load tower proofs' `prod_specs_eval` and `logup_specs_eval` using `hint_load`

[kunxian-xia]

To read N evaluations in the tower proof from hint stream, we currently use a loop to move them ($4N$ felts) from hint stream to heap using ~ $4N$ StoreHintWord, $4N$ ADD and $4N$ BNE (from ZipFor).

This becomes a critical issue since the number $N = \sum_{i < T} L_i * (\textrm{n\_prods[i]} + \textrm{n\_logups}[i])$ is the product of the number of tower proofs $T$ and the number of layers $L$ in each tower.

[kunxian-xia]

circuit 0 r_len: 4, w: 4, lk: 9, n_prods: 8, n_logups: 9, n_layers: 18, n_prod_evals: 288, n_logup_evals: 648
circuit 1 r_len: 3, w: 3, lk: 7, n_prods: 6, n_logups: 7, n_layers: 22, n_prod_evals: 264, n_logup_evals: 616
circuit 2 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 16, n_prod_evals: 256, n_logup_evals: 704
circuit 3 r_len: 3, w: 3, lk: 9, n_prods: 6, n_logups: 9, n_layers: 17, n_prod_evals: 204, n_logup_evals: 612
circuit 4 r_len: 3, w: 3, lk: 13, n_prods: 6, n_logups: 13, n_layers: 17, n_prod_evals: 204, n_logup_evals: 884
circuit 5 r_len: 0, w: 0, lk: 1, n_prods: 0, n_logups: 1, n_layers: 15, n_prod_evals: 0, n_logup_evals: 60
circuit 6 r_len: 3, w: 3, lk: 5, n_prods: 6, n_logups: 5, n_layers: 18, n_prod_evals: 216, n_logup_evals: 360
circuit 7 r_len: 3, w: 3, lk: 8, n_prods: 6, n_logups: 8, n_layers: 11, n_prod_evals: 132, n_logup_evals: 352
circuit 8 r_len: 3, w: 3, lk: 8, n_prods: 6, n_logups: 8, n_layers: 17, n_prod_evals: 204, n_logup_evals: 544
circuit 9 r_len: 3, w: 3, lk: 8, n_prods: 6, n_logups: 8, n_layers: 14, n_prod_evals: 168, n_logup_evals: 448
circuit 10 r_len: 3, w: 3, lk: 8, n_prods: 6, n_logups: 8, n_layers: 20, n_prod_evals: 240, n_logup_evals: 640
circuit 11 r_len: 3, w: 3, lk: 5, n_prods: 6, n_logups: 5, n_layers: 20, n_prod_evals: 240, n_logup_evals: 400
circuit 12 r_len: 4, w: 4, lk: 18, n_prods: 8, n_logups: 18, n_layers: 5, n_prod_evals: 80, n_logup_evals: 360
circuit 13 r_len: 4, w: 4, lk: 16, n_prods: 8, n_logups: 16, n_layers: 6, n_prod_evals: 96, n_logup_evals: 384
circuit 14 r_len: 0, w: 0, lk: 1, n_prods: 0, n_logups: 1, n_layers: 15, n_prod_evals: 0, n_logup_evals: 60
circuit 15 r_len: 0, w: 0, lk: 1, n_prods: 0, n_logups: 1, n_layers: 18, n_prod_evals: 0, n_logup_evals: 72
circuit 21 r_len: 53, w: 53, lk: 1123, n_prods: 106, n_logups: 1123, n_layers: 17, n_prod_evals: 3604, n_logup_evals: 76364
circuit 25 r_len: 36, w: 36, lk: 921, n_prods: 72, n_logups: 921, n_layers: 13, n_prod_evals: 1872, n_logup_evals: 47892
circuit 27 r_len: 20, w: 20, lk: 625, n_prods: 40, n_logups: 625, n_layers: 5, n_prod_evals: 400, n_logup_evals: 12500
circuit 29 r_len: 19, w: 19, lk: 1073, n_prods: 38, n_logups: 1073, n_layers: 14, n_prod_evals: 1064, n_logup_evals: 60088
circuit 31 r_len: 0, w: 1, lk: 0, n_prods: 1, n_logups: 0, n_layers: 20, n_prod_evals: 40, n_logup_evals: 0
circuit 32 r_len: 0, w: 1, lk: 0, n_prods: 1, n_logups: 0, n_layers: 17, n_prod_evals: 34, n_logup_evals: 0
circuit 33 r_len: 2, w: 2, lk: 6, n_prods: 4, n_logups: 6, n_layers: 16, n_prod_evals: 128, n_logup_evals: 384
circuit 34 r_len: 3, w: 3, lk: 9, n_prods: 6, n_logups: 9, n_layers: 18, n_prod_evals: 216, n_logup_evals: 648
circuit 35 r_len: 4, w: 4, lk: 12, n_prods: 8, n_logups: 12, n_layers: 14, n_prod_evals: 224, n_logup_evals: 672
circuit 36 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 21, n_prod_evals: 336, n_logup_evals: 924
circuit 37 r_len: 4, w: 4, lk: 10, n_prods: 8, n_logups: 10, n_layers: 11, n_prod_evals: 176, n_logup_evals: 440
circuit 38 r_len: 4, w: 4, lk: 9, n_prods: 8, n_logups: 9, n_layers: 9, n_prod_evals: 144, n_logup_evals: 324
circuit 40 r_len: 3, w: 3, lk: 8, n_prods: 6, n_logups: 8, n_layers: 15, n_prod_evals: 180, n_logup_evals: 480
circuit 41 r_len: 4, w: 4, lk: 9, n_prods: 8, n_logups: 9, n_layers: 20, n_prod_evals: 320, n_logup_evals: 720
circuit 42 r_len: 1, w: 0, lk: 0, n_prods: 1, n_logups: 0, n_layers: 21, n_prod_evals: 42, n_logup_evals: 0
circuit 43 r_len: 0, w: 0, lk: 1, n_prods: 0, n_logups: 1, n_layers: 15, n_prod_evals: 0, n_logup_evals: 60
circuit 44 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 16, n_prod_evals: 256, n_logup_evals: 704
circuit 47 r_len: 4, w: 4, lk: 15, n_prods: 8, n_logups: 15, n_layers: 15, n_prod_evals: 240, n_logup_evals: 900
circuit 48 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 18, n_prod_evals: 288, n_logup_evals: 792
circuit 49 r_len: 3, w: 3, lk: 9, n_prods: 6, n_logups: 9, n_layers: 8, n_prod_evals: 96, n_logup_evals: 288
circuit 50 r_len: 0, w: 0, lk: 1, n_prods: 0, n_logups: 1, n_layers: 15, n_prod_evals: 0, n_logup_evals: 60
circuit 51 r_len: 0, w: 0, lk: 1, n_prods: 0, n_logups: 1, n_layers: 19, n_prod_evals: 0, n_logup_evals: 76
circuit 52 r_len: 0, w: 1, lk: 0, n_prods: 1, n_logups: 0, n_layers: 6, n_prod_evals: 12, n_logup_evals: 0
circuit 53 r_len: 0, w: 1, lk: 0, n_prods: 1, n_logups: 0, n_layers: 19, n_prod_evals: 38, n_logup_evals: 0
circuit 54 r_len: 0, w: 1, lk: 0, n_prods: 1, n_logups: 0, n_layers: 5, n_prod_evals: 10, n_logup_evals: 0
circuit 56 r_len: 4, w: 4, lk: 16, n_prods: 8, n_logups: 16, n_layers: 6, n_prod_evals: 96, n_logup_evals: 384
circuit 57 r_len: 4, w: 4, lk: 15, n_prods: 8, n_logups: 15, n_layers: 19, n_prod_evals: 304, n_logup_evals: 1140
circuit 58 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 12, n_prod_evals: 192, n_logup_evals: 528
circuit 59 r_len: 4, w: 4, lk: 14, n_prods: 8, n_logups: 14, n_layers: 15, n_prod_evals: 240, n_logup_evals: 840
circuit 60 r_len: 3, w: 3, lk: 12, n_prods: 6, n_logups: 12, n_layers: 18, n_prod_evals: 216, n_logup_evals: 864
circuit 61 r_len: 4, w: 4, lk: 10, n_prods: 8, n_logups: 10, n_layers: 8, n_prod_evals: 128, n_logup_evals: 320
circuit 62 r_len: 3, w: 3, lk: 8, n_prods: 6, n_logups: 8, n_layers: 10, n_prod_evals: 120, n_logup_evals: 320
circuit 63 r_len: 3, w: 3, lk: 8, n_prods: 6, n_logups: 8, n_layers: 14, n_prod_evals: 168, n_logup_evals: 448
circuit 64 r_len: 4, w: 4, lk: 10, n_prods: 8, n_logups: 10, n_layers: 16, n_prod_evals: 256, n_logup_evals: 640
circuit 66 r_len: 3, w: 3, lk: 13, n_prods: 6, n_logups: 13, n_layers: 11, n_prod_evals: 132, n_logup_evals: 572
circuit 67 r_len: 4, w: 4, lk: 14, n_prods: 8, n_logups: 14, n_layers: 15, n_prod_evals: 240, n_logup_evals: 840
circuit 68 r_len: 3, w: 3, lk: 12, n_prods: 6, n_logups: 12, n_layers: 17, n_prod_evals: 204, n_logup_evals: 816
circuit 69 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 18, n_prod_evals: 288, n_logup_evals: 792
circuit 70 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 20, n_prod_evals: 320, n_logup_evals: 880
circuit 71 r_len: 1, w: 1, lk: 0, n_prods: 2, n_logups: 0, n_layers: 17, n_prod_evals: 68, n_logup_evals: 0
circuit 72 r_len: 0, w: 1, lk: 0, n_prods: 1, n_logups: 0, n_layers: 12, n_prod_evals: 24, n_logup_evals: 0
circuit 73 r_len: 4, w: 4, lk: 11, n_prods: 8, n_logups: 11, n_layers: 19, n_prod_evals: 304, n_logup_evals: 836
circuit 74 r_len: 3, w: 3, lk: 9, n_prods: 6, n_logups: 9, n_layers: 14, n_prod_evals: 168, n_logup_evals: 504
circuit 75 r_len: 0, w: 0, lk: 1, n_prods: 0, n_logups: 1, n_layers: 15, n_prod_evals: 0, n_logup_evals: 60
circuit 76 r_len: 11, w: 11, lk: 120, n_prods: 22, n_logups: 120, n_layers: 5, n_prod_evals: 220, n_logup_evals: 2400

[kunxian-xia]

A typical case of $N = 16000 + 224644$, therefore we have to spend at least 2.88M cycles to read these evaluations from hint_stream (input_stream).