Error, Precision & Latency¶
In binary arithmetic, precision is mostly bit-width. In stochastic computing, precision is mostly how long you wait.
Estimator variance (i.i.d. unipolar)¶
If bits are i.i.d. Bernoulli(\(x\)) and \(\hat{x} = N^{-1}\sum_t X_t\), then
\[\mathrm{Var}(\hat{x}) = \frac{x(1-x)}{N}.\]
Worst-case variance is at \(x=0.5\): \(\mathrm{Var}=1/(4N)\). Root-mean-square error therefore scales like \(1/(2\sqrt{N})\).
\[\text{Rough rule: } N \approx \frac{1}{4\,\varepsilon^2} \text{ for RMS }\varepsilon\text{ near }x=\tfrac12\]
| Target RMS \(\varepsilon\) | Ballpark \(N\) (i.i.d., \(x\approx0.5\)) |
|---|---|
| 0.05 | ~100 |
| 0.02 | ~625 |
| 0.01 | ~2500 |
Deterministic / low-discrepancy streams can beat this scaling for structured workloads — at the cost of more sophisticated SNGs.
Latency vs throughput¶
graph TD
N[Stream length N] --> Lat[Latency ≈ N cycles]
N --> Acc[Accuracy improves]
Par[Parallel wires / tiling] --> TP[Throughput up]
Par --> Area[Area up]Designers pick among:
- Long serial streams — tiny area, high latency
- Parallel bitstreams — \(k\) wires ≈ \(k\times\) throughput
- Hybrid stochastic–binary — SC for cheap nonlinearities, binary for precision-critical paths
Error types to separate¶
| Error | Cause | Mitigation |
|---|---|---|
| Sampling variance | Finite \(N\) | Longer streams, low-discrepancy |
| Correlation bias | Dependent operands | Independent SNGs, decorrelation |
| Quantization / clamping | Encoding range | Bipolar vs unipolar choice |
| Circuit faults | SEU, stuck-at | Often softened by density coding |
Reporting hygiene (for readers of papers)¶
When comparing designs via the Paper Map, check whether authors report:
- Stream length \(N\) (or equivalent progressive counter depth)
- RNG / SNG type and sharing
- Application metric and arithmetic MSE
- Technology node and energy measurement method
Without those, “\(10\times\) energy” claims are hard to interpret.