I made a program to use all 16 cogs to search the 16x16x16 space for our made-up xoroshiro32+ PRNG. It's running now, but won't have results for several hours:

dat
'
' Launch cogs 15..0 with xoroshiro32+ search program
'
org
setq ##$1000/4-1 'clear 4096 bytes starting at $1000
wrlong #0,##$1000
.loop coginit cognum,#@search 'last iteration relaunches cog 0
djns cognum,#.loop
cognum long 15
'
' Search 1 x 16 x 16 solutions - each cog
'
org
search cogid t1 't1 is cogid 0..15
mov t2,#0 't2 ranges 0..15
mov t3,#0 't3 ranges 0..15
new mov state,#1 'new test, seed state
mov tests,#0 'reset test counter
loop getword s0,state,#0 'extract s0 and s1 words from state long
getword s1,state,#1
setword s0,s0,#1 'copy bottom words to top words for ROL
setword s1,s1,#1
xor s1,s0 's1 ^= s0
rol s0,t1 's0 = rol(s0,t1) ^ s1 ^ (s1 << t2)
xor s0,s1
mov x,s1
shl x,t2
xor s0,x
rol s1,t3 's1 = rol(s1,t3)
setword state,s0,#0 'pack s0 and s1 back into state long
setword state,s1,#1
ijz tests,#fail 'if 2^32 tests, caught in loop, fail
cmp state,#1 wz 'loop if not returned to seed value
if_nz jmp #loop
ijnz tests,#fail 'if not 2^32-1 iterations, fail
decod x,#12 '2^32-1 iterations!
setnib x,t1,#2
setnib x,t2,#1
setnib x,t3,#0
wrbyte #$FF,x 'mark byte in buffer
fail incmod t2,#15 wc 'try next solution, inc t2/t3
if_c incmod t3,#15 wc
if_nc jmp #new
test t1 wz 'this cog's solution space checked
if_nz cogstop t1 'if not cog0, stop now
'
' Cog 0 reports the results
'
bmask dirb,#15 'output to leds
mov ptra,#0 'reset ptra
show rdbyte x,ptra[##$1000] wz 'get byte[ptra+$1000], 0?
if_nz mov outb,ptra 'show 3 nibbles on leds
if_nz jp #57,#$ 'wait for pb3 cycle
if_nz waitx ##80_000_000/10
if_nz jnp #57,#$
if_nz waitx ##80_000_000/10
incmod ptra,##$FFF 'find and display next solution
jmp #show
'
' Variables
'
state res 1
tests res 1
t1 res 1
t2 res 1
t3 res 1
s0 res 1
s1 res 1
x res 1

int RANDOM () {
static unsigned long register; /*Register must be unsigned so right
shift works properly.*/
/*Register should be initialized with some random value.*/
register = ((((register >> 31) /*Shift the 32nd bit to the first bit*/
^ (register >> 6) /*XOR it with the seventh bit*/
^ (register >> 4) /*XOR it with the fifth bit*/
^ (register >> 2) /*XOR it with the third bit*/
^ (register >> 1) /*XOR it with the second bit*/
^ register) /*and XOR it with the first bit.*/
& 0x0000001) /*Strip all the other bits off and*/
<<31) /*move it back to the 32nd bit.*/
| (register >> 1); /*Or with the register shifted right.*/
return register & 0x00000001; /*Return the first bit.*/
}

As he says:

"One final note of caution: There are many more feedback arrangements for various-length LSFRs that produce maximum-length sequences, but don't fiddle with the feedback sequences without the proper mathematical theory. The particular bits that are XORed together may seem arbitrary, but they are chosen to ensure that the sequence takes 2n-1 bits to repeat. Blindly choosing a different feedback sequence can easily make the output sequence repeat after only a couple of hundred bits, and you would be better off sticking with your store-bought RND function."

That's a simple LFSR. This would be the code for it in PASM:

'
' Iterate lfsr, result in C flag
'
iterate test lfsr,mask wc 'get parity of AND result
_ret_ rcr lfsr,#1 wc 'new bit in, old bit out
lfsr long 1
mask long $8000_0056

I effectively made this and it ran for $7FFF_FFFF iterations before it repeated the seed value of 1:

Still an excellent result for a shot in the dark. I've not tried that test with any of my runs. It does beg the question of how easy it seems to be to get a long repeat sequence. Looks like it happens quite easy.

There is significant variations in the PractRand test results when differing constants are chosen. Here's a collect of Xoroshiro32+ runs I've just tried:
a=11; b=1; c=6: High quality to 256 KBytes.
a=11; b=2; c=6: High quality to 128 KBytes.
a=11; b=3; c=6: High quality to 64 KBytes.
a=11; b=4; c=6: High quality to 64 KBytes.
a=11; b=5; c=6: High quality to 64 KBytes.
a=11; b=1; c=7: High quality to 64 KBytes.
a=11; b=2; c=7: High quality to 128 KBytes.
a=11; b=3; c=7: High quality to 256 KBytes.
a=11; b=4; c=7: High quality to 128 KBytes.
a=11; b=5; c=7: High quality to 32 KBytes.
a=11; b=1; c=8: High quality to 128 KBytes.
a=11; b=2; c=8: High quality to 16 KBytes.
a=11; b=3; c=8: High quality to 32 KBytes.
a=11; b=4; c=8: High quality to 128 KBytes.
a=11; b=5; c=8: High quality to 64 KBytes.
a=11; b=1; c=9: High quality to 64 KBytes.
a=11; b=2; c=9: High quality to 64 KBytes.
a=11; b=3; c=9: High quality to 64 KBytes.
a=11; b=4; c=9: High quality to 128 KBytes.
a=11; b=5; c=9: High quality to 128 KBytes.
a=11; b=1; c=10: High quality to 128 KBytes.
a=11; b=2; c=10: High quality to 64 KBytes.
a=11; b=3; c=10: High quality to 64 KBytes.
a=11; b=4; c=10: High quality to 32 KBytes.
a=11; b=5; c=10: High quality to 256 KBytes.
a=12; b=1; c=6: High quality to 256 KBytes.
a=12; b=2; c=6: *Instant fail!!!
a=12; b=3; c=6: High quality to 128 KBytes.
a=12; b=4; c=6: *Instant fail!!!
a=12; b=5; c=6: High quality to 32 KBytes.
a=12; b=1; c=7: High quality to 256 KBytes.
a=12; b=2; c=7: High quality to 64 KBytes.
a=12; b=3; c=7: High quality to 256 KBytes.
a=12; b=4; c=7: High quality to 32 KBytes.
a=12; b=5; c=7: High quality to 64 KBytes.
a=12; b=1; c=8: High quality to 64 KBytes.
a=12; b=2; c=8: *Instant fail!!!
a=12; b=3; c=8: High quality to 32 KBytes.
a=12; b=4; c=8: *Instant fail!!!
a=12; b=5; c=8: High quality to 128 KBytes.
a=12; b=1; c=9: High quality to 128 KBytes.
a=12; b=2; c=9: High quality to 128 KBytes.
a=12; b=3; c=9: High quality to 64 KBytes.
a=12; b=4; c=9: High quality to 64 KBytes.
a=12; b=5; c=9: High quality to 64 KBytes.
a=12; b=1; c=10: High quality to 32 KBytes.
a=12; b=2; c=10: *Instant fail!!!
a=12; b=3; c=10: High quality to 256 KBytes.
a=12; b=4; c=10: *Instant fail!!!
a=12; b=5; c=10: High quality to 128 KBytes.
a=13; b=1; c=6: High quality to 128 KBytes.
a=13; b=2; c=6: High quality to 128 KBytes.
a=13; b=3; c=6: High quality to 64 KBytes.
a=13; b=4; c=6: High quality to 256 KBytes.
a=13; b=5; c=6: High quality to 64 KBytes.
a=13; b=1; c=7: High quality to 64 KBytes.
a=13; b=2; c=7: High quality to 256 KBytes.
a=13; b=3; c=7: High quality to 32 KBytes.
a=13; b=4; c=7: High quality to 512 KBytes. (Winner #1)
a=13; b=5; c=7: High quality to 64 KBytes.
a=13; b=1; c=8: High quality to 128 KBytes.
a=13; b=2; c=8: High quality to 64 KBytes.
a=13; b=3; c=8: High quality to 256 KBytes. (This one conforms to my earlier description)
a=13; b=4; c=8: High quality to 32 KBytes.
a=13; b=5; c=8: High quality to 8 KBytes.
a=13; b=1; c=9: High quality to 128 KBytes.
a=13; b=2; c=9: High quality to 64 KBytes.
a=13; b=3; c=9: High quality to 32 KBytes.
a=13; b=4; c=9: High quality to 128 KBytes.
a=13; b=5; c=9: High quality to 64 KBytes.
a=13; b=1; c=10: High quality to 64 KBytes.
a=13; b=2; c=10: High quality to 64 KBytes.
a=13; b=3; c=10: High quality to 8 KBytes.
a=13; b=4; c=10: High quality to 8 KBytes.
a=13; b=5; c=10: High quality to 128 KBytes.
a=14; b=1; c=6: High quality to 256 KBytes.
a=14; b=2; c=6: *Instant fail!!!
a=14; b=3; c=6: High quality to 256 KBytes.
a=14; b=4; c=6: *Instant fail!!!
a=14; b=5; c=6: High quality to 128 KBytes.
a=14; b=1; c=7: High quality to 128 KBytes.
a=14; b=2; c=7: High quality to 256 KBytes.
a=14; b=3; c=7: High quality to 256 KBytes.
a=14; b=4; c=7: High quality to 64 KBytes.
a=14; b=5; c=7: High quality to 512 KBytes. (Winner #2)
a=14; b=1; c=8: High quality to 64 KBytes.
a=14; b=2; c=8: *Instant fail!!!
a=14; b=3; c=8: High quality to 64 KBytes.
a=14; b=4; c=8: *Instant fail!!!
a=14; b=5; c=8: High quality to 32 KBytes.
a=14; b=1; c=9: High quality to 64 KBytes.
a=14; b=2; c=9: High quality to 128 KBytes.
a=14; b=3; c=9: High quality to 64 KBytes.
a=14; b=4; c=9: High quality to 128 KBytes. (This is your above constants)
a=14; b=5; c=9: High quality to 16 KBytes.
a=14; b=1; c=10: High quality to 128 KBytes.
a=14; b=2; c=10: *Instant fail!!!
a=14; b=3; c=10: High quality to 64 KBytes.
a=14; b=4; c=10: *Instant fail!!!
a=14; b=5; c=10: High quality to 32 KBytes.
a=15; b=1; c=6: High quality to 256 KBytes.
a=15; b=2; c=6: High quality to 512 KBytes. (Winner #3)
a=15; b=3; c=6: High quality to 128 KBytes.
a=15; b=4; c=6: High quality to 64 KBytes.
a=15; b=5; c=6: High quality to 512 KBytes. (Winner #4)
a=15; b=1; c=7: High quality to 128 KBytes.
a=15; b=2; c=7: High quality to 128 KBytes.
a=15; b=3; c=7: High quality to 128 KBytes.
a=15; b=4; c=7: High quality to 128 KBytes.
a=15; b=5; c=7: High quality to 64 KBytes.
a=15; b=1; c=8: High quality to 256 KBytes.
a=15; b=2; c=8: High quality to 64 KBytes.
a=15; b=3; c=8: High quality to 128 KBytes.
a=15; b=4; c=8: High quality to 64 KBytes.
a=15; b=5; c=8: High quality to 256 KBytes.
a=15; b=1; c=9: High quality to 256 KBytes.
a=15; b=2; c=9: High quality to 64 KBytes.
a=15; b=3; c=9: High quality to 64 KBytes.
a=15; b=4; c=9: High quality to 64 KBytes.
a=15; b=5; c=9: High quality to 64 KBytes.
a=15; b=1; c=10: High quality to 128 KBytes.
a=15; b=2; c=10: High quality to 128 KBytes.
a=15; b=3; c=10: High quality to 64 KBytes.
a=15; b=4; c=10: High quality to 32 KBytes.
a=15; b=5; c=10: High quality to 16 KBytes.

* Note that, no matter the result size, whenever all three constants are even numbers the tests are instant off-the-chart failures for every test of the battery.

Whaaa! I think I've messed up somewhere. I've rebuilt my code and am now getting up to 512 MBytes! You might have to discard most of what I've written there, sorry.

Whaaa! I think I've messed up somewhere. I've rebuilt my code and am now getting up to 512 MBytes! You might have to discard most of what I've written there, sorry.

The even constants rule still applies though.

Crap, it's 5:10AM too!

I went to bed late, too, and now I feel lousy.

When I get into the office today, I'm hoping my 16-cog xoroshiro32+ tester has some full-length results. Then, let's test those. That's when we'll know for sure if we'very got something. Your initial tests showed some promise, anyway.

Cogs are starting to COGSTOP themselves on the xoroshiro32+ search program I started running last night. Maybe in an hour, I'll be able to see if we have any results - that is, $FFFF_FFFF iterations before returning to $0000_0001. If we get some good rotate/shift-value sets, I think Evanh will be able to test their actual quality.

It's looking like these values for t1, t2, and t3 produce maximal-length runs:

10,7,11
11,1,2
11,2,6
11,3,10
11,7,10

There's going to be a bunch more possibilities, but could you please test a few of these and discover if there is any qualitative difference among them?

After running that 16-cog xoroshiro32+ search program for 15 hours, I got this data out.

Note that for each 3-digit hex number, the 1st nibble is t1, the second nibble is t2, and the third nibble is t3. So, for the first data set, $05B, t1=0, t2=5, and t3=11.

I'm hoping Evanh will run a few of these cases through dieharder and find out if we have a quality solution here for a 32-bit state PRNG. I would only bother with sets that have all different digits and no 0's in them.

I've repaired my bug. I wasn't completely taking care of arbitrary length in the rotl() function. I'd suffered from it from day one I think.

#define ACCUM_MASK (((1UL<<RESULT_SIZE)-1)<<1|1) // One more bit than results.
static inline uint64_t rotl(uint64_t value, int shift) {
return (value << shift) | ((value & ACCUM_MASK) >> (RESULT_SIZE + 1 - shift)) & ACCUM_MASK;
}

Right, rerunning last night's effort, my best Xoroshiro32+ high quality run was 1 GByte with constants: 14,4,7. Next best was 14,3,7 with one very suspicious case at 1 GByte. Both of these failed at 2 GByte. Massive fix from last night.

Of the ones you've highlighted:
10,7,11 HQ to 256 MB. Fails at 512 MB.
11,1,2 HQ to 64 MB. Fails at 256 MB.
11,2,6 HQ to 256 MB. Fails at 512 MB.
11,3,10 HQ to 8 MB. Fails at 32 MB.
11,7,10 HQ to 1 MB. Fails at 4 MB.
5,2,6 HQ to 64 MB. Fails at 128 MB.

I've confirmed same result from using Heater's original rotl() but with uint16_t and also plugging in your above routine without any of my auto sizing tricks.

Attached is the 14,4,7 full test output. What's notable now is the solid and dramatic failure case at 2 GBytes. Whereas when I had the bug in my rotl() function, all the higher sized tests failed in the same single test case every time - DC6-9x1Bytes-1.

How many bits are you sampling at each iteration? I would think bits 15..1 of the addition would be the best.

When you say a test failed at 2GB, how many iterations was that? Ideally, we should be getting 4G iterations passing, right? All those patterns should repeat at 4G - 1.

Generator result size is 15 bits and I continuously concatenate those together to feed bytes to the Practrand test suit. So that's just over 1 GSamples at the 2 GByte point.

I'd post my full code now but I'm at work again ...

The author of Xoroshiro says not to expect any relationship between repeat length and quality. Other than, I guess, the quality has to collapse beyond the repeat length.

The author of Xoroshiro says not to expect any relationship between repeat length and quality. Other than, I guess, the quality has to collapse beyond the repeat length.

But what about random seeding? That effectively jumps you ahead to some unknown point, right?

Generator result size is 15 bits and I continuously concatenate those together to feed bytes to the Practrand test suit. So that's just over 1 GSamples at the 2 GByte point.

I'd post my full code now but I'm at work again ...

OK, sounds good. And those 15 bits are the top bits of the sum, right, excluding the LSB?

I wonder why we hit a wall after so many samples, instead of it just going on forever.

You mean multiple reseedings during a single test? That would invalidate the purpose of the test I'd think.

I meant that, in practice, a PRNG may be seeded, and it will iterate from that seed value. If we poop out at 512MB, what would happen to our quality if we seeded with a value from the fail point?

OK, sounds good. And those 15 bits are the top bits of the sum, right, excluding the LSB?

Yep. I right shift the s0 + s1 sum, then mask 0x7fff to be sure.

I wonder why we hit a wall after so many samples, instead of it just going on forever.

The random data stream does go forever. It's just the Practrand test suit is sensitive and self terminates when the accumulated stats go too far out. Makes for easy automation of test combinations.

I meant that, in practice, a PRNG may be seeded, and it will iterate from that seed value. If we poop out at 512MB, what would happen to our quality if we seeded with a value from the fail point?

As things are done at the moment, the test would start afresh and produce another 512 MBytes before failing again. The initial seed value shouldn't have any impact on quality.

The way I think about it is the repeat length doesn't change, same as if the total length of 0 -99 is 100 and, assuming 99 rolls over back to 0, no matter where you start, eg: at 50, it still takes 100 iterations to get back to the starting value of 50 again.

I meant that, in practice, a PRNG may be seeded, and it will iterate from that seed value. If we poop out at 512MB, what would happen to our quality if we seeded with a value from the fail point?

As things are done at the moment, the test would start afresh and produce another 512 MBytes before failing again. The initial seed value shouldn't have any impact on quality.

EDIT: Fixed my fail value.

Okay. Thanks for explaining that. I understand now.

I've confirmed same result from using Heater's original rotl() but with uint16_t and also plugging in your above routine without any of my auto sizing tricks.

Attached is the 14,4,7 full test output. What's notable now is the solid and dramatic failure case at 2 GBytes. Whereas when I had the bug in my rotl() function, all the higher sized tests failed in the same single test case every time - DC6-9x1Bytes-1.

I just realized something... 14,4,7 did not show up in my overnight 16-cog test as a setting which had a full-length period of $FFFF_FFFF iterations before returning to the original seed. That might explain the dramatic fail at 2GB. How did you decide to test that case?

I automated 125 test runs of the a,b,c combinations resulting in 125 Practrand result output files. The longer a particular test lasts the bigger the result file becomes. I examined only the biggest files and posted my favourite from them.

I've since widened the combinations to 500 or so but not got anything better.

## Comments

13,61013,610That's a simple LFSR. This would be the code for it in PASM:

11,046There is significant variations in the PractRand test results when differing constants are chosen. Here's a collect of Xoroshiro32+ runs I've just tried:

a=11; b=1; c=6: High quality to 256 KBytes.

a=11; b=2; c=6: High quality to 128 KBytes.

a=11; b=3; c=6: High quality to 64 KBytes.

a=11; b=4; c=6: High quality to 64 KBytes.

a=11; b=5; c=6: High quality to 64 KBytes.

a=11; b=1; c=7: High quality to 64 KBytes.

a=11; b=2; c=7: High quality to 128 KBytes.

a=11; b=3; c=7: High quality to 256 KBytes.

a=11; b=4; c=7: High quality to 128 KBytes.

a=11; b=5; c=7: High quality to 32 KBytes.

a=11; b=1; c=8: High quality to 128 KBytes.

a=11; b=2; c=8: High quality to 16 KBytes.

a=11; b=3; c=8: High quality to 32 KBytes.

a=11; b=4; c=8: High quality to 128 KBytes.

a=11; b=5; c=8: High quality to 64 KBytes.

a=11; b=1; c=9: High quality to 64 KBytes.

a=11; b=2; c=9: High quality to 64 KBytes.

a=11; b=3; c=9: High quality to 64 KBytes.

a=11; b=4; c=9: High quality to 128 KBytes.

a=11; b=5; c=9: High quality to 128 KBytes.

a=11; b=1; c=10: High quality to 128 KBytes.

a=11; b=2; c=10: High quality to 64 KBytes.

a=11; b=3; c=10: High quality to 64 KBytes.

a=11; b=4; c=10: High quality to 32 KBytes.

a=11; b=5; c=10: High quality to 256 KBytes.

a=12; b=1; c=6: High quality to 256 KBytes.

a=12; b=2; c=6: *Instant fail!!!

a=12; b=3; c=6: High quality to 128 KBytes.

a=12; b=4; c=6: *Instant fail!!!

a=12; b=5; c=6: High quality to 32 KBytes.

a=12; b=1; c=7: High quality to 256 KBytes.

a=12; b=2; c=7: High quality to 64 KBytes.

a=12; b=3; c=7: High quality to 256 KBytes.

a=12; b=4; c=7: High quality to 32 KBytes.

a=12; b=5; c=7: High quality to 64 KBytes.

a=12; b=1; c=8: High quality to 64 KBytes.

a=12; b=2; c=8: *Instant fail!!!

a=12; b=3; c=8: High quality to 32 KBytes.

a=12; b=4; c=8: *Instant fail!!!

a=12; b=5; c=8: High quality to 128 KBytes.

a=12; b=1; c=9: High quality to 128 KBytes.

a=12; b=2; c=9: High quality to 128 KBytes.

a=12; b=3; c=9: High quality to 64 KBytes.

a=12; b=4; c=9: High quality to 64 KBytes.

a=12; b=5; c=9: High quality to 64 KBytes.

a=12; b=1; c=10: High quality to 32 KBytes.

a=12; b=2; c=10: *Instant fail!!!

a=12; b=3; c=10: High quality to 256 KBytes.

a=12; b=4; c=10: *Instant fail!!!

a=12; b=5; c=10: High quality to 128 KBytes.

a=13; b=1; c=6: High quality to 128 KBytes.

a=13; b=2; c=6: High quality to 128 KBytes.

a=13; b=3; c=6: High quality to 64 KBytes.

a=13; b=4; c=6: High quality to 256 KBytes.

a=13; b=5; c=6: High quality to 64 KBytes.

a=13; b=1; c=7: High quality to 64 KBytes.

a=13; b=2; c=7: High quality to 256 KBytes.

a=13; b=3; c=7: High quality to 32 KBytes.

a=13; b=4; c=7: High quality to 512 KBytes. (Winner #1)

a=13; b=5; c=7: High quality to 64 KBytes.

a=13; b=1; c=8: High quality to 128 KBytes.

a=13; b=2; c=8: High quality to 64 KBytes.

a=13; b=3; c=8: High quality to 256 KBytes. (This one conforms to my earlier description)

a=13; b=4; c=8: High quality to 32 KBytes.

a=13; b=5; c=8: High quality to 8 KBytes.

a=13; b=1; c=9: High quality to 128 KBytes.

a=13; b=2; c=9: High quality to 64 KBytes.

a=13; b=3; c=9: High quality to 32 KBytes.

a=13; b=4; c=9: High quality to 128 KBytes.

a=13; b=5; c=9: High quality to 64 KBytes.

a=13; b=1; c=10: High quality to 64 KBytes.

a=13; b=2; c=10: High quality to 64 KBytes.

a=13; b=3; c=10: High quality to 8 KBytes.

a=13; b=4; c=10: High quality to 8 KBytes.

a=13; b=5; c=10: High quality to 128 KBytes.

a=14; b=1; c=6: High quality to 256 KBytes.

a=14; b=2; c=6: *Instant fail!!!

a=14; b=3; c=6: High quality to 256 KBytes.

a=14; b=4; c=6: *Instant fail!!!

a=14; b=5; c=6: High quality to 128 KBytes.

a=14; b=1; c=7: High quality to 128 KBytes.

a=14; b=2; c=7: High quality to 256 KBytes.

a=14; b=3; c=7: High quality to 256 KBytes.

a=14; b=4; c=7: High quality to 64 KBytes.

a=14; b=5; c=7: High quality to 512 KBytes. (Winner #2)

a=14; b=1; c=8: High quality to 64 KBytes.

a=14; b=2; c=8: *Instant fail!!!

a=14; b=3; c=8: High quality to 64 KBytes.

a=14; b=4; c=8: *Instant fail!!!

a=14; b=5; c=8: High quality to 32 KBytes.

a=14; b=1; c=9: High quality to 64 KBytes.

a=14; b=2; c=9: High quality to 128 KBytes.

a=14; b=3; c=9: High quality to 64 KBytes.

a=14; b=4; c=9: High quality to 128 KBytes. (This is your above constants)

a=14; b=5; c=9: High quality to 16 KBytes.

a=14; b=1; c=10: High quality to 128 KBytes.

a=14; b=2; c=10: *Instant fail!!!

a=14; b=3; c=10: High quality to 64 KBytes.

a=14; b=4; c=10: *Instant fail!!!

a=14; b=5; c=10: High quality to 32 KBytes.

a=15; b=1; c=6: High quality to 256 KBytes.

a=15; b=2; c=6: High quality to 512 KBytes. (Winner #3)

a=15; b=3; c=6: High quality to 128 KBytes.

a=15; b=4; c=6: High quality to 64 KBytes.

a=15; b=5; c=6: High quality to 512 KBytes. (Winner #4)

a=15; b=1; c=7: High quality to 128 KBytes.

a=15; b=2; c=7: High quality to 128 KBytes.

a=15; b=3; c=7: High quality to 128 KBytes.

a=15; b=4; c=7: High quality to 128 KBytes.

a=15; b=5; c=7: High quality to 64 KBytes.

a=15; b=1; c=8: High quality to 256 KBytes.

a=15; b=2; c=8: High quality to 64 KBytes.

a=15; b=3; c=8: High quality to 128 KBytes.

a=15; b=4; c=8: High quality to 64 KBytes.

a=15; b=5; c=8: High quality to 256 KBytes.

a=15; b=1; c=9: High quality to 256 KBytes.

a=15; b=2; c=9: High quality to 64 KBytes.

a=15; b=3; c=9: High quality to 64 KBytes.

a=15; b=4; c=9: High quality to 64 KBytes.

a=15; b=5; c=9: High quality to 64 KBytes.

a=15; b=1; c=10: High quality to 128 KBytes.

a=15; b=2; c=10: High quality to 128 KBytes.

a=15; b=3; c=10: High quality to 64 KBytes.

a=15; b=4; c=10: High quality to 32 KBytes.

a=15; b=5; c=10: High quality to 16 KBytes.

* Note that, no matter the result size, whenever all three constants are even numbers the tests are instant off-the-chart failures for every test of the battery.

11,046The even constants rule still applies though.

Crap, it's 5:10AM too!

13,610I went to bed late, too, and now I feel lousy.

When I get into the office today, I'm hoping my 16-cog xoroshiro32+ tester has some full-length results. Then, let's test those. That's when we'll know for sure if we'very got something. Your initial tests showed some promise, anyway.

13,61013,610Here is the algorithm:

It's looking like these values for t1, t2, and t3 produce maximal-length runs:

10,7,11

11,1,2

11,2,6

11,3,10

11,7,10

There's going to be a bunch more possibilities, but could you please test a few of these and discover if there is any qualitative difference among them?

13,610Note that for each 3-digit hex number, the 1st nibble is t1, the second nibble is t2, and the third nibble is t3. So, for the first data set, $05B, t1=0, t2=5, and t3=11.

I'm hoping Evanh will run a few of these cases through dieharder and find out if we have a quality solution here for a 32-bit state PRNG. I would only bother with sets that have all different digits and no 0's in them.

13,610Could you please test this one first:

t1 = 5

t2 = 2

t3 = 6

Thanks. I just did a separate test on that one and it looks really good to me. I wonder what dieharder would say.

11,04613,610Cool.

11,046I've repaired my bug. I wasn't completely taking care of arbitrary length in the rotl() function. I'd suffered from it from day one I think.

Right, rerunning last night's effort, my best Xoroshiro32+ high quality run was 1 GByte with constants: 14,4,7. Next best was 14,3,7 with one very suspicious case at 1 GByte. Both of these failed at 2 GByte. Massive fix from last night.

Of the ones you've highlighted:

10,7,11 HQ to 256 MB. Fails at 512 MB.

11,1,2 HQ to 64 MB. Fails at 256 MB.

11,2,6 HQ to 256 MB. Fails at 512 MB.

11,3,10 HQ to 8 MB. Fails at 32 MB.

11,7,10 HQ to 1 MB. Fails at 4 MB.

5,2,6 HQ to 64 MB. Fails at 128 MB.

11,04611,046Attached is the 14,4,7 full test output. What's notable now is the solid and dramatic failure case at 2 GBytes. Whereas when I had the bug in my rotl() function, all the higher sized tests failed in the same single test case every time - DC6-9x1Bytes-1.

13,610How many bits are you sampling at each iteration? I would think bits 15..1 of the addition would be the best.

When you say a test failed at 2GB, how many iterations was that? Ideally, we should be getting 4G iterations passing, right? All those patterns should repeat at 4G - 1.

11,046I'd post my full code now but I'm at work again ...

11,04613,610But what about random seeding? That effectively jumps you ahead to some unknown point, right?

13,610OK, sounds good. And those 15 bits are the top bits of the sum, right, excluding the LSB?

I wonder why we hit a wall after so many samples, instead of it just going on forever.

11,04613,610I meant that, in practice, a PRNG may be seeded, and it will iterate from that seed value. If we poop out at 512MB, what would happen to our quality if we seeded with a value from the fail point?

11,046The random data stream does go forever. It's just the Practrand test suit is sensitive and self terminates when the accumulated stats go too far out. Makes for easy automation of test combinations.

11,046As things are done at the moment, the test would start afresh and produce another 512 MBytes before failing again. The initial seed value shouldn't have any impact on quality.

EDIT: Fixed my fail value.

10,188This makes rational sense, but I can't seem to visualize it.

Random numbers are weird.

11,04610,18813,61013,610Okay. Thanks for explaining that. I understand now.

13,610I just realized something... 14,4,7 did not show up in my overnight 16-cog test as a setting which had a full-length period of $FFFF_FFFF iterations before returning to the original seed. That might explain the dramatic fail at 2GB. How did you decide to test that case?

My findings skipped over 14,4,7:

$E3D = 14,3,13

$E89 = 13,8,9

11,046I've since widened the combinations to 500 or so but not got anything better.