msieve poly select: choosing Stage1norm

[VBCurtis]

Premise: Stage1Norm (the part computed by the GPU) should be chosen to maximize the number of quality stage 2 hits (the -nps part) per unit time.

I ran 11 days' worth of GPU poly select on the c195 from Aliquot sequence 4788 mentioned in the poly request thread. Each day, I changed the Stage1 norm, recording the number of stage 2 hits produced better than an arbitrarily selected bound expected to be useful for root-opt and polynomial generation.

Conclusion: Best choice of Stage 1 bound was 4 to 4.5 e28, where default is 3e29. This produced quality hits nearly 3 times more rapidly than default stage1 norm.

The data: First column is stage1norm selected. Second is time for the run, third is number of -nps hits better than 2e26, fourth is number of hits better than 1.5e26. I originally recorded hits better than 2.5e26, but they became so numerous I changed the stage2norm to 2e26.

stage1 time 2e26 1.5e26
3e29  24.8 hr  40  19
2e29  24.2 hr  48  23
1e29  12.5 hr  43  15
7e28  31.0 hr  127 51
5e28  24.6 hr  113 41
4.5e28 23.4 hr 124 65
4.5e28 14.8 hr 59  25
4e28  18.2 hr  85  41
3.5e28 23.8 hr 117 39
3e28  20.2 hr  80  31

The last two columns provide a sense of how sensitive the results may be to small changes in desired stage2norm. stage1 norm 4.5e28 was so good the first time that I gave it a second run, and will do yet another run comparing 4.5e28 to 4e28 when I next have GPU time.
On a per-hour basis, note the default norm produces less than 1 hit per hour (last column, 1.5e26) while 4.5e28 produces over 2 hits per hour. For a looser requirement 2e26 for stage 2, 4e28 and 4.5e28 produce triple the hits of the default stage1 norm.
I did not repeat the same coeffs, but the ranges were done sequentially (running from A1 = 500k to 1.2M). Further testing alternative 4e28 and 4.5e28 should reduce the chances of lucky hits distorting results, but a plot of hits per hour vs stage1 bound shows consistently improved production as stage1 is lowered from default toward 5e28.

I plan to repeat this experiment on the next candidate in the 170s I run multiple days on. If anyone else wish to collect such data, please post it! We've been making norm choices based on guesswork, but empirical data can improve our choices and hopefully find better polys.