Equation 6 · 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 equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol T
T is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
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.
Symbol bars
bars is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
How to interpret it
Its accuracy depends on the assumptions and range of use described in the article.
What the article says around this equation
which, if schemas run at roughly a uniform average size , simplifies to T(n) + n . Nothing about this term depends on whether the model calls any tool at all; it is the price of the menu, not the price of the order. And critically, because a stateless inference API resends its full input on every request, T(n) is not paid once at the start of a session — it is paid again, in full, on every single turn for as long as tools remain attached to the request.
Sources cited in the article section
- [4] Pricing ↗
- [6] Function Calling ↗
- [5] Introducing Advanced Tool Use on the Claude Developer Platform ↗
These citations give research context. Read each source to check which claims it supports.
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