New PRNG for calculators

[Namir]

(08-08-2024 05:48 PM)SlideRule Wrote:  Stumbled on Example 4: HP-25 Pocket Calculator, Computational Statistics, 25^th Conference, page 131

"The pseudorandom numbers of the HP-25 pocket calculator are generated
by the algorithm
                               u_i ≡ ( u_i-1+3)⁵    (mod 1)

With a seed u₀, 0 < u₀ < 1, the floating-point arithmetic creates sequences
which behave as demonstrated in Example 1. After a more or less ran-
dom behaviour they enter a short subsequence repeating itself. The HP-25
calculator may enter such a cycle of only 29 numbers.

This effect can be attributed to the rounding errors of the floating-point
arithmetic. The generators of the BASIC interpreters on the Commodore
and Apple microcomputers (CBM PET 2001 Series and Apple II europlus )
are further examples for generators having this defect. They are described
and analysed in Afflerbach (1985) and Ripley ( 1987, p.18 ). In Afferbach
(1985) it is shown that they may reach short cycles of period 202 and 703,
respectively. Examples 3 and 4 elucidate why congruential generators de-
fined by integer calculations ( and division by the modulus ) should be pre-
ferred to congruential generators mod 1. More about the implementation of
pseudorandom number generators can be found in Gentle (1990) and Ripley
( 1990 )."

BEST!9SlideRule
argh - yet another senior moment!

I found my old Matlab code that I had used to test PRNGs for calculators. I adapted the code for testing r=frac((3+r)^5) (as the first of many variants). I also made Matlab round the random numbers to 10 decimals to make then comparable to vintage HP calculators. The results were impressive and better than my previous favorite r=frac(r*997).

So now I will study the variants of that algorithms by changing the integer shift and power values.

Namir