Affine arithmetic (AA) is a model for self-validated numerical analysis. In AA, the quantities of interest are represented as affine combinations (affine forms) of certain primitive variables, which stand for sources of uncertainty in the data or approximations made during the computation. Affine arithmetic is meant to be an improvement on interval arithmetic (IA), and is similar to generalized interval arithmetic, first-order Taylor arithmetic, the center-slope model, and ellipsoid calculus — in the sense that it is an automatic method to derive first-order guaranteed approximations to general formulas. Affine arithmetic is potentially useful in every numeric problem where one needs guaranteed enclosures to smooth functions, such as solving systems of non-linear equations, analyzing dynamical systems, integrating functions, differential equations, etc. Applications include ray tracing, plotting curves, intersecting implicit and parametric surfaces, error analysis (mathematics), process control, worst-case analysis of electric circuits, and more. Definition In affine arithmetic, each input or computed quantity x is represented by a formula where are known floating-point numbers, and are symbolic variables whose values are only known to lie in the range [-1,+1]. Thus, for example, a quantity X which is known to lie in the range [3,7] can be represented by the affine form , for some k. Conversely, the form implies that the corresponding quantity X lies in the range [3,17]. The sharing of a symbol among two affine forms , implies that the corresponding quantities X, Y are partially dependent, in the sense that their joint range is smaller than the Cartesian product of their separate ranges. For example, if and , then the individual ranges of X and Y are [2,18] and [13,27], but the joint range of the pair (X,Y) is the hexagon with corners (2,27), (6,27), (18,19), (18,13), (14,13), (2,21) — which is a proper subset of the rectangle [2,18]×[13,27]. Affine arithmetic operations Affine forms can be combined with the standard arithmetic operations or elementary functions, to obtain guaranteed approximations to formulas. Affine operations For example, given affine forms for X and Y, one can obtain an affine form for Z = X + Y simply by adding the forms — that is, setting for every j. Similarly, one can compute an affine form for Z = X, where is a known constant, by setting for every j. This generalizes to arbitrary affine operations like Z = X + Y + . Non-affine operations A non-affine operation , like