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The Center Manhattan-Distance is the Manhattan-Distance or number of orthogonal King moves on the otherwise empty board from any square to the four squares {d4, d5, e4, e5} in the center of the board. In conjunction with Center Distance a constant array might be considered as the base of Piece-Square tables. Center Manhattan-Distance is used in various evaluation terms, for instance to encourage the king to centralize in the ending, as well in Mop-up evaluation.

Lookup

Rather than to calculate the Center Manhattan-Distance from square coordinates, taking the sum from the file- and rank- Center Distance each, a lookup of a small array is appropriate. This is how the Center Manhattan-Distance might be defined in C or C++:

const int arrCenterManhattanDistance[64] = { // char is sufficient as well, also unsigned
  6, 5, 4, 3, 3, 4, 5, 6,
  5, 4, 3, 2, 2, 3, 4, 5,
  4, 3, 2, 1, 1, 2, 3, 4,
  3, 2, 1, 0, 0, 1, 2, 3,
  3, 2, 1, 0, 0, 1, 2, 3,
  4, 3, 2, 1, 1, 2, 3, 4,
  5, 4, 3, 2, 2, 3, 4, 5,
  6, 5, 4, 3, 3, 4, 5, 6
};

Calculation

Avoid Memory lookup purists may use following calculation. The routine first extracts file and rank. If either is less than 4, its ones' complement is used, to get the file or rank wise Manhattan Center Distance from the two lower bits each (while the third lowest bit 2, is always set).

file = sq  & 7;             //  0  1  2  3  4  5  6  7
if (file < 4) file = ~file; // -1 -2 -3 -4  4  5  6  7  bit 2 is always set
file &= 3;                  //  3  2  1  0  0  1  2  3

The file and rank distances are added for the final Manhattan Center Distance. The slight optimization considers each summand has bit 2 set, to perform the 'add' with a final post-mask 7, rather than to add two pre-masks with 3.

/**
 * manhattanCenterDistance
 * @author Gerd Isenberg
 * @param sq square 0..63
 * @return Manhattan Center Distance
 */
int manhattanCenterDistance(int sq) {
   int file, rank;
   file  = sq  & 7;
   rank  = sq >> 3;
   file ^= (file-4) >> 8;
   rank ^= (rank-4) >> 8;
   return (file + rank) & 7;
}

with following generated x86 Assembly:

?manhattanCenterDistance PROC NEAR
; _sq$ = ecx
  00000    8b d1       mov  edx, ecx
  00002    c1 f9 03    sar  ecx, 3
  00005    8d 41 fc    lea  eax, DWORD PTR [ecx-4]
  00008    c1 f8 08    sar  eax, 8
  0000b    33 c1       xor  eax, ecx
  0000d    83 e2 07    and  edx, 7
  00010    8d 4a fc    lea  ecx, DWORD PTR [edx-4]
  00013    c1 f9 08    sar  ecx, 8
  00016    33 ca       xor  ecx, edx
  00018    03 c1       add  eax, ecx
  0001a    83 e0 07    and  eax, 7
  0001d    c3          ret  0

See also

• Avoiding Branches
• Center Distance
• Corner-Distance in KBNK Endgame
• Manhattan-Distance
• Mop-up Evaluation
• Piece-Square Tables

Forum Posts

• Simple method for simple mates for programs without TBs by J. Wesley Cleveland, CCC, November 25, 2016 » Mop-up Evaluation, KBNK Endgame

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