Floating Point
A binary format representing real numbers as a sign, a fractional mantissa, and an exponent.
Definition
Floating point stores a real number as a sign, a mantissa (significant digits), and an exponent, much like scientific notation in base 2. The IEEE 754 standard defines the common single (32-bit) and double (64-bit) formats.
The standard also defines rounding modes and special results so that computations behave predictably across machines, which is essential for reproducibility. Comparing floating-point numbers for exact equality is nonetheless unreliable; tolerances are used instead.
The gap between representable numbers grows with magnitude, so precision is relative rather than absolute, and adding a tiny number to a huge one can have no effect at all. This behavior, combined with special values like infinity and not-a-number, means robust numerical code must be written with the format's quirks in mind. The IEEE standard's guarantees are what make results consistent and reproducible across different machines.
Consequences
- Wide dynamic range, from tiny to enormous magnitudes.
- Finite precision: most reals are stored approximately.
- Rounding error accumulates over many operations.
- Special values: infinity and not-a-number (NaN).
Why it matters
Floating point is how computers do real-number arithmetic, so its limits underlie every numerical result. Ignoring rounding leads to floating-point error, catastrophic cancellation, and irreproducible results, a central concern in scientific computing.
Fusion connection
Fusion simulations run in double precision because plasma quantities span many orders of magnitude, and Kronos tracks numerical precision as part of verifying that results are trustworthy.