Assisting LLMs with code generation improved their performance on mathematical reasoning tasks. However, the evaluation of code-assisted LLMs is generally restricted to execution correctness, lacking a rigorous evaluation of their generated programs. In this work, we bridge this gap by conducting an in-depth analysis of code-assisted LLMs generated programs in response to math reasoning tasks, with a focus on evaluating the soundness of the underlying reasoning processes. For this purpose, we assess the generations of five LLMs, on several math datasets, both manually and automatically, and propose a taxonomy of generated programs based on their logical soundness. Our findings show that the capabilities of models significantly impact the logic implemented to solve the problem. Closed-source LLMs ground their programs in mathematical concepts, whereas open-source models often resort to unsound reasoning, relying on memorized information and exhaustive searches. Furthermore, increasing the difficulty of problems decreases sound generations for all models, revealing a critical shortcoming of LLMs on complex mathematics, contrary to what accuracy metrics suggest. Our work highlights the need for more holistic evaluations of code-assisted LLMs beyond execution accuracy metrics, toward a better understanding of LLMs' limits in the math domain.