# MathCompile Development Roadmap
## Project Overview
MathCompile is a high-performance mathematical expression compiler that transforms symbolic expressions into optimized machine code. The project combines symbolic computation, automatic differentiation, and just-in-time compilation to achieve maximum performance for mathematical computations.
## Current Status: Phase 3 - Advanced Optimization (100% Complete)
**Last Updated**: May 29, 2025
### ✅ Completed Features
#### Phase 1: Core Infrastructure (100% Complete)
- ✅ **Expression AST**: Complete algebraic data type for mathematical expressions
- ✅ **Basic Operations**: Addition, subtraction, multiplication, division, power
- ✅ **Transcendental Functions**: sin, cos, ln, exp, sqrt with optimized implementations
- ✅ **Variable Management**: Support for named variables and indexed variables
- ✅ **Type Safety**: Generic type system with f64 specialization
#### Phase 2: Compilation Pipeline (100% Complete)
- ✅ **Cranelift Backend**: High-performance JIT compilation to native machine code
- ✅ **Rust Code Generation**: Alternative backend for debugging and cross-compilation
- ✅ **Memory Management**: Efficient variable allocation and stack management
- ✅ **Function Compilation**: Complete pipeline from AST to executable functions
- ✅ **Performance Optimization**: Register allocation and instruction optimization
#### Phase 3: Advanced Optimization (100% Complete)
- ✅ **Symbolic Optimization**: Comprehensive algebraic simplification engine
- ✅ **Automatic Differentiation**: Forward and reverse mode AD with optimization
- ✅ **Egglog Integration**: Equality saturation for advanced symbolic optimization
- ✅ **Egglog Extraction**: Hybrid extraction system combining egglog equality saturation with pattern-based optimization
- ✅ **Advanced Summation Engine**: Multi-dimensional summations with separability analysis
- ✅ **Convergence Analysis**: Infinite series convergence testing with ratio, root, and comparison tests
- ✅ **Pattern Recognition**: Arithmetic, geometric, power, and telescoping series detection
- ✅ **Closed-Form Evaluation**: Automatic conversion to closed-form expressions where possible
- ✅ **Variable System Refactoring**: Replaced global registry with per-function ExpressionBuilder approach for improved thread safety and isolation
- ✅ **Domain Analysis & Abstract Interpretation**: Complete domain-aware symbolic optimization ensuring mathematical transformations are only applied when valid
- ✅ **A-Normal Form (ANF)**: Intermediate representation with scope-aware common subexpression elimination
### 🔄 Recently Completed (Phase 3 Final Features)
#### Domain Analysis & Abstract Interpretation ✅
**Completed**: May 29, 2025
- **✅ Abstract Domain System**: Complete lattice-based domain representation (Bottom, Top, Positive, NonNegative, Negative, NonPositive, Interval, Union, Constant)
- **✅ Domain Analyzer**: Abstract interpretation engine that tracks mathematical domains through expression trees
- **✅ Transformation Validator**: Safety checker ensuring transformations like `exp(ln(x)) = x` are only applied when `x > 0`
- **✅ Symbolic Integration**: Domain validation integrated into `SymbolicOptimizer` for safe transformations
- **✅ Comprehensive Domain Arithmetic**: Full support for domain operations (join, meet, containment checks)
- **✅ Expression Domain Caching**: Performance optimization with cached domain computations
- **✅ Conservative Analysis**: Safe approximations when exact domain analysis is complex
- **✅ Property-Based Testing**: Comprehensive test coverage for all domain operations and transformation rules
- **✅ Working Demo**: Complete example demonstrating domain analysis capabilities
**Technical Architecture**:
```rust
pub enum AbstractDomain {
Bottom, Top, Positive, NonNegative, Negative, NonPositive,
Interval(f64, f64), Union(Vec<AbstractDomain>), Constant(f64),
}
pub struct DomainAnalyzer {
variable_domains: HashMap<usize, AbstractDomain>,
expression_cache: HashMap<String, AbstractDomain>,
}
pub struct TransformationValidator {
analyzer: DomainAnalyzer,
}
```
**Key Capabilities**:
- **Domain Tracking**: Automatically computes domains for complex expressions like `ln(x + y)` where `x > 0, y >= 0`
- **Safe Transformations**: Validates that `exp(ln(x)) = x` only when `x > 0`, preventing domain errors
- **Integration with Optimization**: `SymbolicOptimizer` uses domain analysis to ensure all simplifications are mathematically valid
- **Performance**: Cached domain computations with efficient lattice operations
- **Extensibility**: Easy to add new transformation rules and domain constraints
**Impact**: Eliminates a major source of mathematical errors in symbolic optimization, ensuring that transformations like `sqrt(x^2) = x` are only applied when the domain constraints are satisfied (e.g., `x >= 0`).
**Current Status**: ✅ **FULLY OPERATIONAL** - All tests passing (125/125), demo working, integrated with symbolic optimizer
#### Developer Documentation & Architecture Clarity ✅
**Completed**: January 2025
- **✅ DEVELOPER_NOTES.md**: Comprehensive documentation explaining the different AST and expression types, their roles, and relationships
- **Architecture Overview**: Detailed explanation of the Final Tagless approach and how it solves the expression problem
- **Expression Type Hierarchy**: Complete documentation of all expression types from core traits to concrete implementations
- **Usage Patterns**: Examples showing when and how to use each expression type (`DirectEval`, `ASTEval`, `PrettyPrint`, etc.)
- **Design Benefits**: Clear explanation of performance, type safety, and extensibility advantages
- **Common Pitfalls**: Documentation of potential issues and how to avoid them
#### README Improvements & Tested Examples ✅
**Completed**: January 2025
- **✅ Tested README Examples**: Created `examples/readme.rs` with all README code examples to ensure they actually work
- **✅ Compile-and-Load API**: Implemented `RustCompiler::compile_and_load()` method with auto-generated file paths
- **✅ Working Code Snippets**: All README examples are now tested and functional, copied directly from working code
- **✅ Comprehensive Examples**: Covers symbolic optimization, automatic differentiation, and multiple compilation backends
- **✅ Error-Free Documentation**: No more non-existent functions or incorrect API usage in README
#### Variable System Architecture Overhaul ✅
**Completed**: January 2025
- **Removed Global Variable Registry** to eliminate thread safety issues and test isolation problems
- **Implemented ExpressionBuilder Pattern** with per-function variable registries for better encapsulation
- **Enhanced Thread Safety**: Each ExpressionBuilder maintains its own isolated variable registry
- **Improved Test Reliability**: Eliminated test interference from shared global state
- **Maintained Performance**: Index-based variable access with efficient HashMap lookups
- **Simplified API**: Clean separation between expression building and evaluation phases
- **Real-world Ready**: Designed for concurrent usage in production environments
- **Backend Integration**: ✨ **NEWLY COMPLETED** - Updated Rust and Cranelift backends to use variable registry system
**Technical Details**:
- `ExpressionBuilder` provides isolated variable management per function
- `VariableRegistry` struct with bidirectional name↔index mapping
- Removed all global state dependencies from core modules
- Updated summation engine, symbolic AD, and compilation backends
- **Backend Variable Mapping**: Both Rust codegen and Cranelift backends now use `VariableRegistry` for proper variable name-to-index mapping
- **Improved Code Generation**: Multi-variable functions generate correct parameter extraction from arrays
- **Test Coverage**: All backend tests updated and passing with new variable system
- Comprehensive test coverage with proper isolation
- Zero breaking changes to existing functionality
#### Previously Completed Features
1. **Egglog Extraction System** ✅
- Hybrid approach combining egglog equality saturation with pattern-based extraction
- Comprehensive rewrite rules for algebraic simplification
- Robust fallback mechanisms for complex expressions
- Integration with existing symbolic optimization pipeline
2. **Multi-Dimensional Summation Support** ✅
- `MultiDimRange` for nested summation ranges
- `MultiDimFunction` for multi-variable functions
- Separability analysis for factorizable multi-dimensional sums
- Closed-form evaluation for separable dimensions
- Comprehensive test coverage with 6 new test cases
3. **Convergence Analysis Framework** ✅
- `ConvergenceAnalyzer` with configurable test strategies
- Ratio test, root test, and comparison test implementations
- Support for infinite series convergence determination
- Integration with summation simplification pipeline
4. **A-Normal Form (ANF) Implementation** ✅ **NEWLY COMPLETED**
- **Automatic Common Subexpression Elimination**: ANF transformation automatically introduces let-bindings for shared subexpressions
- **Hybrid Variable Management**: Efficient `VarRef` system distinguishing user variables (`VarRef::User(usize)`) from generated temporaries (`VarRef::Bound(u32)`)
- **Clean Code Generation**: ANF expressions generate readable Rust code with proper let-bindings and variable scoping
- **Type-Safe Conversion**: Generic ANF converter that works with any `NumericType + Clone + Zero`
- **Integration Ready**: Seamlessly integrates with existing `VariableRegistry` system and compilation backends
- **Rigorous PL Foundation**: Based on established programming language theory for intermediate representations
- **Zero String Management Overhead**: Integer-based variable generation avoids string allocation during optimization
- **Comprehensive Test Coverage**: Full test suite demonstrating conversion, code generation, and CSE capabilities
### 🎯 Next Steps (Phase 4: Advanced Integration & Scale)
**Status**: Ready to Begin (May 2025)
With domain analysis now complete, the mathematical expression library has achieved a major milestone in safety and correctness. The next phase focuses on advanced integration, performance optimization, and expanding the ecosystem.
#### 🔥 Immediate Priorities (Q2-Q3 2025)
1. **Enhanced Domain-Aware Optimizations**
- [ ] **Domain-Guided Constant Folding**: Use domain information to safely evaluate more constant expressions
- [ ] **Conditional Transformations**: Apply different optimization rules based on domain constraints
- [ ] **Domain Propagation**: Improve domain inference through complex expression chains
- [ ] **User Domain Hints**: Allow users to specify domain constraints for better optimization
2. **ANF-Domain Integration**
- [ ] **Domain-Aware ANF**: Integrate domain analysis into A-Normal Form transformations
- [ ] **Safe CSE**: Ensure common subexpression elimination respects domain constraints
- [ ] **Domain-Preserving Let-Bindings**: Maintain domain information through ANF transformations
- [ ] **Optimization Metrics**: Track domain safety improvements in ANF pipeline
3. **Advanced Egglog Integration**
- [ ] **Domain-Aware Rewrite Rules**: Enhance egglog rules with domain preconditions
- [ ] **Conditional Rewrites**: Only apply transformations when domain constraints are satisfied
- [ ] **Domain Extraction**: Extract optimal expressions while preserving domain safety
- [ ] **Hybrid Optimization**: Combine egglog equality saturation with domain analysis
4. **Performance & Scalability**
- [ ] **Domain Cache Optimization**: Improve performance of domain computation caching
- [ ] **Parallel Domain Analysis**: Thread-safe domain analysis for concurrent workloads
- [ ] **Incremental Analysis**: Update domains efficiently when expressions change
- [ ] **Memory Management**: Optimize memory usage for large expression trees
#### 🌟 Strategic Goals (Q4 2025 - 2026)
**Production-Ready Mathematical Computing:**
- [ ] **Industrial Applications**: Deploy in scientific computing, finance, and engineering
- [ ] **Language Bindings**: Python, Julia, MATLAB interfaces with domain safety
- [ ] **Framework Integration**: NumPy, SciPy, JAX compatibility with domain awareness
- [ ] **Real-time Systems**: Ultra-low latency compilation with domain validation
**Advanced Mathematical Features:**
- [ ] **Complex Domain Analysis**: Extend to complex numbers and multi-valued functions
- [ ] **Interval Arithmetic**: Rigorous interval-based domain tracking
- [ ] **Symbolic Domain Constraints**: Express domain constraints symbolically
- [ ] **Proof Generation**: Generate mathematical proofs of transformation validity
**Ecosystem Expansion:**
- [ ] **Educational Tools**: Interactive domain analysis for teaching mathematics
- [ ] **Research Platform**: Support for advanced mathematical research
- [ ] **Industry Partnerships**: Collaborate with mathematical software companies
- [ ] **Open Source Community**: Build contributor ecosystem around domain-aware optimization
#### 🔬 Research Directions
**Theoretical Foundations:**
- [ ] **Domain Lattice Theory**: Formal verification of domain operations
- [ ] **Transformation Soundness**: Prove correctness of domain-aware transformations
- [ ] **Completeness Analysis**: Determine optimal domain precision vs. performance trade-offs
- [ ] **Abstract Interpretation Extensions**: Explore advanced abstract domains
**Practical Applications:**
- [ ] **Machine Learning**: Domain-aware automatic differentiation for neural networks
- [ ] **Quantum Computing**: Domain analysis for quantum circuit optimization
- [ ] **Distributed Computing**: Domain-aware expression distribution across clusters
- [ ] **Embedded Systems**: Lightweight domain analysis for resource-constrained environments
#### 📊 Success Metrics
**Technical Metrics:**
- Domain analysis coverage: >95% of mathematical transformations validated
- Performance impact: <5% overhead for domain-aware optimization
- Safety improvement: 100% elimination of domain-related mathematical errors
- Test coverage: >98% for all domain analysis components
**Adoption Metrics:**
- Community engagement: Active contributors and users
- Industrial adoption: Production deployments in scientific computing
- Academic recognition: Publications and citations in mathematical software literature
- Ecosystem growth: Third-party tools and integrations
#### 🛠️ Implementation Strategy
**Phase 4A: Foundation Enhancement (Q2 2025)**
- Complete domain-aware optimizations
- Integrate with ANF and egglog systems
- Establish performance benchmarks
- Create comprehensive documentation
**Phase 4B: Ecosystem Development (Q3 2025)**
- Build language bindings and integrations
- Develop educational and research tools
- Establish industry partnerships
- Create contributor onboarding
**Phase 4C: Advanced Features (Q4 2025)**
- Implement advanced domain theories
- Add proof generation capabilities
- Optimize for production workloads
- Expand to new mathematical domains
**Phase 4D: Community & Scale (2026)**
- Build open source community
- Support large-scale deployments
- Advance theoretical foundations
- Explore new application domains
---
## Recent Achievements ✅
### A-Normal Form (ANF) with Scope-Aware Common Subexpression Elimination
**Status: COMPLETE (December 2024)**
#### What We Built
- **ANF Intermediate Representation**: Complete transformation from `ASTRepr` to A-Normal Form
- **Scope-Aware CSE**: Common subexpression elimination that respects variable lifetimes
- **Hybrid Variable Management**: `VarRef::User(usize)` + `VarRef::Bound(u32)` system
- **Clean Code Generation**: Produces readable, efficient Rust code
- **Property-Based Testing**: Comprehensive test coverage including robustness testing
#### Technical Architecture
**Core Types:**
```rust
pub enum VarRef {
User(usize), // Original variables from VariableRegistry
Bound(u32), // ANF temporary variables (unique IDs)
}
pub enum ANFExpr<T> {
Atom(ANFAtom<T>), // Constants & variables
Let(VarRef, ANFComputation<T>, Box<ANFExpr<T>>), // let var = comp in body
}
pub struct ANFConverter {
binding_depth: u32, // Current nesting level
next_binding_id: u32, // Unique variable generator
expr_cache: HashMap<StructuralHash, (u32, VarRef, u32)>, // CSE cache
}
```
**Key Innovation - Scope-Aware CSE:**
```rust
// Cache entry: (scope_depth, variable, binding_id)
if cached_scope <= self.binding_depth {
return ANFExpr::Atom(ANFAtom::Variable(cached_var)); // Safe to reuse
} else {
self.expr_cache.remove(&structural_hash); // Out of scope, remove
}
```
#### Current Capabilities
- **Basic CSE**: Automatically eliminates common subexpressions
- **Scope Safety**: Variables only referenced within valid binding scope
- **Limited Constant Folding**: Basic arithmetic operations on constants
- **Clean Code Generation**: Produces readable nested let-bindings
- **Property-Based Testing**: Robustness testing with random expressions
#### Current Limitations
- **No dead code elimination**: Unused let-bindings are not removed
- **Limited constant folding**: Only basic arithmetic operations
- **No optimization metrics**: No quantitative analysis of CSE effectiveness
- **Memory growth**: CSE cache grows without bounds
- **Scope management complexity**: Current approach may have edge cases
- **Domain safety**: No validation for transcendental function domains
#### Integration Points
**Existing Systems:**
- ✅ **VariableRegistry**: Seamless user variable management
- ✅ **ASTRepr**: Direct conversion from existing AST
- ✅ **Code Generation**: Produces valid Rust code
- ✅ **Test Infrastructure**: Comprehensive test coverage
**Future Integration Targets:**
- 🔄 **Egglog**: ANF as input for e-graph optimization
- 🔄 **JIT Compilation**: ANF → LLVM IR generation
- 🔄 **Symbolic Differentiation**: ANF-based autodiff
- 🔄 **Advanced Optimizations**: Enhanced constant folding, dead code elimination
## Ongoing Work 🚧
## Roadmap: Generic Numeric Types in Symbolic Optimizer
### Motivation
- Enable support for custom numeric types (e.g., rationals, dual numbers, complex numbers, arbitrary precision, etc.)
- Allow symbolic and automatic differentiation over types other than f64
- Facilitate integration with other math libraries and future-proof the codebase
### Technical Goals
- Make ASTRepr, symbolic optimizer, and all relevant passes generic over T: NumericType (or similar trait)
- Ensure all simplification, constant folding, and codegen logic works for generic T, not just f64
- Add trait bounds and/or specialization for transcendental and floating-point-specific rules
- Maintain performance and ergonomics for the common f64 case
### Considerations
- Some optimizations and simplifications are only valid for floating-point types (e.g., NaN, infinity, ln/exp rules)
- Codegen and JIT backends may need to be specialized or limited to f64 for now
- Test coverage must include both f64 and at least one custom numeric type (e.g., Dual<f64> or BigRational)
### Steps
1. Refactor ASTRepr and all symbolic passes to be generic over T
2. Add NumericType trait (if not already present) with required operations
3. Update tests and property-based tests to use both f64 and a custom type
4. Document which features are only available for f64 (e.g., JIT, codegen)
5. (Optional) Add feature flags for advanced numeric types
---
## Testing: Property-based tests for constant propagation
- Add proptests to ensure that constant folding and propagation in both symbolic and ANF passes are correct and robust.
- These tests should generate random expressions and check that all evaluation strategies (direct, ANF, symbolic) agree on results for all constant subexpressions.
## Domain Awareness
- Symbolic simplification should be domain-aware: only apply rewrites like exp(ln(x)) = x when x > 0.
- Property-based tests (proptests) must filter out invalid domains (e.g., negative values for ln, sqrt, etc.) to avoid spurious failures.
- Long-term: consider encoding domain constraints in the symbolic system and/or test harness.