For accurate line-by-line modeling of molecular cross sections several physical processes “beyond Voigt” have to be considered. For the speed-dependent Voigt and Rautian profiles (SDV, SDR) and the Hartmann-Tran profile the difference $w\left(\mathrm{i}{z}_{-}\right)-w\left(\mathrm{i}{z}_{+}\right)$ of two complex error functions (essentially Voigt functions) has to be evaluated where the function arguments z_± are given by the sum and difference of two square roots. These two terms describing z_± can be huge and the default implementation of the difference can lead to large cancellation errors. First we demonstrate that these problems can be avoided by a simple reformulation of $z_{-}$. Furthermore we show that a single rational approximation of the complex error function valid in the whole complex plane (e.g. by Humlíček, 1979 or Weideman, 1994) allows an evaluation of the SDV and SDR with four significant digits or better. Our benchmarks indicate that the SDV and SDR function evaluations are about a factor 2.2 slower compared to the Voigt function, but for evaluation of molecular cross sections this time lag does not significantly prolong the overall program execution because speed-dependent parameters are available only for a fraction of strong lines.

为了对分子截面进行精确的逐行建模，必须考虑“超越Voigt”的几个物理过程。对于速度相关的Voigt和Rautian配置文件（SDV，SDR）和Hartmann-Tran配置文件，差异$w（\mathrm{一世}{ž}_{-}）-w（\mathrm{一世}{ž}_{+}）$必须评估两个复数误差函数（基本上是Voigt函数）中的一个，其中函数参数z _±由两个平方根的和与差给出。这两个描述z _±的项可能很大，并且差异的默认实现可能会导致较大的取消误差。首先，我们证明，通过简单地重新定义${ž}_{-}$。此外，我们表明，在整个复平面上有效的复数误差函数的单个有理逼近（例如Humlíček，1979或Weideman，1994）允许用四个有效数字或更佳的数字来评估SDV和SDR。我们的基准测试表明，与Voigt函数相比，SDV和SDR函数的评估要慢2.2倍，但是对于分子截面的评估，此时间差不会显着延长整体程序的执行时间，因为速度相关的参数仅适用于强线的分数。