bit操作和数据表示

最新推荐文章于 2022-04-13 22:15:46 发布

fanjingdan

最新推荐文章于 2022-04-13 22:15:46 发布

阅读量2k

点赞数

分类专栏：重读ics和cse

本文链接：https://blog.youkuaiyun.com/fanjingdan/article/details/79924846

版权

重读ics和cse 专栏收录该内容

3 篇文章

订阅专栏

二进制和十进制转换

Bits 01010
Value 0*2^4+1*2^3+0*2^2+1*2^1+0*2^0 = 10
Value 102(1100110)
Bits 102/2 = 51…0 ,个位数=0
51/2 = 25…1,十位数=1
25/2 = 12… 1
12/2 = 6… 0
6/2 = 3…0
3/2 = 1…1
1/2 = 0… 1
十进制类比：十进制的也是/10看余数

Group bits as numbers : Three encodings

1.Unsigned encoding

≥0
UMIN=0
UMAX=2^w-1

2.Two’s-complement encoding

最高位是负的
为什么叫Two’s-complement：
7：0111
-7：1000（complement）+1=1001
~x + 1 == -x
TMAX=7
TMIN=-8
[-8，-1] 8个数字
[0，7] 8个数字，跟unsigned表示是一样的
unsigned转2‘s complement

其他signed

Ones’ Complement:

最高位权重=2^(w-1)-1

Sign-Magnitude

最高位是sign位

C 特性

unsigned可以和Two’s complement互转

Type conversion (implicitly)
Type casting (explicitly)
bit不变，重新翻译

short int           x =  12345;
unsigned short int ux = (unsigned short) x;//ux = 12345,不变
short int           y  = -12345;
unsigned short int uy = (unsigned short) y;//uy = 53191，变大正数

By default constants are considered to be signed integers

Unsigned if have “U” as suffix
0U, 4294967259U

If mix unsigned and signed in single expression

signed 变unsigned
operation： <, >, ==, <=, >=
所以 -1 > 0U -1转unsigned

expand

short转long

short int x =  12345;
int ix = (int) x; 
short int y = -12345;
int iy = (int) y;

最高位是0就加0
最高位是1就加1
expand
1001=-7=1-8
11001=-7=1+（8-16）
111001=-7=1+（8+16-32）
…

truncate

long转short

int       x  = 53191;
short int sx = x;
int       y  = -12345;
Short int sy = y;

truncate
手算十进制结果：
unsigned:mod2^k
signed:转unsigned（bit不变），mod2^k，转回来

+-*/

+-

Overflow and Underflow

在+时只能出现一次
+算出来，位数超了，砍掉超出位，按照Unsigned或者Two’s-complement 翻译，然后就出现了意想不到的效果。。。
overflow
unsigned:大于2^w:-2^w
signed:小于-2^w:+2^w
大于2^w:-2^w

怎么发现overflow

Overflow iff either:
u, v < 0, s ≥ 0 (NegOver)
u, v ≥ 0, s < 0 (PosOver)
ovf = (u<0 == v<0) && (u<0 != s<0);
ovf

additive inverse

Let UCompw (u ) = 2^w – u
UAddw(u , UCompw (u )) = 0

*/

<<算*2^k

3.Floating point encoding

Base-two version of scientific notation for representing real numbers

这里写图片描述
101.112 = 1*22 + 0*21 + 1*20 + 1*2-1 + 1*2-2
=5又3/4
类比十进制，不断*10，取走个位数
0.6875 10进制转二进制：不断*2，取走个位数
res x
1 1.375
0 0.750
1 1.500
1 1.000

0.1011 2进制= 0.6875 10进制

IEEE 754

Designed by W. Kahan for Intel processors (Turing Award 1989)
Based on a small and consistent set of principles, elegant, understandable, hard to make go fast

V = (-1)^s M 2^E
Sign bit s determines whether number is negative or positive
Significand M normally a fractional value in range [1.0,2.0).
Exponent E weights value by power of two
这里写图片描述
s is sign bit S
exp field encodes E
frac field encodes M
Sizes
Single precision (32 bits): 1 s, 8 exp, 23 frac
Double precision (64 bits): 1 s, 11 exp, 52 frac
Condition
exp ≠ 000…0 and exp ≠ 111…1

Exponent coded as biased value
E = exp – Bias
exp : unsigned value denoted by exp
Bias : Bias value
Single prec: 127 (exp: 1… 254, E: -126…127)
Double prec: 1023 (exp: 1…2046, E: -1022…1023)
In general: Bias = 2^(m-1) - 1, where m is the number of exponent bits
Significand coded with implied leading 1
M = 1.xxx…x2
xxx…x: bits of frac
Minimum when 000…0 (M = 1.0)
Maximum when 111…1 (M = 2.0 – )
Get extra leading bit for “free”