feat: use more efficient field merger code

[TomAFrench]

Description

Problem*

Resolves

Summary*

This PR replaces the value merger code reverted in #6673 but solely for fields.

Additional Context

Documentation*

Check one:

• No documentation needed.
• Documentation included in this PR.
• [For Experimental Features] Documentation to be submitted in a separate PR.

PR Checklist*

• I have tested the changes locally.
• I have formatted the changes with Prettier and/or cargo fmt on default settings.

[github-actions]

Peak Memory Sample

Program	Peak Memory	%
keccak256	78.48M	0%
workspace	123.78M	0%
regression_4709	422.91M	0%
ram_blowup_regression	1.58G	0%
rollup-base-public	11.27G	10%
rollup-base-private	9.01G	50%
private-kernel-tail	201.81M	0%
private-kernel-reset	739.74M	3%
private-kernel-inner	291.88M	0%
parity-root	172.14M	0%

[github-actions]

Compilation Report

Program	Compilation Time	%
sha256_regression	1.100s	-9%
regression_4709	0.794s	-7%
ram_blowup_regression	15.140s	-2%
rollup-root	3.760s	0%
rollup-block-merge	4.020s	14%
rollup-base-public	52.100s	66%
rollup-base-private	22.200s	88%
private-kernel-tail	0.975s	-6%
private-kernel-reset	14.260s	64%
private-kernel-inner	1.956s	-8%

[aakoshh]

Related to this I found that when we were casting to Field we saved a lot of range constraints implied on the binary operations performed on Unsigned types. First I thought that must be unsafe as it would mask a potential overflow, but with the c*a + !c*b formula below it seems safe (or actually it has an else_condition below, but assuming that they are mutually exclusive).

I assume you're not doing the c * (a-b) + b trick because of a potential overflow, but isn't it true that mathematically this should be okay, if the cost of the extra Casts to Field would be acceptable?

[TomAFrench]

Yeah, we're not concerned about overflows here as the merged value is constrained to then_value or else_value and any underflow in the difference term will disappear (This is why we needed to cast to Fields as the integer types imply checked arithmetic)

I did this just for fields in this PR as this doesn't need the inputs to be casted to fields and back (as it's already the correct type).

[TomAFrench]

I think we should wait for #6817 before pushing this further due to negligible changes within the test suite.

[github-actions]

Execution Report

Program	Execution Time	%
sha256_regression	0.054s	0%
regression_4709	0.001s	0%
ram_blowup_regression	0.574s	-1%
rollup-root	0.107s	1%
rollup-block-merge	0.107s	1%
rollup-base-public	1.610s	21%
rollup-base-private	0.613s	19%
private-kernel-tail	0.022s	0%
private-kernel-reset	0.484s	26%
private-kernel-inner	0.097s	0%

[github-actions]

Compilation Memory Report

Program	Peak Memory	%
keccak256	78.55M	0%
workspace	123.79M	0%
regression_4709	422.96M	0%
ram_blowup_regression	1.58G	0%
rollup-base-public	2.50G	0%
rollup-base-private	1.12G	0%
private-kernel-tail	200.40M	0%
private-kernel-reset	750.85M	3%
private-kernel-inner	290.46M	0%

[github-actions]

Execution Memory Report

Program	Peak Memory	%
keccak256	74.65M	0%
workspace	123.91M	0%
regression_4709	315.98M	0%
ram_blowup_regression	512.41M	0%
rollup-base-public	796.43M	61%
rollup-base-private	344.13M	3%
private-kernel-tail	181.08M	0%
private-kernel-reset	302.62M	18%
private-kernel-inner	212.78M	0%

[github-actions]

Changes to circuit sizes

Generated at commit: e0d9016217b2b5f0bdf757d499f244754ad0a2d4, compared to commit: 8d7f1f909c50887bca6ff1243d70622c2a3b3ec8

🧾 Summary (10% most significant diffs)

Program	ACIR opcodes (+/-)	%	Circuit size (+/-)	%
array_sort	+2 ❌	+3.77%	+4 ❌	+3.13%
regression_5252	-18 ✅	-0.05%	-276 ✅	-0.62%

---

Full diff report 👇

Program	ACIR opcodes (+/-)	%	Circuit size (+/-)	%
array_sort	55 (+2)	+3.77%	132 (+4)	+3.13%
hashmap	50,846 (+149)	+0.29%	114,992 (+161)	+0.14%
u128	500 (+4)	+0.81%	4,458 (+5)	+0.11%
binary_operator_overloading	125 (+2)	+1.63%	4,265 (+2)	+0.05%
regression_capacity_tracker	117 (+1)	+0.86%	3,972 (+1)	+0.03%
bench_eddsa_poseidon	16,441 (+2)	+0.01%	19,588 (+2)	+0.01%
nested_array_dynamic	3,358 (0)	0.00%	12,755 (+1)	+0.01%
references	7 (-1)	-12.50%	23 (0)	0.00%
slices	758 (+2)	+0.26%	3,901 (-5)	-0.13%
slice_dynamic_index	995 (-7)	-0.70%	6,385 (-20)	-0.31%
regression_5252	32,898 (-18)	-0.05%	44,243 (-276)	-0.62%