fix: better init of spergel HLR

[beckermr]

This PR has a new way to initialize the spergel HLR and a better way to compute autodiff using implicit differentiation.

[codspeed-hq]

Merging this PR will improve performance by ×2.2

⚠️ Unknown Walltime execution environment detected

Using the Walltime instrument on standard Hosted Runners will lead to inconsistent data.

For the most accurate results, we recommend using CodSpeed Macro Runners: bare-metal machines fine-tuned for performance measurement consistency.

⚠️ Different runtime environments detected

Some benchmarks with significant performance changes were compared across different runtime environments,
which may affect the accuracy of the results.

Open the report in CodSpeed to investigate

⚡ 4 improved benchmarks
❌ 3 regressed benchmarks
✅ 29 untouched benchmarks
🆕 2 new benchmarks

Warning

Please fix the performance issues or acknowledge them on CodSpeed.

Performance Changes

	Mode	Benchmark	BASE	HEAD	Efficiency
❌	WallTime	test_benchmark_spergel_init[run]	6.7 µs	58.9 µs	-88.68%
❌	Simulation	test_benchmark_spergel_init[run]	103.7 µs	345.1 µs	-69.95%
❌	WallTime	test_benchmark_moffat_init[run]	50.2 µs	81.2 µs	-38.24%
⚡	Simulation	test_benchmark_spergel_xvalue[run]	34,863.3 ms	313.4 ms	×110
⚡	WallTime	test_benchmark_spergel_xvalue[run]	1,510.6 ms	38.5 ms	×39
⚡	Simulation	test_benchmark_spergel_calcfluxrad[run]	1,091.3 µs	623.8 µs	+74.94%
⚡	WallTime	test_benchmark_spergel_calcfluxrad[run]	181.7 µs	120.6 µs	+50.7%
🆕	Simulation	test_benchmark_spergel_init_float[run]	N/A	104.6 µs	N/A
🆕	WallTime	test_benchmark_spergel_init_float[run]	N/A	7.8 µs	N/A

Tip

Investigate this regression by commenting @codspeedbot fix this regression on this PR, or directly use the CodSpeed MCP with your agent.

---

_{Comparing test-spergel (65db956) with main (040d8b2)}