Tweak Late Move Reduction at root

Maintain best move counter at the root and allow there only moves which has a counter
of zero for Late Move Reduction. For compensation only the first three moves are excluded
from Late Move Reduction per default instead the first four moves.

What we can further do:

- here we use a simple counting scheme but perhaps some aging to fade out early iterations
  could be helpful
- use the best move counter also at inner nodes for LMR and/or pruning

STC:
LLR: 2.95 (-2.94,2.94) [0.50,4.50]
Total: 17414 W: 3984 L: 3733 D: 9697
http://tests.stockfishchess.org/tests/view/5d6234bb0ebc5939d09f2aa2

LTC:
LLR: 2.96 (-2.94,2.94) [0.00,3.50]
Total: 38058 W: 6448 L: 6166 D: 25444
http://tests.stockfishchess.org/tests/view/5d62681a0ebc5939d09f2f27

Closes https://github.com/official-stockfish/Stockfish/pull/2282

Bench: 3568210

src/search.cpp

@@ -472,7 +472,10 @@ void Thread::search() {
                   ++failedHighCnt;
               }
               else
+              {
+                  ++rootMoves[pvIdx].bestMoveCount;
                   break;
+              }
 
               delta += delta / 4 + 5;
 
@@ -1072,7 +1075,8 @@ moves_loop: // When in check, search starts from here
       // Step 16. Reduced depth search (LMR). If the move fails high it will be
       // re-searched at full depth.
       if (    depth >= 3 * ONE_PLY
-          &&  moveCount > 1 + 3 * rootNode
+          &&  moveCount > 1 + 2 * rootNode
+          && (!rootNode || thisThread->best_move_count(move) == 0)
           && (  !captureOrPromotion
               || moveCountPruning
               || ss->staticEval + PieceValue[EG][pos.captured_piece()] <= alpha

src/search.h

@@ -70,6 +70,7 @@ struct RootMove {
   Value previousScore = -VALUE_INFINITE;
   int selDepth = 0;
   int tbRank = 0;
+  int bestMoveCount = 0;
   Value tbScore;
   std::vector<Move> pv;
 };

src/thread.cpp

@@ -52,6 +52,15 @@ Thread::~Thread() {
   stdThread.join();
 }
 
+/// Thread::bestMoveCount(Move move) return best move counter for the given root move
+
+int Thread::best_move_count(Move move) {
+
+  auto rm = std::find(rootMoves.begin() + pvIdx,
+                      rootMoves.begin() + pvLast, move);
+
+  return rm != rootMoves.begin() + pvLast ? rm->bestMoveCount : 0;
+}
 
 /// Thread::clear() reset histories, usually before a new game
 

src/thread.h

@@ -56,6 +56,7 @@ public:
   void idle_loop();
   void start_searching();
   void wait_for_search_finished();
+  int best_move_count(Move move);
 
   Pawns::Table pawnsTable;
   Material::Table materialTable;