如何重新present 0.1在浮点运算及十进制 [英] How To Represent 0.1 In Floating Point Arithmetic And Decimal
问题描述
我想了解浮点运算更好，见过几个环节，以什么每台计算机科学家应该知道关于浮点运算
I am trying to understand floating point arithmetic better and have seen a few links to 'What Every Computer Scientist Should Know About Floating Point Arithmetic'
我还是不明白，一个像0.1或0.5如何存储在花车和小数
I still don't understand how a number like 0.1 or 0.5 is stored in floats and as decimals
能否有人请解释它是如何布局的内存
Can someone please explain how it is laid out is memory
我知道浮动是TWP部分（即一个数的功率）
I know about the float being twp parts (ie a number to the power of)
推荐答案
我一直指出人民对哈拉尔德·施密特在线转换，与它的漂亮的照片的维基百科的文章IEEE754-1985沿
I've always pointed people towards Harald Schmidt's online converter, along with the Wikipedia IEEE754-1985 article with its nice pictures.
有关这两个特定值，你会得到（0.1）：
For those two specific values, you get (for 0.1):
s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm 1/n
0 01111011 10011001100110011001101
| || || || || || +- 8388608
| || || || || |+--- 2097152
| || || || || +---- 1048576
| || || || |+------- 131072
| || || || +-------- 65536
| || || |+----------- 8192
| || || +------------ 4096
| || |+--------------- 512
| || +---------------- 256
| |+------------------- 32
| +-------------------- 16
+----------------------- 2
符号为正，这是pretty容易。
The sign is positive, that's pretty easy.
该指数是 64 + 32 + 16 + 8 + 2 + 1 = 123 - 127偏压= -4 ，所以乘数是 2 -4 或 1/16 。
The exponent is 64+32+16+8+2+1 = 123 - 127 bias = -4, so the multiplier is 2-4 or 1/16.
尾数是矮胖。它由 1 的（隐式底座）加（与每个都是值得所有这些位 1 /（2 N ）为 N 开始于 1 键，向右增加）， {1/2，1/16，1/32，1/256，1/512，1/4096，8192分之1，1/65536，一十三万一千零七十二分之一，一百〇四万八千五百七十六分之一，二百〇九万七千一百五十二分之一，八百三十八万八千六百〇八分之一} 。
The mantissa is chunky. It consists of 1 (the implicit base) plus (for all those bits with each being worth 1/(2n) as n starts at 1 and increases to the right), {1/2, 1/16, 1/32, 1/256, 1/512, 1/4096, 1/8192, 1/65536, 1/131072, 1/1048576, 1/2097152, 1/8388608}.
当您添加所有这些，你得到 1.60000002384185791015625 。
When you add all these up, you get 1.60000002384185791015625.
在乘以乘数，你得到 0.100000001490116119384765625 ，这就是为什么他们说你不能再present 0.1 完全一样的IEEE754浮点，并提供了这么多的机会，就那么对于人回答为什么不0.1 + 0.1 + 0.1 = = 0.3？型问题： - ）
When you multiply that by the multiplier, you get 0.100000001490116119384765625, which is why they say you cannot represent 0.1 exactly as an IEEE754 float, and provides so much opportunity on SO for people answering "why doesn't 0.1 + 0.1 + 0.1 == 0.3?"-type questions :-)
0.5的例子是基本上容易。这是再presented为：
The 0.5 example is substantially easier. It's represented as:
s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm
0 01111110 00000000000000000000000
这意味着它的隐含基地， 1 ，再加上没有其他添加剂（所有的尾数位均为零）。
which means it's the implicit base, 1, plus no other additives (all the mantissa bits are zero).
符号再次阳性。该指数是 64 + 32 + 16 + 8 + 4 + 2 = 126 - 127偏差= -1 。因此，乘数为 2 1 是 1/2 或 0.5 。
The sign is again positive. The exponent is 64+32+16+8+4+2 = 126 - 127 bias = -1. Hence the multiplier is 2-1 which is 1/2 or 0.5.
因此，最终值为 1 乘 0.5 或 0.5 。瞧！
So the final value is 1 multiplied by 0.5, or 0.5. Voila!
我有时会发现更容易十进制的角度来考虑它。
I've sometimes found it easier to think of it in terms of decimal.
数1.345相当于
1 + 3/10 + 4/100 + 5/1000
或
-1 -2 -3
1 + 3*10 + 4*10 + 5*10
同样，IEEE754重新presentation十进制 0.8125 是：
s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm
0 01111110 10100000000000000000000
随着1隐含的基地，这相当于二进制：
With the implicit base of 1, that's equivalent to the binary:
01111110-01111111
1.101 * 2
或
-1
(1 + 1/2 + 1/8) * 2 (no 1/4 since that bit is 0)
变成：
(8/8 + 4/8 + 1/8) * 1/2
和然后的就变成了：
13/8 * 1/2 = 0.8125
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