Use exploration rate for reductions

This patch measures how frequently search is exploring new configurations.
This is done be computing a running average of ttHit. The ttHitAverage rate
is somewhat low (e.g. 30% for startpos) in the normal case, while it can be
very high if no progress is made (e.g. 90% for the fortress I used for testing).

This information can be used to influence search. In this patch, by adjusting
reductions if the rate > 50%. A first version (using a low ttHitAverageResolution
and this 50% threshold) passed  testing:

STC
LLR: 2.96 (-2.94,2.94) [-1.50,4.50]
Total: 26425 W: 5837 L: 5650 D: 14938
http://tests.stockfishchess.org/tests/view/5dcede8b0ebc5902563258fa

LTC
LLR: 2.96 (-2.94,2.94) [0.00,3.50]
Total: 32313 W: 5392 L: 5128 D: 21793
http://tests.stockfishchess.org/tests/view/5dcefb1f0ebc590256325c0e

However, as discussed in pull request 2414, using a larger ttHitAverageResolution
gives a better approximation of the underlying distributions. This needs a slight
adjustment for the threshold as the new distributions are shifted a bit compared
to the older ones, and this threshold seemingly is sensitive (we used 0.53125 here).
https://github.com/official-stockfish/Stockfish/pull/2414

This final version also passed testing, and is used for the patch:

STC
LLR: 2.95 (-2.94,2.94) [-1.50,4.50]
Total: 16025 W: 3555 L: 3399 D: 9071
http://tests.stockfishchess.org/tests/view/5dd070b90ebc5902579e20c2

LTC
LLR: 2.96 (-2.94,2.94) [0.00,3.50]
Total: 37576 W: 6277 L: 5998 D: 25301
http://tests.stockfishchess.org/tests/view/5dd0f58e6f544e798086f224

Closes https://github.com/official-stockfish/Stockfish/pull/2414

Bench: 4989584

src/search.cpp

@@ -61,6 +61,9 @@ namespace {
   // Different node types, used as a template parameter
   enum NodeType { NonPV, PV };
 
+  constexpr uint64_t ttHitAverageWindow     = 4096;
+  constexpr uint64_t ttHitAverageResolution = 1024;
+
   // Razor and futility margins
   constexpr int RazorMargin = 661;
   Value futility_margin(Depth d, bool improving) {
@@ -363,6 +366,7 @@ void Thread::search() {
       multiPV = std::max(multiPV, (size_t)4);
 
   multiPV = std::min(multiPV, rootMoves.size());
+  ttHitAverage = ttHitAverageWindow * ttHitAverageResolution / 2;
 
   int ct = int(Options["Contempt"]) * PawnValueEg / 100; // From centipawns
 
@@ -665,6 +669,9 @@ namespace {
     ttMove =  rootNode ? thisThread->rootMoves[thisThread->pvIdx].pv[0]
             : ttHit    ? tte->move() : MOVE_NONE;
     ttPv = PvNode || (ttHit && tte->is_pv());
+    // thisThread->ttHitAverage can be used to approximate the running average of ttHit
+    thisThread->ttHitAverage =   (ttHitAverageWindow - 1) * thisThread->ttHitAverage / ttHitAverageWindow
+                                + ttHitAverageResolution * ttHit;
 
     // At non-PV nodes we check for an early TT cutoff
     if (  !PvNode
@@ -1082,6 +1089,10 @@ moves_loop: // When in check, search starts from here
       {
           Depth r = reduction(improving, depth, moveCount);
 
+          // Decrease reduction if the ttHit running average is large
+          if (thisThread->ttHitAverage > 544 * ttHitAverageResolution * ttHitAverageWindow / 1024)
+              r--;
+
           // Reduction if other threads are searching this position.
           if (th.marked())
               r++;

src/thread.h

@@ -61,6 +61,7 @@ public:
   Pawns::Table pawnsTable;
   Material::Table materialTable;
   size_t pvIdx, pvLast;
+  uint64_t ttHitAverage;
   int selDepth, nmpMinPly;
   Color nmpColor;
   std::atomic<uint64_t> nodes, tbHits, bestMoveChanges;