math/rand/v2: remove bias in ExpFloat64 and NormFloat64
The original implementation of the ziggurat algorithm was designed for
32-bit random integer inputs. This necessitated reusing some low-order
bits for the slice selection and the random coordinate, which introduces
statistical bias. The result is that PractRand consistently fails the
math/rand normal and exponential sequences (transformed to uniform)
within 2 GB of variates.
This change adjusts the ziggurat procedures to use 63-bit random inputs,
so that there is no need to reuse bits between the slice and coordinate.
This is sufficient for the normal sequence to survive to 256 GB of
PractRand testing.
An alternative technique is to recalculate the ziggurats to use 1024
rather than 128 or 256 slices to make full use of 64-bit inputs. This
improves the survival of the normal sequence to far beyond 256 GB and
additionally provides a 6% performance improvement due to the improved
rejection procedure efficiency. However, doing so increases the total
size of the ziggurat tables from 4.5 kB to 48 kB.
goos: linux
goarch: amd64
pkg: math/rand/v2
cpu: AMD Ryzen 9 7950X 16-Core Processor
│
2703446c2e.amd64 │
e1bbe739fb.amd64 │
│ sec/op │ sec/op vs base │
SourceUint64-32 1.337n ± 1% 1.316n ± 2% ~ (p=0.024 n=20)
GlobalInt64-32 2.225n ± 2% 2.048n ± 1% -7.93% (p=0.000 n=20)
GlobalInt64Parallel-32 0.1043n ± 2% 0.1037n ± 1% ~ (p=0.587 n=20)
GlobalUint64-32 2.058n ± 1% 2.039n ± 2% ~ (p=0.030 n=20)
GlobalUint64Parallel-32 0.1009n ± 1% 0.1013n ± 1% ~ (p=0.984 n=20)
Int64-32 1.719n ± 2% 1.692n ± 2% ~ (p=0.085 n=20)
Uint64-32 1.669n ± 1% 1.643n ± 2% ~ (p=0.049 n=20)
GlobalIntN1000-32 3.321n ± 2% 3.287n ± 1% ~ (p=0.298 n=20)
IntN1000-32 2.479n ± 1% 2.678n ± 2% +8.01% (p=0.000 n=20)
Int64N1000-32 2.477n ± 1% 2.684n ± 2% +8.38% (p=0.000 n=20)
Int64N1e8-32 2.490n ± 1% 2.663n ± 2% +6.99% (p=0.000 n=20)
Int64N1e9-32 2.458n ± 1% 2.633n ± 1% +7.12% (p=0.000 n=20)
Int64N2e9-32 2.486n ± 2% 2.657n ± 1% +6.90% (p=0.000 n=20)
Int64N1e18-32 3.215n ± 2% 3.125n ± 2% -2.78% (p=0.000 n=20)
Int64N2e18-32 3.588n ± 2% 3.476n ± 1% -3.15% (p=0.000 n=20)
Int64N4e18-32 4.938n ± 2% 4.795n ± 1% -2.91% (p=0.000 n=20)
Int32N1000-32 2.673n ± 2% 2.485n ± 2% -7.02% (p=0.000 n=20)
Int32N1e8-32 2.631n ± 2% 2.457n ± 1% -6.63% (p=0.000 n=20)
Int32N1e9-32 2.628n ± 2% 2.452n ± 1% -6.70% (p=0.000 n=20)
Int32N2e9-32 2.684n ± 2% 2.453n ± 1% -8.61% (p=0.000 n=20)
Float32-32 2.240n ± 2% 2.254n ± 1% ~ (p=0.878 n=20)
Float64-32 2.253n ± 1% 2.262n ± 1% ~ (p=0.963 n=20)
ExpFloat64-32 3.677n ± 1% 3.777n ± 2% +2.71% (p=0.004 n=20)
NormFloat64-32 3.761n ± 1% 3.606n ± 1% -4.15% (p=0.000 n=20)
Perm3-32 33.55n ± 2% 33.12n ± 2% ~ (p=0.402 n=20)
Perm30-32 173.2n ± 1% 176.1n ± 1% +1.67% (p=0.000 n=20)
Perm30ViaShuffle-32 115.9n ± 1% 109.3n ± 1% -5.69% (p=0.000 n=20)
ShuffleOverhead-32 101.9n ± 1% 112.5n ± 1% +10.35% (p=0.000 n=20)
Concurrent-32 2.107n ± 6% 2.099n ± 0% ~ (p=0.051 n=20)
goos: darwin
goarch: arm64
pkg: math/rand/v2
cpu: Apple M1
│
2703446c2e.arm64 │
e1bbe739fb.arm64 │
│ sec/op │ sec/op vs base │
SourceUint64-8 2.275n ± 0% 2.290n ± 1% ~ (p=0.044 n=20)
GlobalInt64-8 2.154n ± 1% 2.180n ± 1% ~ (p=0.068 n=20)
GlobalInt64Parallel-8 0.4298n ± 0% 0.4294n ± 0% ~ (p=0.079 n=20)
GlobalUint64-8 2.160n ± 1% 2.170n ± 1% ~ (p=0.129 n=20)
GlobalUint64Parallel-8 0.4286n ± 0% 0.4283n ± 0% ~ (p=0.350 n=20)
Int64-8 2.491n ± 1% 2.481n ± 1% ~ (p=0.330 n=20)
Uint64-8 2.458n ± 0% 2.464n ± 1% ~ (p=0.351 n=20)
GlobalIntN1000-8 2.814n ± 2% 2.814n ± 0% ~ (p=0.325 n=20)
IntN1000-8 2.933n ± 0% 2.934n ± 2% ~ (p=0.079 n=20)
Int64N1000-8 2.962n ± 1% 2.957n ± 1% ~ (p=0.259 n=20)
Int64N1e8-8 2.960n ± 1% 2.935n ± 2% ~ (p=0.276 n=20)
Int64N1e9-8 2.935n ± 2% 2.935n ± 2% ~ (p=0.984 n=20)
Int64N2e9-8 2.934n ± 0% 2.933n ± 4% ~ (p=0.463 n=20)
Int64N1e18-8 3.777n ± 1% 3.781n ± 1% ~ (p=0.516 n=20)
Int64N2e18-8 4.359n ± 1% 4.362n ± 0% ~ (p=0.256 n=20)
Int64N4e18-8 6.536n ± 1% 6.576n ± 1% ~ (p=0.224 n=20)
Int32N1000-8 2.937n ± 0% 2.942n ± 2% ~ (p=0.312 n=20)
Int32N1e8-8 2.937n ± 1% 2.941n ± 1% ~ (p=0.463 n=20)
Int32N1e9-8 2.936n ± 0% 2.938n ± 2% ~ (p=0.044 n=20)
Int32N2e9-8 2.938n ± 2% 2.982n ± 2% ~ (p=0.174 n=20)
Float32-8 3.441n ± 0% 3.441n ± 0% ~ (p=0.064 n=20)
Float64-8 3.441n ± 0% 3.441n ± 0% ~ (p=0.826 n=20)
ExpFloat64-8 4.486n ± 0% 4.472n ± 0% -0.31% (p=0.000 n=20)
NormFloat64-8 4.721n ± 0% 4.716n ± 0% ~ (p=0.051 n=20)
Perm3-8 26.65n ± 0% 26.66n ± 0% ~ (p=0.080 n=20)
Perm30-8 143.2n ± 0% 143.3n ± 0% +0.10% (p=0.000 n=20)
Perm30ViaShuffle-8 143.0n ± 0% 142.9n ± 0% ~ (p=0.642 n=20)
ShuffleOverhead-8 120.6n ± 1% 121.1n ± 1% +0.41% (p=0.010 n=20)
Concurrent-8 2.399n ± 5% 2.379n ± 2% ~ (p=0.365 n=20)
goos: linux
goarch: 386
pkg: math/rand/v2
cpu: AMD Ryzen 9 7950X 16-Core Processor
│
2703446c2e.386 │
e1bbe739fb.386 │
│ sec/op │ sec/op vs base │
SourceUint64-32 2.072n ± 2% 2.087n ± 1% ~ (p=0.440 n=20)
GlobalInt64-32 3.546n ± 27% 3.538n ± 2% ~ (p=0.101 n=20)
GlobalInt64Parallel-32 0.3211n ± 0% 0.3207n ± 1% ~ (p=0.753 n=20)
GlobalUint64-32 3.522n ± 2% 3.543n ± 1% ~ (p=0.071 n=20)
GlobalUint64Parallel-32 0.3172n ± 0% 0.3170n ± 0% ~ (p=0.507 n=20)
Int64-32 2.520n ± 2% 2.548n ± 1% ~ (p=0.267 n=20)
Uint64-32 2.581n ± 1% 2.565n ± 2% ~ (p=0.143 n=20)
GlobalIntN1000-32 6.171n ± 1% 6.300n ± 1% ~ (p=0.037 n=20)
IntN1000-32 4.752n ± 2% 4.750n ± 0% ~ (p=0.984 n=20)
Int64N1000-32 5.429n ± 1% 5.515n ± 2% ~ (p=0.292 n=20)
Int64N1e8-32 5.469n ± 2% 5.527n ± 0% ~ (p=0.013 n=20)
Int64N1e9-32 5.489n ± 2% 5.531n ± 2% ~ (p=0.256 n=20)
Int64N2e9-32 5.492n ± 2% 5.514n ± 2% ~ (p=0.606 n=20)
Int64N1e18-32 8.927n ± 1% 9.059n ± 1% ~ (p=0.229 n=20)
Int64N2e18-32 9.622n ± 1% 9.594n ± 1% ~ (p=0.703 n=20)
Int64N4e18-32 12.03n ± 1% 12.05n ± 2% ~ (p=0.733 n=20)
Int32N1000-32 4.817n ± 1% 4.840n ± 2% ~ (p=0.941 n=20)
Int32N1e8-32 4.801n ± 1% 4.832n ± 2% ~ (p=0.228 n=20)
Int32N1e9-32 4.798n ± 1% 4.815n ± 2% ~ (p=0.560 n=20)
Int32N2e9-32 4.840n ± 1% 4.813n ± 1% ~ (p=0.015 n=20)
Float32-32 10.51n ± 4% 10.90n ± 2% +3.71% (p=0.007 n=20)
Float64-32 20.33n ± 3% 20.32n ± 4% ~ (p=0.566 n=20)
ExpFloat64-32 12.59n ± 2% 12.95n ± 3% +2.86% (p=0.002 n=20)
NormFloat64-32 7.350n ± 2% 7.570n ± 1% +2.99% (p=0.007 n=20)
Perm3-32 39.29n ± 2% 37.80n ± 2% -3.79% (p=0.000 n=20)
Perm30-32 219.1n ± 2% 214.0n ± 1% -2.33% (p=0.002 n=20)
Perm30ViaShuffle-32 189.8n ± 2% 188.7n ± 2% ~ (p=0.147 n=20)
ShuffleOverhead-32 158.9n ± 2% 160.8n ± 1% ~ (p=0.176 n=20)
Concurrent-32 3.306n ± 3% 3.288n ± 0% -0.54% (p=0.005 n=20)
For #61716.
Change-Id: I4c5fe710b310dc075ae21c97d1805bcc20db5050
Reviewed-on: https://go-review.googlesource.com/c/go/+/516275
Auto-Submit: Russ Cox <rsc@golang.org>
LUCI-TryBot-Result: Go LUCI <golang-scoped@luci-project-accounts.iam.gserviceaccount.com>
Reviewed-by: Dmitri Shuralyov <dmitshur@google.com>
Reviewed-by: Rob Pike <r@golang.org>