multiplication or , cannot be performed exactly, since the result would not be an affine form of the . In that case, one should take a suitable affine function G that approximates F to first order, in the ranges implied by and ; and compute , where is an upper bound for the absolute error in that range, and is a new symbolic variable not occurring in any previous form. The form then gives a guaranteed enclosure for the quantity Z; moreover, the affine forms jointly provide a guaranteed enclosure for the point (X,Y,...,Z), which is often much smaller than the Cartesian product of the ranges of the individual forms. Chaining operations Systematic use of this method allows arbitrary computations on given quantities to be replaced by equivalent computations on their affine forms, while preserving first-order correlations between the input and output and guaranteeing the complete enclosure of the joint range. One simply replaces each arithmetic operation or elementary function call in the formula by a call to the corresponding AA library routine. For smooth functions, the approximation errors made at each step are proportional to the square h2 of the width h of the input intervals. For this reason, affine arithmetic will often yield much tighter bounds than standard interval arithmetic (whose errors are proportional to h). Roundoff errors In order to provide guaranteed enclosure, affine arithmetic operations must account for the roundoff errors in the computation of the resulting coefficients . This cannot be done by rounding each in a specific direction, because any such rounding would falsify the dependencies between affine forms that share the symbol . Instead, one must compute an upper bound to the roundoff error of each , and add all those to the coefficient of the new symbol (rounding up). Thus, because of roundoff errors, even affine operations like Z = X and Z = X + Y will add the extra term . The handling of roundoff errors increases the code complexity and execution time of AA operations. In applications where those errors are known to be unimportant (because they are dominated by uncertainties in the input data and/or by the linearization errors), one may use a simplified AA library that does not implement roundoff error control. Affine projection model Affine arithmetic can be viewed in matrix form as follows. Let be all input and computed quantities in use at some point