# Big-O Crossover Calculator

> When does c₁·n log n beat c₂·n²? Crossover points, log-log chart and run times. Compares cost functions such as 50·n log n against 2·n² or 2ⁿ with your constant factors, finds every input size where one overtakes another, draws them on a log-log chart, and converts operation counts into running time at a given number of operations per second.

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://bigocrossovercalculator.usefulweb.site/ (interactive tool at https://bigocrossovercalculator.usefulweb.site/tool)
- JSON API: POST https://bigocrossovercalculator.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://bigocrossovercalculator.usefulweb.site/api/v1/tool)
- OpenAPI: https://bigocrossovercalculator.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://bigocrossovercalculator.usefulweb.site/mcp - tool `big_o_crossover_calculator`

## Inputs
- functions (table, required): Cost functions
- nMin (number): Smallest n
- nMax (number): Largest n
- opsPerSecond (number): Operations per second - For converting cost into time; 10^9 is a rough figure for simple operations on one CPU core.
- sizes (list): Input sizes for the table

## Method
- Each cost is c·f(n) with log₂ for logarithms; n! is evaluated as Γ(n + 1), so every function is continuous in n. All comparisons are done on log10 of the cost, so 2ⁿ or n! for large n never overflow.
- Crossovers: the difference of the log costs is sampled at 2,000 log-spaced sizes between the smallest and largest n; every sign change is refined by 80 bisection steps. Crossovers closer together than the grid spacing can be missed.
- Time = cost ÷ operations per second. Real programs also depend on memory access, caches and constant overheads that a single constant c only approximates.