Improvements eval/interp at geometric progression for unequal parameters

[vneiger]

This improves evaluation / interpolation at geometric progression in case of parameters that are not all essentially identical. This also resorts more directly to the fft_small middle product when possible.

For the moment, evaluation is done, and speedup is when one of these occurs:

• the number of evaluation points is less than the length used for the precomputation data
• the degree of the evaluated polynomial is less than the number of evaluation points
• the valuation of the evaluated polynomial is not zero (which should not happen too often, but maybe some specific usages have this)

[vneiger]

Example timings, when precomputation has been done on 10 * (nb evaluation points). The speed-up factor is almost 10. The interesting column is the one before last, which measures the time for evaluation excluding precomputations.

BEFORE:

==== nb eval points == 1 * (poly length) ====
len     points |       GEOMETRIC PROGRESSION
len     points |iter    fast    w/ prec precomp
1             1|6.1e-03 1.5e-01 5.6e-02 7.4e-01
2             2|1.3e-02 1.8e-01 1.3e-01 1.4e+00
3             3|2.4e-02 2.4e-01 3.4e-01 2.1e+00
4             4|4.2e-02 3.0e-01 4.8e-01 2.6e+00
6             6|9.9e-02 4.3e-01 8.0e-01 3.9e+00
8             8|2.0e-01 5.7e-01 1.2e+00 5.3e+00
10           10|3.4e-01 6.9e-01 1.6e+00 6.5e+00
12           12|5.1e-01 8.9e-01 3.4e+00 7.8e+00
16           16|9.8e-01 1.4e+00 6.1e+00 1.1e+01
20           20|1.6e+00 1.7e+00 1.5e+01 1.3e+01
30           30|3.9e+00 2.7e+00 2.6e+01 2.0e+01
45           45|8.9e+00 4.6e+00 5.2e+01 2.9e+01
70           70|2.2e+01 8.5e+00 9.5e+01 4.3e+01
100         100|4.6e+01 1.4e+01 1.7e+02 6.6e+01
200         200|1.9e+02 2.4e+01 1.1e+02 1.3e+02
400         400|7.3e+02 4.2e+01 2.1e+02 2.5e+02
800         800|2.9e+03 8.5e+01 4.4e+02 4.8e+02
1600       1600|0.0e+00 1.8e+02 9.4e+02 9.6e+02
3200       3200|0.0e+00 3.7e+02 2.0e+03 1.9e+03
6400       6400|0.0e+00 7.4e+02 4.2e+03 3.8e+03
12800     12800|0.0e+00 1.5e+03 9.1e+03 7.6e+03

AFTER:

==== nbits = 62====
==== nb eval points == 1 * (poly length) ====
len     points |       GEOMETRIC PROGRESSION
len     points |iter    fast    w/ prec precomp
1             1|6.2e-03 1.5e-01 3.2e-02 7.4e-01
2             2|1.3e-02 1.8e-01 4.5e-02 1.4e+00
3             3|2.4e-02 2.5e-01 6.0e-02 2.1e+00
4             4|4.1e-02 2.9e-01 7.6e-02 2.7e+00
6             6|9.8e-02 4.2e-01 1.1e-01 4.0e+00
8             8|2.0e-01 5.5e-01 1.5e-01 5.5e+00
10           10|3.4e-01 6.8e-01 1.9e-01 6.4e+00
12           12|5.1e-01 8.9e-01 3.1e-01 7.6e+00
16           16|1.0e+00 1.3e+00 5.6e-01 1.0e+01
20           20|1.6e+00 1.7e+00 7.1e-01 1.3e+01
30           30|3.8e+00 2.7e+00 1.3e+00 1.9e+01
45           45|9.1e+00 4.6e+00 2.5e+00 2.8e+01
70           70|2.2e+01 8.5e+00 5.3e+00 4.3e+01
100         100|4.6e+01 1.5e+01 9.7e+00 6.2e+01
200         200|1.9e+02 2.4e+01 1.5e+01 1.3e+02
400         400|7.5e+02 4.3e+01 2.5e+01 2.5e+02
800         800|2.9e+03 8.5e+01 5.1e+01 5.0e+02
1600       1600|0.0e+00 1.9e+02 1.1e+02 1.0e+03
3200       3200|0.0e+00 3.8e+02 2.2e+02 2.0e+03
6400       6400|0.0e+00 7.5e+02 4.5e+02 3.9e+03
12800     12800|0.0e+00 1.5e+03 9.7e+02 7.8e+03

[vneiger]

Example timings, when the number of evaluations points is larger than the polynomial degree. This is the time for evaluation, excluding precomputation (which is the same before and after).

len    | 2*len points   |  3*len points    |  4*len points
len    |BEFORE  AFTER   |  BEFORE  AFTER   | BEFORE  AFTER
1      |3.9e-02 3.8e-02 |  4.4e-02 4.2e-02 | 4.9e-02 4.4e-02
2      |6.2e-02 5.4e-02 |  7.7e-02 6.2e-02 | 8.8e-02 6.8e-02
3      |1.0e-01 8.2e-02 |  1.4e-01 1.0e-01 | 1.8e-01 1.2e-01
4      |1.4e-01 1.1e-01 |  1.9e-01 1.4e-01 | 2.4e-01 1.6e-01
6      |2.1e-01 1.6e-01 |  3.0e-01 2.1e-01 | 3.8e-01 2.6e-01
8      |2.9e-01 2.2e-01 |  4.2e-01 2.9e-01 | 5.5e-01 3.5e-01
10     |3.8e-01 2.9e-01 |  5.6e-01 3.8e-01 | 7.4e-01 4.7e-01
12     |7.0e-01 5.0e-01 |  1.1e+00 7.1e-01 | 1.4e+00 9.6e-01
16     |1.2e+00 8.8e-01 |  1.9e+00 1.2e+00 | 2.6e+00 1.6e+00
20     |1.6e+00 1.1e+00 |  2.4e+00 1.6e+00 | 3.2e+00 2.0e+00
30     |3.0e+00 2.2e+00 |  4.5e+00 3.0e+00 | 6.2e+00 3.8e+00
45     |5.8e+00 4.2e+00 |  8.8e+00 5.8e+00 | 1.2e+01 7.3e+00
70     |1.2e+01 8.8e+00 |  2.9e+01 1.2e+01 | 3.9e+01 1.6e+01
100    |3.3e+01 1.6e+01 |  5.0e+01 3.3e+01 | 6.7e+01 4.1e+01
200    |2.1e+01 1.9e+01 |  3.4e+01 2.1e+01 | 4.2e+01 2.8e+01
400    |4.2e+01 3.6e+01 |  6.3e+01 4.3e+01 | 8.4e+01 5.7e+01
800    |8.7e+01 7.1e+01 |  1.3e+02 8.7e+01 | 1.7e+02 1.1e+02
1600   |1.8e+02 1.5e+02 |  2.7e+02 1.8e+02 | 3.5e+02 2.3e+02
3200   |3.8e+02 3.0e+02 |  5.6e+02 3.8e+02 | 7.5e+02 4.9e+02
6400   |8.1e+02 6.3e+02 |  1.2e+03 8.1e+02 | 1.6e+03 1.1e+03
12800  |1.7e+03 1.4e+03 |  2.6e+03 1.7e+03 | 3.3e+03 2.2e+03

[vneiger]

And finally, comparison of the previous implementation using mullow versus the use of fft_small midprod directly. This is a bad idea for small lengths, so I inserted some branching condition that seems to be fairly accurate at picking the fastest variant. This is the version present in the commits to be pushed in a minute.

len   |   npoints == len       |    npoints == 2*len     |    npoints == 3*len
len   |mullow  midprod branch  | mullow  midprod branch  | mullow  midprod branch
1     |3.3e-02 3.0e+00 3.3e-02 | 3.8e-02 3.0e+00 3.9e-02 | 4.3e-02 3.0e+00 4.3e-02
2     |4.5e-02 3.1e+00 4.5e-02 | 5.4e-02 3.0e+00 5.5e-02 | 6.2e-02 3.0e+00 6.4e-02
3     |6.0e-02 2.9e+00 6.0e-02 | 8.1e-02 3.0e+00 8.3e-02 | 1.0e-01 3.0e+00 1.2e-01
4     |7.6e-02 2.9e+00 7.7e-02 | 1.1e-01 3.0e+00 1.1e-01 | 1.3e-01 3.1e+00 1.4e-01
6     |1.1e-01 3.0e+00 1.1e-01 | 1.6e-01 3.1e+00 1.6e-01 | 2.1e-01 3.2e+00 2.1e-01
8     |1.5e-01 3.0e+00 1.5e-01 | 2.2e-01 3.2e+00 2.3e-01 | 2.9e-01 3.3e+00 3.0e-01
10    |1.9e-01 3.0e+00 2.0e-01 | 2.9e-01 3.2e+00 2.9e-01 | 3.8e-01 3.4e+00 3.9e-01
12    |3.2e-01 3.1e+00 3.2e-01 | 5.0e-01 3.3e+00 5.1e-01 | 6.8e-01 3.6e+00 7.1e-01
16    |5.5e-01 3.2e+00 5.6e-01 | 8.7e-01 3.4e+00 8.9e-01 | 1.2e+00 3.6e+00 1.3e+00
20    |7.2e-01 3.4e+00 7.2e-01 | 1.1e+00 3.7e+00 1.2e+00 | 1.6e+00 3.8e+00 1.6e+00
30    |1.4e+00 3.6e+00 1.4e+00 | 2.2e+00 4.0e+00 2.3e+00 | 3.0e+00 4.3e+00 3.1e+00
45    |2.6e+00 3.9e+00 2.6e+00 | 4.2e+00 4.5e+00 4.2e+00 | 5.8e+00 5.1e+00 5.1e+00
70    |5.4e+00 4.5e+00 4.6e+00 | 8.8e+00 8.3e+00 8.5e+00 | 1.2e+01 8.2e+00 8.2e+00
100   |9.7e+00 8.0e+00 8.1e+00 | 1.6e+01 8.4e+00 8.7e+00 | 3.4e+01 9.6e+00 9.7e+00
200   |1.5e+01 1.3e+01 1.3e+01 | 1.9e+01 1.9e+01 1.8e+01 | 2.1e+01 2.0e+01 1.9e+01
400   |2.6e+01 2.3e+01 2.3e+01 | 3.6e+01 3.4e+01 3.4e+01 | 4.3e+01 4.1e+01 4.0e+01
800   |5.3e+01 4.7e+01 4.8e+01 | 7.1e+01 6.4e+01 6.6e+01 | 8.9e+01 8.0e+01 8.1e+01
1600  |1.1e+02 9.7e+01 1.0e+02 | 1.5e+02 1.3e+02 1.4e+02 | 1.8e+02 1.7e+02 1.7e+02
3200  |2.3e+02 2.0e+02 2.1e+02 | 3.0e+02 2.8e+02 2.8e+02 | 3.9e+02 3.5e+02 3.5e+02
6400  |4.7e+02 4.3e+02 4.2e+02 | 6.3e+02 5.8e+02 5.8e+02 | 8.1e+02 7.3e+02 7.3e+02
12800 |1.0e+03 8.7e+02 8.7e+02 | 1.4e+03 1.2e+03 1.2e+03 | 1.7e+03 1.6e+03 1.6e+03

[vneiger]

And, to conclude, the improvement compared to what is in main, when the precomputation is the same as len (otherwise, speed-up is greater).

len   |  npoints == len | npts == 2*len   | npts == 3*len
len   | old     new     | old     new     | old     new
1     | 3.8e-02 3.3e-02 | 4.3e-02 3.9e-02 | 4.9e-02 4.3e-02
2     | 5.2e-02 4.5e-02 | 6.8e-02 5.5e-02 | 7.8e-02 6.4e-02
3     | 6.8e-02 6.0e-02 | 1.1e-01 8.3e-02 | 1.5e-01 1.2e-01
4     | 8.6e-02 7.7e-02 | 1.4e-01 1.1e-01 | 2.0e-01 1.4e-01
6     | 1.2e-01 1.1e-01 | 2.2e-01 1.6e-01 | 3.0e-01 2.1e-01
8     | 1.7e-01 1.5e-01 | 3.0e-01 2.3e-01 | 4.3e-01 3.0e-01
10    | 2.1e-01 2.0e-01 | 3.8e-01 2.9e-01 | 5.6e-01 3.9e-01
12    | 3.5e-01 3.2e-01 | 7.4e-01 5.1e-01 | 1.1e+00 7.1e-01
16    | 5.7e-01 5.6e-01 | 1.2e+00 8.9e-01 | 1.9e+00 1.3e+00
20    | 7.4e-01 7.2e-01 | 1.6e+00 1.2e+00 | 2.4e+00 1.6e+00
30    | 1.4e+00 1.4e+00 | 3.0e+00 2.3e+00 | 4.6e+00 3.1e+00
45    | 2.6e+00 2.6e+00 | 5.9e+00 4.2e+00 | 8.9e+00 5.1e+00
70    | 5.4e+00 4.6e+00 | 1.2e+01 8.5e+00 | 3.0e+01 8.2e+00
100   | 9.8e+00 8.1e+00 | 3.3e+01 8.7e+00 | 4.9e+01 9.7e+00
200   | 1.5e+01 1.3e+01 | 2.1e+01 1.8e+01 | 3.3e+01 1.9e+01
400   | 2.7e+01 2.3e+01 | 4.2e+01 3.4e+01 | 6.4e+01 4.0e+01
800   | 5.4e+01 4.8e+01 | 8.8e+01 6.6e+01 | 1.3e+02 8.1e+01
1600  | 1.1e+02 1.0e+02 | 1.8e+02 1.4e+02 | 2.7e+02 1.7e+02
3200  | 2.4e+02 2.1e+02 | 3.8e+02 2.8e+02 | 5.7e+02 3.5e+02
6400  | 4.7e+02 4.2e+02 | 8.1e+02 5.8e+02 | 1.2e+03 7.3e+02
12800 | 1.0e+03 8.7e+02 | 1.7e+03 1.2e+03 | 2.6e+03 1.6e+03

[albinahlback]

Nice!!