Home * Board Representation * Bitboards * BitScan

		BitScan, a function that determines the bit-index of the least significant 1 bit (LS1B) or the most significant 1 bit (MS1B) in an integer such as bitboards. If exactly one bit is set in an unsigned integer, representing a numerical value of a power of two, this is equivalent to a base-2 logarithm. Many implementations have been devised since the advent of bitboards, as described on this page, and some implementation samples of concrete open source engines listed for didactic purpose.
M. C. Escher, Eye, 1946 [1]

Hardware vs. Software

For recent x86-64 architectures like Core 2 duo and K10, one should use the Processor Instructions for Bitscans via intrinsics or inline assembly, see x86-64 timing. P4 and K8 have rather slow bitscan-instructions. K8 uses so called vector path instructions ^[2] with 9 or 11 cycles latency, even blocking other processor resources. For these processors, specially K8 with already fast multiplication, the De Bruijn Multiplication (64-bit mode) or Matt Taylor's Folded 32-bit Multiplication (32-bit mode) might be the right choice. Other routines mentioned might be advantageous on certain architectures, specially with slow integer multiplications.

Non Empty Sets

Bitscan is most often used in serializing bitboards, and is therefor - due to a leading while-condition - not called with empty sets. Until stated otherwise, most mentioned bitscan-routines in C/C++ have the same prototype and assume none empty sets as actual parameter.

Bitscan forward

A bitscan forward is used to find the index of the least significant 1 bit (LS1B).

Trailing Zero Count

Bitscan forward is identical with a Trailing Zero Count for none empty sets, possibly available as machine instruction on some architectures, for instance the x86-64 bit-manipulation expansion set BMI1.

De Bruijn Multiplication

The De Bruijn bitscan was devised in 1997, according to Donald Knuth ^[3] by Martin Läuter, and independently by Charles Leiserson, Harald Prokop and Keith H. Randall a few month later ^{[4] [5]} , to determine the LS1B index by minimal perfect hashing. De Bruijn sequences were named after the Dutch mathematician Nicolaas de Bruijn. Interestingly sequences with the binary alphabet were already investigated by the French mathematician Camille Flye Sainte-Marie in 1894, but later "forgotten" and re-investigated and generalized by De Bruijn and Tanja van Ardenne-Ehrenfest half a century later ^[6] .

A 64-bit De Bruijn sequence contains 64-overlapped unique 6-bit sequences, thus a circle of 64 bits, where five leading zeros overlap five hidden "trailing" zeros. There are 226 = 67108864 odd sequences with 6 leading binary zeros and 226 even sequences with 5 leading binary zeros, which may be calculated from the odd ones by shifting left one.

With isolated LS1B

A multiplication with a power of two value (the isolated LS1B) acts like a left shift by it's exponent. Thus, if we multiply a 64-bit De Bruijn sequence with the isolated LS1B, we get a unique six bit subsequence inside the most significant bits. To obtain the bit-index we need to extract these upper six bits by shifting right the product, to lookup an array.

const int index64[64] = {
    0,  1, 48,  2, 57, 49, 28,  3,
   61, 58, 50, 42, 38, 29, 17,  4,
   62, 55, 59, 36, 53, 51, 43, 22,
   45, 39, 33, 30, 24, 18, 12,  5,
   63, 47, 56, 27, 60, 41, 37, 16,
   54, 35, 52, 21, 44, 32, 23, 11,
   46, 26, 40, 15, 34, 20, 31, 10,
   25, 14, 19,  9, 13,  8,  7,  6
};
 
/**
 * bitScanForward
 * @author Martin Läuter (1997)
 *         Charles E. Leiserson
 *         Harald Prokop
 *         Keith H. Randall
 * "Using de Bruijn Sequences to Index a 1 in a Computer Word"
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of least significant one bit
 */
int bitScanForward(U64 bb) {
   const U64 debruijn64 = C64(0x03f79d71b4cb0a89);
   assert (bb != 0);
   return index64[((bb & -bb) * debruijn64) >> 58];
}

See also how to Generate your "private" De Bruijn Bitscan Routine.

With separated LS1B

Instead of the classical LS1B isolation, Kim Walisch proposed the faster xor with the ones' decrement. The separation bb ^ (bb-1) contains all bits set including and below the LS1B. The 222 (4,194,304) upper De Bruijn sequences of the 226 available leave unique 6-bit indices. Using LS1B separation takes advantage of the x86 lea instruction, which saves the move instruction and unlike negate, has no data dependency on the flag register. Kim reported a 10 to 15 percent faster execution (compilers: g++-4.7 -O2, clang++-3.1 -O2, x86_64) than the traditional 64-bit De Bruijn bitscan on Intel Nehalem and Sandy Bridge CPUs.

const int index64[64] = {
    0, 47,  1, 56, 48, 27,  2, 60,
   57, 49, 41, 37, 28, 16,  3, 61,
   54, 58, 35, 52, 50, 42, 21, 44,
   38, 32, 29, 23, 17, 11,  4, 62,
   46, 55, 26, 59, 40, 36, 15, 53,
   34, 51, 20, 43, 31, 22, 10, 45,
   25, 39, 14, 33, 19, 30,  9, 24,
   13, 18,  8, 12,  7,  6,  5, 63
};
 
/**
 * bitScanForward
 * @author Kim Walisch (2012)
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of least significant one bit
 */
int bitScanForward(U64 bb) {
   const U64 debruijn64 = C64(0x03f79d71b4cb0a89);
   assert (bb != 0);
   return index64[((bb ^ (bb-1)) * debruijn64) >> 58];
}

Matt Taylor's Folding trick

A 32-bit friendly implementation to find the the bit-index of LS1B by Matt Taylor ^[7]. The xor with the ones' decrement, bb ^ (bb-1) contains all bits set including and below the LS1B. The 32-bit xor-difference of both halves yields either the complement of the upper half, or the lower half otherwise. Some samples:

ls1b		bb ^ (bb-1)		folded
63		0xffffffffffffffff		0x00000000
62		0x7fffffffffffffff		0x80000000
59		0x0fffffffffffffff		0xf0000000
32		0x00000001ffffffff		0xfffffffe
31		0x00000000ffffffff		0xffffffff
30		0x000000007fffffff		0x7fffffff
0		0x0000000000000001		0x00000001

Even if this folded "LS1B" contains multiple consecutive one-bits, the multiplication is De Bruijn like. There are only two magic 32-bit constants with the combined property of 32- and 64-bit De Bruijn sequences to apply this minimal perfect hashing:

const int lsb_64_table[64] =
{
   63, 30,  3, 32, 59, 14, 11, 33,
   60, 24, 50,  9, 55, 19, 21, 34,
   61, 29,  2, 53, 51, 23, 41, 18,
   56, 28,  1, 43, 46, 27,  0, 35,
   62, 31, 58,  4,  5, 49, 54,  6,
   15, 52, 12, 40,  7, 42, 45, 16,
   25, 57, 48, 13, 10, 39,  8, 44,
   20, 47, 38, 22, 17, 37, 36, 26
};
 
/**
 * bitScanForward
 * @author Matt Taylor (2003)
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of least significant one bit
 */
int bitScanForward(U64 bb) {
   unsigned int folded;
   assert (bb != 0);
   bb ^= bb - 1;
   folded = (int) bb ^ (bb >> 32);
   return lsb_64_table[folded * 0x78291ACF >> 26];
}

A slightly modified version may take one x86-register less in 32-bit mode, but calculates bb-1 twice:

int bitScanForwardM(BitBoard bb) {
   unsigned int folded;
   assert (bb != 0);
   folded  = (int)((bb ^ (bb-1)) >> 32);
   folded ^= (int)( bb ^ (bb-1)); // lea
   return lsb_64_table[folded * 0x78291ACF >> 26];
}

with this VC6 generated x86 assembly to compare:

bitScanForward PROC NEAR                   bitScanForwardM PROC NEAR
   mov  ecx, DWORD PTR _bb$[esp-4]            mov  eax, DWORD PTR _bb$[esp-4]
   mov  eax, DWORD PTR _bb$[esp]              mov  ecx, eax
   mov  edx, ecx                              add  ecx, -1
   push esi                                   mov  ecx, DWORD PTR _bb$[esp]
   add  edx, -1                               mov  edx, ecx
   mov  esi, eax                              adc  edx, -1
   adc  esi, -1                               xor  edx, ecx
   xor  ecx, edx                              lea  ecx, DWORD PTR [eax-1]
   xor  eax, esi                              xor  edx, ecx
   pop  esi
   xor  eax, ecx                              xor  edx, eax
   imul eax, 78291acfH                        imul edx, 78291acfH
   shr  eax, 26                               shr  edx, 26
   mov  eax, DWORD PTR _lsb_64_table[eax*4]   mov  eax, DWORD PTR _lsb_64_table[edx*4]
   ret  0                                     ret  0
bitScanForward ENDP                        bitScanForward ENDP

Walter Faxon's magic Bitscan

Walter Faxon's 32-bit friendly magic bitscan ^[8] uses a fast none minimal perfect hashing function:

const char LSB_64_table[154] =
{
#define __ 0
   22,__,__,__,30,__,__,38,18,__, 16,15,17,__,46, 9,19, 8, 7,10,
   0, 63, 1,56,55,57, 2,11,__,58, __,__,20,__, 3,__,__,59,__,__,
   __,__,__,12,__,__,__,__,__,__, 4,__,__,60,__,__,__,__,__,__,
   __,__,__,__,21,__,__,__,29,__, __,37,__,__,__,13,__,__,45,__,
   __,__, 5,__,__,61,__,__,__,53, __,__,__,__,__,__,__,__,__,__,
   28,__,__,36,__,__,__,__,__,__, 44,__,__,__,__,__,27,__,__,35,
   __,52,__,__,26,__,43,34,25,23, 24,33,31,32,42,39,40,51,41,14,
   __,49,47,48,__,50, 6,__,__,62, __,__,__,54
#undef __
};
 
/**
 * bitScanForward
 * @author Walter Faxon, slightly modified
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of least significant one bit
 */
int bitScanForward(U64 bb)
{
   unsigned int t32;
   assert(bb);
   bb  ^= bb - 1;
   t32  = (int)bb ^ (int)(bb >> 32);
   t32 ^= 0x01C5FC81;
   t32 +=  t32 >> 16;
   t32 -= (t32 >> 8) + 51;
   return LSB_64_table [t32 & 255]; // 0..63
}

A slightly modified version may take one x86-register less in 32-bit mode, but calculates bb-1 twice:

int bitScanForward(U64 bb)
{
   int t32 = 0x01C5FC81;
   assert(bb);
   t32 ^= (int)((bb ^ (bb-1)) >> 32);
   t32 ^= (int)( bb ^ (bb-1)); // lea
   t32 += t32 >> 16;
   t32 -=(t32 >>  8) + 51;
   return LSB_64_table [t32 & 255];
}

The initial LS1B separation by bb ^ (bb-1) and folding is equivalent to Matt's,

ls1b		bb ^ (bb-1)		folded
63		0xffffffffffffffff		0x00000000
62		0x7fffffffffffffff		0x80000000
59		0x0fffffffffffffff		0xf0000000
32		0x00000001ffffffff		0xfffffffe
31		0x00000000ffffffff		0xffffffff
30		0x000000007fffffff		0x7fffffff
0		0x0000000000000001		0x00000001

while Walter originally resets the LS1B, yielding in a cyclic index wrap:

LS1B		(bb & (bb-1)) ^ (bb-1)		folded
0		0x0000000000000000		0x00000000
63		0x7fffffffffffffff		0x80000000
60		0x0fffffffffffffff		0xf0000000
33		0x00000001ffffffff		0xfffffffe
32		0x00000000ffffffff		0xffffffff
31		0x000000007fffffff		0x7fffffff
1		0x0000000000000001		0x00000001

Bitscan by Modulo

Another idea is to apply a modulo (remainder of a division) operation of the isolated LS1B by the prime number 67 ^{[9] [10]} . The remainder 0..66 can be used to perfectly hash the bit-index table. Three gaps are 0, 17, and 34, so the mod 67 can make a branchless trailing zero count:

Bit-Index	Bitboard	mod 67
-	0x0000000000000000	0
0	0x0000000000000001	1
1	0x0000000000000002	2
2	0x0000000000000004	4
3	0x0000000000000008	8
4	0x0000000000000010	16
5	0x0000000000000020	32
6	0x0000000000000040	64
7	0x0000000000000080	61
8	0x0000000000000100	55
9	0x0000000000000200	43
10	0x0000000000000400	19
11	0x0000000000000800	38
12	0x0000000000001000	9
13	0x0000000000002000	18
14	0x0000000000004000	36
15	0x0000000000008000	5
16	0x0000000000010000	10
17	0x0000000000020000	20
18	0x0000000000040000	40
19	0x0000000000080000	13
20	0x0000000000100000	26
21	0x0000000000200000	52
22	0x0000000000400000	37
23	0x0000000000800000	7
24	0x0000000001000000	14
25	0x0000000002000000	28
26	0x0000000004000000	56
27	0x0000000008000000	45
28	0x0000000010000000	23
29	0x0000000020000000	46
30	0x0000000040000000	25
31	0x0000000080000000	50
32	0x0000000100000000	33
33	0x0000000200000000	66
34	0x0000000400000000	65
35	0x0000000800000000	63
36	0x0000001000000000	59
37	0x0000002000000000	51
38	0x0000004000000000	35
39	0x0000008000000000	3
40	0x0000010000000000	6
41	0x0000020000000000	12
42	0x0000040000000000	24
43	0x0000080000000000	48
44	0x0000100000000000	29
45	0x0000200000000000	58
46	0x0000400000000000	49
47	0x0000800000000000	31
48	0x0001000000000000	62
49	0x0002000000000000	57
50	0x0004000000000000	47
51	0x0008000000000000	27
52	0x0010000000000000	54
53	0x0020000000000000	41
54	0x0040000000000000	15
55	0x0080000000000000	30
56	0x0100000000000000	60
57	0x0200000000000000	53
58	0x0400000000000000	39
59	0x0800000000000000	11
60	0x1000000000000000	22
61	0x2000000000000000	44
62	0x4000000000000000	21
63	0x8000000000000000	42

/**
 * trailingZeroCount
 * @param bb bitboard to scan
 * @return index (0..63) of least significant one bit, 64 if bb is zero
 */
int trailingZeroCount(U64 bb) {
   static const int lookup67[67+1] = {
      64,  0,  1, 39,  2, 15, 40, 23,
       3, 12, 16, 59, 41, 19, 24, 54,
       4, -1, 13, 10, 17, 62, 60, 28,
      42, 30, 20, 51, 25, 44, 55, 47,
       5, 32, -1, 38, 14, 22, 11, 58,
      18, 53, 63,  9, 61, 27, 29, 50,
      43, 46, 31, 37, 21, 57, 52,  8,
      26, 49, 45, 36, 56,  7, 48, 35,
       6, 34, 33, -1 };
   return lookup67[(bb & -bb) % 67];
}

Since div/mod is an expensive instruction, a modulo by a constant is likely replaced by reciprocal fixed point multiplication to get the quotient and a second multiplication and difference to get the remainder. Compared with De Bruijn multiplication it is still too slow.

Divide and Conquer

This is a broad group of bitscans that test in succession, like the trailing zero count based on Reinhard Scharnagl's proposal ^[11] :

/**
 * trailingZeroCount
 *  like bitScanForward for none empty sets
 * @author Reinhard Scharnagl
 * @param bb bitboard to scan
 * @return index (0..64)
 */
unsigned char lsbRS[256] = {
    8, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    7, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0,
    4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0
};
 
int trailingZeroCount(U64 b) {
  unsigned buf;
  int acc = 0;
 
  if ((buf = (unsigned)b) == 0) {
    buf = (unsigned)(b >> 32);
    acc = 32;
  }
  if ((unsigned short)buf == 0) {
    buf >>= 16;
    acc += 16;
  }
  if ((unsigned char)buf == 0) {
    buf >>= 8;
    acc += 8;
  }
  return acc + lsbRS[buf & 0xff];
}

What about direct calculation? On x86 this is a chain of test, set and lea instructions:

/**
 * bitScanForward
 * @author Gerd Isenberg
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of least significant one bit
 */
int bitScanForward(U64 bb) {
   unsigned int lsb;
   assert (bb != 0);
   bb &= -bb; // LS1B-Isolation
   lsb = (unsigned)bb
       | (unsigned)(bb>>32);
   return (((((((((((unsigned)(bb>>32) !=0)  * 2)
                 + ((lsb & 0xffff0000) !=0)) * 2)
                 + ((lsb & 0xff00ff00) !=0)) * 2)
                 + ((lsb & 0xf0f0f0f0) !=0)) * 2)
                 + ((lsb & 0xcccccccc) !=0)) * 2)
                 + ((lsb & 0xaaaaaaaa) !=0);
}

Double conversion of LS1B

Assuming 64-bit doubles and little-endian structure (not portable). We convert the isolated LS1B to a double and interprete the exponent:

/**
 * bitScanForward
 * @author Gerd Isenberg
 * @param bb bitboard to scan
 * @return index (0..63) of least significant one bit
 *         -1023 if passing zero
 */
int bitScanForward(U64 bb)
{
   union {
      double d;
      struct {
         unsigned int mantissal : 32;
         unsigned int mantissah : 20;
         unsigned int exponent : 11;
         unsigned int sign : 1;
      };
   } ud;
   ud.d = (double)(bb & -bb); // isolated LS1B to double
   return ud.exponent - 1023;
}

Index of LS1B by Popcount

If we have a fast population-count instruction, we can count the trailing zeros of LS1B after subtracting one:

// precondition bb != 0
int bitScanForward(U64 bb) {
   assert (bb != 0);
   return popCount( (bb & -bb) - 1 );
}

Bitscan reverse

A bitscan reverse is used to find the index of the most significant 1 bit (MS1B). For non empty sets it is equivalent to floor of the base-2 logarithm. MS1B isolalation or separation is more expensive than LS1B isolalation or separation, due to the LS1B related Two's complement tricks are not applicable. However, beside Divide and Conquer and Double conversion, Bitscan reverse with MS1B separation is mentioned.

Divide and Conquer

As introduced by Eugene Nalimov in 2000, for an IA-64 version of Crafty ^{[12] [13]}

/**
 * bitScanReverse
 * @author Eugene Nalimov
 * @param bb bitboard to scan
 * @return index (0..63) of most significant one bit
 */
int bitScanReverse(U64 bb)
{
   int result = 0;
   if (bb > 0xFFFFFFFF) {
      bb >>= 32;
      result = 32;
   }
   if (bb > 0xFFFF) {
      bb >>= 16;
      result += 16;
   }
   if (bb > 0xFF) {
      bb >>= 8;
      result += 8;
   }
   return result + ms1bTable[bb];
}

Tribute to Frank Zappa

A branchless and little bit obfuscated version of the devide and conquer bitScanReverse with in-register-lookup ^[14] - as tribute to Frank Zappa with identifiers from Freak Out! (1966), Hot Rats (1969), Waka/Jawaka (1972), Sofa (1975), One Size Fits All (1975), Sheik Yerbouti (1979), and Jazz from Hell (1986):

typedef unsigned __int64 OneSizeFits;
typedef unsigned int HotRats;
const HotRats s      =   0;
const HotRats heik   = 457;
const HotRats y      =   1;
const HotRats e      =   2;
const HotRats r      =   3;
const HotRats b      =   4;
const HotRats o      =   5;
const HotRats u      =   8;
const HotRats t      =  16;
const HotRats i      =  32;
const HotRats     ka = (1<< 4)-1;
const HotRats   waka = (1<< 8)-1;
const HotRats jawaka = (1<<16)-1;
const HotRats jazzFromHell = 0-(16*3*heik);
 
HotRats freakOut(OneSizeFits all) {
   HotRats so,fa;
   fa   = (HotRats)(all >> i);
   so   = (fa!=s)       << o;
   fa  ^= (HotRats) all & (fa!=s)-y;
   so  ^= (jawaka < fa) << b;
   fa >>= (jawaka < fa) << b;
   so  ^= (  waka - fa) >> t    & u;
   fa >>= (  waka - fa) >> t    & u;
   so  ^= (    ka - fa) >> u    & b;
   fa >>= (    ka - fa) >> u    & b;
   so  ^=  jazzFromHell >> e*fa & r;
   return so;
}

De Bruijn Multiplication

While the tribute to Frank Zappa is quite 32-bit friendly ^[15], Kim Walisch suggested to use the parallel prefix fill for a MS1B separation with the same De Bruijn multiplication and lookup as in his bitScanForward routine with separated LS1B, with less instructions in 64-bit mode. A log base 2 method was already devised by Eric Cole on January 8, 2006, and shaved off rounded up to one less than the next power of 2 by Mark Dickinson ^[16] on December 10, 2009, as published in Sean Eron Anderson's Bit Twiddling Hacks for 32-bit integers ^[17].

const int index64[64] = {
    0, 47,  1, 56, 48, 27,  2, 60,
   57, 49, 41, 37, 28, 16,  3, 61,
   54, 58, 35, 52, 50, 42, 21, 44,
   38, 32, 29, 23, 17, 11,  4, 62,
   46, 55, 26, 59, 40, 36, 15, 53,
   34, 51, 20, 43, 31, 22, 10, 45,
   25, 39, 14, 33, 19, 30,  9, 24,
   13, 18,  8, 12,  7,  6,  5, 63
};
 
/**
 * bitScanReverse
 * @authors Kim Walisch, Mark Dickinson
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of most significant one bit
 */
int bitScanReverse(U64 bb) {
   const U64 debruijn64 = C64(0x03f79d71b4cb0a89);
   assert (bb != 0);
   bb |= bb >> 1; 
   bb |= bb >> 2;
   bb |= bb >> 4;
   bb |= bb >> 8;
   bb |= bb >> 16;
   bb |= bb >> 32;
   return index64[(bb * debruijn64) >> 58];
}

Double conversion

Assuming 64-bit doubles and little-endian structure (not portable!). Conversion to a double, interpreting the exponent. To avoid possible rounding errors, some lower bits may be cleared.

/**
 * bitScanReverse
 * @author Gerd Isenberg
 * @param bb bitboard to scan
 * @return index (0..63) of most significant one bit
 *         -1023 if passing zero
 */
int bitScanReverse(U64 bb)
{
   union {
      double d;
      struct {
         unsigned int mantissal : 32;
         unsigned int mantissah : 20;
         unsigned int exponent : 11;
         unsigned int sign : 1;
      };
   } ud;
   ud.d = (double)(bb & ~(bb >> 32));  // avoid rounding error
   return ud.exponent - 1023;
}

Leading Zero Count

Some processors have a fast leading zero count instruction. The Motorola 68020 has a bit field find first one instruction (BFFFO), which actually performs an up to 32-bit Leading Zero Count ^[18] . x86-64 AMD K10 has lzcnt as part of the SSE4a extension ^{[19] [20]} , BMI1 has lzcnt as well, while AVX-512CD even features leading zero count on vectors of eight bitbaords.

One can replace bitScanReverse of non empty sets by leadingZeroCount xor 63. Like trailing zero count, it returns 64 for empty sets, and might therefor save the leading condition in some applications.

Bitscan versus Zero Count

While the presented bitscan routines are suited to work only on none empty sets and return a value-range from 0 to 63 as bit-index, leading or trailing zero-count instructions or routines leave 64 for empty sets. Zero-counting has a immanent property of dealing correctly with empty sets - while it likely takes a conditional branch to implement this semantic in bit-scanning.

int trailingZeroCount(U64 bb) {
    if ( bb )
       return bitScanForward(bb);
    return 64;
}
 
int leadingZeroCount(U64 bb) {
    if ( bb )
       return bitScanReverse(bb) ^ 63;
    return 64;
}

Bitscan with Reset

While traversing sets, one may combine bitscanning with reset found bit. That implies passing the bitboard per reference or pointer, and tends to confuse compilers to keep all inside registers inside a typical serialization loop ^[21] .

int bitScanForwardWithReset(U64 &bb) { // also called dropForward
    int idx = bitScanForward(bb);
    bb &= bb - 1; // reset bit outside
    return idx;
}

Generalized Bitscan

This generalized bitscan uses a boolean parameter to scan reverse or forward. It relies on bitScanReverse, but conditionally masks the LS1B in case of scanning forward. It might be used in the classical approach to get positive or negative ray directions with one generalized routine.

 /**
 * generalized bitScan
 * @author Gerd Isenberg
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @param reverse, true bitScanReverse, false bitScanForward
 * @return index (0..63) of least/most significant one bit
 */
 int bitScan(U64 bb, bool reverse) {
    U64 rMask;
    assert (bb != 0);
    rMask = -(U64)reverse;
    bb &= -bb | rMask;
    return bitScanReverse(bb);
 }

Processor Instructions for Bitscans

x86

x86-64 processors have bitscan instructions and can be accessed with compilers today through either inline assembly or compiler intrinsics. For the Microsoft/Intel C compiler, the intrinsics can be accessed by including and using the instructions _BitScanForward64 ^[22] , _BitScanReverse64 ^[23] or _lzcnt64 ^[24] .

unsigned char_BitScanForward64(unsigned long * Index,  unsigned __int64 Mask);
unsigned char _BitScanReverse64(unsigned long * Index,  unsigned __int64 Mask);
unsigned __int64 __lzcnt64(unsigned __int64 value); // AMD K10 only see CPUID

Linux provides library functions ^[25] , find first bit set (ffsll) in a word leaves an index of 1..64, and zero of no bit is set ^[26] . GCC 4.4.5 further has the Built-in Function _builtin_ffsll for finding the least significant one bit, _builtin_ctzll for trailing, and _builtin_clzll for leading zero count ^[27] :

/* Returns one plus the index of the least significant 1-bit of x, or if x is zero, returns zero */
int __builtin_ffsll (unsigned long long);
 
/* Returns the number of trailing 0-bits in x, starting at the least significant bit position.
   If x is 0, the result is undefined */
int __builtin_ctzll (unsigned long long);
 
/* Returns the number of leading 0-bits in x, starting at the most significant bit position.
   If x is 0, the result is undefined */
int __builtin_clzll (unsigned long long);

Emulating Intrinsics

For the GNU C compiler, the intrinsics can be emulated with inline assembly ^[28] .

//These processor instructions work only for 64-bit processors
#ifdef _MSC_VER
    #include <intrin.h>
    #ifdef _WIN64
        #pragma intrinsic(_BitScanForward64)
        #pragma intrinsic(_BitScanReverse64)
        #define USING_INTRINSICS
    #endif
#elif defined(__GNUC__) && defined(__LP64__)
    static INLINE unsigned char _BitScanForward64(unsigned long* Index, U64 Mask)
    {
        U64 Ret;
        __asm__
        (
            "bsfq %[Mask], %[Ret]"
            :[Ret] "=r" (Ret)
            :[Mask] "mr" (Mask)
        );
        *Index = (unsigned long)Ret;
        return Mask?1:0;
    }
    static INLINE unsigned char _BitScanReverse64(unsigned long* Index, U64 Mask)
    {
        U64 Ret;
        __asm__
        (
            "bsrq %[Mask], %[Ret]"
            :[Ret] "=r" (Ret)
            :[Mask] "mr" (Mask)
        );
        *Index = (unsigned long)Ret;
        return Mask?1:0;
    }
    #define USING_INTRINSICS
#endif

Intrinsics versus asm

Alternatively, rather than to emulate the intrinsics one might use the standard prototype, by using intrinsics or inline assembly for GCC ^[29] :

#ifdef USE_X86INTRINSICS
#include <intrin.h>
#pragma intrinsic(_BitScanForward64)
#pragma intrinsic(_BitScanReverse64)
 
/**
 * bitScanForward
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of least significant one bit
 */
int bitScanForward(U64 x) {
   unsigned long index;
   assert (x != 0);
   _BitScanForward64(&index, x);
   return (int) index;
}
 
/**
 * bitScanReverse
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of most significant one bit
 */
int bitScanReverse(U64 x) {
   unsigned long index;
   assert (x != 0);
   _BitScanReverse64(&index, x);
   return (int) index;
}
#else
 
/**
 * bitScanForward
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of least significant one bit
 */
int bitScanForward(U64 x) {
   assert (x != 0);
   asm ("bsfq %0, %0" : "=r" (x) : "0" (x));
   return (int) x;
}
 
/**
 * bitScanReverse
 * @param bb bitboard to scan
 * @precondition bb != 0
 * @return index (0..63) of most significant one bit
 */
int bitScanReverse(U64 x) {
   assert (x != 0);
   asm ("bsrl %0, %0" : "=r" (x) : "0" (x));
   return (int) x;
}
#endif

Bsf/Bsr x86-64 Timings

The instruction latency and reciprocal throughput ^[30] heavily differs between various x86-64 architectures:

Architecture Stepping	Instruction(s)	Latency / Cycles	Reciprocal Throughput
AMD
K8 [31]	BSF reg16/32/64, mreg16/32/64	Vector Path 8/8/9	8/8/9
	BSR reg16/32/64, mreg16/32/64	Vector Path 11	11
K10 [32]	BSF reg, reg	Vector Path 4	4
	BSR reg, reg	Vector Path 4	4
	LZCNT reg, reg	Direct Path single 2	1
Intel [33]
ATOM	BSF/BSR	16	15
NetBurst 0F_3H	BSF/BSR	16	4
NetBurst 0F_2H	BSF/BSR	8	2
Core 06_0EH	BSF/BSR	2	1
65 nm Intel Core 06_0FH	BSF/BSR	2	1
Enhanced Intel Core 06_17H	BSF/BSR	1	1
Enhanced Intel Core 06_1DH	BSF/BSR	1	1
Nehalem 06_1AH	BSF/BSR	3	1
Sandy Bridge Ivy Bridge Haswell [34]	BSF/BSR	3	1
	LZCNT	3	1
	TZCNT	3	1

Bsf/Bsr behavior with zero source

Intel and AMD specify different behavior. In praxis there seems no difference so far. However, as long as Intel docs explicitly state content undefined, it is recommend to don't rely on a pre-initialized content of that target register, if the source is zero.

• Intel : If the content of the source operand is 0, the content of the destination operand is undefined. ^[35]
• AMD: If the second operand contains 0, the instruction sets ZF to 1 and does not change the contents of the destination register. ^[36]

ARM

ARM has CLZ (Count Leading Zeros) instruction for 32-bit integers. ARM instruction is available in ARMv5 and above, 32-bit Thumb instruction is available in ARMv6T2 and ARMv7 ^[37] , the C-intrinsic is called _builtin_clz ^{[38] [39] [40]} .

Engine Samples

• BitScan in Amundsen
• BitScan in Chess 0.5
• BitScan in CookieCat
• BitScan in Crafty (23.5)
• BitScan in Gibbon
• BitScan in Gk
• BitScan in HeavyChess
• BitScan in Kurt
• BitScan in Murka
• BitScan in Prophet
• BitScan in RedQueen
• BitScan in Spector
• BitScan in Tucano

See also

• Bitboard Serialization
• BITSCAN, a C++ library for bitstrings by Pablo San Segundo
• Bit-Twiddling
• De Bruijn Sequence Generator
• Java-Bitscan
• Population Count

Publications

• 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
• Charles E. Leiserson, Harald Prokop and Keith H. Randall (1998). Using de Bruijn Sequences to Index a 1 in a Computer Word, pdf
• Pablo San Segundo, Ramón Galán (2005). Bitboards: A New Approach. AIA 2005
• Donald Knuth (2009). The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise tricks & techniques, as Pre-Fascicle 1a postscript, p. 10
• Andreas Stiller (2013). Spezialkommando - Bits setzen, abfragen, scannen und mehr. c't Magazin für Computertechnik 7/2013, p. 186 (German)

Forum Posts

1996 ...

• Bitboards: speeding up FirstOne by Laurent Desnogues, rgcc, April 10, 1996 » Othello
• bitboard 2^i mod 67 is unique by Stefan Plenkner, rgcc, August 6, 1996
• bitboard 2^i mod 67 is unique by Stefan Plenkner, rgcc, August 7, 1996
• bitboard 2^i mod 67 is unique by Joël Rivat, rgcc, September 2, 1996
• Question to Bob: Crafty , Alpha and FindBit() by Guido Schimmels, CCC, June 05, 1998
• To Nalimov and other programmers about BSF/BSR in VC by Dezhi Zhao, CCC, January 16, 1999

2000 ...

• Re: TASM 5.0 versus BSF by Frans Morsch, comp.lang.asm.x86, March 28, 2000
• Will the Itanium have a BSF or BSR instruction? by Larry Griffiths, CCC, August 15, 2000
  Re: Will the Itanium have a BSF or BSR instruction? by Eugene Nalimov, CCC, August 16, 2000
• Binary question by Severi Salminen, CCC, October 19, 2000
• Bitboards and Piece Lists by Dann Corbit, CCC, June 14, 2001
• FirstBit() in assembler by David Rasmussen, CCC, January 13, 2002
• Reply from Intel about BSF/BSR by Severi Salminen, CCC, January 31, 2002
• "Using de Bruijn Sequences to Index a 1 in a Computer Word" by Oliver Roese, CCC, February 08, 2002
• Another hacky method for bitboard bit extraction by Walter Faxon, CCC, November 17, 2002
• Modulo verus BitScan and MMX-PopCount by Gerd Isenberg, CCC, November 29, 2002
• Fast 3DNow! BitScan by Gerd Isenberg, CCC, December 01, 2002
• Bitscan Conclusions by Matt Taylor, CCC, January 05, 2003
• Bitscan by Matt Taylor, CCC, February 11, 2003
• FirstOne for Linux by Sune Fischer, CCC, March 29, 2003
• Bit magic by Matt Taylor, comp.lang.asm.x86, June 26, 2003
• Re: De Bruijn Sequence Generator by Dieter Bürssner, CCC, December 30, 2003 » De Bruijn Sequence Generator
• Determining location of LSB/MSB by Renze Steenhuisen, CCC, February 09, 2004
• Nalimov: bsf/bsr intrinsics implementation still not optimal by Dezhi Zhao, CCC, September 22, 2004
  Re: Nalimov: bsf/bsr intrinsics implementation still not optimal by Eugene Nalimov, CCC, September 23, 2004

2005 ...

• A data point for PowerPC bitboard program authors by Steven Edwards, CCC, May 09, 2005 » PowerPC
• Best BitBoard LSB funktion? by Reinhard Scharnagl, Winboard Programming Forum, July 20, 2005
• Fastest bitboard compress routine when you can't use ASM by mambofish, CCC, May 31, 2007
• Bit twiddling question, part 2: arbitrary bitscan order by Zach Wegner, CCC, August 11, 2009
• 32 bit versions for bitscan64 by Michael Hoffmann, CCC, August 21, 2009
• 64-bit intrinsic performance by Nathan Thom, CCC, October 27, 2009
• Bit Scan (equivalent to ASM instructions bsr and bsf) by Pascal Georges, CCC, December 24, 2009

2010 ...

• bitScanReverse32 by Luca Hemmerich, CCC, January 25, 2010
• Introduction and (hopefully) contribution - bitboard methods by Alcides Schulz, CCC, June 03, 2011 » Population Count
• Leading Zero Count Question by Matthew R. Brades, CCC, September 16, 2012
• Optimizing bitboards for ARM by Martin Sedlak, CCC, November 17, 2012
• Symmetric move generation using bitboards by Lasse Hansen, CCC, December 20, 2014
• Stockfish 32-bit and hardware instructions on MSVC++ by Syed Fahad, CCC, December 30, 2014 » Stockfish, BitScan, Population Count

2015 ...

• Fun with De Bruijn by Henk van den Belt, CCC, August 27, 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 » Population Count

External Links

• Find first set from Wikipedia
• The Aggregate Magic Algorithms by Hank Dietz
• Bit Twiddling Hacks by Sean Eron Anderson
• An Efficient Bit-Reversal Sorting Algorithm for the Fast Fourier Transform by Jennifer Elaan, January 16, 2005
• Efficient bit scan mechanism - United States Patent 6172623 from FreePatentsOnline.com
• Frank Zappa & the Mothers - King Kong BBC Studio Recording 1968, YouTube Video
  

References

1. ^ Picture gallery "Back in Holland 1941 - 1954" from The Official M.C. Escher Website
2. ^ Chip Architect: Detailed Architecture of AMD's Opteron - 1.3 A third class of Instructions by Hans de Vries
3. ^ Donald Knuth (2009). The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise tricks & techniques, as Pre-Fascicle 1a postscript, p 10
4. ^ Charles E. Leiserson, Harald Prokop and Keith H. Randall (1998). Using de Bruijn Sequences to Index a 1 in a Computer Word, pdf
5. ^ "Using de Bruijn Sequences to Index a 1 in a Computer Word" discussion in CCC, February 08, 2002
6. ^ N. G. de Bruijn (1975). Acknowledgement of priority to C. Flye Sainte-Marie on the counting of circular arrangements of 2n zeros and ones that show each n-letter word exactly once. Technical Report, Technische Hogeschool Eindhoven, available as pdf reprint
7. ^ Bit magic by Matt Taylor, comp.lang.asm.x86, June 26, 2003
8. ^ Another hacky method for bitboard bit extraction by Walter Faxon, CCC, November 17, 2002
9. ^ bitboard 2^i mod 67 is unique by Stefan Plenkner, rgcc, August 6, 1996
10. ^ Pablo San Segundo, Ramón Galán (2005). Bitboards: A New Approach. AIA 2005
11. ^ Best BitBoard LSB funktion? by Reinhard Scharnagl, Winboard Programming Forum, July 20, 2005
12. ^ Re: Will the Itanium have a BSF or BSR instruction? by Eugene Nalimov, CCC, August 16, 2000
13. ^ ms1bTable array in Eugene Nalimovs bitScanReverse by Stef Luijten, CCC, April 17, 2011
14. ^ just another reverse bitscan by Gerd Isenberg, CCC, December 22, 2005
15. ^ final version - homage to FZ by Gerd Isenberg, CCC, December 23, 2005
16. ^ EuroPython 2012: Florence, July 2–8 | Mark Dickinson
17. ^ Find the log base 2 of an N-bit integer in O(lg(N)) operations with multiply and lookup from Bit Twiddling Hacks by Sean Eron Anderson
18. ^ 68020 Bit Field Instructions
19. ^ SSE4a from Wikipedia
20. ^ __lzcnt16, __lzcnt, __lzcnt64 Visual C++ Language Reference
21. ^ Bitscan by Matt Taylor, CCC, February 11, 2003
22. ^ _BitScanForward, _BitScanForward64 Visual C++ Language Reference
23. ^ _BitScanReverse, _BitScanReverse64 Visual C++ Language Reference
24. ^ __lzcnt16, __lzcnt, __lzcnt64 Visual C++ Language Reference
25. ^ Section 3: library functions - Linux man pages
26. ^ ffsll(3): find first bit set in word - Linux man page
27. ^ Other Builtins - Using the GNU Compiler Collection (GCC)
28. ^ Re: Nalimov: bsf/bsr intrinsics implementation still not optimal by Eugene Nalimov, CCC, September 23, 2004
29. ^ Matters Computational - ideas, algorithms, source code (pdf) Ideas and Source Code by Jörg Arndt
30. ^ Instruction tables, Lists of instruction latencies, throughputs and microoperation breakdowns for Intel and AMD CPU's (pdf) by Agner Fog
31. ^ Software Optimization Guide for AMD64 Processors
32. ^ Software Optimization Guide for AMD Family 10h and 12h Processors
33. ^ Intel 64 and IA32 Architectures Optimization Reference Manual
34. ^ Haswell Instructions Latency
35. ^ Intel® 64 and IA-32 Architectures Software Developer’s Manual Volume 2A: Instruction Set Reference, A-M (pdf) BSF—Bit Scan Forward 3-87
36. ^ AMD64 Architecture Programmer’s Manual Volume 3: General-Purpose and System Instructions (pdf) Bit Scan Forward pg. 111
37. ^ ARM Information Center > General data processing instructions > CLZ
38. ^ ARM Information Center > Instruction intrinsics > __builtin_clz
39. ^ Other Builtins - Using the GNU Compiler Collection (GCC)
40. ^ Bit Scan (equivalent to ASM instructions bsr and bsf) by Pascal Georges, CCC, December 24, 2009

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