Test and bench for eval_polynomial and compute_inner_product

[ashWhiteHat]

No description provided.

[ashWhiteHat]

It seems there are more efficient optimization in following comment.
#548 (comment)
I reverted my change so this is just bench and test for eval_polynomial and compute_inner_product.

[daira]

Quote-replying just to preserve the original comment, which does not describe the current PR contents.

Test and bench for eval_polynomial and compute_inner_product

Found more efficient solution and reverted Prev PR. This is just test and bench for eval_polynomial and compute_inner_product

Prev PR

# Parallelize `compute_inner_product`

Hi there.
I introduced parallelize to `compute_inner_product`.

## What I did

### Introduce parallelize

Introduce parallelize using rayon `join` and recursive function.

### Test

Write the test which checks `compute_inner_product` works fine.

### Bench

Add `compute_inner_product` bench which checks performance.
I would like to check exact performance so I set the `profile.bench` and closer to production environment.

### Other

I told that `all recursive` is the best option but it seems wrong.
The `rayon::join` uses the [`work-stealing`](https://en.wikipedia.org/wiki/Work_stealing#:~:text=In%20parallel%20computing%2C%20work%20stealing,of%20processors%20(or%20cores).) and the efficiency is related to `task size` and `multiplicity`.
For small task size, the shallower multiplicity, the faster performance.
For larger task size, the deeper multiplicity, the faster performance.

The `mul` operation on Field is small task so I bundle some tasks to out at a higher n as following.
https://github.com/NoCtrlZ/halo2/blob/feat/coeffs-arithmetic-optimization/halo2_proofs/src/arithmetic.rs#L287

## Bench

The optimized `compute_inner_product` us faster for almost all k except `12` and `13`.
I benched it locally with thread num = 8 and following is the result.

| k | default | recursive |
| ---- | ---- | ---- |
|3| 91.496 us 92.714 us 94.384 us | 88.904 us 89.356 us 89.869 us |
|4| 182.11 us 184.99 us 188.17 us | 176.85 us 177.94 us 179.15 us |
|5| 365.83 us 372.23 us 379.90 us | 351.58 us 352.67 us 353.86 us |
|6| 706.55 us 712.62 us 719.54 us | 701.19 us 702.64 us 704.16 us |
|7| 1.4597 ms 1.4862 ms 1.5121 ms | 1.4050 ms 1.4095 ms 1.4144 ms |
|8| 2.9451 ms 3.0102 ms 3.1018 ms | 2.8082 ms 2.8175 ms 2.8274 ms |
|9| 5.7104 ms 5.7810 ms 5.8706 ms | 5.7026 ms 5.7158 ms 5.7300 ms |
|10| 11.682 ms 11.904 ms 12.171 ms | 11.401 ms 11.439 ms 11.484 ms |
|11| 22.588 ms 22.683 ms 22.778 ms | 22.631 ms 22.682 ms 22.737 ms |
|12| 44.691 ms 44.839 ms 44.992 ms | 45.028 ms 45.111 ms 45.196 ms |
|13| 90.120 ms 90.559 ms 91.058 ms | 89.995 ms 90.650 ms 91.797 ms |
|14| 182.29 ms 186.16 ms 191.74 ms | 179.63 ms 179.96 ms 180.33 ms |
|15| 381.50 ms 387.02 ms 393.56 ms | 359.97 ms 360.73 ms 361.59 ms | 
|16| 761.80 ms 769.39 ms 777.99 ms | 718.81 ms 720.44 ms 722.32 ms |
|17| 1.5566 s 1.5844 s 1.6162 s | 1.4663 s 1.4800 s 1.4959 s |
|18| 3.0584 s 3.0874 s 3.1217 s | 2.9104 s 2.9301 s 2.9554 s |

## Uncertainty

I think `best_fft` perform more complicated operation than `compute_inner_product` but `best_fft` is far [faster](https://github.com/zcash/halo2/pull/543#issuecomment-1104646161) even same k.

I would appreciate it if you confirm. Thank you!

[daira]

Please force-push this PR so that it only touches the test and benchmarking code.

[ashWhiteHat]

Hi @daira
I fixed the PR according to your review.
I would appreciate it if you would confirm.
Thank you!

[therealyingtong]

utACK