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Equation 12 · Part 14 · The Real Cost of Running Claude in Production, Beyond the Per-Token Price

fraction

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}
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

The passage around this formula

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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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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