Introduce mean squared score for delta adjustments

This patch introduces the value `meanSquaredScore`, which makes the
initial delta sensitive to unstable iterative deepening scores.

Passed STC:
https://tests.stockfishchess.org/tests/view/66fed74286d5ee47d953bb42
LLR: 2.98 (-2.94,2.94) <0.00,2.00>
Total: 71104 W: 18635 L: 18262 D: 34207
Ptnml(0-2): 234, 8365, 17993, 8714, 246

Passed LTC:
https://tests.stockfishchess.org/tests/view/6700088e86d5ee47d953bbe9
LLR: 2.95 (-2.94,2.94) <0.50,2.50>
Total: 212544 W: 54238 L: 53560 D: 104746
Ptnml(0-2): 120, 23093, 59172, 23763, 124

closes official-stockfish#5627

Bench: 1395505

src/search.cpp

@@ -312,8 +312,8 @@ void Search::Worker::iterative_deepening() {
             selDepth = 0;
 
             // Reset aspiration window starting size
+            delta     = 5 + std::abs(rootMoves[pvIdx].meanSquaredScore) / 13797;
             Value avg = rootMoves[pvIdx].averageScore;
-            delta     = 5 + avg * avg / 11797;
             alpha     = std::max(avg - delta, -VALUE_INFINITE);
             beta      = std::min(avg + delta, VALUE_INFINITE);
 
@@ -1065,7 +1065,7 @@ Value Search::Worker::search(
             // (alpha, beta), then that move is singular and should be extended. To
             // verify this we do a reduced search on the position excluding the ttMove
             // and if the result is lower than ttValue minus a margin, then we will
-            //  extend the ttMove. Recursive singular search is avoided.
+            // extend the ttMove. Recursive singular search is avoided.
 
             // Note: the depth margin and singularBeta margin are known for having
             // non-linear scaling. Their values are optimized to time controls of
@@ -1265,6 +1265,10 @@ Value Search::Worker::search(
             rm.averageScore =
               rm.averageScore != -VALUE_INFINITE ? (value + rm.averageScore) / 2 : value;
 
+            rm.meanSquaredScore = rm.meanSquaredScore != -VALUE_INFINITE * VALUE_INFINITE
+                                  ? (value * std::abs(value) + rm.meanSquaredScore) / 2
+                                  : value * std::abs(value);
+
             // PV move or new best move?
             if (moveCount == 1 || value > alpha)
             {

src/search.h

@@ -91,15 +91,16 @@ struct RootMove {
         return m.score != score ? m.score < score : m.previousScore < previousScore;
     }
 
-    uint64_t          effort          = 0;
-    Value             score           = -VALUE_INFINITE;
-    Value             previousScore   = -VALUE_INFINITE;
-    Value             averageScore    = -VALUE_INFINITE;
-    Value             uciScore        = -VALUE_INFINITE;
-    bool              scoreLowerbound = false;
-    bool              scoreUpperbound = false;
-    int               selDepth        = 0;
-    int               tbRank          = 0;
+    uint64_t          effort           = 0;
+    Value             score            = -VALUE_INFINITE;
+    Value             previousScore    = -VALUE_INFINITE;
+    Value             averageScore     = -VALUE_INFINITE;
+    Value             meanSquaredScore = -VALUE_INFINITE * VALUE_INFINITE;
+    Value             uciScore         = -VALUE_INFINITE;
+    bool              scoreLowerbound  = false;
+    bool              scoreUpperbound  = false;
+    int               selDepth         = 0;
+    int               tbRank           = 0;
     Value             tbScore;
     std::vector<Move> pv;
 };