Home * Board Representation * Bitboards * BitScan
Eye.jpg

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


See also


Publications


Forum Posts

1996 ...

2000 ...

2005 ...

2010 ...

2015 ...


External Links


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

What links here?


Up one Level