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Published equation contexts

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}

Why this formula appears here

Collect the pieces above into one expression and the 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 was neither truncated at the maximum-tokens cap nor lost to a timeout, an overload error, or a rate limit. Cost per accepted…

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pinp_{\mathrm{in}}

Symbol p_in

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

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pcwp_{\mathrm{cw}}

Symbol p_cw

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

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pcrp_{\mathrm{cr}}

Symbol p_cr

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

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poutp_{\mathrm{out}}

Symbol p_out

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

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

Numerator: p_in(n + τ) + p_cw w + p_cr r + p_out(y + z)

The complete quantity above the fraction bar.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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}

Equation 12 · Foundation Models

The Real Cost of Running Claude in Production, Beyond the Per-Token Price

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Collect the pieces above into one expression and the 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 was neither truncated at the maximum-tokens cap nor lost to a timeout, an overload error, or a rate limit. Cost per accepted…

Meanings in this article

  • τ\tau: the fixed tool and system-prompt tokens added regardless of content.
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