How does this code involving xor actually works?

Question

I have a variable that represents the XOR of 2 numbers. For example: int xor = 7 ^ 2;
I am looking into a code that according to comments finds the rightmost bit that is set in XOR:

int rightBitSet = xor & ~(xor - 1);

I can't follow how exactly does this piece of code work. I mean in the case of 7^2 it will indeed set rightBitSet to 0001 (in binary) i.e. 1. (indeed the rightmost bit set)
But if the xor is 7^3 then the rightBitSet is being set to 0100 i.e 4 which is also the same value as xor (and is not the rightmost bit set).
The logic of the code is to find a number that represents a different bit between the numbers that make up xor and although the comments indicate that it finds the right most bit set, it seems to me that the code finds a bit pattern with 1 differing bit in any place.
Am I correct? I am not sure also how the code works. It seems that there is some relationship between a number X and the number X-1 in its binary representation?
What is this relationship?

but 7^2 inverts the second to reightmost bit. doesnt touch other bits. 7 is 111 2 is 010 so we have 101 as result — huseyin tugrul buyukisik, Aug 04 '12 at 08:27
and you may have guessed 7^7 subtracts 7 from 7 which result 0 — huseyin tugrul buyukisik, Aug 04 '12 at 08:33
Intersting.I hadn't notice that.But how does this help in my OP? — Cratylus, Aug 04 '12 at 08:36
possible duplicate of [Unset the rightmost set bit](http://stackoverflow.com/questions/4703964/unset-the-rightmost-set-bit) — Paul R, Aug 04 '12 at 08:36
@PaulR:I don't think it is a duplicate.1)I don't want to unset a bit.I am interested in understanding how does this code give a differing bit between the numbers that make up the `xor` 2)I don't see in the thread you link some explanation on the relationship of `n` and `n-1` in their bitwise representation. — Cratylus, Aug 04 '12 at 08:44
@tugrul:What do you mean by subtraction? `18^7` is not `11`. — Cratylus, Aug 04 '12 at 15:22
you cannot show 18 in one bcd digit. you need two. but 7 is one — huseyin tugrul buyukisik, Aug 04 '12 at 15:36
for example: 18^13 both fits in two bcd digits and answer is 5 — huseyin tugrul buyukisik, Aug 04 '12 at 15:37
So you are saying that xor is a subtraction for numbers of the same bitcore representation length? — Cratylus, Aug 04 '12 at 18:29

score 3 · Accepted Answer · answered Aug 04 '12 at 08:39

3

The effect of subtracting 1 from a binary number is to replace the least significant 1 in it with a 0, and set all the less significant bits to 1. For example:

5 - 1 = 101 - 1 = 100 = 4
4 - 1 = 100 - 1 = 011 = 3
6 - 1 = 110 - 1 = 101 = 5

So in evaluating x & ~(x - 1): above x's least significant 1, ~(x - 1) has the same set bits as ~x, so above x's least significant 1, x & ~(x-1) has no 1 bits. By definition, x has a 1 bit at its least significant 1, and as we saw above ~(x - 1) will, too, but ~(x - 1) will have 0s below that point. Therefore, x & ~(x - 1) will have only one 1 bit, at the least significant bit of x.

answered Aug 04 '12 at 08:39

Louis Wasserman

191,574
25
345
413

+1 for the first part.I did not understand the second paragraph at all.`above x's least significant 1, x & ~(x-1) has no 1 bits`. Why? – Cratylus Aug 04 '12 at 08:52
Rephrase: the _only_ effect of subtracting 1 from a binary number is to replace the least significant 1 in it with a 0. So `x - 1` has the same bits as `x`, above the ls1 of `x`, so `x & ~(x - 1)` is all 0s above x's ls1. – Louis Wasserman Aug 04 '12 at 09:00
I think I got it. So essentially we get a representation that is the least significant bit set (above and bellow bits are clear) that the numbers making up the xor defer?So the comments actually mean he least significant bit? – Cratylus Aug 04 '12 at 09:04
Just curious.How do you remember these stuff?Do you write code manipulating bits often?I am asking because I am always interested in what other do in case it can help me improve – Cratylus Aug 04 '12 at 09:08
"Do you write code manipulating bits often?" [Yes](http://docs.guava-libraries.googlecode.com/git-history/release/javadoc/src-html/com/google/common/math/IntMath.html). – Louis Wasserman Aug 04 '12 at 09:44

How does this code involving xor actually works?

1 Answers1