x_{1} is the generalized arithmetic-geometric mean of the elements of the 2-element set x_{2} (set; cardinality must be 2) of order x_{3} (either single extended-real number xor an unordered pair/2-element set of extended-real numbers).

Elements of x_{2} must have the same units/dimensionality; the result has the same units/dimensionality as them. If x_{3}=p for a single extended-real number p, then x_{3}=(p,p-1) also. If x_{3 }= (p,q) then the algorithm uses the pth-power mean (cnanfadi) and the qth-power mean; thus x_{3 }= (1,0) corresponds to the standard arithmetic-geometric mean, x_{3 }= (0, -1) corresponds to the geometric-harmonic mean. A poor choice of x_{3} will lead to non-convergence of the sequences produced by the algorithm and, thus, leave x_{1} undefined (NAN error). x_{1} is the value to which each of the sequences produced by the algorithm converge (iff these values are mutually equal). Let M_{i} denote the unweighted ith-power mean for all i and let x_{2 }= (a_{0,g} x_{3 }= (p,q); then the algorithm produces potentially infinite sequences a = (a_{0, a}= (g_{0, g}, where a_{n }= M_{p(a}= M_{q(a} for all positive integers n. Notice that this word is symmetric (x_{1} remains constant and none of the sumti change in meaning) under an internal permutation of the elements/entries of x_{2} and/or of x_{3} (when taken as a pair or set), separately. See also: gau'i'o.

- gau'i'o
*(exp!)* - digit/number: Gauss' arithmetic-geometric mean of 1 and √(2)=sqrt(2) constant G ≈ .8346268…