Float Cascade to model multi-component floats

Let’s walk through how this FloatCascade<N> building block architecture naturally extends to support adaptive precision priest. The key insight is that priest uses a variable-length cascade that can grow and shrink as needed.

Here’s how the FloatCascade<N> building block architecture supports adaptive precision priest:

Key Architectural Elements:

1. VariableCascade - Dynamic Version of FloatCascade

• Uses std::vector<double> instead of std::array<double, N>

• Can grow/shrink during computation

• Same component ordering principles as fixed-size cascades

• Provides seamless conversion to/from FloatCascade<N>

2. Bidirectional Conversion Between Fixed and Adaptive

// Fixed → Adaptive (promotion)
dd d1(1.0);
priest p1(d1);  // Seamless conversion

// Adaptive → Fixed (truncation/extraction)
priest result = some.complex_calculation();
dd final.dd = result.to_dd();  // Extract to double-double
td final.td = result.to_td();  // Extract to triple-double

3. Adaptive Precision Control

The priest class uses AdaptivePrecision config to control:

• Termination criteria: When to stop iterating (1/3 problem solved!)
• Component limits: Max/min cascade length
• Tolerance: Absolute vs relative error bounds
• Performance: Early termination, auto-compression

4. The 1/3 Solution in Action

struct AdaptivePrecision {
    double relative_tolerance;
    unsigned max_components;
    unsigned max_iterations;
};

AdaptivePrecision high_precision{
    .relative_tolerance = 1e-25,  // Very tight tolerance
    .max_components = 20,         // Allow up to 20 components
    .max_iterations = 50          // Limit iterations
};

priest one(1.0, high.precision);
priest three(3.0, high.precision);
priest one.third = one / three;  // Adapts precision automatically!

The division algorithm:

1. Starts with hardware precision estimate
2. Iterates using Newton-Raphson refinement
3. Terminates when error tolerance is met OR max iterations reached
4. Controls component growth to prevent explosion
5. Compresses result to remove unnecessary precision

5. Seamless Workflow

// Start with fixed precision (fast)
dd quick.calc = dd(1.0) / dd(3.0);  // Limited precision

// Need more precision? Promote seamlessly
priest precise.calc = priest(quick_calc);
precise.calc = priest(1.0) / priest(3.0);  // Now adaptive!

// Back to fixed for performance-critical code
td final.result = precise_calc.to_td();

6. Shared ExpansionOps Engine

All types (dd, td, qd, priest) use the same core algorithms:

• two_sum, fast_two_sum for basic operations
• Expansion addition/multiplication algorithms
• Same mathematical foundations, different precision strategies

Benefits of This Architecture:

1. No Precision Walls: Smooth upgrade path from dd → td → qd → priest
2. Performance Flexibility: Use fixed precision when speed matters, adaptive when accuracy matters
3. Building Block Reuse: Same FloatCascade principles throughout
4. Controlled Iteration: Solves the “infinite 1/3” problem with configurable termination
5. Universal Compatibility: All types can interchange via FloatCascade extraction

This design gives us what Priest intended: adaptive precision that does only as much work as necessary to guarantee a correct result, while maintaining Universal’s principle of independent, interoperable number systems!