[Bug-gnubg] Errors in evaluation of races with backgammons

[Philippe Michel]

It looks like there are some bugs in the evaluation of backgammon chances 
in races, in raceBGprob() in eval.c . For instance, this position :

    +24-23-22-21-20-19------18-17-16-15-14-13-+  O: gnubg (Cube: 2)
    | O  O     O  O  O |   |    O             |  0 points
    |                O |   |                  |
    |                O |   |                  |
    |                  |   |                  |
    |                  |   |                  |
    |                  |BAR|                  |v
  X | 8                |   |                  |
  X | X                |   |                  |
  X | X  O             |   |                  |
 XX | X  O           O |   |                  |  On roll
 XX | X  O        O  O |   | O                |  0 points
    +-1--2--3--4--5--6-------7--8--9-10-11-12-+  X: pm

evaluates as :

Position ID:    pU4AWhz/AAAAAA 
Match ID:       QQkAAAAAAAAA

Evaluator:      Race

        Win     W(g)    W(bg)   L(g)    L(bg)   Equity    Cubeful 
static: 1.000   1.000   0.678   0.000   0.000    +2.678    +2.678

The backgammon ratio is wrong by about 0.1.

It is supposed to be computed exactly (not by a net) as the winning chance 
of a 8 vs 2 1 0 0 3 cubeless bearoff, either by not-particularly-obvious 
internal functions or with the bearoff databases.

In the example above, it is the internal functions that are used and there 
may be a bug in them or, perhaps more likely, they are not used properly.

If I twist the code in eval.c to only use the databases, I get :

        Win     W(g)    W(bg)   L(g)    L(bg)   Equity    Cubeful 
static: 1.000   1.000   0.784   0.000   0.000    +2.784    +2.784
 1 ply: 1.000   1.000   0.784   0.000   0.000    +2.784    +2.784
 2 ply: 1.000   1.000   0.744   0.000   0.000    +2.744    +2.744
 3 ply: 1.000   1.000   0.744   0.000   0.000    +2.744    +2.744
 4 ply: 1.000   1.000   0.709   0.000   0.000    +2.709    +2.709

Now the bg rate is accurate for 0 and 1 ply, but not for deeper searches. 
The diminished backgammons surely don't come from misplays by X so there 
must be another problem here, either in shorter races or in the collating 
process to get the result from higher plies.