update csapp 2.3 and 2.2

content/csapp/2.2.md

@@ -232,5 +232,3 @@ $x_7=1$
 
 | 1   | 1   | 1   | 1   | 1   | 1   | 1   | 1   | $x_7$ | $x_6$ | $x_5$ | $x_4$ | $x_3$ | $x_2$ | $x_1$ | $x_0$ |
 |:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|:-----:|
-
-

content/csapp/2.3.md

@@ -10,15 +10,17 @@ Integer Arithmetic(整数运算)
 
 <!--more-->
 
-## Unsigned Addition
+## Unsigned
+
+### Addition
 
 For x and y $0 \leq x \leq 2^w$ , $0 \leq y \leq 2^w$
 
 $$
 x+_{w}^{u}y= \begin{cases} x+y, & \text {x + y < $2^w$} \\\\ 3n+1, & \text{$2^{w}\leq{x+y}{\leq}2^{w+1}$}  \end{cases}
 $$
 
-## Additive Inverse
+### Inverse
 
 For x，$0 \leq x \leq 2^w$
 
@@ -40,15 +42,28 @@ $$
 -_{w}^{u}y= \begin{cases} x, & \text {x = 0} \\\\ 2^{w}-x, & \text{x $\geq$ 0}  \end{cases}
 $$
 
-## Two's Complement Addition
+### Multiplication
+
+| x             | $x_{w-1},x_{w-2}...x_{0}$                |
+|:-------------:| ---------------------------------------- |
+| y             | $y_{w-1},y_{w-2}...y_{0}$                |
+| $z=x{\cdot}y$ | $\underbrace{Z_{w-1},Z_{w-2}...Z_0}_{w}$ |
+
+$$
+x *_{w}^{u}y= (x{\cdot}y) mod 2^w
+$$
+
+## Two's Complement
+
+### Addition
 
 For x and y $-2^{w-1}{\leq}{x}{\leq}2^{w-1}-1, -2^{w-1}{\leq}{y}{\leq}2^{w-1}-1$
 
 $$
 x+_{w}^{t}y= \begin{cases} x+y-2^{w}, & \text {$2^{w-1}{\leq}{x+y}$(Positive overflow)} \\\\ x+y, & \text{$-2^{w-1}\leq{x+y}{<}2^{w-1}$} \\\\ x+y+2^{w}, & \text{x+y<$-2^{w-1}$(Negative overflow)}  \end{cases}
 $$
 
-## Two's Complement Addition Code
+### Code
 
 ```c
 #include <stdio.h>
@@ -62,7 +77,7 @@ int main() {
 }
 ```
 
-## Two's  Complement Negation
+### Negation
 
 For x $-2^{w-1}{\leq}x<2^{w-1}-1$
 
@@ -82,4 +97,55 @@ $$
 Tmin_w+_{w}^{t}Tmain_w = -2^w+2^w=0
 $$
 
+### Multiplication
+
+$$
+x *_{w}^{t}y=U2T_w((x{\cdot}y)\mod{2^w})
+$$
+
+### Three-bit Multiplication Example
+
+| Mode             | x         | y         | ${x}{\cdot}{y}$ | Truncated ${x}{\cdot}{y}$ |
+|:----------------:|:--------- |:--------- |:--------------- |:------------------------- |
+| Unsigned         | $5[101]$  | $3[011]$  | $15[001 111]$   | $7[111]$                  |
+| Two's Complement | $-3[101]$ | $3[011]$  | $-9[110 111]$   | $-1[111]$                 |
+| Unsigned         | $4[100]$  | $7[111]$  | $28[001 100]$   | $4[100]$                  |
+| Two's Complement | $-4[100]$ | $-1[111]$ | $4[000 100]$    | $-4[100]$                 |
+| Unsigned         | $3[011]$  | $3[011]$  | $9[001 001]$    | $1[001]$                  |
+| Two's Complement | $3[011]$  | $3[011]$  | $9[001 001]$    | $1[001]$                  |
+
+For signed integer x, y and unsigned integer $x^{'}, y^{'}$
+
+$$
+B2U_w(x^{'})=x_{w-1}{\cdot}2^{w-1}+x_{w-2}{\cdot}2^{w-2}+...+x_{0}{\cdot}2^{0} \\\\
+B2T_w(x)=x_{w-1}{\cdot}-2^{w-1}+x_{w-2}{\cdot}2^{w-2}+...+x_{0}{\cdot}2^{0} \\\\
+B2U_w(x^{'})-B2T_w(x)=x^{w-1}{\cdot}2^{w} \\\\
+x^{'} = x + x_{w-1}{\cdot}2^{w} \\\\
+y^{'}=y+{y_{w-1}}{\cdot}2^w \\\\
+({x^{'}}{\cdot}{y^{'}})\mod{2^w} = [(x+x_{w-1}{\cdot}{2^w}){\cdot}(y+{y_{w-1}}{\cdot}{2^w})] \mod 2^w \\\\
+({x^{'}}{\cdot}{y^{'}})\mod{2^w} = [x{\cdot}y+(x_{w-1}y+y_{w-1}x)2^w]+x_{w-1}y_{w-1}2^{2w}] \mod 2^w \\\\
+({x^{'}}{\cdot}{y^{'}})\mod{2^w} = (x{\cdot}y\mod2^w)
+$$
+
+## Multiplying by Constants
+
+| $x{\cdot}2$     | $x<<1$ |
+| --------------- | ------ |
+| $x{\cdot}4$     | $x<<2$ |
+| ...             | ...    |
+| $x{\cdot}2^{k}$ | $x<<k$ |
+
+For unsigned integer x
+
+$$
+x_{w-1}|x_{w-2}|...|x_0 \\\\
+B2U_w(x)=x_{w-1}{\cdot}2^{w-1}+x_{w-2}{\cdot}2^{w-2}+...+x_0{\cdot}2^0=\sum_{i=0}^{w-1}x_{i}2^{i} \\\\
+x_{w-1}|x_{w-2}|...|x_0|\underbrace{0|...|0}_{k} \\\\
+$$
+
+$$
+B2U_{w+k}(x^{'})=x_{w-1}{\cdot}2^{w-1+k}+x_{w-2}{\cdot}2^{w-2+k}+...+x_0{\cdot}2^k+0{\cdot}2^{k-1}+0{\cdot}2^{k-2}+...+0{\cdot}2^{0} \\\\
+B2U_{w+k}(x^{'})=\sum_{i=0}^{w-1}x_{i}2^{i}{\cdot}2^k=x2^k
+$$
+