# Floating-Point Explorer

> See how IEEE-754 half/single/double store a number: bits, exact value, ULP, rounding error. Shows the sign, exponent and fraction bits of a decimal number in IEEE-754 binary16, binary32 or binary64, the exact decimal value that is actually stored (computed with big integers), the rounding error, the ULP and the neighbouring representable numbers. Evaluates expressions such as 0.1 + 0.2 step by step and explains subnormals, infinities and NaN.

Part of usefulweb.site (https://usefulweb.site/). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://floatingpointexplorer.usefulweb.site/ (interactive tool at https://floatingpointexplorer.usefulweb.site/tool)
- JSON API: POST https://floatingpointexplorer.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://floatingpointexplorer.usefulweb.site/api/v1/tool)
- OpenAPI: https://floatingpointexplorer.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://floatingpointexplorer.usefulweb.site/mcp - tool `floating_point_explorer`

## Inputs
- value (text, required): Number, expression or bit pattern - A decimal (0.1, -6.02e23), an expression with + − * / and parentheses, NaN / Infinity, or a bit pattern 0x… / 0b….
- format (select): Format

## Method
- Decimal input is parsed as an exact fraction and rounded to the nearest representable value of the chosen format (round half to even), with big-integer arithmetic, so there is no double rounding via JavaScript numbers.
- A normal number is (−1)^sign × 1.fraction × 2^(exponent − bias); subnormals (exponent field 0) are (−1)^sign × 0.fraction × 2^(1 − bias). The exact decimal of the stored value always terminates because the denominator is a power of two.
- ULP is the gap to the next representable number away from zero; the neighbours are the bit patterns ±1 (crossing between ±0 and the smallest subnormals).
- Expressions are evaluated twice: exactly with fractions, and in the chosen format with rounding after every literal and every operation (IEEE-754 semantics, like JavaScript for binary64 or C float for binary32).