# Matt Taylor

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Matt Taylor was involved in forum discussions on low level optimization and x86 assembly language issues. Beside other things Matt was busy to optimize a minimal perfect hashing scheme for bitscan purposes.

# BitScan

Based on ideas of Walter Faxon and De Bruijn Multiplication, Matt came up with a genius folding trick as a quintessence [1] [2]:

```For 64-bits, I do things a little differently simply because a 64-bit multiply is slow. I start out with x ^= (x - 1) just like normal which generates a key equal to 2^n - 1 (where n is the index of the LSB, 1 is index 0). Now fold the 64-bit key into 32-bits by xoring the high 32-bits with the low 32-bits.
```
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];
}
```

# Forum Posts

Re: Bit magic by Matt Taylor, comp.lang.asm.x86, June 29, 2003