How long does a float32 twin of a chaotic system shadow the float64 one, and do its statistics survive?
research/free-f32shadow
How long does a float32 twin of a chaotic system shadow the float64 one, and do its statistics survive? Lorenz and Rössler: float32 departs from float64 after a median 12.3 and 164 time units, like a float64 start error of about 5×10⁻⁶ (100× the float32 roundoff), but the Lyapunov exponent and time-averaged statistics agree within sampling error (stylised).
What it found
Lorenz-63 and Rössler integrated with the same RK4 scheme in float32 and float64 from the same start. Float32 leaves the float64 trajectory (distance > 1) after a median 12.3 time units on Lorenz and 164 on Rössler, which matches a float64 start error of about 5×10−6, roughly 100 times the float32 unit roundoff: departure time follows ln(1/δ0)/λ with per-step rounding setting the effective error, and that error grows slowly with the number of steps (4.5×10−6 to 1.3×10−5 for 8× the steps). The Lyapunov exponent (0.8995 vs 0.8997) and long-run time averages agree within sampling error (|z| < 1.2, 200 members), and a float16 run showed no detectable bias on 100 members either. Stylised: two autonomous flows, CPU numpy, no contact or policy.
A figure from its own output
Paper, code and results
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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.