1 Overview

This HW will give you a chance to practice using binary and bit-wise operators. You’ll likely find Booleans §4 a useful reference.

2 Task

Visit https://kytos.cs.virginia.edu/cso1/pa01.php and complete the four problems listed below. The text boxes want lightweight code using just operators and assignments, like

x = 0x20
y = b + x

The goal is to end up with one variable having a particular value, based on other variables that are provided with new values in each test case. Do not add conditionals, loops, subroutines, etc.

2.1 The Tasks

We want you to do four of the problems. There are others puzzles on the site as well if you want more practice, but the only four we grade are:

subtract

Given x and y, set z to x - y without using - or multi-bit constants.

For full credit, use ≤ 10 operations from {!, ~, +, <<, >>, &, ^, |}.

bottom

Given b, set the low-order b bits of x to 1; the others to 0. For example, if b is 3, x should be 7. Pay special attention to the edge cases: if b is 32 x should be −1; if b is 0 x should be 0. Do not use - in your solution.

For full credit, use ≤ 40 operations from {!, ~, +, <<, >>, &, ^, |}.

anybit

Given x, set y to 1 if any bit in x is 1; set y to 0 if x is all 0s.

For full credit, use ≤ 40 operations from {~, +, -, <<, >>, &, ^, |}.

bitcount

Given x, set y to the number of bits in x that are 1.

For full credit, use ≤ 40 operations from {!, ~, +, -, <<, >>, &, ^, |}.

3 Collaboration

You may work with other students in this class on this assignment, but only in the following two ways:

You worked together from the beginning, solving the problem as a team, with each person contributing.

Each teammate should cite this in each problem with a C-style comment at the top of each solution and also cite the originator of any single-person contributions where they appear, like
```
// Part of a team with mst3k and lat7h
x = -y
w = -x // lat7h came up with this line
z = x + y
```
You helped someone with a task you’d already finished, helping them think through their incorrect solution and not giving them or trying to lead them to your solution.

The helper should acknowledge they did this by returning to their previously-submitted solutions and re-submitting them with an added comment at the top, like
```
// I helped tj1a
x = -y
w = -x
```
The helpee should acknowledge they got this by adding a comment at the top, like
```
// tj1a helped me
x = -y
w = -x
```

In all cases, include computing IDs in your citations to streamline our automated tools that assist with collaboration checking.

4 Hints

If needed, we have some hints you can look at.

subtract

Consider the definition of two’s compliment.

bottom

The obvious solution ~(0xFFFFFFFF << b) won’t work. Bit shifts always do a modulo on their right-hand operand, so a << b does the same thing as a << (b % (8*sizeof(a)). Thus, << 32 and << 0 do the same thing.

anybit

The easy solution would be y = !!x but we don’t allow !. Nor do we allow enough operations to do a loop-like solution.

You can divide and conquer. Try defining x1 where if any bit anywhere in x was 1, some bit in the bottom 16 bits of x1 is 1. The given task is see if any 1 bit is in the 32 bits of x. How could you reduce it to see if any 1 bit is in the bottom 16 bits of x1?

bitcount

The obvious solution would be something like

ans = 0;
for(int i=0; i<32; i+=1) {
    a += x&1;
    x >>=1;
}

We don’t allow for loops, but even if you replace it with 32 copies that’s still 96 operations, and we only allow 40 for this task.

The trick is to do things in parallel, treating a number like a vector of smaller numbers. Suppose I wanted to count the bits of an 8-bit number with bits abcdefgh. With a little shifting and masking I could make three numbers

0b00e00h
0a00d00g
0000c00f

and add them to get xx0yy0zz where xx = a+b, yy = c+d+e, and zz = f+g+h.

Extending this trick to several rounds on 32 bits will solve this problem.