Calculating the inverse kinematics (IK) is a fundamental challenge in robotics. Compared to numerical or learning-based approaches, analytical IK provides higher efficiency and accuracy. However, existing analytical approaches are difficult to use in most applications, as they require human ingenuity in the derivation process, are numerically unstable, or rely on time-consuming symbolic manipulation. In contrast, we propose a method that, for the first time, enables an analytical IK derivation and computation in less than a millisecond in total. Our work is based on an automatic online decomposition of the IK into pre-solved, numerically stable subproblems via a kinematic classification of the respective manipulator. In numerical experiments, we demonstrate that our approach is orders of magnitude faster in deriving the IK than existing tools that employ symbolic manipulation. Following this one-time derivation, our method matches and often surpasses baselines, such as IKFast, in terms of speed and accuracy during the computation of explicit IK solutions. Finally, we provide an open-source C++ toolbox with Python wrappers that substantially reduces the entry barrier to using analytical IK in applications like rapid prototyping and kinematic robot design.