EmbeddingGemma 2: an open, lightweight multimodal embedding model
deepmind.google·6d ago
AI Engineering
Lobsters·Aug 20·5 min read
Double-double: 31 digits of precision without leaving the FPU
TL;DR
A technique called double-double combines two double-precision floating-point numbers to achieve approximately 31 decimal digits of precision, significantly improving numerical accuracy without the overhead of arbitrary-precision libraries. This method is particularly useful for applications like deep zooming into fractals, where standard double precision falls short.
✦ Why It Matters
Consider implementing double-double for precision-critical applications to avoid performance penalties from arbitrary-precision libraries.
Key Takeaways
- Use double-double for high precision in floating-point calculations.
- Avoid heap allocation by implementing double-double directly.
- Leverage error-free transformations to maintain accuracy.
- Benchmark double-double against traditional methods for performance insights.
How It Works
The double-double technique stores two doubles, treating them as one number to capture more significant digits. It maintains an invariant where the high part represents the rounded value, while the low part captures the rounding error.
This allows for precise arithmetic operations using error-free transformations, such as Knuth's two-sum and two-product algorithms, which recover lost precision during calculations.