (* 100 (Double/parseDouble x99))]]
  (do (prn (str x33 "->" tx33))
      (prn (str x99 "->" tx99))))

对于 Java 人员 - 这会遍历 java.lang.Long-s 的范围，通过分别向其添加 0.33 和 0.99 为每个(x33 和 x99)创建两个 java.lang.String-s。它将 Strings 精简解析为 Doubles，并将结果乘以 100，然后打印映射。观察结果(最后的nil无关紧要):

"0.33->33.0"
"0.99->99.0"
"1.33->133.0"
"1.99->199.0"
"2.33->233.0"
"2.99->299.0"
"3.33->333.0"
"3.99->399.0"
"4.33->433.0"
"4.99->499.0"
"5.33->533.0"
"5.99->599.0"
"6.33->633.0"
"6.99->699.0"
"7.33->733.0"
"7.99->799.0"
"8.33->833.0"
"8.99->899.0"
"9.33->933.0"
"9.99->999.0"
"10.33->1033.0"
"10.99->1099.0"
"11.33->1133.0"
"11.99->1199.0"
"12.33->1233.0"
"12.99->1299.0"
"13.33->1333.0"
"13.99->1399.0"
"14.33->1433.0"
"14.99->1499.0"
"15.33->1533.0"
"15.99->1599.0"
"16.33->1632.9999999999998"
"16.99->1698.9999999999998"
"17.33->1732.9999999999998"
"17.99->1798.9999999999998"
"18.33->1832.9999999999998"
"18.99->1898.9999999999998"
"19.33->1932.9999999999998"
"19.99->1998.9999999999998"
"20.33->2032.9999999999998"
"20.99->2099.0"
"21.33->2133.0"
"21.99->2199.0"
"22.33->2233.0"
"22.99->2299.0"
"23.33->2333.0"
"23.99->2399.0"
"24.33->2433.0"
"24.99->2499.0"
"25.33->2533.0"
"25.99->2599.0"
"26.33->2633.0"
"26.99->2699.0"
"27.33->2733.0"
"27.99->2799.0"
"28.33->2833.0"
"28.99->2899.0"
"29.33->2933.0"
"29.99->2999.0"
"30.33->3033.0"
"30.99->3099.0"
"31.33->3133.0"
"31.99->3199.0"
"32.33->3233.0"
"32.99->3299.0"
"33.33->3333.0"
"33.99->3399.0"
"34.33->3433.0"
"34.99->3499.0"
"35.33->3533.0"
"35.99->3599.0"
"36.33->3633.0"
"36.99->3699.0"
"37.33->3733.0"
"37.99->3799.0"
"38.33->3833.0"
"38.99->3899.0"
"39.33->3933.0"
"39.99->3999.0"
"40.33->4033.0"
"40.99->4099.0"
"41.33->4133.0"
"41.99->4199.0"
"42.33->4233.0"
"42.99->4299.0"
"43.33->4333.0"
"43.99->4399.0"
"44.33->4433.0"
"44.99->4499.0"
"45.33->4533.0"
"45.99->4599.0"
"46.33->4633.0"
"46.99->4699.0"
"47.33->4733.0"
"47.99->4799.0"
"48.33->4833.0"
"48.99->4899.0"
"49.33->4933.0"
"49.99->4999.0"
"50.33->5033.0"
"50.99->5099.0"
"51.33->5133.0"
"51.99->5199.0"
"52.33->5233.0"
"52.99->5299.0"
"53.33->5333.0"
"53.99->5399.0"
"54.33->5433.0"
"54.99->5499.0"
"55.33->5533.0"
"55.99->5599.0"
"56.33->5633.0"
"56.99->5699.0"
"57.33->5733.0"
"57.99->5799.0"
"58.33->5833.0"
"58.99->5899.0"
"59.33->5933.0"
"59.99->5999.0"
"60.33->6033.0"
"60.99->6099.0"
"61.33->6133.0"
"61.99->6199.0"
"62.33->6233.0"
"62.99->6299.0"
"63.33->6333.0"
"63.99->6399.0"
"64.33->6433.0"
"64.99->6498.999999999999"
"65.33->6533.0"
"65.99->6598.999999999999"
"66.33->6633.0"
"66.99->6698.999999999999"
"67.33->6733.0"
"67.99->6798.999999999999"
"68.33->6833.0"
"68.99->6898.999999999999"
"69.33->6933.0"
"69.99->6998.999999999999"
"70.33->7033.0"
"70.99->7098.999999999999"
"71.33->7133.0"
"71.99->7198.999999999999"
"72.33->7233.0"
"72.99->7298.999999999999"
"73.33->7333.0"
"73.99->7398.999999999999"
"74.33->7433.0"
"74.99->7498.999999999999"
"75.33->7533.0"
"75.99->7598.999999999999"
"76.33->7633.0"
"76.99->7698.999999999999"
"77.33->7733.0"
"77.99->7798.999999999999"
"78.33->7833.0"
"78.99->7898.999999999999"
"79.33->7933.0"
"79.99->7998.999999999999"
"80.33->8033.0"
"80.99->8098.999999999999"
"81.33->8133.0"
"81.99->8199.0"
"82.33->8233.0"
"82.99->8299.0"
"83.33->8333.0"
"83.99->8399.0"
"84.33->8433.0"
"84.99->8499.0"
"85.33->8533.0"
"85.99->8599.0"
"86.33->8633.0"
"86.99->8699.0"
"87.33->8733.0"
"87.99->8799.0"
"88.33->8833.0"
"88.99->8899.0"
"89.33->8933.0"
"89.99->8999.0"
"90.33->9033.0"
"90.99->9099.0"
"91.33->9133.0"
"91.99->9199.0"
"92.33->9233.0"
"92.99->9299.0"
"93.33->9333.0"
"93.99->9399.0"
"94.33->9433.0"
"94.99->9499.0"
"95.33->9533.0"
"95.99->9599.0"
"96.33->9633.0"
"96.99->9699.0"
"97.33->9733.0"
"97.99->9799.0"
"98.33->9833.0"
"98.99->9899.0"
"99.33->9933.0"
"99.99->9999.0"
nil

如您所见，有两个这样的 double 集群，它们的行为与其他 double 不同。对我来说是个谜……天知道通往无限的道路上还有多少。任何人都可以阐明这一点吗？谢谢!我知道并非所有有理数都可以用 float 表示，但我认为这是另外一回事。

最佳答案

IEEE 754 二进制 float 应该被认为是在 binades 中组织的，间隔范围从 2 的幂到下一个，其中所有数字由相同的最小间隔 (ULP) 分隔。

"15.99->1599.0""16.33->1632.9999999999998"..."20.33->2032.9999999999998""20.99->2099.0"

The above cluster corresponds to a more-inaccurate-than-usual computation when the origin value is in the 16-32 binade and the result is in the 1024-2048 binade.

"63.99->6399.0""64.33->6433.0""64.99->6498.999999999999""65.33->6533.0""65.99->6598.999999999999""66.33->6633.0""66.99->6698.999999999999"..."80.33->8033.0""80.99->8098.999999999999""81.33->8133.0""81.99->8199.0"

For some reason, there is another cluster where one computation out of two is more inaccurate than usual, when the origin value is in the 64-128 binade and the result in the 4096-8192 binade.

You can observe what happens more precisely in each of these two clusters by looking at the computations at the binary level. Computations for values where the input value and the output value are respectively in the same binades will look very similar (and that's how they are either all accurate or all inaccurate at the same time).

The C99 program below makes the computation appear as regular as it is, within the first cluster (this program should be trivial to translate to Clojure, as long as you can make use of Java's printing functions and of the %a format for hexadecimal):

#include <stdio.h>

int main(int argc, char** argv) 
{  
  printf("%a -> %.13a (exact result %a)\n", 15.33, 15.33 * 100.0, 1533.0);
  printf("%a -> %.13a (exact result %a)\n", 16.33, 16.33 * 100.0, 1633.0);
  printf("%a -> %.13a (exact result %a)\n", 17.33, 17.33 * 100.0, 1733.0);
  printf("%a -> %.13a (exact result %a)\n", 18.33, 18.33 * 100.0, 1833.0);
  printf("...\n");
  printf("%a -> %.13a (exact result %a)\n", 20.33, 20.33 * 100.0, 2033.0);
  printf("%a -> %.13a (exact result %a)\n", 21.33, 21.33 * 100.0, 2133.0);
}

输出是:

0x1.ea8f5c28f5c29p+3 -> 0x1.7f40000000000p+10 (exact result 0x1.7f4p+10)0x1.0547ae147ae14p+4 -> 0x1.983ffffffffffp+10 (exact result 0x1.984p+10)0x1.1547ae147ae14p+4 -> 0x1.b13ffffffffffp+10 (exact result 0x1.b14p+10)0x1.2547ae147ae14p+4 -> 0x1.ca3ffffffffffp+10 (exact result 0x1.ca4p+10)...0x1.4547ae147ae14p+4 -> 0x1.fc3ffffffffffp+10 (exact result 0x1.fc4p+10)0x1.5547ae147ae14p+4 -> 0x1.0aa0000000000p+11 (exact result 0x1.0aap+11)

上面，一个数字的十六进制表示中的 p+4 指数表示该数字在 16-32 二进制中。如您所见，此 binade 中的所有 X.33 数字都具有相似的表示形式。对它们的计算会得出相似的精确结果，但当结果低于 2048 时，它必须与精确结果高于 2048 时不同地四舍五入。

将以 ...ae147ae14 结尾的数字乘以 100 会产生以 ...fffffffd 结尾的精确结果，或二进制形式的 ...11111111101。

  • 当乘以 100 的数为16.33时，需要截去最后两位二进制数进行四舍五入。计算的确切结果以二进制 …11111111101 结尾，与前一个可表示数比下一个可表示数更接近，因此乘法结果四舍五入为以 …二进制为 111111111，或十六进制为 …fff(或十进制为 …99998)。

  • 当乘以100的数是21.33时，因为这次乘法的结果落在下一个二进制数中，所以最后三个二进制数需要被切断四舍五入。被截断的最后三位二进制数字是 101。它们代表超过一半的 ULP，因此最接近的可表示值是高于精确值的可表示值。该值由 FPU 选择作为乘法结果，以 …00000 结尾，恰好是 2133.0 的浮点表示。

关于java - 关于乘法的奇怪的 java.lang.Double 簇，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/25630224/

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