Equation 24 · The Token Tax of Giving a Model More Tools
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol n
n is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
How to interpret it
Read this expression with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
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…
Read the full surrounding passage
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 ] .
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.