## Advanced bit manipulation-fu

There are lots of things I think a good programmer should understand, and understand well. One, of course, is that OOP amounts to BSE for programmers. Another, and the topic of today, is that good programmers should know how to manipulate bits in their sleep. (No, not __those__ bits!)

It goes without saying that in order even to be called a programmer one must know the truth tables of the basic operations of Boolean algebra (**and**, **or**, **xor**, and **negation** in particular) backwards and forwards. What I’m thinking of here though is that programmers should be able to do bitwise arithmetic on two’s complement numbers as if their lives depended on it. (Which it does, for e.g. graphics programmers, as low-level twiddling and other clever optimization is their livelihood.)

### Basic bit manipulations

Before moving on to some more advanced stuff, let’s start out with the very basics of bit manipulation. Given a binary 2’s-complement number **x**, we have the following useful identities:

Identity |
Example |
Identity and explanation |

x | 00101100 | x, the original value |

~x | 11010011 | ~x, complement of x |

-x | 11010100 | -x = ~x+1, negation of x (this you MUST know about 2’s-complement arithmetic!) |

x & -x | 00000100 | x & -x, extract lowest bit set |

x | -x | 11111100 | x | -x, mask for bits above (AND including) lowest bit set |

x ^ -x | 11111000 | x ^ -x, mask for bits above (NOT including) lowest bit set |

x & (x - 1) | 00101000 | x & (x - 1), strip off lowest bit set |

x | (x - 1) | 00101111 | x | (x - 1), fill in all bits below lowest bit set |

x ^ (x - 1) | 00000111 | x ^ (x - 1) = ~x ^ -x, mask for bits below (AND including) lowest bit set |

~x & (x - 1) | 00000011 | ~x & (x - 1) = ~(x | -x) = (x & -x) - 1, mask for bits below (NOT including) lowest bit set |

x | (x + 1) | 00101101 | x | (x + 1), fills in lowest zero bit |

x / (x & -x) | 00001011 | x / (x & -x), shift number right so lowest bit set ends up at bit 0 |

If you’re not used to twiddling bits these identities may not seem immediately obvious, but a few minutes with pen and paper and you should be able to see how any one of them works. Several of these should be committed to memory, including, at the very least, how to extract or strip off the lowest bit set.

From the above identities we can derive some truly fabulous expressions, like e.g. **-~x** for **x + 1**, and **~-x** for **x - 1**! (These are cool in a geeky way, but please don’t use these expressions in your code, unless you’re writing an entry for IOCCC.) In addition to geeking out, we can also do some really useful stuff with these identities, as I will show next.

### A problem requiring advanced bit-fu

Once you know a large number of the identities above (and you should), you are equipped to solve problems that would otherwise be nontrivial. Let us consider the interesting problem of creating a function ** uint NextKBitNumber(uint)** which given an unsigned integer with K bits set, returns the immediately larger unsigned integer also with K bits set.

“Huh,” you say? Well, as always, the first thing to do is to understand the problem, so let’s study some data. Let’s start with a 5-bit number, with 2 bits set. From combinatorics we know there should be (5 choose 2) = 10 alternatives. We can easily list them by hand:

0 0 0 1|1

0 0 1|0 1

0 0 1|1 0

0 1|0 0 1

0 1|0 1 0

0 1|1 0 0

1|0 0 0 1

1|0 0 1 0

1|0 1 0 0

1|1 0 0 0

As you can see, I inserted a vertical bar on each line to highlight an obvious recurring pattern: the lower bit travels from right to left until it becomes adjacent with the upper bit. At that point we bump the upper bit one position to the left, and reset the lower bit back to the rightmost position. Let’s verify this pattern for three bits set. Here we have 7-bit numbers, with 3 bits set:

0 0 0 0 1|1|1

0 0 0 1|0 1|1

0 0 0 1|1|0 1

0 0 0 1|1|1 0

0 0 1|0 0 1|1

0 0 1|0 1|0 1

0 0 1|0 1|1 0

0 0 1|1|0 0 1

0 0 1|1|0 1 0

0 0 1|1|1 0 0

0 1|0 0 0 1|1

0 1|0 0 1|0 1

0 1|0 0 1|1 0

0 1|0 1|0 0 1

0 1|0 1|0 1 0

0 1|0 1|1 0 0

0 1|1|0 0 0 1

0 1|1|0 0 1 0

0 1|1|0 1 0 0

0 1|1|1 0 0 0

1|0 0 0 0 1|1

1|0 0 0 1|0 1

1|0 0 0 1|1 0

1|0 0 1|0 0 1

1|0 0 1|0 1 0

1|0 0 1|1 0 0

1|0 1|0 0 0 1

1|0 1|0 0 1 0

1|0 1|0 1 0 0

1|0 1|1 0 0 0

1|1|0 0 0 0 1

1|1|0 0 0 1 0

1|1|0 0 1 0 0

1|1|0 1 0 0 0

1|1|1 0 0 0 0

OK, so the low bit still travels from right to left and gets bumped back to bit 0 after it becomes adjacent with another bit, but there’s clearly more going on here: sometimes we have two bits bumped back. Studying the bit patterns for a while should make it apparent that what happens (conceptually) is that once the lowest bit has become adjacent with a sequence of one or more bits it “jumps” those bits to the next higher zero bit and all the bits that were jumped are bumped back to bit 0.

Followed that? Good! Let’s see how we can implement this logic using the bit manipulations from above.

### First attempt

We can accomplish the left-shifting of the lowest bit by extracting the lowest bit set (using **b = x & -x**) and adding this onto the original number (**t = x + b**). This also takes care of the “jumping over” part, but in doing so it clears out all the bits we jumped past so we need to add those back in. We can create those bits by taking the lowest bit in **t** (using **c = t & -t**), dividing it by **b** to shift it down, and then creating the appropriate mask **m = (c / b >> 1) - 1** by shifting by two (as we counted one bit too many) and subtracting one (to form the mask). The final result is then **r = t | m**. That was a mouthful, so let’s look at an example:

Operation |
Example |
Operation and explanation |

x | 01011100 | x, the original value |

b = x & -x | 00000100 | b = x & -x, extract lowest set bit b in x |

t = x + b | 01100000 | t = x + b, shift lowest set bit to the left, here causing “jump” past two adjacent one bits |

c = t & -t | 00100000 | c = t & -t, extract lowest set bit c in t |

c / b | 00001000 | c / b, shifts c down by factor of b |

m = (c / b >> 1) - 1 | 00000011 | m = (c / b >> 1) - 1, form bitmask for the (two) bits we jumped past |

r = t | m | 01100011 | r = t | m, form final result |

That took 9 operations. Now that we know how to do it, can we do better? Let’s see.

### Second attempt

It should be clear that we have to perform the division to shift **c** down to form the mask for the bits we jumped past, but can we perhaps somehow avoid the shift and subtract part in forming the mask? Let’s look at a different way of computing **c** and **m**. Going back to the table at the top of the article, we see that we can try **c = t ^ (t - 1)** instead. That forms all bits below the lowest bit set in **t** so after dividing and shifting, we no longer need to subtract one in forming **m**. (But we do have to shift by two instead of by one after making this change.) Here’s the updated table of operations:

Operation |
Example |
Operation and explanation |

x | 01011100 | x, the original value |

b = x & -x | 00000100 | b = x & -x, extract lowest set bit b in x |

t = x + b | 01100000 | t = x + b, shift lowest set bit to the left, here causing “jump” past two adjacent one bits |

c = t ^ (t - 1) | 00111111 | c = t ^ (t - 1), mask for bits below (AND including) lowest bit set in t |

m = (c >> 2) / b | 00000011 | m = (c >> 2) / b, form bitmask for the (two) bits we jumped past |

r = t | m | 01100011 | r = t | m, form final result |

Now we’re down to 8 operations, can we do better?

### Third and final attempt

Turns out we **can** do better! We don’t really need to create one bits all the way down to bit 0 as we did with **c = t ^ (t - 1)** in the previous attempt. We only need to create enough one bits so that after we shift them down to reside at bit position 0, we have the same number of bits as we jumped past. This realization allows us to use **c = x ^ t** instead, which shaves off yet another operation, resulting in the final code:

Operation |
Example |
Operation and explanation |

x | 01011100 | x, the original value |

b = x & -x | 00000100 | b = x & -x, extract lowest set bit b in x |

t = x + b | 01100000 | t = x + b, shift lowest set bit to the left, here causing “jump” past two adjacent one bits |

c = x ^ t | 00111100 | c = x ^ t, form “sufficiently long” sequence of one bits, for shifting down |

m = (c >> 2) / b | 00000011 | m = (c >> 2) / b, form bitmask for the (two) bits we jumped past |

r = t | m | 01100011 | r = t | m, form final result |

That’s 7 operations. Can we do better? I don’t think so, but feel free, go ahead, prove me wrong!

### Improving your bit-fu

The above problem was just a brief taste of the nifty stuff you can do once you’ve truly grokked bit manipulation. And if you think “bit tricks” are just esoterica, think again. For example, the vector units on the PlayStation 2 do not have an **xor** instruction. In the traditional discrete math, logic, or abstract algebra class you probably learned that (A xor B) = ((not(A) and B) or (A and not(B)) so you could synthesize xor in 5 instructions. Ah, but no! The experienced bit manipulator knows that you can do it in 3 instructions as (A xor B) = ((A or B) − (A and B)). (That’s the kind of stuff you pull to get PS2 VU code fast enough to do games like God of War and God of War 2.)

You can also find lots of applications of these sort of “tricks” in my **book**, used to optimize e.g. spatial partitioning methods to make them industrial strength (for collision detection, visibility determination, and other applications).

If you want to become one with the bit, other than lots of practical workouts in the bit-gym, the best printed resource is unarguably **Henry Warren Jr’s Hacker’s Delight** which I highly recommend (also listed on my recommended list). BTW, the above problem is discussed on page 14 of Hacker’s Delight (though not quite in this much detail.)

## roe said,

August 25, 2008 @ 12:38 am

Finally someone who’s as much into bit-twiddling tricks as I am. Excellent post!

## Jon said,

August 25, 2008 @ 9:44 pm

Coincidentally, the last time I had to do any bit twiddling of that sort was to implement an xor instruction on VU1!

## smcameron said,

August 26, 2008 @ 8:58 am

Cool.

Do you have any tips for counting set (or unset) bits? Even a way that works only with contiguously set (or unset) bits would be good enough for my purposes.

I was writing an allocator for fixed chunks of memory, and using the bits within uint32_t’s to mark used/free chunks. Then I loop through all my uint32_t’s looking for something that’s not 0xffffffff (meaning at least one bit is “free”), but then I was using a naive loop to shift bits one by one to find the “free” bit within that uint32_t. (had tried a binary search of the 32 bits, but that turned out to be slower than a loop.)

I see how using these techniques you’ve described, this finding of a free bit can be improved. But, then I still have to know which bit it is, (that is, convert it to a number between 0-31). That I can do in a naive way (looping and bit shifting as before) or perhaps in a machine specific way, but I’m wondering if there’s some cool hardware independent way to do it.

## fatlimey said,

August 26, 2008 @ 1:27 pm

My favorite bit-twiddling is generating the Halton and Hammersley low discrepency quasirandom points: take a fixed-point value and flip the bits around the “decimal” point. Such a simple idea that leads to amazingly robust sampling patterns. Now *that* is deep magic.

## ARBaboon said,

August 28, 2008 @ 5:16 pm

Any thoughts for a faster int log2? I seems like there should be a faster, non-branching way perhaps using sign extension.

## ARBaboon said,

August 28, 2008 @ 5:29 pm

@smcameron

your asking for popcount (population count) which is a single instruction for many architectures (popcount/ctlz/cttx). GCC will detect certain idioms for popcount.

## christer said,

August 28, 2008 @ 9:17 pm

smcameron, as ARBaboon pointed out, what you’re looking for is typically called “popcount” and there are tons of variants. Most approaches use sort of a “reverse divide-and-conquer” approach, by adding (in parallel) pairs of single bits to form two-bit sums, then pairs of two-bit sums are added to form three-bit sums, etc. You can find a lot of versions, with good descriptions, on Sean Anderson’s Bit Twiddling Hacks page, which is a collection of hacks from many different sources.

Anderson’s page also outlines a number of approaches for integer log2() operations.

## DJohn said,

August 29, 2008 @ 2:15 am

Count leading zeros / find first one would be more useful for smcameron, I think. Population count is different, and, in my experience, not as widely available (or as useful).

And Christer: are you really advocating the use of integer division in performance-critical code? Are there processors where it isn’t painfully slow?

Count leading zeros is common enough that it should be treated as the useful primitive that it is. Not existing in C is no excuse for its neglect.

## christer said,

August 31, 2008 @ 6:28 pm

Hi DJohn (Dijon? Mustard fan?)

I consider the

x / (x & -x)expression a portable way of writingx >> BitIndex(x & -x)(or any such similar expression). I recommend people use whichever equivalent expression is most suited for their target machine and application (sometimes, though rarely, would that expression be the one using an integer division).BTW, on the slowness of integer operations… in all the time I’ve done programming (which is a lot of years), integer and floating point operations have leapfrogged each other several times in terms of performance. The lesson learned is that just because one is slower than the other right now, that doesn’t mean it’ll remain that way forever! (And the corollary: always know all tricks!)

smcameron would probably be served by any of

PopCount(),BitIndex(), orCountLeadingZeroes()asPopCount((1 << K) - 1) = BitIndex(1 << K) = 31 - CountLeadingZeroes(1 << K)(for reasonable values ofK).I agree, it would be nice to always have one of these as a hardware instruction!

## christer said,

September 1, 2008 @ 12:06 am

BTW, it might be interesting to note that the

NextKBitNumber()function that I discussed above can be used to solve Cedric’s coding challenge as discussed here and here.Andit would run better than any of those presented solutions. ;)## Kreation’s Edge » Blog Archive » Bit Manipulation said,

September 1, 2008 @ 10:03 pm

[…] brings up an interesting point on Bit Manipulation. Yep, it’s a very important skill to have and it’s not a bad idea to keep reviewing at […]

## fatlimey said,

September 2, 2008 @ 11:26 am

A faster log2?

BitLog is the integer log2 of an integer value

Given an N-bit value, locate the leftmost nonzero bit.

b = the bitwise position of this bit, where 0 = LSB.

n = the NEXT three bits (ignoring the highest 1)

bitlog(x) = 8 x (b-1) + n

Bitlog is exactly 8 times larger than log2(x)-1

## gowrikumar said,

September 22, 2008 @ 10:09 pm

Recently I prepared a presentation explaining the next kth number:

http://www.slideshare.net/gkumar007/bits-next-higher-presentation/

## Norm said,

September 22, 2008 @ 11:56 pm

Hi there,

Can anyone direct me to e-books, pdfs, articles, etc. on low level language tricks? Like, I read an article some while back about doing subtraction with clever addition and making use of the overflow. I lost the link to that article though..

I am interested in that kind of stuff. Mostly about tricks that are employed in writing code for embedded systems where every byte counts.

Thanks,

Norm

## christer said,

September 23, 2008 @ 12:43 am

Norm, This page is a summary of well-known “tricks” gathered from usenet, HAKMEM, and many other sources. Jörg Arndt’s FXT book also contains a lot of info.

All those aside, I highly recommend Henry Warren Jr’s book

Hacker’s Delightas the ultimate resource. It iswellworth the ~$40 it costs.## Norm said,

September 23, 2008 @ 1:31 am

Thanks christer. I want to be good enough to produce something for pouet someday. It seems like this sort of low level tuning has become a hobby as computing power has become expendable for simplicity, but I have a passion for it, nevertheless.

## The More You Know said,

September 23, 2008 @ 5:26 am

[…] In truth most programmers know that bit manipulation more akin to a peasant flipping some switches. Even so, having those switches flipped at the right moment sure is useful. This was taken orginally in part from http://realtimecollisiondetection.net/blog/?p=78 […]

## jtlien said,

September 23, 2008 @ 8:01 am

One of the best sights for bit manipulation routines is Georg Arnt’s web page and book at

http://www.jjj.de/bitwizardry/bitwizardrypage.html

## Matrix67: My Blog » Blog Archive » ??????????????k?1????? said,

September 23, 2008 @ 9:25 pm

[…] ???http://realtimecollisiondetection.net/blog/?p=78 Posted in Program Impossible Tags: ??, ??, ??, ???Trackback: […]

## LordHavoc said,

February 8, 2009 @ 3:39 am

In response to ARBaboon’s question about a faster int log2 - how fast is your current one?

The method I use (for unsigned int) is branch based at the C code level but the compiler tends to use conditional moves for it on x86, here it is for reference:

#define log2i(n) ((((n) & 0xAAAAAAAA) != 0 ? 1 : 0) | (((n) & 0xCCCCCCCC) != 0 ? 2 : 0) | (((n) & 0xF0F0F0F0) != 0 ? 4 : 0) | (((n) & 0xFF00FF00) != 0 ? 8 : 0) | (((n) & 0xFFFF0000) != 0 ? 16 : 0))

I’d be curious to see a truly branchless version, although I think that one is fine if the compiler does a decent job :)

## LordHavoc said,

February 8, 2009 @ 3:43 am

Note: my above log2i macro requires that n be a power of 2, it only identifies which one.