Expansion of Exp in Riemannian manifold

Question

Let $M$ be a smooth manifold (could be $\mathbb{R}^n$) and $g_1(\cdot,\cdot),g_2(\cdot,\cdot)$ two inner products. Let $p \in M$ and denote by $\exp_1$ and $\exp_2$ the corresponding exponential maps based at $p$: $\exp_i: T_pM \to M$; both are local diffeomorphism mapping $0 \in T_pM$ to $p\in M$. Then $\exp_1^{-1}\circ \exp_2: T_pM \to T_pM$ is well-defined locally around $0$. Is anything known about this function? For example, is its Taylor series known?