浮点数介绍
浮点数（Floating Point）是计算机中用二进制表示实数的一种近似表示方式。就是用科学记数法来表示一个实数，分为3个部分，第一个部分是符号位，第二个部分是指数，第三个部分是尾数。
IEEE 754是IEEE的计算机表示浮点数的标准。它包含下面几种：
| Type |
Sign |
Exponent |
Significand field |
Total bits |
Exponent bias |
Bits precision |
Number of decimal digits |
| Half (IEEE 754-2008) |
1 |
5 |
10 |
16 |
15 |
11 |
~3.3 |
| Single |
1 |
8 |
23 |
32 |
127 |
24 |
~7.2 |
| Double |
1 |
11 |
52 |
64 |
1023 |
53 |
~15.9 |
| x86 extended precision |
1 |
15 |
64 |
80 |
16383 |
64 |
~19.2 |
| Quad |
1 |
15 |
112 |
128 |
16383 |
113 |
~34.0 |
最常见的就是单精度（Single）和双精度（Double），就是C++和C#中的float和double。
先举个用float来表示实数的例子，比如我们要表示100，换算成2进制就是1100100，那么就是1.1001*10^6。所以符号位是0，指数是6加上Exponent bias（127）是133，换算成2进制是10000101，尾数是1001。那么拼一起就是0 10000101 10010000000000000000000。
浮点数中的特殊值
接着我们看看根据定义，float中那些特殊值都是啥。看如下的C++程序，把2进制的表示转换成float数值并且打印出来。
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
| int binaryPresentation;
float floatPresentation;
binaryPresentation = 0x42C80000;
floatPresentation = *(float *)&binaryPresentation;
printf("100: %g\n", floatPresentation);
binaryPresentation = 0x00000000;
floatPresentation = *(float *)&binaryPresentation;
printf("Positive ZERO: %g\n", floatPresentation);
binaryPresentation = 0x80000000;
floatPresentation = *(float *)&binaryPresentation;
printf("Negative ZERO: %g\n", floatPresentation);
binaryPresentation = 0x00000001;
floatPresentation = *(float *)&binaryPresentation;
printf("Positive Min Denorm Float: %g\n", floatPresentation);
binaryPresentation = 0x007fffff;
floatPresentation = *(float *)&binaryPresentation;
printf("Positive Max Denorm Float: %g\n", floatPresentation);
binaryPresentation = 0x00800001;
floatPresentation = *(float *)&binaryPresentation;
printf("Positive Min Normalize float: %g\n", floatPresentation);
binaryPresentation = 0x7f7fffff;
floatPresentation = *(float *)&binaryPresentation;
printf("Positive Max Normalize float: %g\n", floatPresentation);
binaryPresentation = 0x7F800000;
floatPresentation = *(float *)&binaryPresentation;
printf("Positive infinite: %g\n", floatPresentation);
binaryPresentation = 0xFF800000;
floatPresentation = *(float *)&binaryPresentation;
printf("Negative infinite: %g\n", floatPresentation);
|
输出如下：
1
2
3
4
5
6
7
8
9
| 100: 100
Positive ZERO: 0
Negative ZERO: -0
Positive Min Denorm Float: 1.4013e-045
Positive Max Denorm Float: 1.17549e-038
Positive Min Normalize float: 1.17549e-038
Positive Max Normalize float: 3.40282e+038
Positive infinite: 1.#INF
Negative infinite: -1.#INF
|
C++中的浮点数特殊值
接着看看C++中的浮点数特殊值是怎么定义的：
1
2
3
4
5
6
7
8
9
| printf("denorm_min: %g\n", std::numeric_limits<float>::denorm_min());
printf("epsilon: %g\n", std::numeric_limits<float>::epsilon());
printf("infinity: %g\n", std::numeric_limits<float>::infinity());
printf("lowest: %g\n", std::numeric_limits<float>::lowest());
printf("max: %g\n", std::numeric_limits<float>::max());
printf("min: %g\n", std::numeric_limits<float>::min());
printf("quiet_NaN: %g\n", std::numeric_limits<float>::quiet_NaN());
printf("signaling_NaN: %g\n", std::numeric_limits<float>::signaling_NaN());
printf("round_error: %g\n", std::numeric_limits<float>::round_error());
|
输出如下：
1
2
3
4
5
6
7
8
9
| denorm_min: 1.4013e-045
epsilon: 1.19209e-007
infinity: 1.#INF
lowest: -3.40282e+038
max: 3.40282e+038
min: 1.17549e-038
quiet_NaN: 1.#QNAN
signaling_NaN: 1.#QNAN
round_error: 0.5
|
CSharp中的浮点数特殊值
再来看看C#中的浮点数特殊值是怎么定义的：
1
2
3
4
5
6
| Console.WriteLine("Epsilon:" + float.Epsilon);
Console.WriteLine("MaxValue:" + float.MaxValue);
Console.WriteLine("MinValue:" + float.MinValue);
Console.WriteLine("NaN:" + float.NaN);
Console.WriteLine("PositiveInfinity:" + float.PositiveInfinity);
Console.WriteLine("NegativeInfinity:" + float.NegativeInfinity);
|
输出如下：
1
2
3
4
5
6
| Epsilon:1.401298E-45
MaxValue:3.402823E+38
MinValue:-3.402823E+38
NaN:NaN
PositiveInfinity:Infinity
NegativeInfinity:-Infinity
|
可以看出来C#和C++中关于Epsilon和MinValue的定义是不一样的。
