Allow for general transposition table sizes. (#1341)

For efficiency reasons current master only allows for transposition table sizes that are N = 2^k in size, the index computation can be done efficiently as (hash % N) can be written instead as (hash & 2^k - 1). On a typical computer (with 4, 8... etc Gb of RAM), this implies roughly half the RAM is left unused in analysis.

This issue was mentioned on fishcooking by Mindbreaker:
http://tests.stockfishchess.org/tests/view/5a3587de0ebc590ccbb8be04

Recently a neat trick was proposed to map a hash into the range [0,N[ more efficiently than (hash % N) for general N, nearly as efficiently as (hash % 2^k):

https://lemire.me/blog/2016/06/27/a-fast-alternative-to-the-modulo-reduction/

namely computing (hash * N / 2^32) for 32 bit hashes. This patch implements this trick and now allows for general hash sizes. Note that for N = 2^k this just amounts to using a different subset of bits from the hash. Master will use the lower k bits, this trick will use the upper k bits (of the 32 bit hash).

There is no slowdown as measured with [-3, 1] test:

http://tests.stockfishchess.org/tests/view/5a3587de0ebc590ccbb8be04
LLR: 2.96 (-2.94,2.94) [-3.00,1.00]
Total: 128498 W: 23332 L: 23395 D: 81771

There are two (smaller) caveats:

1) the patch is implemented for a 32 bit hash (so that a 64 bit multiply can be used), this effectively limits the number of clusters that can be used to 2^32 or to 128Gb of transpostion table. That's a change in the maximum allowed TT size, which could bother those using 256Gb or more regularly.

2) Already in master, an excluded move is hashed into the position key in rather simple way, essentially only affecting the lower 16 bits of the key. This is OK in master, since bits 0-15 end up in the index, but not in the new scheme, which picks the higher bits. This is 'fixed' by shifting the excluded move a few bits up. Eventually a better hashing scheme seems wise.

Despite these two caveats, I think this is a nice improvement in usability.

Bench: 5346341

src/search.cpp

@@ -557,7 +557,7 @@ namespace {
     // search to overwrite a previous full search TT value, so we use a different
     // position key in case of an excluded move.
     excludedMove = ss->excludedMove;
-    posKey = pos.key() ^ Key(excludedMove);
+    posKey = pos.key() ^ Key(excludedMove << 16); // isn't a very good hash
     tte = TT.probe(posKey, ttHit);
     ttValue = ttHit ? value_from_tt(tte->value(), ss->ply) : VALUE_NONE;
     ttMove =  rootNode ? thisThread->rootMoves[thisThread->PVIdx].pv[0]

src/tt.cpp

@@ -33,7 +33,7 @@ TranspositionTable TT; // Our global transposition table
 
 void TranspositionTable::resize(size_t mbSize) {
 
-  size_t newClusterCount = size_t(1) << msb((mbSize * 1024 * 1024) / sizeof(Cluster));
+  size_t newClusterCount = mbSize * 1024 * 1024 / sizeof(Cluster);
 
   if (newClusterCount == clusterCount)
       return;

src/tt.h

@@ -104,9 +104,9 @@ public:
   void resize(size_t mbSize);
   void clear();
 
-  // The lowest order bits of the key are used to get the index of the cluster
+  // The 32 lowest order bits of the key are used to get the index of the cluster
   TTEntry* first_entry(const Key key) const {
-    return &table[(size_t)key & (clusterCount - 1)].entry[0];
+    return &table[(uint32_t(key) * uint64_t(clusterCount)) >> 32].entry[0];
   }
 
 private:

src/ucioption.cpp

@@ -55,7 +55,8 @@ bool CaseInsensitiveLess::operator() (const string& s1, const string& s2) const
 
 void init(OptionsMap& o) {
 
-  const int MaxHashMB = Is64Bit ? 1024 * 1024 : 2048;
+  // at most 2^32 clusters.
+  const int MaxHashMB = Is64Bit ? 131072 : 2048;
 
   o["Debug Log File"]        << Option("", on_logger);
   o["Contempt"]              << Option(0, -100, 100);