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  "body": "\n## Definition\n\nAn *affine form* is an expression\n$$\\hat x = x_0 + \\sum_{i=1}^{k} x_i \\varepsilon_i,$$\nwith real coefficients $x_0, \\ldots, x_k$ and *noise symbols* $\\varepsilon_1, \\ldots, \\varepsilon_k$.\nThe noise symbols stand for unknown reals in $[-1, 1]$, shared by all affine forms of a computation.\nIts *range* is the [[interval|0Q1D85B8G6]] $[x_0 - r, x_0 + r]$ with radius $r = \\sum_{i=1}^{k} \\abs{x_i}$.\n*Affine arithmetic* computes with affine forms by the following rules.\n\n1. 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 $c, d$.\n2. 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 $[-1, 1]$.\n3. 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 $\\hat x, \\hat y$.\n\n## Remarks\n\nExample: $\\hat x = 2 + \\varepsilon_1$ represents $x \\in [1, 3]$, and $\\hat x - \\hat x = 0$ by rule 1, while [[interval arithmetic|QPKKTE1ZY0]] gives $[1, 3] - [1, 3] = [-2, 2]$.\nExample: $\\hat x \\cdot \\hat x = 4 + 4\\varepsilon_1 + \\varepsilon_2$ by rule 3, with range $[-1, 9]$, which contains the true range $[1, 9]$ of $x^2$.\nNon-example: replacing $x^2$ by its tangent $4 + 4\\varepsilon_1$ alone, without a new noise term, violates rule 2: at $\\varepsilon_1 = 1$ it gives $8$, but $x^2 = 9$.\nIn a computer the coefficients are rounded outward, and the rounding errors are also absorbed into a new noise term.\n",
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