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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>
This commit is contained in:
Branden Brown
Gopher Robot
parent
ecda959b99
commit
488e2a56b9
@ -83,17 +83,17 @@ func Example_rand() {
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// Perm generates a random permutation of the numbers [0, n).
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show("Perm", r.Perm(5), r.Perm(5), r.Perm(5))
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// Output:
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// Float32 0.73793465 0.38461488 0.9940225
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// Float64 0.6919607852308565 0.29140004584133117 0.2262092163027547
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// ExpFloat64 0.10400903165715357 0.28855743344575835 0.20489656480442942
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// NormFloat64 -0.5602299711828513 -0.9211692958208376 -1.4262061075859056
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// Int32 1817075958 91420417 1486590581
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// Int64 5724354148158589552 5239846799706671610 5927547564735367388
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// Uint32 2295813601 961197529 3493134579
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// IntN(10) 4 5 1
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// Int32N(10) 8 5 4
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// Int64N(10) 2 6 3
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// Perm [3 4 2 1 0] [4 1 2 0 3] [0 2 1 3 4]
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// Float32 0.73793465 0.38461488 0.9940225
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// Float64 0.6919607852308565 0.29140004584133117 0.2262092163027547
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// ExpFloat64 0.27263589649304043 1.3214739789908194 2.223639057715668
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// NormFloat64 -0.09361151905162404 -1.3531915625472757 0.03212053591352371
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// Int32 1824388269 1817075958 91420417
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// Int64 3546343826724305832 5724354148158589552 5239846799706671610
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// Uint32 1380114714 2295813601 961197529
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// IntN(10) 8 4 5
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// Int32N(10) 1 8 5
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// Int64N(10) 4 2 6
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// Perm [0 4 1 3 2] [4 0 1 3 2] [4 1 3 0 2]
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}
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func ExamplePerm() {
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@ -29,8 +29,9 @@ const (
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// sample = ExpFloat64() / desiredRateParameter
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func (r *Rand) ExpFloat64() float64 {
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for {
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j := r.Uint32()
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i := j & 0xFF
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u := r.Uint64()
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j := uint32(u)
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i := uint8(u >> 32)
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x := float64(j) * float64(we[i])
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if j < ke[i] {
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return x
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@ -36,8 +36,9 @@ func absInt32(i int32) uint32 {
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// sample = NormFloat64() * desiredStdDev + desiredMean
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func (r *Rand) NormFloat64() float64 {
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for {
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j := int32(r.Uint32()) // Possibly negative
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i := j & 0x7F
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u := r.Uint64()
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j := int32(u) // Possibly negative
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i := u >> 32 & 0x7F
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x := float64(j) * float64(wn[i])
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if absInt32(j) < kn[i] {
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// This case should be hit better than 99% of the time.
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@ -225,26 +225,26 @@ func replace(t *testing.T, file string, new []byte) {
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}
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var regressGolden = []any{
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float64(0.1835616265352068), // ExpFloat64()
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float64(0.1747899228736829), // ExpFloat64()
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float64(2.369801563222863), // ExpFloat64()
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float64(1.8580757676846802), // ExpFloat64()
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float64(0.35731123690292155), // ExpFloat64()
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float64(0.5998175837039783), // ExpFloat64()
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float64(0.466149534807967), // ExpFloat64()
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float64(1.333748223451787), // ExpFloat64()
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float64(0.05019983258513916), // ExpFloat64()
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float64(1.4143832256421573), // ExpFloat64()
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float64(0.7274094466687158), // ExpFloat64()
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float64(0.9595398235158843), // ExpFloat64()
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float64(1.3010086894917756), // ExpFloat64()
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float64(0.8678483737499929), // ExpFloat64()
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float64(0.7958895614497015), // ExpFloat64()
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float64(0.12235329704897674), // ExpFloat64()
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float64(1.1625413819613253), // ExpFloat64()
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float64(1.2603945934386542), // ExpFloat64()
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float64(0.22199446394172706), // ExpFloat64()
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float64(2.248962105270165), // ExpFloat64()
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float64(0.018945741402288857), // ExpFloat64()
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float64(0.13829043737893842), // ExpFloat64()
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float64(1.1409883497761604), // ExpFloat64()
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float64(1.2449542292186253), // ExpFloat64()
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float64(0.4849966704675476), // ExpFloat64()
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float64(0.08948056191408837), // ExpFloat64()
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float64(0.41380878045769276), // ExpFloat64()
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float64(0.31325729628567145), // ExpFloat64()
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float64(0.23118058048615886), // ExpFloat64()
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float64(0.2090943007446), // ExpFloat64()
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float64(2.6861652769471456), // ExpFloat64()
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float64(1.3811947596783387), // ExpFloat64()
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float64(1.5595976199841015), // ExpFloat64()
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float64(2.3469708688771744), // ExpFloat64()
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float64(0.5882760784580738), // ExpFloat64()
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float64(0.33463787922271115), // ExpFloat64()
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float64(0.8799304551478242), // ExpFloat64()
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float64(1.616532211418378), // ExpFloat64()
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float64(0.09548420514080316), // ExpFloat64()
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float64(2.448910012295588), // ExpFloat64()
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float32(0.39651686), // Float32()
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float32(0.38516325), // Float32()
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@ -414,26 +414,26 @@ var regressGolden = []any{
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int64(339542337), // IntN(1000000000)
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int64(701992307), // IntN(1073741824)
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float64(0.6694336828657225), // NormFloat64()
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float64(0.7506128421991493), // NormFloat64()
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float64(-0.5466367925077582), // NormFloat64()
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float64(-0.8240444698703802), // NormFloat64()
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float64(0.11563765115029284), // NormFloat64()
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float64(-1.3442355710948637), // NormFloat64()
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float64(-1.0654999977586854), // NormFloat64()
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float64(0.15938628997241455), // NormFloat64()
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float64(-0.8046314635002316), // NormFloat64()
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float64(0.8323920113630076), // NormFloat64()
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float64(1.0611019472659846), // NormFloat64()
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float64(-0.8814992544664111), // NormFloat64()
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float64(0.9236344788106081), // NormFloat64()
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float64(-1.2854378982224413), // NormFloat64()
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float64(0.4683572952232405), // NormFloat64()
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float64(-0.5065217527091702), // NormFloat64()
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float64(-0.6460803205194869), // NormFloat64()
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float64(0.7913615856789362), // NormFloat64()
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float64(-1.6119549224461807), // NormFloat64()
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float64(0.16216183438701695), // NormFloat64()
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float64(0.06909351197715208), // NormFloat64()
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float64(0.5938704963270934), // NormFloat64()
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float64(1.306028863617345), // NormFloat64()
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float64(1.4117443127537266), // NormFloat64()
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float64(0.15696085092285333), // NormFloat64()
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float64(1.360954184661658), // NormFloat64()
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float64(0.34312984093649135), // NormFloat64()
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float64(0.7340067314938814), // NormFloat64()
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float64(0.22135434353553696), // NormFloat64()
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float64(-0.15741313389982836), // NormFloat64()
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float64(-1.080896970111088), // NormFloat64()
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float64(-0.6107370548788273), // NormFloat64()
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float64(-2.3550050260853643), // NormFloat64()
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float64(1.8363976597396832), // NormFloat64()
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float64(-0.7167650947520989), // NormFloat64()
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float64(0.6860847654927735), // NormFloat64()
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float64(0.3403802538398155), // NormFloat64()
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float64(-1.3884780626234523), // NormFloat64()
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float64(0.14097321427512907), // NormFloat64()
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float64(-1.032800550788109), // NormFloat64()
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[]int{}, // Perm(0)
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[]int{0}, // Perm(1)
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