Compare air_public_inputs against python vm + Fix how public memory is built

[fmoletta]

Fixes various bugs related to Public Memory
Adds small modifications to make air public input files equivalent to the ones outputted by cairo-lang
Adds targets to output and compare air_public_input files across vms
Missing in this PR: Adding air_public_input file comparison to CI (This will be added in a later PR due to the complicated nature of the current workflows)

Checklist

• Linked to Github Issue
• Unit tests added
• Integration tests added.
• This change requires new documentation.
  • Documentation has been added/updated.
  • CHANGELOG has been updated.

[codecov]

Codecov Report

Merging #1391 (e860e0e) into main (2aed1ba) will increase coverage by 0.00%.
The diff coverage is 100.00%.

@@           Coverage Diff           @@
##             main    #1391   +/-   ##
=======================================
  Coverage   96.96%   96.96%           
=======================================
  Files          94       94           
  Lines       38645    38662   +17     
=======================================
+ Hits        37473    37490   +17     
  Misses       1172     1172           

Files Changed	Coverage Δ
vm/src/air_public_input.rs	96.92% <100.00%> (ø)
vm/src/vm/runners/cairo_runner.rs	98.04% <100.00%> (+<0.01%)	⬆️
vm/src/vm/vm_core.rs	97.61% <100.00%> (+<0.01%)	⬆️
vm/src/vm/vm_memory/memory_segments.rs	99.02% <100.00%> (+0.01%)	⬆️

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[github-actions]

Benchmark Results for unmodified programs 🚀

Command	Mean [s]	Min [s]	Max [s]	Relative
base big_factorial	4.060 ± 0.011	4.045	4.080	1.00 ± 0.00
head big_factorial	4.040 ± 0.010	4.026	4.056	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base big_fibonacci	3.421 ± 0.005	3.416	3.428	1.01 ± 0.00
head big_fibonacci	3.401 ± 0.008	3.393	3.416	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base blake2s_integration_benchmark	9.601 ± 0.054	9.554	9.703	1.01 ± 0.01
head blake2s_integration_benchmark	9.520 ± 0.043	9.499	9.640	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base compare_arrays_200000	3.527 ± 0.057	3.498	3.689	1.01 ± 0.02
head compare_arrays_200000	3.491 ± 0.015	3.478	3.517	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base dict_integration_benchmark	2.334 ± 0.006	2.326	2.344	1.01 ± 0.00
head dict_integration_benchmark	2.318 ± 0.005	2.312	2.324	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base field_arithmetic_get_square_benchmark	1.718 ± 0.004	1.711	1.723	1.00 ± 0.01
head field_arithmetic_get_square_benchmark	1.713 ± 0.011	1.704	1.737	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base integration_builtins	9.425 ± 0.018	9.393	9.455	1.00 ± 0.00
head integration_builtins	9.415 ± 0.030	9.389	9.481	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base keccak_integration_benchmark	9.822 ± 0.288	9.716	10.640	1.02 ± 0.03
head keccak_integration_benchmark	9.663 ± 0.008	9.653	9.677	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base linear_search	3.505 ± 0.009	3.493	3.522	1.01 ± 0.00
head linear_search	3.483 ± 0.008	3.472	3.493	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base math_cmp_and_pow_integration_benchmark	2.579 ± 0.056	2.553	2.736	1.01 ± 0.02
head math_cmp_and_pow_integration_benchmark	2.551 ± 0.003	2.547	2.556	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base math_integration_benchmark	2.291 ± 0.005	2.285	2.302	1.01 ± 0.00
head math_integration_benchmark	2.276 ± 0.006	2.270	2.286	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base memory_integration_benchmark	2.034 ± 0.010	2.025	2.060	1.01 ± 0.01
head memory_integration_benchmark	2.016 ± 0.004	2.011	2.023	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base operations_with_data_structures_benchmarks	2.241 ± 0.004	2.236	2.251	1.01 ± 0.00
head operations_with_data_structures_benchmarks	2.227 ± 0.010	2.216	2.250	1.00

Command	Mean [ms]	Min [ms]	Max [ms]	Relative
base pedersen	798.2 ± 1.5	796.4	800.7	1.01 ± 0.00
head pedersen	792.8 ± 1.5	790.6	794.9	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base poseidon_integration_benchmark	1.489 ± 0.005	1.479	1.499	1.01 ± 0.00
head poseidon_integration_benchmark	1.471 ± 0.002	1.468	1.476	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base secp_integration_benchmark	2.828 ± 0.005	2.821	2.837	1.00 ± 0.00
head secp_integration_benchmark	2.825 ± 0.005	2.818	2.834	1.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base set_integration_benchmark	1.622 ± 0.004	1.618	1.630	1.00
head set_integration_benchmark	1.697 ± 0.004	1.692	1.705	1.05 ± 0.00

Command	Mean [s]	Min [s]	Max [s]	Relative
base uint256_integration_benchmark	6.312 ± 0.130	6.251	6.680	1.01 ± 0.02
head uint256_integration_benchmark	6.248 ± 0.016	6.235	6.288	1.00

[pefontana]

Nice one @fmoletta !
Can we add the *.air_public_input to the .gitignore file?