Affine arithmetic

Definition

An affine form is an expression $$\hat x = x_0 + \sum_{i=1}^{k} x_i \varepsilon_i,$$ with real coefficients $x_0, \ldots, x_k$ and noise symbols $\varepsilon_1, \ldots, \varepsilon_k$. The noise symbols stand for unknown reals in $[-1, 1]$, shared by all affine forms of a computation. Its range is the interval $[x_0 - r, x_0 + r]$ with radius $r = \sum_{i=1}^{k} \abs{x_i}$. Affine arithmetic computes with affine forms by the following rules.

1. Affine operations are exact: $\hat x \pm \hat y = (x_0 \pm y_0) + \sum (x_i \pm y_i) \varepsilon_i$ and $c \hat x + d = (c x_0 + d) + \sum c x_i \varepsilon_i$ for reals $c, d$.
2. A nonlinear operation $g$ is replaced by an affine form $\hat z = z_0 + \sum z_i \varepsilon_i + z_{k+1} \varepsilon_{k+1}$ with a new noise symbol $\varepsilon_{k+1}$. It is chosen so that $g(x, y, \ldots) \in \hat z$ for all values of $\varepsilon_1, \ldots, \varepsilon_k$ in $[-1, 1]$.
3. For multiplication one may take $\hat x \hat y = x_0 y_0 + \sum_{i=1}^{k} (x_0 y_i + y_0 x_i) \varepsilon_i + r_x r_y \, \varepsilon_{k+1}$, where $r_x, r_y$ are the radii of $\hat x, \hat y$.