The end of this great forum as we know is near. I hope and wish that some members finally achieve the transition to a new server and all topics can be saved. Thanks to everybody that make this forum that great.

I started posting in this forum many years ago with an estimate of Perft(13) and just in case something goes wrong, I want that one of my last posts in the current forum to be about the same, other perft estimate. Here I go:

There is at least one Perft(16) estimate that is very good and can be found at other site: 6.507e+22. It is dated before I registered into TalkChess. I wondered how much could be improved, so I have been running MC samples of Perft(16) of the starting position for two and a half years more less, with GNU 5.50 x64 running in the background. I had not said anything before in the spirit of a secret war or a secret mission in order to get narrower and narrower intervals before posting anything, but the end of the current TalkChess forum has prompted me to share my results as they are right now.

Here is a typical output of GNU 5.50 x64 with the command

**perftmc 16**:

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```
perftmc 16
m=6.518181e+022 sd=2.875989e+020 ci(99%)=[6.444096e+022,6.592267e+022] n=4750894 sdn=6.268663e+023 t=12.53s
m=6.507558e+022 sd=1.971200e+020 ci(99%)=[6.456780e+022,6.558336e+022] n=7126351 sdn=5.262162e+023 t=19.12s
m=6.505622e+022 sd=1.407234e+020 ci(99%)=[6.469371e+022,6.541872e+022] n=9501734 sdn=4.337782e+023 t=25.39s
[...]
```

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```
m := arithmetic mean
sd := standard deviation (formula dividing by the number of iterations)
ci(99%) := 99% confidence interval
n := nodes
sdn = sd·sqrt(n) // sdn is a measure of efficiency, less is better.
t := time
```

My best run in terms of lowest standard deviation up to now was this one:

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```
[...]
m=6.507024e+022 sd=9.432087e+017 ci(99%)=[6.506781e+022,6.507267e+022] n=1617983415 sdn=3.779099e+023 t=280802.70s
Interrupted!
```

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```
m=
Each occurrence is one iteration except the first one, where there are two. So there must be added 1 in each run:
{Runᵢ} ——> Iterationsᵢ = Occurrencesᵢ + 1
Total number of iterations after 1 125 runs: Σ(Iterationsᵢ) = 8 752 634
ⁱ
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
Runᵢ Iterationsᵢ
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
0001 9375
0002 22178
0003 22169
0004 22160
0005 22139
0006 5345
0007 5344
[...]
1116 7403
1117 7404
1118 7346
1119 7417
1120 7422
1121 6726
1122 6714
1123 6726
1124 6728
1125 6735
```

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```
__________________________________________________
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
Perft(16) Montecarlo estimates using GNU 5.50 x64.
__________________________________________________
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
Average Montecarlo estimates like averaging normal distributions with nodes as the weight factor := ωᵢ = (nᵢ) / [Σ(nᵢ)]
ⁱ
Nodes are counted with 32-bit integers and overflows happen. To get the correct number of nodes from the incorrect output n*ᵢ:
n*ᵢ = [ (nᵢ - 2³¹) mod 2³² ] - 2³¹ ; (nᵢ - 2³¹) mod 2³² = 2³¹ + n*ᵢ
sdnᵢ = sdᵢ × sqrt(nᵢ) ; nᵢ = [ (sdnᵢ) / (sdᵢ) ]²
k = round[ (nᵢ - n*ᵢ) / 2³² ]
nᵢ = k × 2³² + n*ᵢ
nᵢ = ( round{ [ (sdnᵢ) / (sdᵢ) ]² - n*ᵢ } / 2³² ) × 2³² + n*ᵢ
_
Average mean := m ≡ <m> = Σ(ωᵢ × mᵢ) = [ Σ(nᵢ × mᵢ) ] / [ Σ(nᵢ) ]
ⁱ __ ⁱ ⁱ
Average standard deviation := sd ≡ <sd> = sqrt{ Σ[(ωᵢ × sdᵢ)²] } = sqrt( { Σ[(nᵢ × sdᵢ)²] } / [Σ(nᵢ)]² ) = ( sqrt{ Σ[(nᵢ × sdᵢ)²] } ) / Σ(nᵢ)
ⁱ ⁱ ⁱ ⁱ ⁱ
Two ways to compute Σ[(nᵢ × sdᵢ)²] :
ⁱ
a) Σ[(nᵢ × sdᵢ)²] = Σ[(nᵢ)² × (sdᵢ)²]
ⁱ ⁱ
b) sdnᵢ = sdᵢ × sqrt(nᵢ) ; (sdnᵢ)² = (sdᵢ)² × nᵢ ; Σ[(nᵢ × sdᵢ)²] = Σ[nᵢ × (sdnᵢ)²]
ⁱ ⁱ
Option a) is chosen.
Solving z₉₅ and z₉₉ in:
erf[ z₉₅ / sqrt(2) ] = 0.95 (95% confidence level in a normal distribution).
erf[ z₉₉ / sqrt(2) ] = 0.99 (99% confidence level in a normal distribution).
z₉₅ ≈ 1.959963984540054236 (rounding up to the 18th decimal at https://oeis.org/A220510 'Decimal expansion of the standard normal deviate for a 95% confidence interval').
z₉₉ ≈ 2.575829303548900761 (rounding up to the 18th decimal at https://oeis.org/A329283 'Decimal expansion of the quantile z_0.995 of the standard normal distribution').
<m> ± z₉₅ × <sd> ∈ [ <m> - 1.959963984540054236 × <sd> , <m> + 1.959963984540054236 × <sd> ]
<m> ± z₉₉ × <sd> ∈ [ <m> - 2.575829303548900761 × <sd> , <m> + 2.575829303548900761 × <sd> ]
95% confidence interval of Perft(16) after 1 125 runs:
[ 65 069 225 156 643 592 417 223 ,
65 069 550 400 274 919 421 543 ]
99% confidence interval of Perft(16) after 1 125 runs:
[ 65 069 174 057 165 657 416 504 ,
65 069 601 499 752 854 422 262 ]
Total CPU time after 1 125 runs: Σ(tᵢ) = 42 755 300.13 seconds (494d 20h 28' 20"13).
ⁱ
```

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```
<m> ~ 65069387778459255919383
<sd> ~ 82971838741039268
```

*sine die*until the computation of Perft(16) or loss of interest, though I prefer to wait and see the immediate future of TalkChess to make a decision.

Regards from Spain.

Ajedrecista.