  Gradient approximations are a class of numerical approximation techniques
that are of central importance in numerical optimization. In derivative-free
optimization, most of the gradient approximations, including the simplex
gradient, centred simplex gradient, and adapted centred simplex gradient, are
in the form of simplex derivatives. Owing to machine precision, the
approximation accuracy of any numerical approximation technique is subject to
the influence of floating point errors. In this paper, we provide a general
framework for floating point error analysis of simplex derivatives. Our
framework is independent of the choice of the simplex derivative as long as it
satisfies a general form. We review the definition and approximation accuracy
of the generalized simplex gradient and generalized centred simplex gradient.
We define and analyze the accuracy of a generalized version of the adapted
centred simplex gradient. As examples, we apply our framework to the
generalized simplex gradient, generalized centred simplex gradient, and
generalized adapted centred simplex gradient. Based on the results, we give
suggestions on the minimal choice of approximate diameter of the sample set.
