Author: Gerd Isenberg
Date: 15:34:44 01/14/03
Go up one level in this thread
On January 14, 2003 at 17:13:29, Gerd Isenberg wrote: >On January 14, 2003 at 16:09:57, Dann Corbit wrote: > >>[SNIP] >>>the x^x-2 trick is absolutely great. >> >>Can someone give me the ten cent explanation of how a parabola is useful? >> >>I want to get either a mathematical or geometric understanding. > >Hi Dann, > >One of the miracles of two's complement ;-) >Complementing bits in MSB direction until the first "one" is complemented. > >One sample of all rook attacks in right board direction: >(bottom left is bit 0) >X - any Piece >R - Rook (or Queen) > >- - - - - - - - >- - - - - - - - >- - - - - - - - >- X - R - - X - >- - - - - - - - >- R X - R - X - >- - - - - - - - >R - - - - - - - > >BitBoards: - is zero for better reading. All operations byte- or rankwise. >0 1 2 3 4 5 6 7 > > Occupied Rooks Occupied-Rooks Occupied-2Rooks...^Occupied >- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - - >- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - - >- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - -|- - - - - - - - >- 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 > x 1 x-1 x-2 x^(x-2) > The first subtraction "resets" the rook bits. The second subtraction toggles the bits from the rook bit positions until the first occupied is toggled and becomes zero. The final xor resets the remaining rook bits, but sets all occupied bits that became previously reset by the second subtraction. For single attackers the trick also works in other positive (or reversed negative) directions with some additional ray masks and 64-bit arithmetic. > >Cheers, >Gerd
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