Equation 12 · Part 17 · The Real Cost of Running Claude in Production, Beyond the Per-Token Price

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

C = \frac{p_{\mathrm{in}}(n + \tau) + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}}(y + z)}{a}

p_{\mathrm{in}}(n + \tau) + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}}(y + z)

What this part means

The complete quantity above the fraction bar.

Its job in the formula

$p_i$ n(n + τ) + $p_c$ w w + $p_c$ r r + $p_o$ ut(y + z) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Full expressionNumerator: p_in(n + τ) + p_cw w + p_cr r + p_out(y + z)Article meaning

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 $\tau$ the fixed tool and system-prompt tokens added regardless of content. Let $p_{\mathrm{in}}$ , $p_{\mathrm{cw}}$ , $p_{\mathrm{cr}}$ , and $p_{\mathrm{out}}$ 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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