Population Count

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Population count, an operation to determine the cardinality of a bitboard, also called Hamming weight or sideways sum. How many one bits exists in a 64-bit computer word? In computer chess, population count is used to evaluate the mobility of pieces from their attack sets, as already applied in Chess 4.6 on the CDC 6600 and CDC Cyber.

Recent x86-64 processors (since AMD K10 with SSE4a, Intel Nehalem with SSE4.2) provide a 64-bit popcount instruction, available via C++ compiler intrinsic or inline assembly. Despite different Intrinsic prototypes (_mm_popcnt_u64 vs. popcnt64), Intel and AMD popcnt instructions are binary compatible, have same encoding (F3 [REX] 0F B8 /r), and both require bit 23 set in RCX of the CPUID function 0000_0001h. Code samples are in C / C++, see Defining Bitboards

=Recurrence Relation= The recursive recurrence relation of population counts can be transformed to iteration as well, but lacks an arithmetical sum-formula: popcnt(0) = 0 popcnt(n) = popcnt(n &#247; 2) + (n mod 2) However, it is helpful to initialize a lookup table, i.e. for bytes: unsigned char popCountOfByte256[];

void initpopCountOfByte256 {  popCountOfByte256[0] = 0; for (int i = 1; i < 256; i++) popCountOfByte256[i] = popCountOfByte256[i / 2] + (i & 1); } =Empty or Single?= Often one has to deal with sparsely populated or even empty bitboards. To determine whether a bitboard is empty or a single populated power of two value, one may use simple boolean statements rather than a complete population count.

Empty Bitboards
To test a bitboard is empty, one compares it with zero, or use the logical not operator: if ( x == 0 ) -> bitboard is empty if ( !x )    -> bitboard is empty The inverse condition (not empty) is of course if ( x != 0 ) -> bitboard is not empty if ( x )     -> bitboard is not empty

Single Populated Bitboards
If the bitboard is not empty, we can extract the LS1B to look whether it is equal with the bitboard. Alternatively and faster, we can reset the LS1B to look whether the bitboard becomes empty. if ( x != 0 && (x & (x-1)) == 0 ) -> population count is one, power of two value One can skip the leading x != 0 condition to test popcount <= 1: if ( (x & (x-1)) == 0 ) -> population count is less or equal than one Again the inverse relation tests, whether a bitboard has more than one bit set: if ( x & (x-1) ) -> population count is greater than one An alternative approach to determine single populated sets, aka power of two values is based on Inclusive LS1B separation divided by two equals the ones' decrement : if ( ((x ^ (x-1)) >> 1) == (x-1) ) -> population count is one, power of two value

=Loop-Approaches=

Too Slow
Brute force adding all 64-bits int popCount (U64 x) { int count = 0; for (int i = 0; i < 64; i++, x >>= 1) count += (int)x & 1; return count; } Of course, this is a slow algorithm, which might be improved by testing x not empty rather than i < 64. But unrolled in parallel prefix manner it already reminds on this one.

Brian Kernighan's way
Consecutively reset LS1B in a loop body and counting loop cycles until the bitset becomes empty. Brian Kernighan mentioned the trick in his and Ritchie's book The C Programming_Language, 2nd Edition 1988, exercise 2-9. However, the method was first published in 1960 by Peter Wegner, discovered independently by Derrick Henry Lehmer, published in 1964 : int popCount (U64 x) { int count = 0; while (x) { count++; x &= x - 1; // reset LS1B }  return count; } Despite branches - this is still one of the fastest approaches for sparsely populated bitboards. Of course the more bits that are set, the longer it takes.

=Lookup= Of course we can not use the whole bitboard as index to a lookup table - since it's size would be 18,446,744,073,709,551,616 bytes! If it is about the population count of a byte, we can simply construct a table lookup with 256 elements. For a bitboard that takes eight byte lookups we can add together: unsigned char popCountOfByte256[];

void initpopCountOfByte256 {  popCountOfByte256[0] = 0; for (int i = 1; i < 256; i++) popCountOfByte256[i] = popCountOfByte256[i / 2] + (i & 1); }

int popCount (U64 x) { return popCountOfByte256[ x       & 0xff] + popCountOfByte256[(x >> 8) & 0xff] + popCountOfByte256[(x >> 16) & 0xff] + popCountOfByte256[(x >> 24) & 0xff] + popCountOfByte256[(x >> 32) & 0xff] + popCountOfByte256[(x >> 40) & 0xff] + popCountOfByte256[(x >> 48) & 0xff] + popCountOfByte256[ x >> 56]; } Looks quite expensive - one may use four 16-bit word-lookups with a pre-calculated 64KByte table though, but that pollutes the memory caches quite a bit. One can also treat the bitboard as array of bytes or words in memory, since endian issues don't care here - that safes all the shifts and 'ands', but has to read byte for byte from memory. int popCount (U64 x) { unsigned char * p = (unsigned char *) &x; return popCountOfByte256[p[0]] + popCountOfByte256[p[1]] + popCountOfByte256[p[2]] + popCountOfByte256[p[3]] + popCountOfByte256[p[4]] + popCountOfByte256[p[5]] + popCountOfByte256[p[6]] + popCountOfByte256[p[7]]; } =SWAR-Popcount= The divide and conquer SWAR-approach deals with counting bits of duos, to aggregate the duo-counts to nibbles and bytes inside one 64-bit register in parallel, to finally sum all bytes together. According to Donald Knuth, a parallel population count routine was already introduced in 1957 due to Donald B. Gillies and Jeffrey C. P. Miller in the first textbook on programming, second edition: The Preparation of Programs for an Electronic Digital Computer, by Maurice Wilkes, David Wheeler and Stanley Gill, pages 191–193.

Counting Duo-Bits
A bit-duo (two neighboring bits) can be interpreted with bit 0 = a, and bit 1 = b as duo := 2b + a The duo population is popcnt(duo) := b + a which can be archived by (2b + a) - (2b + a) &#247; 2 or (2b + a) - b The bit-duo has up to four states with population count from zero to two as demonstrated in following table with binary digits: Only the lower bit is needed from x div 2 - and one don't has to worry about borrows from neighboring duos. SWAR-wise, one needs to clear all "even" bits of the div 2 subtrahend to perform a 64-bit subtraction of all 32 duos: x = x - ((x >> 1) & 0x5555555555555555); Note that the popcount-result of the bit-duos still takes two bits.

Counting Nibble-Bits
The next step is to add the duo-counts to populations of four neighboring bits, the 16 nibble-counts, which may range from zero to four. SWAR-wise it is done by masking odd and even duo-counts to add them together: x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333); Note that the popcount-result of the nibbles takes only three bits, since 100B is the maximum population (of the nibble 1111B).

Byte-Counts
You already got the idea? Now it is about to get the byte-populations from two nibble-populations. Maximum byte-population of 1000B only takes four bits, so it is safe to mask all those four bits of the sum, rather than to mask the summands: x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0f;

Parallel Prefix Adds
We may continue with mask-less parallel prefix SWAR-adds for byte-counts, word-counts and finally double-word-counts, to mask the least significant 8 (or 7) bits for final result in the 0..64 range: x += (x >> 8); x += (x >> 16); x += (x >> 32); return x & 255;

Multiplication
With todays fast 64-bit multiplication one can multiply the vector of 8-byte-counts with 0x0101010101010101 to get the final result in the most significant byte, which is then shifted right: x = (x * 0x0101010101010101) >> 56;

Casting out
Interestingly, there is another approach to add the bytes together. As demonstrated with decimal digits (base 10) and Casting out nines, casting out by modulo base minus one is equivalent to taking the digit sum, which might be applied here with low range 0..8 "base 256" digits: x = x % 255; However, since division and modulo are usually slow instructions and modulo by constant is likely replaced by reciprocal multiplication anyway by the compiler, the multiplication by 0x0101010101010101 aka the 2-adic fraction -1/255 is the preferred method.

The Constants
Putting all together, the various SWAR-Masks and factors as defined by Donald Knuth as 2-adic fractions : const U64 k1 = C64(0x5555555555555555); /* -1/3   */ const U64 k2 = C64(0x3333333333333333); /* -1/5   */ const U64 k4 = C64(0x0f0f0f0f0f0f0f0f); /* -1/17  */ const U64 kf = C64(0x0101010101010101); /* -1/255 */ represented as bitboards: k1 -1/3            k2  -1/5            k4  -1/17           kf  -1/255 0x5555555555555555 0x3333333333333333  0x0f0f0f0f0f0f0f0f  0x0101010101010101 1 . 1 . 1 . 1 .    1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . . .   1 . 1 . 1 . 1 .     1 1 . . 1 1 . .     1 1 1 1 . . . .     1 . . . . . ..

popCount
This is how the complete routine looks in C: int popCount (U64 x) { x = x       - ((x >> 1)  & k1); /* put count of each 2 bits into those 2 bits */ x = (x & k2) + ((x >> 2) & k2); /* put count of each 4 bits into those 4 bits */ x = (x      +  (x >> 4)) & k4 ; /* put count of each 8 bits into those 8 bits */ x = (x * kf) >> 56; /* returns 8 most significant bits of x + (x<<8) + (x<<16) + (x<<24) + ... */   return (int) x; }

Advantage: no branches, no memory lookups, constant runtime - independent from population Drawback: dependency chain, not much parallel speedup

For likely sparsely populated bitboards, the loop-wise Brian Kernighan's way might be the faster one.

HAKMEM 169
A similar technique was proposed by Bill Gosper et al. from Massachusetts Institute of Technology, as published 1972 in Memo 239 (HAKMEM), to add bit-trio- rather than duo populations consecutively, and the 32 bit version relies on casting out 63. Note that the constants in the code below have octal (base-8) digits, originally written in Assembly for the PDP-6. An expanded 64-bit version, casting out 511 or 4095, is slightly less efficient than the binary SWAR version above. int hakmem169_32(unsigned int x) { x = x - ((x >> 1)  & 033333333333) - ((x >> 2) & 011111111111); x = (x + (x >> 3)) & 030707070707 ; return x % 63; /* casting out 63 */ }

=Miscellaneous=

Cardinality of Multiple Sets
If we like to count arrays of sets, we can reduce 2^N-1 popcounts to N popcounts, by applying the odd-major-trick. For three sets to count we safe one, with five additional cheap instructions. odd  =  (x ^ y)  ^ z;  major = ((x ^ y ) & z) | (x & y);

popCount(x) + popCount(y) + popCount(z) == 2*popCount(major) + popCount(odd) The combined popCount3 likely gains more parallel speedup, since there are two independent chains to calculate. Possible Application is to pass the union of both bishops (since usually bishops cover disjoint sets due to different square colors) plus the up to two knight move-target sets. // return popCount(x) + popCount(y) + popCount(z) int popCount3 (U64 x, U64 y, U64 z) { U64 maj = ((x ^ y ) & z) | (x & y); U64 odd = ((x ^ y ) ^ z); maj = maj - ((maj >> 1) & k1 ); odd = odd - ((odd >> 1) & k1 ); maj = (maj & k2) + ((maj >> 2) & k2); odd = (odd & k2) + ((odd >> 2) & k2); maj = (maj + (maj >> 4)) & k4; odd = (odd + (odd >> 4)) & k4; odd = ((maj + maj + odd) * kf ) >> 56; return (int) odd; }

Odd and Major 7-15
Of course - reducing seven popcount to three, or even 15 popcounts to four sounds even more promising. For N = 2^n - 1 it takes N - n odd-major pairs. Thus one add-major pair - five instructions - per saved popCount.

That is 7 - 3 = 4 pairs: one1,two1 := oddMaj(x1,x2,x3) one2,two2 := oddMaj(x4,x5,x6) ones,two3 := oddMaj(x7,one1,one2) twos,four := oddMaj(two1,two2,two3) Or 15 - 4 = 11 pairs: one1,two1 := oddMaj(x1,x2,x3) one2,two2 := oddMaj(x4,x5,x6) one3,two3 := oddMaj(x7,x8,x9) one4,two4 := oddMaj(x10,x11,x12) one5,two5 := oddMaj(x13,x14,x15) one6,two6 := oddMaj(one1,one2,one3) ones,two7 := oddMaj(one4,one5,one6) two8,four1 := oddMaj(two1,two2,two3) two9,four2 := oddMaj(two4,two5,two6) twos,four3 := oddMaj(two7,two8,two9) four,eight := oddMaj(four1,four2,four3)

Odd and Major Digit Counts
Odd-Major is probably also useful to determine digit count sets of attacks or other stuff: U64 odd(U64 x, U64 y, U64 z) {return x^y^z;} U64 maj(U64 x, U64 y, U64 z) {return ((x^y)&z)|(x&y);}

void attackCounts(U64 t[3], const U64 s[7]) { one1 = odd(s[0], s[1], s[2]); two1 = maj(s[0], s[1], s[2]); one2 = odd(s[3], s[4], s[5]); two2 = maj(s[3], s[4], s[5]); t[0] = odd(s[6], one1, one2); two3 = maj(s[6], one1, one2); t[1] = odd(two1, two2, two3); t[2] = maj(two1, two2, two3); } with following semantics: exactly7attacks :=  t[2] &  t[1] &  t[0] exactly6attacks :=  t[2] &  t[1] & ~t[0] exactly5attacks :=  t[2] & ~t[1] &  t[0] exactly4attacks :=  t[2] & ~t[1] & ~t[0] exactly3attacks := ~t[2] &  t[1] &  t[0] exactly2attacks := ~t[2] &  t[1] & ~t[0] exactly1attack :=  ~t[2] & ~t[1] &  t[0] noAttack       :=  ~t[2] & ~t[1] & ~t[0]

atLeast4attacks :=                  t[2] atLeast2attacks := atLeast4attacks | t[1] atLeast1attack := atLeast2attacks | t[0] noAttack       := ~atLeast1attack exactly1attack :=  atLeast1attack  ^ atLeast2attacks

Popcount as log2 of LS1B
Assuming an architecture has a fast popcount-instruction (but no bitscan). One can isolate the LS1B, decrement it and count the remaining trailing "ones" to perform the logarithm dualis: log2(LS1B) = popCount( LS1B - 1 ); bitIndexOfLS1B(x) = popCount( (x & -x) - 1 );

For instance, LS1B is 2^44, decrementing leaves a below LSB1 mask with exactly 44 bits set: 0x0000100000000000  0x00000FFFFFFFFFFF . . . . . . . .     . . . . . . . . . . . . . . . .      . . . . . . . . . . . . 1 . . .      1 1 1 1 . . . . . . . . . . . .      1 1 1 1 1 1 1 1 . . . . . . . .      1 1 1 1 1 1 1 1 . . . . . . . .      1 1 1 1 1 1 1 1 . . . . . . . .      1 1 1 1 1 1 1 1 . . . . . . . .      1 1 1 1 1 1 1 1

Hamming Distance
The hamming distance of two words is defined as the number of corresponding different bits. int hammingDistance (U64 a, U64 b) {return popcnt( a ^ b);} Hamming distance greater than one or two is an important property of codes to detect or even correct one-bit errors.

The minimum and average hamming distance over all Zobrist keys was considered as "quality"-measure of the keys. However, as long the minimum hamming distance is greater zero, linear independence (that is a small subset of all keys doesn't xor to zero), is much more important than hamming distance. Maximizing the minimal hamming distance leads to very poor Zobrist keys.

Weighted PopCount
For a SIMD-wise kind of weighted population count, see the SSE2 dot-product.

Pre-calculated Mobility
Similar to Attacks by Occupancy Lookup to determine attack sets of sliding pieces, we may use pre-calculated population count or even center-weighted population count as a rough estimate on piece mobility. It may not consider subsets of let say safe target squares.

Piece Attacks Count
As pointed out by Marco Costalba, specialized routines to count the population (Mobility) of attack sets of king, knight and line-wise sub-sets of sliding pieces can be done more efficiently than the general SWAR-Popcount. This is similar to Flipping Mirroring and Rotating the whole bitboard versus Rank, File and Diagonal, and is based on mapping the up to eight scattered occupied bits to one byte, to perform a single byte lookup. For various mapping techniques, see:
 * Hashing Multiple Bits from Bitboard Serialization
 * Rank, File and Diagonal from Flipping Mirroring and Rotating
 * Occupancy of any Line

=Popcount in Hardware=
 * Ferranti Mark 1
 * CDC 6600
 * CDC Cyber
 * SSE4.2, Intel x86, x86-64
 * SSE4a, AMD x86, x86-64

=See also=
 * Assembly Popcounts
 * Bit-Twiddling
 * Greater One Sets from General Setwise Operations
 * libpopcnt by Kim Walisch
 * MMX Popcount
 * Mobility in Chess 4.6 on the CDC Cyber
 * SIMD and SWAR Techniques
 * SSE2 Population Count
 * SSSE3 Population Count

=Publications=

1949 ...

 * Alan Turing (1949). Alan Turing's Manual for the Ferranti Mk. I. transcribed in 2000 by Robert Thau, pdf from The Computer History Museum, 9.4 The position of the most significant digit » Ferranti Mark 1
 * Maurice Wilkes, David Wheeler, Stanley Gill (1957). The Preparation of Programs for an Electronic Digital Computer. Addison-Wesley Press; 2nd edition, amazon.com, Donald B. Gillies and Jeffrey C. P. Miller on SWAR-Popcount, pages 191–193
 * Peter Wegner (1960). A technique for counting ones in a binary computer. Communications of the ACM, Volume 3, 1960
 * Michael Beeler, Bill Gosper, Rich Schroeppel (1972). HAKMEM, Memo 239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology
 * David A. Wagner, Steven M. Bellovin (1994). A Programmable Plaintext Recognizer.

2000 ...

 * Simon Y. Berkovich, Gennadi M. Lapir, Marilyn Mack (2000). A Bit-Counting Algorithm Using the Frequency Division Principle. Software, Practice and Experience Vol. 30, No. 14, 2000, pp. 1531-1540
 * Eyas El-Qawasmeh (2001). Beating the Popcount. International Journal of Information Technology, Singapore, Vol. 9. No. 1
 * Henry S. Warren, Jr. (2002). Hacker's Delight. Addison-Wesley Professional
 * Eyas El-Qawasmeh, Wafa'a Al-Qarqaz (2006). Reducing Lookup Table Size used for Bit-Counting Algorithm. Computer Science Dept. Jordan University of Science and Technology, pdf
 * Henry S. Warren, Jr. (2007). The Quest for an Accelerared Population Count. in Andy Oram & Greg Wilson (eds.) (2007).  Beautiful code: Leading Programmers Explain How They Think. O'Reilly, amazon.com
 * Donald Knuth (2009). The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise tricks & techniques, as Pre-Fascicle 1a postscript, Sideways addition, pp 11

2010 ...

 * Henry S. Warren, Jr. (2012). Hacker's Delight, 2nd Edition.  Addison-Wesley Professional, More coverage of population count and counting leading zeros, Array population count
 * Andreas Stiller (2013). Spezialkommando - Intrinsic __popcnt zählt die Einsen. c't Magazin für Computertechnik 5/2013, p. 180 (German)
 * Wojciech Muła, Nathan Kurz, Daniel Lemire (2016). Faster Population Counts Using AVX2 Instructions. arXiv:1611.07612 » AVX2, AVX-512

=Postings=

1998 ...

 * Bean counters Part1 by Peter Fendrich, CCC, August 19, 1998
 * Bean counters Part2 by Peter Fendrich, CCC, August 19, 1998
 * Countbits Function by Roberto Waldteufel, CCC, January 03, 1999
 * Sideways Add / Population Count by Jitze Couperus, Steve Bellovin and Axel H. Horns, cryptography@c2.net, January 28, 1999 » CDC 6600

2000 ...

 * fast bit counting by Flemming Rodler, CCC, April 19, 2000
 * Bit counting revisited by Flemming Rodler, CCC, April 19, 2000
 * PowerPC BitCounting Functions Speed by William Bryant, CCC, April 20, 2000 » PowerPC
 * Counting the number of bits in a 32-bit word by George Marsaglia, comp.lang.c, December 7, 2000
 * Re: Chezzz 1.0.1 - problem solved - for David Rasmussen by David Rasmussen, CCC, February 05, 2003
 * Counting bits by Andreas Herrmann, CCC, April 17, 2003
 * Hamming distance and lower hash table indexing by Tom Likens, CCC, September 02, 2003
 * PopCount optimization by milix, CCC, March 11, 2004

2005 ...

 * Population count in SSE2, again by James Van Buskirk, comp.lang.asm.x86, April 12, 2008
 * core2 popcnt by Frank Phillips, CCC, February 13, 2009
 * Piece attacks count by Marco Costalba, CCC, May 18, 2009 » Attack and Defend Maps
 * Bit twiddlement question: greater of two popcounts by Zach Wegner, CCC, August 06, 2009

2010 ...

 * Stockfish POPCNT support with gcc by Marco Costalba, CCC, January 31, 2010
 * Yet another handmade POPCNT by hopcode, comp.lang.asm.x86, January 05, 2011
 * A brief history of the popcnt instruction by Steven Edwards, CCC, March 22, 2011
 * Introduction and (hopefully) contribution - bitboard methods by Alcides Schulz, CCC, June 03, 2011 » BitScan
 * using Popcount and Prefetch with SSE4 hardware support by Engin Üstün, CCC, May 19, 2012 » Memory, SSE4
 * 64 bits for 64 squares ? by Thomas Petzke, mACE Chess, April 28, 2013
 * Stockfish 32-bit and hardware instructions on MSVC++ by Syed Fahad, CCC, December 30, 2014 » Stockfish, BitScan, Population Count

2015 ...

 * Re: Linux Version of Maverick 1.5 by Michael Dvorkin, CCC, November 12, 2015 » OS X, Maverick
 * syzygy users (and Ronald) by Robert Hyatt, CCC, September 29, 2016 » BitScan

=External Links=
 * Hamming weight from Wikipedia
 * Population count (POPCNT) - CompArch
 * Crazy On Tap - Secret Opcodes
 * Blender: POPCNT for counting bits
 * HAKMEMC -- HAKMEM Programming hacks in C by Alan Mycroft
 * popcount C samples from Henry S. Warren, Jr. (2002, 2012). Hacker's Delight. Addison-Wesley
 * The Aggregate Magic Algorithms -Population Count (Ones Count) by Hank Dietz
 * Counting bits set from Bit Twiddling Hacks by Sean Eron Anderson
 * Optimising Bit Counting using Iterative, Data-Driven Development from Necessary and Sufficient by Damien Wintour
 * Count bits set in parallel a.k.a. Population Count from the bit twiddler by Stephan Brumme
 * Benchmarking CRC32 and PopCnt instructions - strchr.com by Peter Kankowski
 * SSSE3: fast popcount by Wojciech Muła, May 24, 2008 » SSSE3
 * Speeding up bit-parallel population count by Wojciech Muła, April 13, 2015
 * Population count using XOP instructions by Wojciech Muła, December 16, 2016 » XOP
 * GitHub - WojciechMula/sse-popcount: SIMD (SSE) population count by Wojciech Muła
 * GitHub - kimwalisch/libpopcnt: Fast C/C++ bit population count library » libpopcnt by Kim Walisch
 * Census from Wikipedia
 * John Abercrombie 4tet - One, one, one + Spring song, Subway, Cologne, April 12, 1999, 3sat broadcast, YouTube Video
 * John Abercrombie, Bobo Stenson, Lars Danielsson, Jon Christensen

=References=

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