  Deep Neural Networks (DNN) represent a performance-hungry application.
Floating-Point (FP) and custom floating-point-like arithmetic satisfies this
hunger. While there is need for speed, inference in DNNs does not seem to have
any need for precision. Many papers experimentally observe that DNNs can
successfully run at almost ridiculously low precision.
  The aim of this paper is two-fold: first, to shed some theoretical light upon
why a DNN's FP accuracy stays high for low FP precision. We observe that the
loss of relative accuracy in the convolutional steps is recovered by the
activation layers, which are extremely well-conditioned. We give an
interpretation for the link between precision and accuracy in DNNs.
  Second, the paper presents a software framework for semi-automatic FP error
analysis for the inference phase of deep-learning. Compatible with common
Tensorflow/Keras models, it leverages the frugally-deep Python/C++ library to
transform a neural network into C++ code in order to analyze the network's need
for precision. This rigorous analysis is based on Interval and Affine
arithmetics to compute absolute and relative error bounds for a DNN. We
demonstrate our tool with several examples.
