Performance profiling

V3 replaces V2's Eigen with my own spatial algebra on xsimd, and stores every field as a structure of arrays: each component of each field, say the x angular velocity of every joint, is its own contiguous, aligned array. Joints are sorted by their depth in the tree, and each group of SIMD lanes holds joints at the same depth. With many copies of one robot, that puts the same joint of different robots side by side, so one instruction can load, update or store a quantity for eight robots at once (eight lanes with AVX2, four in the default SSE2 build). A Valgrind trace of the compiled binary confirms it: 88.9% of all accesses to simulation data are whole, aligned 8-lane vector loads or stores. The payoff is that filling the lanes is nearly free. On a seven-link arm, one robot takes 3.1 µs per step, while eight robots in the 8-lane build take 4.3 µs, 5.7× the robot-steps per second.

To see where V3 stood, I benchmarked it against MuJoCo 3.15 on the same arm scenes, converted from the same .arow files, at the same 2 ms step, single-threaded on my laptop (Intel Core Ultra 9 285H). Batched across SIMD lanes, Achilles steps a seven-link arm 3.3× faster than MuJoCo's best single-threaded mode, reaching 5.45 million robot-steps per second at 64 robots. In double precision, MuJoCo's own, Achilles is still 1.5–1.6× faster at most batch sizes, and its results match MuJoCo's to about one part in 10¹¹. The one loss is a single seven-link arm, where most SIMD lanes sit empty (0.97×).