Equation 4 · Astra's 63%-Cheaper Claim Has to Beat a Price Disadvantage Nobody Mentions
What does this equation mean?
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
Symbol s
s is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subtraction
Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.
Denominator: (1-s)(10) + s(0.25)
The complete quantity below the fraction bar; it must be nonzero for this division.
Denominator: 10 - 9.75s
The complete quantity below the fraction bar; it must be nonzero for this division.
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.
What the article says around this equation
Call the cache-hit share s — the fraction of input tokens that are cache reads rather than fresh reads. The blended input-token cost per model, in dollars per million tokens, is (1-s) 10 + s r , where r is each model’s cache-read rate: $1.00 for Astra, $0.25 for Fable 5.1. The ratio of Astra’s blended input cost to Fable 5.1’s, at a given cache-hit share, is: . Run that ratio across a stated bracket rather than picking one number and presenting it as fact:
For background, read the article’s source list.
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