update csapp 2.3

content/csapp/2.3.md

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+---
+title: "Integer Arithmetic"
+tags: ["csapp"]
+categories: ["csapp"]
+date: 2021-06-04T17:17:17+07:00 
+draft: false
+---
+
+Integer Arithmetic(整数运算)
+
+<!--more-->
+
+## 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
+
+For x，$0 \leq x \leq 2^w$
+
+$$
+x + x^{'} = x^{'} + x = 0
+$$
+
+$$
+y - x => y + x^{'}
+$$
+
+For $x, x^{'} => 0 \leq x \leq 2^{w}, 0 \leq x^{'} < 2^{w}$
+
+$$
+x + x^{'} = 2^{w} = 0
+$$
+
+$$
+-_{w}^{u}y= \begin{cases} x, & \text {x = 0} \\\\ 2^{w}-x, & \text{x $\geq$ 0}  \end{cases}
+$$
+
+## 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
+
+```c
+#include <stdio.h>
+
+int main() {
+    char x = 127;
+    char y = 1;
+    char z = x + y;
+    printf("z=%d",z);
+    return 0;
+}
+```
+
+## Two's  Complement Negation
+
+For x $-2^{w-1}{\leq}x<2^{w-1}-1$
+
+$$
+-_{w}^{t}x= \begin{cases} -x, & \text {$x > TMin_w$} \\\\ TMin_{w}, & \text{$x=TMin_{w}$}  \end{cases}
+$$
+
+$$
+|Tmin_x| = |Tmax_w| +1
+$$
+
+$$
+Tmin_w + Tmin_w = -2^{w-1}+(-2^{w-1})=-2^{w}
+$$
+
+$$
+Tmin_w+_{w}^{t}Tmain_w = -2^w+2^w=0
+$$
+
+