What does this equation mean?

Anthropic’s Tool Search Tool is a direct, documented implementation of that idea. The mechanism is described plainly: “You provide all your tool definitions to the API, but mark tools with defe$r_l$oading: true to make them discoverable on-demand,” with a lightweight search tool — built-in options include BM25-based and regex-based search — querying an index of names and descriptions, and the full definition of a matching tool injected into context only once the model actually requests it [ 5 ] . The search tool itself is not free, but its overhead is small and fixed rather than proportional to catalogue size: “Only the Tool Search Tool loaded upfront (~500 tokens)” [ 5 ] . Applied to the same 58-tool, roughly 55,000-token example described above, that reframes the earlier formula: instead of the full T(n) $\approx$ $\tau_0$ + n$\bar{s}$ , a session that only ever actually needs m of the n available tools pays approximately $\tau_0$ + $\sigma$ + m$\bar{s}$ , where $\sigma$ is the search tool’s own fixed overhead. The fraction of the original tax avoided is $\rho(n, m) \;=\; 1 - \frac{\sigma + m\bar{s}}{\tau_0 + n\bar{s}}$. which grows toward its ceiling as m shrinks relative to n . Anthropic reports a measured outcome consistent with that shape at their own scale: “Tool Search Tool preserves 191,300 tokens of context compared to 122,800 with Claude’s traditional approach,” described as an “85% reduction in token usage while maintaining access to your full tool library” [ 5 ] . Against a standard 200,000-token context window, those two preserved-context figures imply roughly 8,700 tokens actually consumed by tool definitions with search enabled against roughly 77,200 tokens without it — an arithmetic inference this article draws from the published figures, not one stated as such in the source. The same documentation reports the mechanism also improved accuracy rather than only cost, on internal MCP evaluations: Opus 4 rose from 49% to 74%, and Opus 4.5 from 79.5% to 88.1%, with Tool Search Tool enabled [ 5 ] .