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Equation 12 · Part 4 · The Real Cost of Running Claude in Production, Beyond the Per-Token Price

Symbol τ

C=pin(n+τ)+pcww+pcrr+pout(y+z)aC = \frac{p_{\mathrm{in}}(n + \tau) + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}}(y + z)}{a}
τ\tau

What this part means

the fixed tool and system-prompt tokens added regardless of content.

Its job in the formula

τ occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Where the article explains it

For a single request, let n be new input tokens, w cache-write tokens, r cache-read tokens, y visible output tokens, z billed-but-invisible thinking tokens, and τ\tau the fixed tool and system-prompt tokens added regardless of content.

The passage around this formula

…gap between a rate card and an invoice becomes explicit. For a single request, let n be new input tokens, w cache-write tokens, r cache-read tokens, y visible output tokens, z billed-but-invisible thinking tokens, and τ\tau the fixed tool and system-prompt tokens added regardless of content. Let pinp_{\mathrm{in}} , pcwp_{\mathrm{cw}} , pcrp_{\mathrm{cr}} , and poutp_{\mathrm{out}} be the published rates, and let a ∈\in (0, 1] be the fraction of attempts an application accepts without needing to reissue the request, because it…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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