读书笔记：Representing and manipulation information
1. 信息的存储
a)
|
|
32 bit | 64 bit |
|---|---|---|
| char | 1 | 1 |
| short int | 2 | 2 |
| int | 4 | 4 |
| long int | 4 | 8 |
| long long int | 8 | 8 |
| char* | 4 | 8 |
| float | 4 | 4 |
| double | 8 | 8 |
b)
0x01234567
地址： 0x100 0x101 0x102 0x103
little endian: ... 67 45 23 01
c)
// Code to print the byte representation of program objects.
#include <stdio.h>
typedef unsigned char *byte_pointer; 4
void show_bytes(byte_pointer start, int len) {
int i;
for (i = 0; i < len; i++)
printf(" %.2x", start[i]);
printf("\n");
}
void show_int(int x) {
show_bytes((byte_pointer) &x, sizeof(int));
}
void show_float(float x) {
show_bytes((byte_pointer) &x, sizeof(float));
}
void show_pointer(void *x) {
show_bytes((byte_pointer) &x, sizeof(void *));
}
2. 整数的表达
正负数的表达，负数是正数的补集（two's complement），
B2T4([0001]) = −0*2^3+0*2^2+0*2^1+1*2^0 =0+0+0+1=1
B2T4([0101]) = −0*2^3+1*2^2+0*2^1+1*2^0 =0+4+0+1=5
B2T4([1011]) = −1*2^3+0*2^2+1*2^1+1*2^0 = −8+0+2+1 = −5
B2T4([1111]) = −1*2^3+1*2^2+1*2^1+1*2^0 = −8+4+2+1 = −1
usigned int, 最高位是1的时候变成负数，最高位是0的时候是负数。
3. 整数的运算
4. 浮点数
| 类型 | 存储位数 | 偏移值 | ||||
| 数符(s) | 阶码(E) | 尾数(M) | 总位数 | 十六进制 | 十进制 | |
| 短实数(Single,Float) | 1位 | 8位 | 23位 | 32位 | 0x7FH | +127 |
| 长实数(Double) | 1位 | 11 位 | 52位 | 64位 | 0x3FFH | +1023 |
| 临时实数（延伸双精确度，不常用） | 1位 | 15位 | 64位 | 80位 | 0x3FFFH | +16383 |
F=1.M(二进制)
在单精度时： V=(-1)^s*2^(E-127)*F (127: 255的一半，一半用来表达小于0，一半用来大于0)
在双精度时： V=(-1)^s*2^(E-1023)*F
由于float的有无数个但是能表达的只有有限个， float在坐标轴上分布极其不均匀，越小越精确，越大越不精确，这也是浮点计算遇到大数更容易产生误差的原因，IEEE 定义了两大类五小类的近似标准。
Roundings to nearest
- Round to nearest, ties to even – rounds to the nearest value; if the number falls midway it is rounded to the nearest value with an even (zero) least significant bit, which occurs 50% of the time; this is the default for binary floating-point and the recommended default for decimal.
- Round to nearest, ties away from zero – rounds to the nearest value; if the number falls midway it is rounded to the nearest value above (for positive numbers) or below (for negative numbers); this is intended as an option for decimal floating point.
Directed roundings
- Round toward 0 – directed rounding towards zero (also known astruncation).
- Round toward +∞ – directed rounding towards positive infinity (also known asrounding up orceiling).
- Round toward −∞ – directed rounding towards negative infinity (also known asrounding down orfloor).
Setting the hardware rounding mode to double precision prevents thisfrom happening:
$ gcc -Wall -DDOUBLE
1. Normalized
s≠0 &≠255f
2. Denormalized
s00000000 f
3a. Infinity
s1111111100000000000000000000000
3b. NaN
s11111111≠0
For argument f , it returns ±0 if f is denormalized (preserving the sign of f ) and returns f otherwise.
/* If f is denorm, return 0. Otherwise, return f */
float_bits float_denorm_zero(float_bits f) {
/* Decompose bit representation into parts */
unsigned sign = f>>31;
unsigned exp = f>>23 & 0xFF;
unsigned frac = f & 0x7FFFFF;
if (exp == 0) {
/* Denormalized. Set fraction to 0 */
frac = 0; }
/* Reassemble bits */
return (sign << 31) | (exp << 23) | frac;
}
具体的float如何运算，exception处理，和intel采用的是什么round方式，留以后研究。