Normalization and Anchoring as a Design Space: Five Closed-Form Approximations of the Arithmetic-Geometric Mean

The arithmetic-geometric mean AGM(1,x) is used as a worked example to illustrate a normalization-and-anchoring technique for building closed-form approximations, and to make explicit the choices the technique presents. The method first normalizes AGM(1,x) to expose its asymptotic structure, then fixes the free constants of a chosen functional form by matching the exact boundary data of the complete elliptic integral K at its two limiting regimes, with no fitted parameters. The construction has two dials. The first is the normalization: at its simpler level it cancels the leading logarithmic divergence, giving c(x) = 2ln(4/x)·AGM(1,x), a smooth curve interpolating π as x→0 and 4ln2 at x=1; at its richer level it additionally absorbs the exact next-order term. The second is the order of the anchored ansatz, each further power closing one more endpoint derivative. Developing each dial to two levels yields five cases, which occupy distinct positions on a multi-criteria surface — accuracy, the direction and scope of any error bound, the admissibility of the constants, evaluation cost, and symbolic integrability — on which no single case is best. The two dials are coupled: absorbing the next-order term forces a higher-order ansatz. Four of the five cases are proved to be one-sided bounds on the entire open interval (0,1), certified by continuum interval arithmetic on the interior, extended by dyadic refinement and closed at each endpoint by a remainder-dominance argument. The highest-order case is a lower bound everywhere accurate to within 0.0008% across the whole interval; paired with a bound of opposite sign, it encloses the AGM rigorously between elementary functions. All constants are composed of π and ln2, save the algebraic coefficients of one case and the leading exponent of the two highest-order cases, which is fixed by an exact anchor condition on the boundary asymptotics rather than by fitting. This version supersedes and contains the earlier single-approximation result. Part of a series: Ellipse perimeter · Nonlinear pendulum · Asymmetric cubic oscillator · Relativistic harmonic oscillator