Core Module

The core module of tax provides truncated multivariate Taylor polynomials as first-class C++ objects. A single evaluation pass through any supported mathematical expression yields the function value and all partial derivatives up to a compile-time truncation order \(N\). The library uses expression templates for lazy evaluation: arithmetic and transcendental operations build a lightweight expression tree that is materialized only on assignment, eliminating intermediate temporary objects and enabling automatic sum/product flattening.

The central type, TruncatedTaylorExpansionT<T, N, M>, stores \(\binom{N+M}{M}\) Taylor coefficients in a fixed-size std::array with zero heap allocation. Coefficients follow graded-lexicographic ordering -- grouped first by total degree, then lexicographically within each grade. A comprehensive set of mathematical functions (trigonometric, hyperbolic, transcendental, algebraic, and special functions) is implemented via degree-by-degree recurrence relations, supporting both univariate and multivariate expansions.

Key Headers

Header	Contents
tax/tax.hpp	Umbrella header -- the only include users need
tax/tte.hpp	Core TruncatedTaylorExpansionT<T, N, M> class with constructors, variable factories, coefficient/derivative access, evaluation, differentiation, integration, norms, and in-place operators
tax/operators.hpp	Facade that pulls in all free-function operators and math functions
tax/kernels.hpp	Facade that pulls in all series computation kernels (recurrence relations)
tax/expr/	Expression template nodes: Expr<> CRTP base, binary/unary/sum/product expressions, arithmetic and function call nodes
tax/kernels/	Degree-by-degree recurrence implementations: cauchy.hpp (Cauchy product), algebra.hpp (reciprocal, sqrt, cbrt), trigonometric.hpp (sin, cos, tan, sinh, cosh, tanh and inverses), transcendental.hpp (exp, log, pow, erf, asin, acos, atan, inverse hyperbolics)
tax/utils/	Combinatorics, multi-index enumeration, flat-index mapping, type traits