Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order $\overline{d}-1$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function $|\cdot |$ apart from arithmetic operations and $\overline{d}$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of $O\left({\overline{d}}^2\right)$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in $\overline{d}$. As already conjectured, we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains.

汤姆·斯特劳贝尔（Tom Streubel）观察到，对于abs正规形式的函数，任意阶的广义泰勒展开 $\overline{d}-\mathrm{1个}$可以通过算法分段微分生成。绝对值形式表示实值或向量值函数是由涉及绝对值函数的评估程序定义的$|\cdot |$ 除了算术运算和 $\overline{d}$时间连续可微的单变量内在函数。Streubel展开中的加法项是abs多项式，即既不涉及除法，也不涉及内在函数。在绝对值不出现绝对值的情况下，可以使用Moore递归通过评估程序传播单变量泰勒多项式，而计算量为$Ø（{\overline{d}}^{\mathrm{2个}}）$，前提是所有单变量内在函数均定义为线性ODE的解。这种规律性假设适用于所有标准内在函数，但对于不规则的初等函数，必须诉诸于Faa di Bruno公式，该公式具有指数复杂性。$\overline{d}$。正如已经推测的那样，我们证明了Moore递归可以适用于abs-normal情况的常规内在函数。最后，我们观察到内在函数是真正的解析函数，则展开式可以扩展为绝对收敛于球域上的无限级数。