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Equation 22 · Part 7 · The Token Tax of Giving a Model More Tools

=

ρ(n,m)  =  1−σ+msˉτ0+nsˉ\rho(n, m) \;=\; 1 - \frac{\sigma + m\bar{s}}{\tau_0 + n\bar{s}}
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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 deferlr_loading: 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…

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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.