New PRNG method

[Albert Chan]

Because New PRNG only do shift + addition, it is not very random.

Chi Square percentage number are extremely weak. (i.e. not random)
We want 10% to 90%, close to 50% is excellent (see ent.exe Chi-square Test)

I reuse ent.lua to test randomness
This use 10-digits integer random state, to avoid binary to decimal float conversion errors.

Code:

local s = floor(math.random()*1e10)
local k = floor((pi-3)*1e10)
local r = function(n) 
    local t = s + k
    s = t % 1e5
    s = (t-s + s*1e5 + k) % 1e10
    return floor(n * s/1e10)
end

Note: seed = 0 --> s = floor(0.8438413544573831 * 1e10) = 8438413544

C:\LuaJIT\bit>run ent.lua 0 | ent
seed = 0
Entropy = 7.999983 bits per byte.

Optimum compression would reduce the size
of this 1000000 byte file by 0 percent.

Chi square distribution for 1000000 samples is 23.65, and randomly
would exceed this value more than than 99.99 percent of the times.

Arithmetic mean value of data bytes is 127.5018 (127.5 = random).
Monte Carlo value for Pi is 3.281245125 (error 4.45 percent).
Serial correlation coefficient is 0.091133 (totally uncorrelated = 0.0).

---

I don't quite understand your proposed algorithm.

(s + pi) % 1 = 0.xxxxxyyyyy
next(s) = (0.xxxxx + 0.yyyyy + pi) % 1

Above next(s) least sig. digts always matched pi's. Perhaps more mixing is better?

next(s) = (0.xxxxxyyyyy + 0.yyyyy + pi) % 1

Patch is trivial

Code:

< s = (t-s + s*1e5 + k) % 1e10
> s = (t + s*1e5 + k) % 1e10

C:\LuaJIT\bit>run ent.lua 0 | ent
seed = 0
Entropy = 7.999923 bits per byte.

Optimum compression would reduce the size
of this 1000000 byte file by 0 percent.

Chi square distribution for 1000000 samples is 106.69, and randomly
would exceed this value more than than 99.99 percent of the times.

Arithmetic mean value of data bytes is 127.4946 (127.5 = random).
Monte Carlo value for Pi is 3.140748563 (error 0.03 percent).
Serial correlation coefficient is -0.000728 (totally uncorrelated = 0.0).