Numerical instability is a well-known problem that may cause serious runtime failures. This paper discusses the reason of instability in software development process, and presents a toolchain that not only detects the potential instability in software, but also diagnoses the reason for such instability. We classify the reason of instability into two categories. When it is introduced by software requirements, we call the instability caused by problem. In this case, it cannot be avoided by improving software development, but requires inspecting the requirements, especially the underlying mathematical properties. Otherwise, we call the instability caused by practice. We design our toolchain as 4 loosely-coupled tools, which combine stochastic arithmetic with infinite-precision testing. Each tool in our toolchain can be configured with different strategies according to the properties of the analyzed software. We evaluate our toolchain on subjects from literature. The results show that it effectively detects and separates the instabilities caused by problems from others. We also conduct an evaluation on the latest version of GNU Scientific Library, and the toolchain finds a few real bugs in the well-maintained and widely deployed numerical library. With the help of our toolchain, we report the details and fixing advices to the GSL buglist.