Bit Fiddling
This HW will give you a chance to practice using binary and bitwise operators. You’ll likely find Booleans a useful reference.
Task
Visit https://kytos02.cs.virginia.edu/cso1/bitfiddle.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
That is, set x to be the hex value 0x20
, then set y to be the value of b
plus x
.
The goal is to end up with one variable having a particular value, as defined in the instructions, based on other variables that are provided with new values in each test case. Do not add conditionals, loops, subroutines, etc.
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:
 isequal
 Given
x
andy
, setz
to1
ifx == y
; otherwise setz
to0
without using==
or multibit constants.For full credit, use ≤ 5 operations from {
!
,~
,+
,<<
,>>
,&
,^
,
} and up to 1bit constants.  allevenbits
 Given
x
, sety
to1
if all the evennumbered bits ofx
are set to1
(where bit0
is the least significant bit); otherwise sety
to0
.For full credit, use ≤ 12 operations from {
!
,~
,+
,
,<<
,>>
,&
,^
,
} and up to 8bit constants.  bottom
 Given
b
, set the loworderb
bits ofx
to 1; the others to 0. For example, ifb
is 3,x
should be 7. Pay special attention to the edge cases: ifb
is 32x
should be −1; ifb
is 0x
should be 0. Do not use
in your solution.For full credit, use ≤ 40 operations from {
!
,~
,+
,<<
,>>
,&
,^
,
}.  bitcount
 Given
x
, sety
to the number of bits inx
that are1
.For full credit, use ≤ 40 operations from {
!
,~
,+
,
,<<
,>>
,&
,^
,
}.
Examples
We will do one example in class when we reach the HW1 material. However, if you’d like to practice beforehand, here is the description of subtract
:
 subtract (inclass example)
 Given
x
andy
, setz
tox  y
without using
or multibit constants.For full credit, use ≤ 10 operations from {
!
,~
,+
,<<
,>>
,&
,^
,
}.
Collaboration
This Homework ONLY: 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 Cstyle comment at the top of each solution and also cite the originator of any singleperson contributions where they appear, like
// Part of a team with mst3k and jh2jf x = y w = x // jh2jf 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 previouslysubmitted solutions and resubmitting 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.
Hints
If needed, we have some hints you can look at.
subtract
Consider the definition of two’s compliment.
isequal
Consider an operation that results in a single bit value..
allevenbits
You can divide and conquer. Try defining x1
where if all the even bits anywhere in x
was 1
, some specific bits in the bottom 16 bits of x1
are 1
. How could you reduce it to “see if all even bits in the bottom 16 bits of x1
are 1
”?
bottom
The obvious solution ~(0xFFFFFFFF << b)
won’t work. Bit shifts always do a modulo on their righthand operand, so a << b
does the same thing as a << (b % (8*sizeof(a))
. Thus, << 32
and << 0
do the same thing.
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 8bit 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.