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Equation 6 · Part 3 · Actually Building with the Grok API

Symbol p_i^L

Cost(ni,no)={ni piL+no poL,ni<Tni piH+no poH,ni≥T\text{Cost}(n_i, n_o) = \begin{cases} n_i\, p_i^{L} + n_o\, p_o^{L}, & n_i < T \\ n_i\, p_i^{H} + n_o\, p_o^{H}, & n_i \geq T \end{cases}
piLp_i^{L}

What this part means

the low-tier prices.

Its job in the formula

piLp_i^L is one of the signed contributions combined to compute the quantity on the left.

Where the article explains it

Written as a cost function of input tokens nin_i and output tokens non_o against a threshold T = 200{,}000 , with low-tier prices piLp_i^{L}, poLp_o^{L} and high-tier prices piHp_i^{H}, poHp_o^{H} : Cost(ni,no)={ni piL+no poL,ni<Tni piH+no poH,ni≥T\text{Cost}(n_i, n_o) = \begin{cases} n_i\, p_i^{L} + n_o\, p_o^{L}, & n_i < T \\ n_i\, p_i^{H} + n_o\, p_o^{H}, & n_i \geq T \end{cases}.

The passage around this formula

…— and the higher rate applies to the entire request, not just the tokens past the line [ 1 ] . Written as a cost function of input tokens nin_i and output tokens non_o against a threshold T = 200{,}000 , with low-tier prices piLp_i^{L}, poLp_o^{L} and high-tier prices piHp_i^{H}, poHp_o^{H} : Cost(ni,no)={ni piL+no poL,ni<Tni piH+no poH,ni≥T\text{Cost}(n_i, n_o) = \begin{cases} n_i\, p_i^{L} + n_o\, p_o^{L}, & n_i < T \\ n_i\, p_i^{H} + n_o\, p_o^{H}, & n_i \geq T \end{cases}. For grok-4.6 that means a 199,999-token prompt is billed at two dollars per million input tokens, and a 200,001-token prompt — one token longer — is billed at four dollars per million input tokens and twelve dollars per…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the surrounding passage

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