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\catcode`@ 11
\long\def\@gobble#1{}

%\long\def\JFB@stop@firstoftwo#1#2{\z@#1}
\long\def\JFB@stop@secondoftwo#1#2{\z@#2}
\long\def\JFB@stop@thirdoffour#1#2#3#4{\z@#3}

\long\def\JFB@CheckWhetherBrace#1{%
  \romannumeral
  \expandafter\@gobble\expandafter

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#!/bin/env/python

"""
This is occupied with finding impact on non-uniformity of evaluating

R times 0,X1Y1Y0X0 in an approximate way which neglects the lowest
order term R0 * X0

Compared to absolute bound 2**(-24) of exact formula near 2**32,
we can not make it much worse because the worst we found is

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>>> test_highbit_tex(12345, 100000, 129)
TeX with seed 12345 and runs of length 129
skipping first 54
{0: 49272, 1: 50728}
TeX: observed probability of more 1s after 100000 batches of 129 is 0.50728
Power_divergenceResult(statistic=21.19936, pvalue=4.1390270194375e-06)
>>> test_highbit_py(82297, 100000, 129)
Python with seed 82297, runs of length 129, 100000 times
{0: 50120, 1: 49880}
Python: observed probability of more 1s after 100000 batches of 129 is 0.4988

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#!/bin/env/py
"""
Dimanche 06 mai 2018 à 22:18:52

Pour comparaison entre \pdfuniformdeviate 2 et randrange(2)
"""

import random

from scipy.stats import chisquare

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Examining batches of 165 bits from \pdfuniformdeviate 2
If more 1s than 0s count 1, else count 0.

There seems to be preponderance of case "more 1s than 0s" in case of seed 08035872,
situation looks more random for 10427991, when averaging over all possible
initial shifts.


+++++++++++++++++++++++
10000 runs of 165 draws

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HIGH BIT (\pdfuniformdeviate 2)

In successive batches of 165 draws, are there are more 1s than 0s
The answer is Yes on average for all initial shifts and the two
randomly chosen seeds 149847074 and 256844633

(I should have taken the first one < 2**27, too late)

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HIGH BIT SEEDS 0 AND 1

It appears batches of 165 contains more 1s than 0s signifcantly,
and the effect is more pronounced for seed = 1.

It looks more random for batches of 55 and seed = 0, but again
seed = 1 gives more 1s

+++++++++++++++++++++++
10000 runs of 165 draws

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\newcount\cnta
\newcount\cntb
\newcount\cntc
\newcount\cntd

\newcount\maxrand
\maxrand "10000000

% \the\maxrand

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\newcount\cnta
\newcount\cntb
\newcount\cntc
\newcount\cntd

\newcount\maxrand
\maxrand "10000000

% \the\maxrand

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+++++++++++++++++++++++
10000 runs of 165 draws
+++++++++++++++++++++++

(shift=1, seed=0) 0:4666, 1:5334
(shift=1, seed=1) 0:4723, 1:5277

(shift=2, seed=0) 0:4635, 1:5365
(shift=2, seed=1) 0:4735, 1:5265