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#### Bit puzzle help

Can you explain how the following two "bit puzzles" achieve what the title of the problems say they do.

1. Absolute Value
int abs(int x){
int sign = x >> 31;
int neg_x = ~x+1; //two's compliment
return (neg_x & sign) | (x & ~sign);
}

// assumes a + b + c + d <= 127
// and that a,b,c,d are all positive
// returns a + b + c + d using only 2 additions
int add4(int a, int b, int c, int d){
x = a << 8 | b;
y = c << 8 | d;
int q = x + y;
return (q & 0xff) + (q >> 8);
}

All the explanation in the world would be extremely helpful and appreciated!! Thank you!
2. There are more here, with explanations.
bit hacks
3. Cute stuff. HINTS:

1. Print out the results of (neg_x & sign) and (x & ~sign) separately. Then convert them into binary and you'll get an idea of what the | does (it is pretty easy once you see the two results in binary to figure out what is going on)

2. The reason why this works is because initially the numbers all occupy one byte of (an assumed 32 bit) int. i.e. the numbers are of the format 000a, 000b, 000c, 000d. By bit shifting the bits, it essentially makes two variables
X = 00ab
Y = 00cd
where each variable occupies one byte of the int (this is why there is a restriction that a, b, c and d are <= 127) Again, printing x, y in binary along with a and n will tell you exactly what is happening

Now it adds the numbers, which will add the numbers on individual bytes like this:
Q = 00(a+c)(b+d)
Again, if you print Q in binary, you'll see what is going on. Also note here the restriction that 0 <= a, b, c, d <= 127. Because if the result of (b+d) is > 255, then it will overflow into the calculation of (a+c). Overflows cannot happen if both numbers are <= 127.

The final step is to separate byte (b+d) and byte (a+c) and then add them together to get the final result.