Equation 1 · Building on Frontier Model APIs: Effort, Context, Caching, and Failure Handling

What does this equation mean?

c_{\mathrm{req}} = c_{\mathrm{in}}\,(t_{\mathrm{fresh}} + \alpha\, t_{\mathrm{cached}}) + c_{\mathrm{reason}}\, t_{\mathrm{reason}} + c_{\mathrm{out}}\, t_{\mathrm{out}},

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

c_{\mathrm{req}}

Symbol c_req

$c_r$ eq is part of the quantity the equation computes from the expression on the right.

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c_{\mathrm{in}}

Symbol c_in

$c_i$ n is one of the signed contributions combined to compute the quantity on the left.

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t_{\mathrm{fresh}}

Symbol t_fresh

$t_f$ resh is one of the signed contributions combined to compute the quantity on the left.

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\alpha

Symbol α

the discount applied to cached prefix tokens.

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t_{\mathrm{cached}}

Symbol t_cached

$t_c$ ached is one of the signed contributions combined to compute the quantity on the left.

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c_{\mathrm{reason}}

Symbol c_reason

$c_r$ eason is one of the signed contributions combined to compute the quantity on the left.

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t_{\mathrm{reason}}

Symbol t_reason

$t_r$ eason is one of the signed contributions combined to compute the quantity on the left.

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c_{\mathrm{out}}

Symbol c_out

$c_o$ ut is one of the signed contributions combined to compute the quantity on the left.

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t_{\mathrm{out}}

Symbol t_out

$t_o$ ut is one of the signed contributions combined to compute the quantity on the left.

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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addition

Add the term after the plus sign to the term or group before it.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

Write the per-request cost as three separate terms rather than one blended figure: $c_{\mathrm{req}} = c_{\mathrm{in}}\,(t_{\mathrm{fresh}} + \alpha\, t_{\mathrm{cached}}) + c_{\mathrm{reason}}\, t_{\mathrm{reason}} + c_{\mathrm{out}}\, t_{\mathrm{out}}$ . where $\alpha$ is the discount applied to cached prefix tokens. Three things become obvious the moment it is written this way.

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Sources cited in the article section

[1] Model guidance ↗

These citations give research context. Read each source to check which claims it supports.

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