Roundoff errors cannot be avoided when implementing numerical programs with finite precision. This problem becomes challenging when the program does not employ solely linear operations as non-linearities are inherent to many computational problems in real-world applications. To ensure safety of such non-linear programs, there is an obvious need for methods which output certificates that can be validated inside a proof assistant.

We present a framework to provide upper and lower bounds of absolute roundoff errors. This framework is based on optimization techniques employing robust semidefinite programming (SDP) and sparse sums of squares certificates, which can be formally checked inside the Coq theorem prover.

Our tool covers a wide range of nonlinear programs, including polynomials and transcendental operations as well as conditional statements. We illustrate the efficiency and precision of this tool on non-trivial programs coming from biology, optimization and space control.