Hackathon Team Lambda Optimizations

[mowdl]

Description

This PR optimizes the automatic differentiation tape system in three key ways:

1. Pre-allocates memory for expressions to reduce allocation overhead
2. Combines multiplier and slot data into a single container for better cache performance
3. Simplifies derivative calculations by using direct multiplication instead of FMA

The changes significantly improve memory usage and computational speed while maintaining correctness.

Testing shows performance improvements with no impact on accuracy.

Type of change

• Performance improvement

[CLAassistant]

All committers have signed the CLA.

[github-actions]

Test Results

    29 files      29 suites   14m 30s ⏱️
 1 320 tests  1 320 ✅ 0 💤 0 ❌
48 395 runs  48 395 ✅ 0 💤 0 ❌

Results for commit a6d1312.

♻️ This comment has been updated with latest results.

[coveralls]

Pull Request Test Coverage Report for Build 11864682645

Details

• 38 of 38 (100.0%) changed or added relevant lines in 4 files are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage increased (+0.04%) to 98.767%

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Totals
Change from base Build 11864608732:	0.04%
Covered Lines:	2403
Relevant Lines:	2433

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💛 - Coveralls

[auto-differentiation-dev]

Results:

App	Reference	Optimised	Improvement
Monte-Carlo Swaption Pricer (App 1)	13.061s	6.814s	47.829%
Bermudan Swaption Pricer (App 2)	36.46156ms	21.606425ms	40.742%
QuantLib Overhead (App 3)	4000769ms	3052805ms	23.695%

[auto-differentiation-dev]

Closing this PR as key changes have been incorporated into #150, which consolidates contributions from multiple submissions. Thank you all for your efforts!