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Published equation contexts

c=pinn+pcww+pcrr+pout(y+z)c = p_{\mathrm{in}} n + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}} \left( y + z \right)

Why this formula appears here

Instrument each request to record five quantities: new input tokens, cache-write tokens, cache-read tokens, visible output tokens, and reasoning tokens. The cost of a single request is then c=pinn+pcww+pcrr+pout(y+z)c = p_{\mathrm{in}} n + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}} \left( y + z \right). where n , w , r , y and z are the five counts in that order and each p is the corresponding published rate. The important feature is the last term: visible output and reasoning share one price, and the reasoning count is the only one of the five you cannot recover by inspecting what came back.

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pinp_{\mathrm{in}}

Symbol p_in

pip_in is one of the signed contributions combined to compute the quantity on the left.

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pcwp_{\mathrm{cw}}

Symbol p_cw

pcp_cw is one of the signed contributions combined to compute the quantity on the left.

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pcrp_{\mathrm{cr}}

Symbol p_cr

pcp_cr is one of the signed contributions combined to compute the quantity on the left.

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poutp_{\mathrm{out}}

Symbol p_out

pop_out is one of the signed contributions combined to compute the quantity on the left.

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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=pinn+pcww+pcrr+pout(y+z)c = p_{\mathrm{in}} n + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}} \left( y + z \right)

Equation 1 · Inference Economics

The Unit Economics of a Token

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

Instrument each request to record five quantities: new input tokens, cache-write tokens, cache-read tokens, visible output tokens, and reasoning tokens. The cost of a single request is then c=pinn+pcww+pcrr+pout(y+z)c = p_{\mathrm{in}} n + p_{\mathrm{cw}} w + p_{\mathrm{cr}} r + p_{\mathrm{out}} \left( y + z \right). where n , w , r , y and z are the five counts in that order and each p is the corresponding published rate. The important feature is the last term: visible output and reasoning share one price, and the reasoning count is the only one of the five you cannot recover by inspecting what came back.

Meanings in this article

  • nn: the five counts in that order and each p is the corresponding published rate.
  • ww: the five counts in that order and each p is the corresponding published rate.
  • rr: the five counts in that order and each p is the corresponding published rate.
  • yy: the five counts in that order and each p is the corresponding published rate.
  • zz: the five counts in that order and each p is the corresponding published rate.
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