[Guide] Tapered Eval

As I just mentioned here, game phase interpolation (or tapered eval) is a way of avoiding sharp differences between evaluations in different phases of the game.

The following explanation is based on how it's done in Fruit, and there are similar implementations in Crafty and Stockfish (and probably many many other engines as well).

The scaling looks like this:

eval = ((opening * (256 - phase)) + (endgame * phase)) / 256

Where opening is the evaluation of the position with middle game in mind (e.g. keep kings protected behind their pawn covers) and endgame is the evaluation with endgame in mind (e.g. activate the kings). Both these evaluations are done in parallel when evaluating a position.

The phase is evaluated like this (code specifics left out):

PawnPhase = 0
KnightPhase = 1
BishopPhase = 1
RookPhase = 2
QueenPhase = 4
TotalPhase = PawnPhase*16 + KnightPhase*4 + 
             BishopPhase*4 + RookPhase*4 + QueenPhase*2
 
phase = TotalPhase
 
phase -= wp * PawnPhase // Number of white pawns 
phase -= wn * Knight    // White knights
...
phase -= br * RookPhase
phase -= bq * QueenPhase
 
phase = (phase * 256 + (TotalPhase / 2)) / TotalPhase

Example
White and black each has NNBR and the evaluation for opening is +100 and endgame is +300

According to the above numbers we then get:

phase = (14 * 256 + (24 / 2)) / 24 = 149
## Where 14 is 24-1-1-1-2-1-1-1-2 (TotalPhase - phases of all pieces)

eval = ((100 * (256 - 149)) + (300 * 149)) / 256 = 216 tapered eval

Chess programming wiki - Tapered Eval

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Quite ingenious. Funny how the decision on game phase in current version of Mediocre is extremely close to this (without interpolating on it of course), I wonder if I stole that from Fruit way back when I implemented it. :)