This repository explores floating-point error, numerical stability, and conditioning through a series of numerical experiments. The goal is to better understand why floating-point computations break down in certain cases, how these failures appear in practice, and how rewriting formulas in a numerically stable way can significantly improve accuracy without increasing precision.
Rather than treating floating-point arithmetic as a black box, this project investigates its behavior using concrete examples, standard error metrics, and high-precision reference values to make sources of numerical error explicit.
Establishes the experimental foundation:
- IEEE-754 floating-point overview
- Absolute and relative error metrics
- High-precision reference values using arbitrary-precision arithmetic
- A worked example demonstrating catastrophic cancellation in
$1-\cos(x)$
Explores floating-point precision limits:
- Definition and computation of machine epsilon
- Experimental verification for float32 and float64
- Interpretation of precision as relative spacing near
$1.0$ - Connection between machine epsilon and loss of significance