# BS3 vs DP5 Benchmark Results

Generated: 2026-09-29 (re-measured; first run 2025-10-23)

## Summary

Comprehensive performance comparison between **BS3** (Bogacki-Shampine 3rd order) and **DP5** (Dormand-Prince 5th order) integrators across various test problems and tolerances.

## Key Findings

### Overall Performance Comparison

**DP5 is consistently faster than BS3 across all tested scenarios**, typically by a factor of **1.6x to 4.8x**.

This might seem counterintuitive since BS3 uses fewer stages (4 vs 7), but several factors explain DP5's superior performance:

1. **Higher Order = Larger Steps**: DP5's 5th order accuracy allows larger timesteps while maintaining the same error tolerance
2. **Optimized Implementation**: DP5 has been highly optimized in the existing codebase
3. **Smoother Problems**: The test problems are relatively smooth, favoring higher-order methods

### When to Use BS3

Despite being slower in these benchmarks, BS3 still has value:
- **Lower memory overhead**: Simpler dense output (4 values vs 5 for DP5)
- **Moderate accuracy needs**: For tolerances around 1e-3 to 1e-5 where speed difference is smaller
- **Educational/algorithmic diversity**: Different method characteristics
- **Specific problem types**: May perform better on less smooth or oscillatory problems

## Detailed Results

### 1. Exponential Decay (`y' = -0.5y`, tolerance 1e-5)

| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 2.96 µs | 2.08x slower |
| **DP5** | 1.43 µs | baseline |

Simple 1D problem with smooth exponential solution.

### 2. Harmonic Oscillator (`y'' + y = 0`, tolerance 1e-5)

| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 27.45 µs | 2.26x slower |
| **DP5** | 12.16 µs | baseline |

2D conservative system with periodic solution.

### 3. Nonlinear Pendulum (tolerance 1e-6)

| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 113.66 µs | 3.65x slower |
| **DP5** | 31.15 µs | baseline |

Nonlinear 2D system with trigonometric terms.

### 4. Orbital Mechanics (6D, tolerance 1e-6)

| Method | Time | Ratio |
|--------|------|-------|
| **BS3** | 102.04 µs | 1.56x slower |
| **DP5** | 65.52 µs | baseline |

Higher-dimensional problem with gravitational dynamics.

### 5. Interpolation Performance

| Method | Time (solve + 100 interpolations) | Ratio |
|--------|-----------------------------------|-------|
| **BS3** | 16.28 µs | 4.70x slower |
| **DP5** | 3.46 µs | baseline |

BS3 uses cubic Hermite interpolation, DP5 uses optimized 5th order interpolation.

### 6. Tolerance Scaling

Performance across different tolerance levels (`y' = -y` problem):

| Tolerance | BS3 Time | DP5 Time | Ratio (BS3/DP5) |
|-----------|----------|----------|-----------------|
| 1e-3 | 1.15 µs | 0.92 µs | 1.25x |
| 1e-4 | 2.01 µs | 1.19 µs | 1.69x |
| 1e-5 | 3.41 µs | 1.45 µs | 2.36x |
| 1e-6 | 6.72 µs | 1.92 µs | 3.50x |
| 1e-7 | 13.55 µs | 2.82 µs | 4.80x |

Both 1e-6 and 1e-7 are now measured. The earlier run estimated 1e-6 from the
trend and did not complete 1e-7.

**Observation**: The performance gap widens as tolerance tightens, because DP5's higher order allows it to take larger steps while maintaining accuracy.

## Conclusions

### Performance Characteristics

1. **DP5 is the better default choice** for most problems requiring moderate to high accuracy
2. **Performance gap increases** with tighter tolerances (favoring DP5)
3. **Higher dimensions** slightly favor BS3 relative to DP5 (1.56x vs 3.65x slowdown)
4. **Interpolation** strongly favors DP5 (4.7x faster)

### Implementation Quality

Both integrators pass all accuracy and convergence tests:
- ✅ BS3: 3rd order convergence rate verified
- ✅ DP5: 5th order convergence rate verified (existing implementation)
- ⚠️ Both: FSAL is a property of the *methods*, but neither implementation
  exploits it - each step recomputes `k[0] = f(t, y)` rather than reusing the
  previous step's last stage. See roadmap feature 36.
- ✅ Both: Dense output accurate to specified order

### Future Optimizations

Potential improvements to BS3 performance:
1. **Specialized dense output**: Implement the optimized BS3 interpolation from the 1996 paper
2. **SIMD optimization**: Vectorize stage computations
3. **Memory layout**: Optimize cache usage for k-value storage
4. **Inline hints**: Add compiler hints for critical paths

Even with optimizations, DP5 will likely remain faster for these problem types due to its higher order.

## Recommendations

- **Use DP5**: For general-purpose ODE solving, especially for smooth problems
- **Use BS3**: When you specifically need:
  - Lower memory usage
  - A 3rd order reference implementation
  - Comparison with other 3rd order methods

## Methodology

- **Tool**: Criterion.rs v0.7.0
- **Samples**: 100 per benchmark
- **Warmup**: 3 seconds per benchmark
- **Optimization**: Release mode with full optimizations
- **Platform**: Linux x86_64
- **Compiler**: rustc (specific version from build)

All benchmarks use `std::hint::black_box()` to prevent compiler optimizations from affecting timing.

## Reproducing Results

```bash
cargo bench --bench bs3_vs_dp5
```

Detailed plots and statistics are available in `target/criterion/`.