Forecast horizon under precision

Measured

• f32 diverges at t ≈ 21.5, f64 at t ≈ 44.5, Float106 beyond T = 60
• the ε-law puts the same horizons near 17.6 / 39.8 / 81.3
• Lyapunov λ ≈ 0.906 on the Lorenz attractor

This example runs no flow solver. It measures the effect of scalar precision on forecast horizon in a chaotic system.

Every example carries a type alias:

pub type FloatType = f64;   // or f32, or Float106

The physics is written once against the scalar bound. Changing that one line re-instantiates the entire computation at a different precision. Here, the same Lorenz rate field is instantiated three times and each forecast is run until it diverges from a reference trajectory.

Horizon scaling

A chaotic system amplifies perturbations exponentially at a rate set by the leading Lyapunov exponent, here λ ≈ 0.906. Roundoff at machine epsilon is such a perturbation, and an initial error of size ε grows as e^{λt} until it reaches the scale of the attractor.

The resulting horizon scales as ln(1/ε)/λ:

f32       (ε ≈ 1.2e-7)    t ≈ 17.6
f64       (ε ≈ 2.2e-16)   t ≈ 39.8
Float106  (ε ≈ 1.0e-32)   t ≈ 81.3

And the measured horizons:

f32        t ≈ 21.5
f64        t ≈ 44.5
Float106   beyond T = 60 here; the law puts it near t ≈ 81

The two sets differ by roughly four time units. The README attributes the difference to bursty local stretching: the local expansion rate departs from the global average, so the measured crossing varies around the law’s prediction. The two sets measure different quantities and should not be mixed in a single citation.

Nine orders of magnitude of additional precision extend the horizon by roughly a factor of two, consistent with logarithmic scaling.

Divergence over time

  t        f32 vs f64      f64 vs Float106
  5.0        2.69e-5            2.31e-14
 25.0        1.99e1             4.01e-8
 45.0        1.71e1             2.00e0

At t = 25 the f32 divergence of 19.9 is the scale of the attractor itself, so the two trajectories share no information. At the same time f64 tracks Float106 to eight decimal places. By t = 45 the f64 trajectory has also diverged.

The f32 value at t = 45 is smaller than at t = 25. That decline is not recovery: two uncorrelated trajectories on a bounded attractor fluctuate around a typical separation. Past its horizon the error figure carries no information, and only the first threshold crossing is meaningful.

Chain composition

The run is composed as a causal chain:

CausalFlow::effect()
    .next(simulate)
    .next(analyse)
    .run(...)

PropagatingEffect short-circuits if a trajectory goes non-finite, so a diverged forecast stops the chain rather than feeding NaN into the analysis stage.

Applicability

For most CFD the precision floor is not the limiting term. Discretization and modelling error exceed f64 roundoff by many orders of magnitude.

Chaotic flow and long marched trajectories that feed a decision are the cases where it matters. The blackout corridor records a separate failure mode: at f32 it does not run, because a term in the ionization kernel evaluates to 4.4e-67 and underflows the exponent range at step 1.

No output.txt is committed for this example; the figures above come from its README. Its documented command also omits --release.

Stated limitation

For most CFD the precision floor is not the limiting term; discretization and modelling error exceed f64 roundoff. No output.txt is committed for this example; figures come from its README.