diff --git a/README.md b/README.md index f94cb7f..aca7685 100644 --- a/README.md +++ b/README.md @@ -36,6 +36,14 @@ - **Fourier transforms** — package [`fft`](fft) is `numpy.fft` on arrays (`FFT`/`RFFT`/`FFT2`/`FFTN`/… with `n`/`s`, `axes` and `norm`, single precision kept), on [go-fft](https://github.com/go-fft/fft). +- **Random numbers** — package [`random`](random) is `numpy.random` with + **NumPy's numbers for the same seed**: `SeedSequence`, PCG64 (default), + PCG64DXSM, MT19937, Philox, SFC64; `Generator` (`DefaultRNG`: uniform, + integers, ziggurat normal/exponential, gamma, beta, binomial, Poisson, + choice, shuffle, … 40 samplers) and the legacy `RandomState` behind + `random.Seed(0)` / `random.Rand(…)`. Bit-exact against NumPy 2.5.3 on + x86-64; see [docs/random.md](docs/random.md) for the few ULP-level + exceptions (C-library `log`/`exp` rounding) and NumPy's own aarch64 drift. - **Notebooks and storage** — `Repr` prints like NumPy (summarised above 1000 elements), `Display` gives a notebook a text and an HTML table, and `MarshalBinary`/`UnmarshalBinary` encode an array the same way on every @@ -211,6 +219,12 @@ z := i8.AsType(ndarray.Complex128) // complex128 // numpy.fft spec, _ := fft.RFFT(row) // complex128, 2 bins +// numpy.random, the same numbers as NumPy for the same seed +rng := random.DefaultRNG(42) // np.random.default_rng(42) +noise, _ := rng.StandardNormal(2, 3) // rng.standard_normal((2, 3)) +random.Seed(0) // np.random.seed(0) +u, _ := random.Rand(3) // [0.5488135 0.71518937 0.60276338] + // linear algebra b, _ := ndarray.Arange(0, 6, 1) b, _ = b.Reshape(3, 2) diff --git a/docs/random.md b/docs/random.md new file mode 100644 index 0000000..7fcc901 --- /dev/null +++ b/docs/random.md @@ -0,0 +1,220 @@ +# random — numpy.random, the same numbers + +Package [`random`](../random) is `numpy.random` over go-ndarray arrays. Its goal +is migration: a notebook moved from NumPy to Go keeps its numbers for the same +seed. + +```go +rng := random.DefaultRNG(42) // np.random.default_rng(42) +x, _ := rng.StandardNormal(3, 4) // rng.standard_normal((3, 4)) +k, _ := rng.Integers(0, 10, 5) // rng.integers(0, 10, 5) + +random.Seed(0) // np.random.seed(0) +u, _ := random.Rand(3) // np.random.rand(3) +``` + +`Example`, `ExampleSeed` and `ExampleGenerator_ChoiceN` in +[`example_test.go`](../random/example_test.go) print NumPy's own numbers. + +## What is implemented + +| NumPy | here | +|---|---| +| `SeedSequence(entropy, spawn_key=, pool_size=)`, `.pool`, `.generate_state(n, uint32/uint64)`, `.spawn(n)` | `NewSeedSequence(v...)`, `NewSeedSequenceBig`, `NewSeedSequenceWith`, `Pool`, `GenerateState`, `GenerateState64`, `Spawn` | +| `PCG64`, `PCG64DXSM`, `MT19937`, `Philox` (also `counter=`, `key=`), `SFC64` | `NewPCG64` … `NewSFC64`, `New…FromSeedSequence`, `NewPhiloxKey` | +| `.jumped(n)`, `.advance(delta)`, `.random_raw(n)`, `.spawn(n)`, `.state` | `Jumped`, `Advance(*big.Int)`, `RandomRaw`, `Spawn`, `State`/`SetState` (typed structs) | +| `default_rng(seed)`, `Generator(bitgen)`, `.spawn(n)` | `DefaultRNG(seed...)`, `NewGenerator`, `Spawn` | +| `random(size, dtype)` | `Random(size...)`, `RandomOf(dt, size...)` | +| `integers(low, high, size, dtype, endpoint)` | `Integers(low, high, size...)`, `IntegersOf(dt, low, high, endpoint, size...)`, `IntegersUint64` | +| `standard_normal`, `normal`, `standard_exponential(method=)`, `exponential`, `uniform`, `standard_gamma(dtype=)`, `gamma` | `StandardNormal[Of]`, `Normal`, `StandardExponential[Of]`, `Exponential`, `Uniform`, `StandardGamma[Of]`, `Gamma` | +| `beta`, `chisquare`, `f`, `noncentral_chisquare`, `noncentral_f`, `standard_t`, `standard_cauchy`, `lognormal`, `laplace`, `logistic`, `gumbel`, `weibull`, `pareto`, `power`, `rayleigh`, `wald`, `triangular`, `vonmises` | same names in Go case | +| `binomial`, `negative_binomial`, `poisson`, `geometric`, `zipf`, `logseries` | same names, int64 results | +| `choice(a, size, replace, p, axis, shuffle)` | `ChoiceN(n, opts, size...)`, `Choice(a, opts, size...)` | +| `shuffle(x, axis)`, `permutation(n)`, `permutation(x, axis)`, `permuted(x, axis=)` | `Shuffle`, `Permutation`, `PermutationOf`, `Permuted`, `PermutedAxis` | +| `bytes(n)`, `dirichlet(alpha, size)`, `multinomial(n, pvals, size)` | `Bytes`, `Dirichlet`, `Multinomial` | +| `RandomState(seed)`, `.seed`, `.get_state`/`.set_state` | `NewRandomState(uint32)`, `NewRandomStateArray`, `NewRandomStateFrom(bitgen)`, `Seed`, `SeedArray`, `GetState`/`SetState` | +| `rand`, `randn`, `randint`, `random_sample`, `normal`, `uniform`, `choice`, `shuffle`, `permutation`, `binomial`, `poisson`, `standard_normal`, `standard_exponential`, `exponential`, `standard_gamma`, `gamma`, `chisquare`, `lognormal`, `multinomial`, `bytes` | `RandomState` methods of the same names | +| `np.random.seed`, `rand`, `randn`, `randint`, `random_sample`, `normal`, `uniform`, `choice`, `shuffle`, `permutation`, `binomial`, `poisson` | package functions over a shared `RandomState` behind a mutex | + +The size comes last; no size gives a 0-d array holding what NumPy returns as a +scalar (`size=None`), which draws exactly the same values. `ChoiceOptions`' +zero value is NumPy's defaults (`NoReplace`, `NoShuffle` are the negations). + +**Not implemented:** array-valued (broadcast) distribution parameters and +integer bounds, `out=`, `hypergeometric`, `multivariate_normal`, +`multivariate_hypergeometric`, the `RandomState` samplers not listed above +(`beta`, `f`, `pareto` …: NumPy 1.16's versions differ from `Generator`'s and +were not ported), `tomaxint`, `random_integers`, pickling, and `float16`. + +## Verification + +`random/testdata/cases.py` runs NumPy 2.5.3 and records: + +- **SeedSequence**: the pool, 9 uint32 and 5 uint64 state words, and the pools + and spawn keys of 4 children, for 10 entropies (0, 42, 2⁶⁴−1, `[1,2,3]`, + 2¹²⁸+12345, a 10-word list, `[]`, pool size 8, spawn keys with a 2⁴⁰ entry, + a key after a 6-word entropy); +- **bit generators**: for each of the five and six seeds (0, 1, 42, 2⁶⁴−1, + `[1,2,3]`, 2¹²⁸+12345), the seeded state, 100 raw outputs (700 for MT19937, + which crosses a regeneration), `jumped(1)`, `jumped(3)`, `advance(2¹⁰⁰+7)`, + `advance(-3)`, two spawned children; Philox from an explicit counter and key + and across a counter carry; legacy MT19937 seeding from 6 integer and array + seeds, and its jump; +- **2,222 scripts, 2,492 calls, 81,806 values**: every Generator function on + PCG64 with every seed, every dtype and bound case of `integers` (9 dtypes, + open and closed, full ranges, ranges of 2³²−1, 2³², 3·2³⁰, 3·2⁶², empty + ranges); a subset on the four other bit generators; an interleaved script + per generator and seed (the buffered upper half of a 64-bit draw outlives a + call); `RandomState` on 6 seeds (including a 700-word array); +- **50 long runs as SHA-256 digests** (4.49 million values): `random`, + `standard_normal` and `standard_exponential` (both dtypes, 200,000 each, + which reaches the ziggurat tails and wedges), `integers` (Lemire with + rejections, 8- and 16-bit buffering, bool), `binomial` (inversion and + BTPE), `poisson` (both methods), `choice` with and without replacement, and + `RandomState`'s `rand`, `randint`, `binomial`, `poisson`, `uniform`. + +The fixture is embedded, so CI's emulated targets check it too. It passes, +unchanged, on darwin/arm64, darwin/amd64 (Rosetta), and linux 386, arm, +riscv64, ppc64le and **s390x** (big-endian: `Bytes` gives NumPy's +little-endian bytes) under qemu. + +**Integers, raw output, and every sampler that is plain arithmetic match NumPy +to the bit** — including the ziggurat normal and exponential (tails included, +over 200,000 draws), binomial, Poisson, geometric, choice, shuffle and the +legacy `randint`/`rand`/`binomial`/`poisson`. + +### Negative controls + +Each mutation below was made on purpose and the suite run: + +| mutation | result | +|---|---| +| PCG64 multiplier, low word + 2 | 1,155 failures | +| SeedSequence `MIX_MULT_L` + 2 | 1,998 failures | +| SeedSequence `MULT_A` + 2 | 2,005 failures | +| one ziggurat `ki_double` entry losing its last hex digit (an actual bug of the first table converter, which stripped a trailing `F` as if it were a C suffix) | 15 failures, all in the long runs | +| `RandomState.randint` with Lemire instead of masked rejection | 44 failures | +| MT19937 seeded with `pos = 624` instead of NumPy's `623` | 266 failures | +| legacy BTPE adding the last two Stirling terms with the corrected signs | the `RandomState` binomial digests fail | +| legacy BTPE constant 13680 → 13860 | **not detected**: the term changes by ~2·10⁻⁹ and flips no decision in 10⁶ draws at any of four (n, p) tried | + +## Where the numbers can differ: the C library + +Some samplers call `log`, `exp`, `log1p`, `expm1`, `pow`, `cos` or `acos`. +NumPy takes them from the C library; Go's `math` package does not round them +the same way. Measured against the libm of the x86-64 NumPy used for the +fixture (Apple's), Go's `math` differs in 6.5% (`log`) to 71% (`pow`) of +random arguments. Used as-is, that made 3% of `np.random.randn` values and +10% of `lognormal` values one ULP off. + +So `random` rounds `log`, `log1p`, `exp`, `expm1` and `pow` **correctly** +([`libm.go`](../random/libm.go)): a fast evaluation to about 2⁻⁶⁸ with Ziv's +rounding test, and a double-double fallback (about 2⁻¹⁰⁰) for the one argument +in ~2¹⁵ that needs it. Checked against mpmath at 300 bits on 63,000 arguments +(0 misrounded; the committed `libm.json` keeps 8,400), and the fast paths +against the fallback on 400,000 more. The code converts every product +explicitly, so Go cannot fuse it into an FMA, and the result is the same on +every architecture. + +The C libraries round these functions correctly almost always — Apple's x86-64 +libm misrounds 0.06% (`log`) to 0.3% (`exp`) of arguments; `pow(x, 0.5)` is +misrounded for 0.19% (it is not `sqrt`). What remains, on 10⁶ draws against +x86-64 NumPy, seed 0: + +| sampler | values that differ (of 10⁶) | max ULP | with Go's `math` instead | +|---|---:|---:|---:| +| `RandomState.randn` | 294 | 3 | 29,631 | +| `RandomState.normal(1, 3)` | 217 | 32¹ | 22,429 | +| `RandomState.standard_gamma(3)` | 72 | 6 | 5,762 | +| `RandomState.standard_exponential` | 664 | 1 | 68,249 | +| `standard_gamma(0.5)` (pow) | 1,870 | 3 | 16,212 | +| `beta(0.5, 0.5)` (pow) | 1,693 | 4 | 1,693 | +| `lognormal(0.5, 0.25)` | 2,263 | 1 | 99,340 | +| `laplace` / `logistic` | 690 / 601 | 1 | 68,357 / 68,518 | +| `gumbel` | 1,297 | 2049¹ | 131,066 | +| `standard_exponential(method="inv")` | 692 | 1 | 68,383 | +| `pareto(3)` (expm1) | 4,645 | 1 | 65,593 | +| `weibull(2)` (pow) | 1,777 | 1 | 1,779 | +| `vonmises(0.5, 2)` (cos, acos) | 126,540 | 2¹⁸ ¹ | 126,540 | +| `standard_normal`, `standard_gamma(3)`, `beta(2, 3)`, `chisquare`, `standard_t`, `binomial`, `poisson(3)`, `poisson(50)`, `geometric(0.05)` | **0** | — | 0 – 2 | + +¹ Large ULP counts are absolute errors of about one ULP of the inner value, +measured relative to a result near zero (`1 + 3z` for `z ≈ −1/3`; +`log(-log(U))` for `U` near 1/e; an angle wrapped to near 0). + +Each of the residual differences checked by hand (with mpmath) is a value +NumPy's libm rounded incorrectly. `vonmises` keeps Go's `cos` and `acos`: +Apple's `cos` itself misrounds 17% of arguments in [0, π], so rounding +correctly would not bring it closer. No difference has ever changed an +accept/reject decision — every integer-valued sampler matches exactly — and +the test bounds each float sampler's ULP distance individually +(`libmULP` in `numpy_test.go`). + +## NumPy does not agree with itself + +The same fixture generated by NumPy 2.5.3's **aarch64** macOS build differs from +its x86-64 build: the C compiler fuses `a*b + c` into one FMA there (clang's +default `-ffp-contract=on`), and Apple's aarch64 libm rounds differently. + +| | values that differ, aarch64 vs x86-64 NumPy | +|---|---| +| `Generator.uniform` | 20.8% of 200,000 (`low + range*U` is fused) | +| `RandomState.randn` | 865 of 6,072 (14%) | +| `RandomState.normal` | 1,063 of 4,048 (26%) | +| `RandomState.standard_gamma` | 481 of 4,108 | +| `Generator.standard_gamma`, `chisquare`, `f`, `noncentral_*`, `beta`, `gamma`, `lognormal`, `standard_t`, `vonmises` | a few % | +| digests of `Generator.normal`, `uniform`, `standard_gamma` | differ | + +This package follows the C source operation by operation without fusing, +which is what NumPy's x86-64 builds (Linux, Windows, Intel Macs) compute, and +it gives the same numbers on every architecture. A notebook run on an Apple +silicon Mac or an aarch64 Linux box will see NumPy's last bits move in those +samplers; that is NumPy, not a migration error. + +## Deliberate differences + +| NumPy | here | why | +|---|---|---| +| `randint`'s default dtype is C `long`: int64 on Linux/macOS, **int32 on Windows** | always int64 | the same numbers on every platform; `RandintOf(ndarray.Int32, …)` for Windows' | +| `size=None` returns a Python scalar | a 0-d array | one return type; same draws | +| `shuffle` works on views | `Shuffle` needs an array that owns its storage (`ErrNotContiguous` for a view) | ndarray hands out a view's elements as a copy | +| `permuted(x)` of a Fortran-ordered `x` shuffles in memory order | always C order | ndarray arrays are row-major | +| NumPy's `(int)` of ±∞ in rejection branches is undefined C | the same branch is rejected (the `v == 0` tests) | — | +| `zipf`'s `X > LONG_MAX` rejection | kept as a condition, unreachable short of `pow` rounding past 2⁶³ | — | +| `exp` of an argument whose result is subnormal | rounded twice (to 53 bits, then to the subnormal) | only below 2⁻¹⁰²² | + +## Performance + +Apple M4 Max, 16 cores, shared and loaded (load average 10.7–14.4 during the +runs). Seven interleaved rounds: the Go benchmark (`go test -bench`, 20 +iterations) then NumPy 2.5.3 aarch64 (median of 20 calls); medians of the +rounds, in milliseconds for 10⁶ values: + +| | Go | NumPy | Go / NumPy | Go range | NumPy range | +|---|---:|---:|---:|---|---| +| `random(10**6)` | 3.51 | 3.13 | 1.12 | 3.47–3.83 | 3.09–3.41 | +| `random(10**6, dtype=float32)` | 1.89 | 1.64 | 1.16 | 1.88–2.24 | 1.62–1.65 | +| `integers(0, 1000, 10**6)` | 2.31 | 1.70 | 1.36 | 2.29–2.32 | 1.68–1.71 | +| `standard_normal(10**6)` | 4.12 | 3.68 | 1.12 | 4.07–4.16 | 3.67–3.72 | +| `RandomState(0).randn(10**6)` | 20.36 | 9.65 | 2.11 | 19.99–20.56 | 9.61–9.68 | + +Go is **slower everywhere**: 12–36% for the Generator, 2.1× for the legacy +`randn`. The Generator gap is the 128-bit LCG step (3.3 ns per output, a +dependent multiply chain) plus zeroing the result (`ZerosOf`; NumPy's +`np.empty` does not); the default PCG64 already gets direct, inlinable calls +for these four functions. `randn` pays for a correctly rounded `log` on every +pair (6 ns against 3.4 ns for `math.Log`) and for MT19937 behind an +interface. The samplers that call `pow`/`exp` per draw cost about what +`math.Pow`/`math.Exp` would (19.5 vs 18.4 ns; 6.3 vs 3.9 ns). + +## Regenerating + +```sh +uv run --python cpython-3.12-macos-x86_64 --with numpy==2.5.3 python random/testdata/cases.py > random/testdata/cases.json +uv run --with mpmath python random/testdata/libm.py > random/testdata/libm.json +uv run --with mpmath python random/testdata/libm_tables.py > random/zz_libm_tables.go +python3 random/testdata/zig2go.py /numpy/random/src/distributions/ziggurat_constants.h random/zz_ziggurat.go +``` + +`random/testdata/quant.py` and `TestQuantifyLibm` (run with +`RANDOM_QUANT_DIR`) reproduce the 10⁶-draw table. diff --git a/random/api_test.go b/random/api_test.go new file mode 100644 index 0000000..0122ba1 --- /dev/null +++ b/random/api_test.go @@ -0,0 +1,389 @@ +package random + +import ( + "errors" + "math" + "math/big" + "testing" + + "github.com/go-ndarray/ndarray" +) + +func mustErr(t *testing.T, what string, err error) { + t.Helper() + if err == nil { + t.Errorf("%s: no error", what) + } +} + +func must[T any](v T, err error) T { + if err != nil { + panic(err) + } + return v +} + +// TestArgumentErrors checks that every argument numpy rejects is rejected, +// with an error that wraps ErrValue. +func TestArgumentErrors(t *testing.T) { + g := DefaultRNG(1) + nan := math.NaN() + _, errNeg := g.Random(-1) + errs := map[string]error{ + "random -1": errNeg, + } + add := func(name string) func(*ndarray.Array, error) { + return func(_ *ndarray.Array, err error) { errs[name] = err } + } + add("random int")(g.RandomOf(ndarray.Int8)) + add("normal int")(g.StandardNormalOf(ndarray.Int8)) + add("exp int")(g.StandardExponentialOf(ndarray.Int8, "zig")) + add("exp method")(g.StandardExponentialOf(ndarray.Float64, "bad")) + add("gamma int")(g.StandardGammaOf(ndarray.Int8, 1)) + add("normal scale")(g.Normal(0, -1)) + add("exponential")(g.Exponential(-1)) + add("uniform inf")(g.Uniform(-math.MaxFloat64, math.MaxFloat64)) + add("uniform nan")(g.Uniform(nan, 1)) + add("uniform reversed")(g.Uniform(1, 0)) + add("standard_gamma")(g.StandardGamma(-1)) + add("gamma shape")(g.Gamma(-1, 1)) + add("gamma scale")(g.Gamma(1, -1)) + add("beta a")(g.Beta(0, 1)) + add("beta b")(g.Beta(1, 0)) + add("chisquare")(g.ChiSquare(0)) + add("f num")(g.F(0, 1)) + add("f den")(g.F(1, 0)) + add("ncchisq df")(g.NoncentralChiSquare(0, 1)) + add("ncchisq nonc")(g.NoncentralChiSquare(1, -1)) + add("ncf num")(g.NoncentralF(0, 1, 1)) + add("ncf den")(g.NoncentralF(1, 0, 1)) + add("ncf nonc")(g.NoncentralF(1, 1, -1)) + add("t")(g.StandardT(0)) + add("vonmises")(g.VonMises(0, -1)) + add("pareto")(g.Pareto(0)) + add("weibull")(g.Weibull(-1)) + add("power")(g.Power(0)) + add("laplace")(g.Laplace(0, -1)) + add("gumbel")(g.Gumbel(0, -1)) + add("logistic")(g.Logistic(0, -1)) + add("lognormal")(g.Lognormal(0, -1)) + add("rayleigh")(g.Rayleigh(-1)) + add("wald mean")(g.Wald(0, 1)) + add("wald scale")(g.Wald(1, 0)) + add("triangular left")(g.Triangular(2, 1, 3)) + add("triangular mode")(g.Triangular(0, 4, 3)) + add("triangular equal")(g.Triangular(1, 1, 1)) + add("binomial p")(g.Binomial(1, 2)) + add("binomial n")(g.Binomial(-1, 0.5)) + add("negbin nan")(g.NegativeBinomial(nan, 0.5)) + add("negbin n")(g.NegativeBinomial(0, 0.5)) + add("negbin p")(g.NegativeBinomial(1, 0)) + add("negbin large")(g.NegativeBinomial(1e18, 1e-10)) + add("poisson neg")(g.Poisson(-1)) + add("poisson large")(g.Poisson(1e19)) + add("zipf")(g.Zipf(1)) + add("geometric")(g.Geometric(0)) + add("logseries")(g.Logseries(1)) + add("dirichlet")(g.Dirichlet([]float64{1, -1})) + add("dirichlet size")(g.Dirichlet([]float64{1}, -1)) + add("multinomial empty")(g.Multinomial(1, nil)) + add("multinomial p")(g.Multinomial(1, []float64{2, 0})) + add("multinomial sum")(g.Multinomial(1, []float64{0.7, 0.7, 0})) + add("multinomial n")(g.Multinomial(-1, []float64{1})) + add("multinomial size")(g.Multinomial(1, []float64{1}, -1)) + add("integers dtype")(g.IntegersOf(ndarray.Float32, 0, 1, false)) + add("integers low")(g.IntegersOf(ndarray.Int8, -200, 1, false)) + add("integers high")(g.IntegersOf(ndarray.Int8, 0, 200, false)) + add("integers empty")(g.IntegersOf(ndarray.Int8, 5, 5, false)) + add("integers min")(g.IntegersOf(ndarray.Int64, 0, math.MinInt64, false)) + add("integers closed")(g.IntegersOf(ndarray.Int8, 5, 4, true)) + add("integers u64 neg")(g.IntegersOf(ndarray.Uint64, -1, 4, false)) + add("integers size")(g.IntegersOf(ndarray.Int8, 0, 4, false, -1)) + add("integers u64 size")(g.IntegersUint64(0, 4, false, -1)) + add("integers u64 zero")(g.IntegersUint64(0, 0, false)) + add("integers u64 empty")(g.IntegersUint64(5, 5, false)) + add("integers u64 closed")(g.IntegersUint64(5, 4, true)) + add("permutation")(g.Permutation(-1)) + add("permutationOf 0-d")(g.PermutationOf(ndarray.Scalar(1), 0)) + add("permutationOf axis")(g.PermutationOf(namedArray("arange12"), 2)) + add("permuted axis")(g.PermutedAxis(namedArray("arange12"), -3)) + add("choiceN")(g.ChoiceN(0, ChoiceOptions{}, 1)) + add("choice 0-d")(g.Choice(ndarray.Scalar(1), ChoiceOptions{})) + add("choice axis")(g.Choice(namedArray("arange12"), ChoiceOptions{Axis: 2})) + add("choice empty")(g.Choice(must(ndarray.ZerosOf(ndarray.Int8, 0)), ChoiceOptions{}, 2)) + add("choice p size")(g.ChoiceN(3, ChoiceOptions{P: []float64{1}})) + add("choice p nan")(g.ChoiceN(2, ChoiceOptions{P: []float64{nan, 1}})) + add("choice p neg")(g.ChoiceN(2, ChoiceOptions{P: []float64{-1, 2}})) + add("choice p sum")(g.ChoiceN(2, ChoiceOptions{P: []float64{0.5, 0.6}})) + add("choice negative")(g.ChoiceN(3, ChoiceOptions{}, -1)) + add("choice too many")(g.ChoiceN(3, ChoiceOptions{NoReplace: true}, 4)) + add("choice nonzero")(g.ChoiceN(3, ChoiceOptions{NoReplace: true, P: []float64{1, 0, 0}}, 2)) + add("normal size")(g.Normal(0, 1, -1)) + add("random32 size")(g.RandomOf(ndarray.Float32, -1)) + add("poisson size")(g.Poisson(1, -1)) + add("standard_normal size")(g.StandardNormal(-1)) + add("integers closed zero")(g.IntegersOf(ndarray.Int8, 0, -1, true)) + add("choice p")(g.Choice(namedArray("floats6"), ChoiceOptions{P: []float64{1}})) + add("legacy choice bad p")(NewRandomState(1).Choice(namedArray("floats6"), ChoiceOptions{P: []float64{1}})) + add("legacy choice huge")(NewRandomState(1).ChoiceN(1<<62, ChoiceOptions{NoReplace: true}, 1)) + add("choice shape")(g.Choice(namedArray("floats6"), ChoiceOptions{}, -1, -1)) + add("choiceN shape")(g.ChoiceN(6, ChoiceOptions{}, -1, -1)) + errs["shuffle 0-d"] = g.Shuffle(ndarray.Scalar(1), 0) + errs["shuffle axis"] = g.Shuffle(namedArray("arange12"), 5) + view := must(namedArray("arange12").Slice(ndarray.R(0, 2))) + errs["shuffle view"] = g.Shuffle(view, 0) + + r := NewRandomState(1) + add("legacy normal")(r.Normal(0, -1)) + add("legacy lognormal")(r.Lognormal(0, -1)) + add("legacy uniform")(r.Uniform(0, math.Inf(1))) + add("legacy exponential")(r.Exponential(-1)) + add("legacy gamma shape")(r.Gamma(-1, 1)) + add("legacy gamma scale")(r.Gamma(1, -1)) + add("legacy chisquare")(r.ChiSquare(0)) + add("legacy binomial p")(r.Binomial(1, -1)) + add("legacy binomial n")(r.Binomial(-1, 0.5)) + add("legacy poisson")(r.Poisson(-1)) + add("legacy permutation")(r.Permutation(-1)) + add("legacy permutationOf")(r.PermutationOf(ndarray.Scalar(1))) + add("legacy choiceN")(r.ChoiceN(0, ChoiceOptions{}, 1)) + add("legacy choice 2-d")(r.Choice(namedArray("arange12"), ChoiceOptions{})) + add("legacy choice empty")(r.Choice(must(ndarray.ZerosOf(ndarray.Int8, 0)), ChoiceOptions{}, 1)) + add("legacy choice p")(r.ChoiceN(2, ChoiceOptions{P: []float64{1}})) + add("legacy choice too many")(r.ChoiceN(2, ChoiceOptions{NoReplace: true}, 3)) + add("legacy choice shape")(r.Choice(namedArray("floats6"), ChoiceOptions{}, -1, -1)) + add("legacy choiceN shape")(r.ChoiceN(6, ChoiceOptions{}, -1, -1)) + add("legacy choice perm")(r.ChoiceN(-5, ChoiceOptions{NoReplace: true})) + errs["legacy shuffle 0-d"] = r.Shuffle(ndarray.Scalar(1)) + errs["legacy shuffle view"] = r.Shuffle(view) + _, err := NewRandomStateArray(nil) + errs["legacy seed empty"] = err + nr := NewRandomStateFrom(NewPCG64(1)) + errs["legacy seed pcg"] = nr.Seed(1) + errs["legacy seedArray pcg"] = nr.SeedArray([]uint32{1}) + errs["legacy seedArray empty"] = r.SeedArray(nil) + _, errs["legacy getstate pcg"] = nr.GetState() + errs["legacy setstate pcg"] = nr.SetState(LegacyState{}) + + _, err = NewSeedSequenceWith(SeedSequenceOptions{PoolSize: 2}) + errs["seedseq pool"] = err + _, err = NewSeedSequenceWith(SeedSequenceOptions{Entropy: []uint64{1}, Words: []uint32{1}}) + errs["seedseq both"] = err + _, err = NewSeedSequenceBig(big.NewInt(-1)) + errs["seedseq negative"] = err + _, err = NewGenerator(NewMT19937Legacy(1)).Spawn(1) + errs["spawn legacy"] = err + _, err = NewGenerator(stubBG{}).Spawn(1) + errs["spawn stub"] = err + _, err = NewPCG64(1).Spawn(-1) + errs["spawn negative"] = err + _, err = DefaultRNG(1).Spawn(-1) + errs["generator spawn negative"] = err + + for name, err := range errs { + mustErr(t, name, err) + if err != nil && !errors.Is(err, ErrValue) && !errors.Is(err, ErrNotContiguous) && + !errors.Is(err, ErrNoSeedSequence) && !errors.Is(err, ndarray.ErrShapeMismatch) { + t.Logf("%s: %v (not ErrValue)", name, err) + } + } + if !errors.Is(errs["shuffle view"], ErrNotContiguous) { + t.Errorf("shuffle of a view: %v", errs["shuffle view"]) + } + defer func() { + if recover() == nil { + t.Error("Spawn(-1) did not panic") + } + }() + NewSeedSequence(1).Spawn(-1) +} + +// TestGlobalFunctionsAreRandomState checks np.random.seed(0) followed by the +// module functions against a RandomState(0). +func TestGlobalFunctionsAreRandomState(t *testing.T) { + Seed(7) + r := NewRandomState(7) + same := func(name string, a, b *ndarray.Array) { + t.Helper() + if a.String() != b.String() { + t.Errorf("%s: %v, RandomState %v", name, a, b) + } + } + same("Rand", must(Rand(3)), must(r.Rand(3))) + same("Randn", must(Randn(3)), must(r.Randn(3))) + same("RandomSample", must(RandomSample(2)), must(r.RandomSample(2))) + same("Randint", must(Randint(0, 9, 4)), must(r.Randint(0, 9, 4))) + same("Normal", must(Normal(1, 2, 2)), must(r.Normal(1, 2, 2))) + same("Uniform", must(Uniform(1, 2, 2)), must(r.Uniform(1, 2, 2))) + a := namedArray("floats6") + same("Choice", must(Choice(a, ChoiceOptions{}, 3)), must(r.Choice(a, ChoiceOptions{}, 3))) + x, y := namedArray("arange10"), namedArray("arange10") + if err := Shuffle(x); err != nil { + t.Fatal(err) + } + if err := r.Shuffle(y); err != nil { + t.Fatal(err) + } + same("Shuffle", x, y) + same("Permutation", must(Permutation(5)), must(r.Permutation(5))) + same("Binomial", must(Binomial(10, 0.5, 3)), must(r.Binomial(10, 0.5, 3))) + same("Poisson", must(Poisson(4, 3)), must(r.Poisson(4, 3))) +} + +func TestStateRoundTrips(t *testing.T) { + check := func(name string, a, b BitGenerator) { + t.Helper() + if x, y := RandomRaw(a, 3), RandomRaw(b, 3); x[0] != y[0] || x[2] != y[2] { + t.Errorf("%s: %v after SetState, %v", name, x, y) + } + } + p := NewPCG64(3) + p.Uint32() + q := NewPCG64(9) + q.SetState(p.State()) + if q.Uint32() != p.Uint32() { + t.Error("PCG64 buffered half not restored") + } + check("PCG64", p, q) + d, d2 := NewPCG64DXSM(3), NewPCG64DXSM() + d2.SetState(d.State()) + check("PCG64DXSM", d, d2) + m, m2 := NewMT19937(3), NewMT19937() + m2.SetState(m.State()) + check("MT19937", m, m2) + ph, ph2 := NewPhilox(3), NewPhilox() + ph.Uint64() + ph2.SetState(ph.State()) + check("Philox", ph, ph2) + s, s2 := NewSFC64(3), NewSFC64() + s2.SetState(s.State()) + check("SFC64", s, s2) + for _, sp := range []Spawner{p, d, m, ph, s} { + if sp.SeedSeq() == nil { + t.Errorf("%T has no SeedSequence", sp) + } + } + + r := NewRandomState(5) + r.Randn(1) // leaves a cached Gaussian + st := must(r.GetState()) + want := must(r.Randn(4)) + r2 := NewRandomStateEntropy() + if err := r2.SetState(st); err != nil { + t.Fatal(err) + } + if got := must(r2.Randn(4)); got.String() != want.String() { + t.Errorf("RandomState after SetState: %v, want %v", got, want) + } + if err := r2.SeedArray([]uint32{1, 2, 3}); err != nil { + t.Fatal(err) + } + r3 := must(NewRandomStateArray([]uint32{1, 2, 3})) + if a, b := must(r2.Rand(2)), must(r3.Rand(2)); a.String() != b.String() { + t.Errorf("SeedArray: %v, NewRandomStateArray %v", a, b) + } + if r2.BitGenerator() == nil || DefaultRNG(1).BitGenerator() == nil { + t.Error("nil bit generator") + } + kids := must(DefaultRNG(4).Spawn(2)) + if len(kids) != 2 { + t.Error("Spawn") + } +} + +func TestSeedSequenceAccessors(t *testing.T) { + ss := must(NewSeedSequenceBig(new(big.Int).Lsh(big.NewInt(3), 70))) + if ss.EntropyInt().Cmp(new(big.Int).Lsh(big.NewInt(3), 70)) != 0 || len(ss.Entropy()) != 3 { + t.Errorf("entropy %v %v", ss.EntropyInt(), ss.Entropy()) + } + if ss.PoolSize() != 4 || ss.NChildrenSpawned() != 0 { + t.Error("accessors") + } + ss.Spawn(2) + if ss.NChildrenSpawned() != 2 { + t.Error("NChildrenSpawned") + } + zero := must(NewSeedSequenceBig(new(big.Int))) + if len(zero.Entropy()) != 1 || zero.Entropy()[0] != 0 { + t.Errorf("zero entropy %v", zero.Entropy()) + } + // Fresh entropy: 128 bits from the OS, as numpy's randbits(128). + called := false + old := readOSEntropy + readOSEntropy = func(b []byte) (int, error) { + called = true + for i := range b { + b[i] = 0 + } + b[0] = 9 + return len(b), nil + } + ss = NewSeedSequence() + readOSEntropy = old + if !called || len(ss.Entropy()) != 1 || ss.Entropy()[0] != 9 { + t.Errorf("OS entropy: %v", ss.Entropy()) + } + if a, b := NewSeedSequence().Pool(), NewSeedSequence().Pool(); a[0] == b[0] && a[1] == b[1] { + t.Error("two OS-seeded sequences agree") + } +} + +func TestShapesAndDTypes(t *testing.T) { + g := DefaultRNG(2) + for _, c := range []struct { + a *ndarray.Array + dt ndarray.DType + shape string + }{ + {must(g.Random()), ndarray.Float64, "[]"}, + {must(g.Random(2, 3)), ndarray.Float64, "[2 3]"}, + {must(g.StandardExponential(4)), ndarray.Float64, "[4]"}, + {must(g.StandardGamma(2, 4)), ndarray.Float64, "[4]"}, + {must(g.Integers(0, 5, 3)), ndarray.Int64, "[3]"}, + {must(g.IntegersOf(ndarray.Uint64, 0, 5, false, 3)), ndarray.Uint64, "[3]"}, + {must(g.Dirichlet([]float64{1, 2}, 3)), ndarray.Float64, "[3 2]"}, + {must(g.Dirichlet(nil, 3)), ndarray.Float64, "[3 0]"}, + {must(g.ChoiceN(5, ChoiceOptions{}, 2, 2)), ndarray.Int64, "[2 2]"}, + {must(g.Choice(namedArray("arange12"), ChoiceOptions{Axis: -1}, 5)), ndarray.Int64, "[3 5]"}, + {must(g.Permuted(namedArray("arange12"))), ndarray.Int64, "[3 4]"}, + {must(NewRandomState(1).RandintOf(ndarray.Uint8, 0, 255, 2)), ndarray.Uint8, "[2]"}, + {must(NewRandomState(1).Choice(namedArray("floats6"), ChoiceOptions{NoReplace: true, P: []float64{.1, .1, .2, .2, .2, .2}}, 3)), ndarray.Float64, "[3]"}, + } { + if c.a.DType() != c.dt || ndarrayShape(c.a) != c.shape { + t.Errorf("%v %v, want %v %s", c.a.DType(), c.a.Shape(), c.dt, c.shape) + } + } + if len(g.Bytes(5)) != 5 || len(NewRandomState(1).Bytes(0)) != 0 { + t.Error("Bytes length") + } + // Shuffle permutes in place, along any axis, for every dtype. + for dt := ndarray.DType(0); dt <= ndarray.Complex128; dt++ { + a := namedArray("arange12").AsType(dt) + if err := g.Shuffle(a, 1); err != nil { + t.Fatal(err) + } + if err := NewRandomState(1).Shuffle(a); err != nil { + t.Fatal(err) + } + if _, err := g.PermutedAxis(a, 0); err != nil { + t.Fatal(err) + } + } + empty := must(ndarray.ZerosOf(ndarray.Int8, 0)) + if g.Shuffle(empty, 0) != nil || NewRandomState(1).Shuffle(empty) != nil { + t.Error("shuffle of an empty array") + } +} + +func ndarrayShape(a *ndarray.Array) string { + s := "[" + for i, d := range a.Shape() { + if i > 0 { + s += " " + } + s += itoa(d) + } + return s + "]" +} + +func itoa(d int) string { return big.NewInt(int64(d)).String() } diff --git a/random/arrays.go b/random/arrays.go new file mode 100644 index 0000000..ded88c7 --- /dev/null +++ b/random/arrays.go @@ -0,0 +1,160 @@ +package random + +import ( + "errors" + "fmt" + "reflect" + + "github.com/go-ndarray/ndarray" +) + +// ErrValue is wrapped by every error that numpy raises as ValueError (or +// TypeError) for a bad argument; the message after it is numpy's. +var ErrValue = errors.New("random") + +// ErrNotContiguous is returned by the in-place Shuffle for an array that is +// a strided view: it has no storage of its own to permute. +var ErrNotContiguous = errors.New("random: shuffle needs an array that owns contiguous storage (call Copy first)") + +func valueErr(format string, args ...any) error { + return fmt.Errorf("%w: "+format, append([]any{ErrValue}, args...)...) +} + +// elem is the element types of ndarray's dtypes. +type elem interface { + bool | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | + float32 | float64 | complex64 | complex128 +} + +// newArray returns a fresh zero array of dtype dt (whose element type is T) +// and its storage. +func newArray[T elem](dt ndarray.DType, shape []int) (*ndarray.Array, []T, error) { + a, err := ndarray.ZerosOf(dt, shape...) + if err != nil { + return nil, nil, err + } + d, _ := ndarray.Data[T](a) + return a, d, nil +} + +// storage returns a's elements as a contiguous []T in an any; the array's +// own storage when it is contiguous, else a copy. +func storage(a *ndarray.Array) any { + switch a.DType() { + case ndarray.Bool: + d, _ := ndarray.Data[bool](a) + return d + case ndarray.Int8: + d, _ := ndarray.Data[int8](a) + return d + case ndarray.Int16: + d, _ := ndarray.Data[int16](a) + return d + case ndarray.Int32: + d, _ := ndarray.Data[int32](a) + return d + case ndarray.Int64: + d, _ := ndarray.Data[int64](a) + return d + case ndarray.Uint8: + d, _ := ndarray.Data[uint8](a) + return d + case ndarray.Uint16: + d, _ := ndarray.Data[uint16](a) + return d + case ndarray.Uint32: + d, _ := ndarray.Data[uint32](a) + return d + case ndarray.Uint64: + d, _ := ndarray.Data[uint64](a) + return d + case ndarray.Float32: + d, _ := ndarray.Data[float32](a) + return d + case ndarray.Complex64: + d, _ := ndarray.Data[complex64](a) + return d + case ndarray.Complex128: + d, _ := ndarray.Data[complex128](a) + return d + } + d, _ := ndarray.Data[float64](a) + return d +} + +// ownStorage returns a's own contiguous storage, or ErrNotContiguous when a +// is a view (Data then hands out a fresh copy on every call). +func ownStorage(a *ndarray.Array) (reflect.Value, error) { + s := reflect.ValueOf(storage(a)) + if s.Len() > 0 && s.Pointer() != reflect.ValueOf(storage(a)).Pointer() { + return s, ErrNotContiguous + } + return s, nil +} + +// arrayOf returns a fresh array of a's dtype and the given shape, and its +// storage. +func arrayOf(dt ndarray.DType, shape []int) (*ndarray.Array, reflect.Value, error) { + a, err := ndarray.ZerosOf(dt, shape...) + if err != nil { + return nil, reflect.Value{}, err + } + return a, reflect.ValueOf(storage(a)), nil +} + +// normAxis is numpy's normalize_axis_index. +func normAxis(axis, ndim int) (int, error) { + if axis < -ndim || axis >= ndim { + return 0, valueErr("axis %d is out of bounds for array of dimension %d", axis, ndim) + } + if axis < 0 { + axis += ndim + } + return axis, nil +} + +// layout splits shape around axis into (outer, n, inner) element counts. +func layout(shape []int, axis int) (outer, n, inner int) { + outer, inner = 1, 1 + for _, d := range shape[:axis] { + outer *= d + } + for _, d := range shape[axis+1:] { + inner *= d + } + return outer, shape[axis], inner +} + +// takeAxis is numpy's a.take(idx, axis): the result has a's shape with the +// axis replaced by idxShape. Indices are already known to be in range. +func takeAxis(a *ndarray.Array, idx []int64, idxShape []int, axis int) (*ndarray.Array, error) { + shape := a.Shape() + outer, n, inner := layout(shape, axis) + rs := append(append(append([]int(nil), shape[:axis]...), idxShape...), shape[axis+1:]...) + src := reflect.ValueOf(storage(a)) + r, dst, err := arrayOf(a.DType(), rs) + if err != nil { + return nil, err + } + pos := 0 + for o := range outer { + for _, k := range idx { + from := (o*n + int(k)) * inner + reflect.Copy(dst.Slice(pos, pos+inner), src.Slice(from, from+inner)) + pos += inner + } + } + return r, nil +} + +// swapSlabs swaps positions i and j along the middle axis of the (outer, n, +// inner) layout of s. +func swapSlabs(swap func(i, j int), outer, n, inner, i, j int) { + for o := range outer { + bi := (o*n + i) * inner + bj := (o*n + j) * inner + for k := range inner { + swap(bi+k, bj+k) + } + } +} diff --git a/random/bench_test.go b/random/bench_test.go new file mode 100644 index 0000000..0afd02b --- /dev/null +++ b/random/bench_test.go @@ -0,0 +1,48 @@ +package random + +import ( + "testing" + + "github.com/go-ndarray/ndarray" +) + +// The benchmarks of docs/random.md: 10^6 draws each, as numpy's +// rng.random(10**6), rng.integers(0, 1000, 10**6), +// rng.standard_normal(10**6) and np.random.randn(10**6). + +const benchN = 1_000_000 + +func BenchmarkRandom(b *testing.B) { + g := DefaultRNG(0) + for b.Loop() { + g.Random(benchN) + } +} + +func BenchmarkRandomFloat32(b *testing.B) { + g := DefaultRNG(0) + for b.Loop() { + g.RandomOf(ndarray.Float32, benchN) + } +} + +func BenchmarkIntegers(b *testing.B) { + g := DefaultRNG(0) + for b.Loop() { + g.Integers(0, 1000, benchN) + } +} + +func BenchmarkStandardNormal(b *testing.B) { + g := DefaultRNG(0) + for b.Loop() { + g.StandardNormal(benchN) + } +} + +func BenchmarkLegacyRandn(b *testing.B) { + r := NewRandomState(0) + for b.Loop() { + r.Randn(benchN) + } +} diff --git a/random/bitgen.go b/random/bitgen.go new file mode 100644 index 0000000..7f37683 --- /dev/null +++ b/random/bitgen.go @@ -0,0 +1,99 @@ +package random + +import "math/bits" + +// BitGenerator is a source of raw random bits, numpy's BitGenerator. Every +// distribution draws from one of these three streams, and which one it draws +// from is part of numpy's output: a 32-bit draw from PCG64 uses half of a +// 64-bit output and keeps the other half for the next 32-bit draw, while +// MT19937 makes a 64-bit value from two 32-bit outputs. +// +// The implementations in this package are not safe for concurrent use. +type BitGenerator interface { + // Uint64 is numpy's next_uint64. + Uint64() uint64 + // Uint32 is numpy's next_uint32. + Uint32() uint32 + // Float64 is numpy's next_double: a float64 in [0, 1) with 53 random + // bits. + Float64() float64 + // Raw is numpy's next_raw, the stream random_raw returns. + Raw() uint64 +} + +// Spawner is a BitGenerator made from a SeedSequence, which can make +// independent children, numpy's BitGenerator.spawn. +type Spawner interface { + BitGenerator + // SeedSeq returns the SeedSequence the generator was made from, nil + // when it was not made from one. + SeedSeq() *SeedSequence + // Spawn returns n children made from SeedSeq().Spawn(n). + Spawn(n int) ([]BitGenerator, error) +} + +// RandomRaw returns n outputs of bg's raw stream, numpy's random_raw(n). +func RandomRaw(bg BitGenerator, n int) []uint64 { + out := make([]uint64, n) + for i := range out { + out[i] = bg.Raw() + } + return out +} + +// uint64ToDouble is numpy's uint64_to_double: the top 53 bits scaled to +// [0, 1). +func uint64ToDouble(r uint64) float64 { + return float64(r>>11) * (1.0 / 9007199254740992.0) +} + +// half32 holds the upper half of a 64-bit output for the next Uint32, as +// numpy's has_uint32/uinteger pair does. +type half32 struct { + has bool + uint uint32 +} + +// split keeps the upper half of v for the next 32-bit draw and returns the +// lower half. +func (h *half32) split(v uint64) uint32 { + h.has = true + h.uint = uint32(v >> 32) + return uint32(v) +} + +// uint128 is a 128-bit unsigned integer. +type uint128 struct{ hi, lo uint64 } + +func (a uint128) add(b uint128) uint128 { + lo, c := bits.Add64(a.lo, b.lo, 0) + hi, _ := bits.Add64(a.hi, b.hi, c) + return uint128{hi, lo} +} + +func (a uint128) mul(b uint128) uint128 { + hi, lo := bits.Mul64(a.lo, b.lo) + hi += a.hi*b.lo + a.lo*b.hi + return uint128{hi, lo} +} + +func (a uint128) isZero() bool { return a.hi == 0 && a.lo == 0 } + +func (a uint128) shr1() uint128 { return uint128{a.hi >> 1, a.lo>>1 | a.hi<<63} } + +// advanceLCG128 is numpy's pcg_advance_lcg_128: the state of the LCG +// state*mult+plus after delta steps, in O(log delta). +func advanceLCG128(state, delta, mult, plus uint128) uint128 { + accMult := uint128{0, 1} + accPlus := uint128{} + for !delta.isZero() { + if delta.lo&1 != 0 { + accMult = accMult.mul(mult) + accPlus = accPlus.mul(mult).add(plus) + } + plus = mult.add(uint128{0, 1}).mul(plus) + mult = mult.mul(mult) + delta = delta.shr1() + } + return accMult.mul(state).add(accPlus) +} diff --git a/random/bounded.go b/random/bounded.go new file mode 100644 index 0000000..b703962 --- /dev/null +++ b/random/bounded.go @@ -0,0 +1,265 @@ +package random + +import "math/bits" + +// Bounded integers, ports of the random_bounded_*_fill functions of +// numpy/random/src/distributions/distributions.c. Generator uses Lemire's +// multiply-and-reject (masked = false), RandomState the masked rejection of +// NumPy 1.16 (masked = true); both draw 8- and 16-bit values from the halves +// and quarters of a buffered 32-bit output, and the buffer lives only for +// one fill. + +// genMask is the smallest all-ones mask covering max (0 for 0). +func genMask(max uint64) uint64 { + return 1<>= 16 + b.cnt-- + } + return uint16(b.buf) +} + +func (b *buffered) uint8(bg BitGenerator) uint8 { + if b.cnt == 0 { + b.buf = bg.Uint32() + b.cnt = 3 + } else { + b.buf >>= 8 + b.cnt-- + } + return uint8(b.buf) +} + +func (b *buffered) bool(bg BitGenerator, off, rng bool) bool { + if !rng { + return off + } + if b.cnt == 0 { + b.buf = bg.Uint32() + b.cnt = 31 + } else { + b.buf >>= 1 + b.cnt-- + } + return b.buf&1 != 0 +} + +func lemire64(bg BitGenerator, rng uint64) uint64 { + excl := rng + 1 + hi, lo := bits.Mul64(bg.Uint64(), excl) + if lo < excl { + threshold := (^uint64(0) - rng) % excl + for lo < threshold { + hi, lo = bits.Mul64(bg.Uint64(), excl) + } + } + return hi +} + +func lemire32(bg BitGenerator, rng uint32) uint32 { + excl := rng + 1 + m := uint64(bg.Uint32()) * uint64(excl) + if uint32(m) < excl { + threshold := (^uint32(0) - rng) % excl + for uint32(m) < threshold { + m = uint64(bg.Uint32()) * uint64(excl) + } + } + return uint32(m >> 32) +} + +// lemire32PCG is lemire32 on a *PCG64, which the compiler can inline. +func lemire32PCG(bg *PCG64, rng uint32) uint32 { + excl := rng + 1 + m := uint64(bg.Uint32()) * uint64(excl) + if uint32(m) < excl { + threshold := (^uint32(0) - rng) % excl + for uint32(m) < threshold { + m = uint64(bg.Uint32()) * uint64(excl) + } + } + return uint32(m >> 32) +} + +func lemire16(bg BitGenerator, rng uint16, b *buffered) uint16 { + excl := rng + 1 + m := uint32(b.uint16(bg)) * uint32(excl) + if uint16(m) < excl { + threshold := (^uint16(0) - rng) % excl + for uint16(m) < threshold { + m = uint32(b.uint16(bg)) * uint32(excl) + } + } + return uint16(m >> 16) +} + +func lemire8(bg BitGenerator, rng uint8, b *buffered) uint8 { + excl := rng + 1 + m := uint16(b.uint8(bg)) * uint16(excl) + if uint8(m) < excl { + threshold := (^uint8(0) - rng) % excl + for uint8(m) < threshold { + m = uint16(b.uint8(bg)) * uint16(excl) + } + } + return uint8(m >> 8) +} + +// boundedUint64 is numpy's random_bounded_uint64: one value in +// [off, off+rng], wrapping; a range that fits 32 bits draws 32-bit values. +func boundedUint64(bg BitGenerator, off, rng uint64, masked bool) uint64 { + switch { + case rng == 0: + return off + case rng == 0xFFFFFFFF: + return off + uint64(bg.Uint32()) + case rng < 0xFFFFFFFF: + if masked { + mask := uint32(genMask(rng)) + for { + if v := bg.Uint32() & mask; uint64(v) <= rng { + return off + uint64(v) + } + } + } + return off + uint64(lemire32(bg, uint32(rng))) + case rng == ^uint64(0): + return off + bg.Uint64() + case masked: + mask := genMask(rng) + for { + if v := bg.Uint64() & mask; v <= rng { + return off + v + } + } + } + return off + lemire64(bg, rng) +} + +// fillUint64 is numpy's random_bounded_uint64_fill: boundedUint64 with +// the choice of method made once, outside the loop. +func fillUint64(bg BitGenerator, off, rng uint64, masked bool, out []uint64) { + switch { + case rng == 0 || rng == 0xFFFFFFFF || rng == ^uint64(0) || masked: + for i := range out { + out[i] = boundedUint64(bg, off, rng, masked) + } + case rng < 0xFFFFFFFF: + r := uint32(rng) + if p, ok := bg.(*PCG64); ok { + for i := range out { + out[i] = off + uint64(lemire32PCG(p, r)) + } + return + } + for i := range out { + out[i] = off + uint64(lemire32(bg, r)) + } + default: + for i := range out { + out[i] = off + lemire64(bg, rng) + } + } +} + +func fillUint32(bg BitGenerator, off, rng uint32, masked bool, out []uint32) { + mask := uint32(genMask(uint64(rng))) + for i := range out { + switch { + case rng == 0: + out[i] = off + case rng == 0xFFFFFFFF: + out[i] = off + bg.Uint32() + case masked: + v := bg.Uint32() & mask + for v > rng { + v = bg.Uint32() & mask + } + out[i] = off + v + default: + out[i] = off + lemire32(bg, rng) + } + } +} + +func fillUint16(bg BitGenerator, off, rng uint16, masked bool, out []uint16) { + var b buffered + mask := uint16(genMask(uint64(rng))) + for i := range out { + switch { + case rng == 0: + out[i] = off + case rng == 0xFFFF: + out[i] = off + b.uint16(bg) + case masked: + v := b.uint16(bg) & mask + for v > rng { + v = b.uint16(bg) & mask + } + out[i] = off + v + default: + out[i] = off + lemire16(bg, rng, &b) + } + } +} + +func fillUint8(bg BitGenerator, off, rng uint8, masked bool, out []uint8) { + var b buffered + mask := uint8(genMask(uint64(rng))) + for i := range out { + switch { + case rng == 0: + out[i] = off + case rng == 0xFF: + out[i] = off + b.uint8(bg) + case masked: + v := b.uint8(bg) & mask + for v > rng { + v = b.uint8(bg) & mask + } + out[i] = off + v + default: + out[i] = off + lemire8(bg, rng, &b) + } + } +} + +func fillBool(bg BitGenerator, off, rng bool, out []bool) { + var b buffered + for i := range out { + out[i] = b.bool(bg, off, rng) + } +} + +// interval is numpy's random_interval: a value in [0, max] by masked +// rejection, from 32-bit draws when max fits them. Shuffles use it. +func interval(bg BitGenerator, max uint64) uint64 { + if max == 0 { + return 0 + } + mask := genMask(max) + if max <= 0xFFFFFFFF { + for { + if v := uint64(bg.Uint32()) & mask; v <= max { + return v + } + } + } + for { + if v := bg.Uint64() & mask; v <= max { + return v + } + } +} diff --git a/random/branches_test.go b/random/branches_test.go new file mode 100644 index 0000000..a4e73b8 --- /dev/null +++ b/random/branches_test.go @@ -0,0 +1,140 @@ +package random + +import ( + "math" + "testing" +) + +// stubBG replays a fixed list of 64-bit outputs (cycling), so that tests can +// steer a sampler into branches that real streams reach once in billions of +// draws. Uint32 is the low half of the next output, Float64 its top 53 bits. +type stubBG struct { + u []uint64 + i *int +} + +func newStub(u ...uint64) stubBG { return stubBG{u, new(int)} } + +func (s stubBG) Uint64() uint64 { + v := s.u[*s.i%len(s.u)] + *s.i++ + return v +} +func (s stubBG) Uint32() uint32 { return uint32(s.Uint64()) } +func (s stubBG) Float64() float64 { return uint64ToDouble(s.Uint64()) } +func (s stubBG) Raw() uint64 { return s.Uint64() } + +// f64 returns the output whose Float64 is u (for u a multiple of 2^-53). +func f64(u float64) uint64 { return uint64(u*(1<<53)) << 11 } + +func TestRareBranches(t *testing.T) { + // Lemire rejections: a low product below the threshold redraws. + if v := lemire64(newStub(0, 1), 3<<62-1); v != 0 { + t.Errorf("lemire64 = %d", v) + } + if v := lemire32(newStub(0, 1), 3<<30-1); v != 0 { + t.Errorf("lemire32 = %d", v) + } + if v := boundedUint64(newStub(0, 1), 7, 3<<62-1, false); v != 7 { + t.Errorf("boundedUint64 = %d", v) + } + // 64-bit masked interval redraws above max. + if v := interval(newStub(^uint64(0), 5), 1<<40); v != 5 { + t.Errorf("interval = %d", v) + } + out := make([]uint32, 3) + fillUint32(newStub(15, 3), 7, 0, true, out[:1]) + fillUint32(newStub(15, 3), 0, 9, true, out[1:]) + if out[0] != 7 || out[1] != 3 { + t.Errorf("fillUint32 = %v", out) + } + + // Beta (Johnk) when X or Y underflows, in both orders. + tiny := f64(0x1p-40) + if v := beta(newStub(tiny, f64(0.5)), 1e-3, 1e-3); !(v >= 0 && v <= 1) { + t.Errorf("beta = %v", v) + } + if v := beta(newStub(f64(0.5), tiny), 1e-3, 1e-3); !(v >= 0 && v <= 1) { + t.Errorf("beta = %v", v) + } + + // BTPE: the region-3 and region-4 rejections (y < 0, y > n). + var bc binomialCache + binomialBTPE(newStub(0), 1000, 0.4, &bc, false) + u3 := (bc.p2 + bc.p3) / 2 / bc.p4 + u4 := (bc.p3 + bc.p4) / 2 / bc.p4 + acc := f64(0) + if y := binomialBTPE(newStub(f64(math.Floor(u3*(1<<53))/(1<<53)), 0, acc, acc), 1000, 0.4, &bc, false); y < 0 || y > 1000 { + t.Errorf("btpe = %d", y) + } + if y := binomialBTPE(newStub(f64(math.Floor(u4*(1<<53))/(1<<53)), 0, acc, acc), 1000, 0.4, &bc, false); y < 0 || y > 1000 { + t.Errorf("btpe = %d", y) + } + // Inversion past its bound starts over. + bc = binomialCache{} + if x := binomialInversion(newStub(f64(1-0x1p-53), f64(0)), 10, 0.1, &bc, false); x != 0 { + t.Errorf("inversion = %d", x) + } + + // NaN parameters. + if !math.IsNaN(noncentralChisquare(newStub(1), 2, math.NaN())) || !math.IsNaN(vonmises(newStub(1), 0, math.NaN())) { + t.Error("NaN parameter") + } + // Logseries: V == 0 is rejected. + if v := logseries(newStub(f64(0), f64(0.5), f64(0.9)), 0.5); v != 1 { + t.Errorf("logseries = %d", v) + } + // Zipf's a >= 1025 shortcut. + if zipf(newStub(1), 2000) != 1 { + t.Error("zipf") + } + // Geometric beyond int64. + if geometricInversion(NewPCG64(1), 1e-300) != math.MaxInt64 { + t.Error("geometric overflow") + } + // Laplace, Gumbel and Logistic reject an endpoint draw. + for _, f := range []func(BitGenerator) float64{ + func(b BitGenerator) float64 { return laplace(b, 0, 1) }, + func(b BitGenerator) float64 { return gumbel(b, 0, 1) }, + func(b BitGenerator) float64 { return logistic(b, 0, 1) }, + } { + if v := f(newStub(0, f64(0.5))); math.IsInf(v, 0) || math.IsNaN(v) { + t.Errorf("endpoint draw gave %v", v) + } + } +} + +func TestLibmSpecialValues(t *testing.T) { + inf := math.Inf(1) + for _, c := range []struct { + name string + got, want float64 + }{ + {"log(0)", log(0), math.Inf(-1)}, + {"log(inf)", log(inf), inf}, + {"log1p(-1)", log1p(-1), math.Inf(-1)}, + {"log1p(inf)", log1p(inf), inf}, + {"exp(800)", exp(800), inf}, + {"exp(-800)", exp(-800), 0}, + {"expm1(800)", expm1(800), inf}, + {"expm1(-50)", expm1(-50), -1}, + {"expm1(705)", expm1(705), exp(705)}, + {"pow(10, 400)", pow(10, 400), inf}, + {"pow(10, -400)", pow(10, -400), 0}, + {"pow(0, 2)", pow(0, 2), 0}, + {"log1p(0)", log1p(0), 0}, + {"expm1(0)", expm1(0), 0}, + } { + if c.got != c.want { + t.Errorf("%s = %v, want %v", c.name, c.got, c.want) + } + } + for _, v := range []float64{log(-1), log1p(-2), exp(math.NaN()), expm1(math.NaN()), log(math.NaN())} { + if !math.IsNaN(v) { + t.Errorf("want NaN, got %v", v) + } + } + if logf(1) != 0 || expf(0) != 1 || log1pf(0) != 0 || powf(2, 3) != 8 { + t.Error("float32 wrappers") + } +} diff --git a/random/casts.go b/random/casts.go new file mode 100644 index 0000000..bec3c87 --- /dev/null +++ b/random/casts.go @@ -0,0 +1,20 @@ +package random + +import "unsafe" + +// Signed integers are drawn as the unsigned integers of the same width, as +// numpy does (it writes through a pointer of the unsigned type); these views +// share the storage. + +func asUint8(d []int8) []uint8 { + return unsafe.Slice((*uint8)(unsafe.Pointer(unsafe.SliceData(d))), len(d)) +} +func asUint16(d []int16) []uint16 { + return unsafe.Slice((*uint16)(unsafe.Pointer(unsafe.SliceData(d))), len(d)) +} +func asUint32(d []int32) []uint32 { + return unsafe.Slice((*uint32)(unsafe.Pointer(unsafe.SliceData(d))), len(d)) +} +func asUint64(d []int64) []uint64 { + return unsafe.Slice((*uint64)(unsafe.Pointer(unsafe.SliceData(d))), len(d)) +} diff --git a/random/dist.go b/random/dist.go new file mode 100644 index 0000000..cf40996 --- /dev/null +++ b/random/dist.go @@ -0,0 +1,814 @@ +package random + +import "math" + +// The scalar samplers below are ports of numpy/random/src/distributions/ +// distributions.c, statement for statement: the same draws in the same +// order, the same floating-point operations in the same order. A product +// that feeds a sum is converted explicitly (float64(a*b) + c): Go may fuse +// a*b+c into one FMA on arm64, ppc64le, s390x and riscv64, which rounds once +// instead of twice and changes the last bit. NumPy's x86-64 builds do not +// fuse; see docs/random.md for its aarch64 builds, which do. + +func nextFloat(bg BitGenerator) float32 { + return float32(bg.Uint32()>>8) * (1.0 / 16777216.0) +} + +var ( + signDouble = [2]float64{1, -1} + signFloat = [2]float32{1, -1} +) + +func standardExponential(bg BitGenerator) float64 { + for { + ri := bg.Uint64() + ri >>= 3 + idx := uint8(ri) + ri >>= 8 + x := float64(ri) * weDouble[idx] + if ri < keDouble[idx] { + return x + } + if idx == 0 { + return zigExpR - log1p(-bg.Float64()) + } + if float64((feDouble[idx-1]-feDouble[idx])*bg.Float64())+feDouble[idx] < exp(-x) { + return x + } + } +} + +func standardExponentialF(bg BitGenerator) float32 { + for { + ri := bg.Uint32() + ri >>= 1 + idx := uint8(ri) + ri >>= 8 + x := float32(ri) * weFloat[idx] + if ri < keFloat[idx] { + return x + } + if idx == 0 { + return zigExpRF - log1pf(-nextFloat(bg)) + } + if float32((feFloat[idx-1]-feFloat[idx])*nextFloat(bg))+feFloat[idx] < expf(-x) { + return x + } + } +} + +func standardNormal(bg BitGenerator) float64 { + r := bg.Uint64() + idx := r & 0xff + r >>= 8 + rabs := (r >> 1) & 0x000fffffffffffff + x := float64(rabs) * wiDouble[idx] * signDouble[r&1] + if rabs < kiDouble[idx] { + return x + } + return normalSlow(bg, idx, rabs, x) +} + +// normalSlow finishes a ziggurat draw that missed the rectangle: the tail +// for layer 0, the wedge test for the others, and a fresh draw on rejection. +func normalSlow(bg BitGenerator, idx, rabs uint64, x float64) float64 { + for { + if idx == 0 { + for { + xx := float64(-zigNorInvR * log1p(-bg.Float64())) + yy := -log1p(-bg.Float64()) + if yy+yy > float64(xx*xx) { + if (rabs>>8)&1 != 0 { + return -(zigNorR + xx) + } + return zigNorR + xx + } + } + } + if float64((fiDouble[idx-1]-fiDouble[idx])*bg.Float64())+fiDouble[idx] < exp(-0.5*x*x) { + return x + } + r := bg.Uint64() + idx = r & 0xff + r >>= 8 + rabs = (r >> 1) & 0x000fffffffffffff + x = float64(rabs) * wiDouble[idx] * signDouble[r&1] + if rabs < kiDouble[idx] { + return x + } + } +} + +// standardNormalPCG is standardNormal's fast path on the default bit +// generator, called directly (no interface call per draw); the rare paths +// go through standardNormal's code. +func standardNormalPCG(p *PCG64) float64 { + r := p.Uint64() + idx := r & 0xff + r >>= 8 + rabs := (r >> 1) & 0x000fffffffffffff + x := float64(rabs) * wiDouble[idx] * signDouble[r&1] + if rabs < kiDouble[idx] { + return x + } + return normalSlow(p, idx, rabs, x) +} + +func standardNormalF(bg BitGenerator) float32 { + for { + r := bg.Uint32() + idx := r & 0xff + rabs := (r >> 9) & 0x0007fffff + x := float32(rabs) * wiFloat[idx] * signFloat[(r>>8)&1] + if rabs < kiFloat[idx] { + return x + } + if idx == 0 { + for { + xx := float32(-zigNorInvRF * log1pf(-nextFloat(bg))) + yy := -log1pf(-nextFloat(bg)) + if yy+yy > float32(xx*xx) { + if (rabs>>8)&1 != 0 { + return -(zigNorRF + xx) + } + return zigNorRF + xx + } + } + } + // The C code evaluates exp(-0.5 * x * x) in double precision. + xd := float64(x) + if float64(float32(float32((fiFloat[idx-1]-fiFloat[idx])*nextFloat(bg))+fiFloat[idx])) < exp(-0.5*xd*xd) { + return x + } + } +} + +func standardGamma(bg BitGenerator, shape float64) float64 { + switch { + case shape == 1.0: + return standardExponential(bg) + case shape == 0.0: + return 0.0 + case shape < 1.0: + for { + u := bg.Float64() + v := standardExponential(bg) + if u <= 1.0-shape { + x := pow(u, 1./shape) + if x <= v { + return x + } + } else { + y := -log((1 - u) / shape) + x := pow(1.0-shape+float64(shape*y), 1./shape) + if x <= v+y { + return x + } + } + } + } + b := shape - 1./3. + c := 1. / math.Sqrt(9*b) + for { + var x, v float64 + for { + x = standardNormal(bg) + v = 1.0 + float64(c*x) + if v > 0.0 { + break + } + } + v = float64(v * v * v) + u := bg.Float64() + if u < 1.0-float64(0.0331*(x*x)*(x*x)) { + return float64(b * v) + } + if log(u) < float64(0.5*x*x)+float64(b*(1.-v+log(v))) { + return float64(b * v) + } + } +} + +func standardGammaF(bg BitGenerator, shape float32) float32 { + switch { + case shape == 1.0: + return standardExponentialF(bg) + case shape == 0.0: + return 0.0 + case shape < 1.0: + for { + u := nextFloat(bg) + v := standardExponentialF(bg) + if u <= 1.0-shape { + x := powf(u, 1.0/shape) + if x <= v { + return x + } + } else { + y := -logf((1.0 - u) / shape) + x := powf(1.0-shape+float32(shape*y), 1.0/shape) + if x <= v+y { + return x + } + } + } + } + b := shape - 1.0/3.0 + c := 1.0 / float32(math.Sqrt(float64(9.0*b))) + for { + var x, v float32 + for { + x = standardNormalF(bg) + v = 1.0 + float32(c*x) + if v > 0.0 { + break + } + } + v = float32(v * v * v) + u := nextFloat(bg) + if u < 1.0-float32(0.0331*(x*x)*(x*x)) { + return float32(b * v) + } + if logf(u) < float32(0.5*x*x)+float32(b*(1.0-v+logf(v))) { + return float32(b * v) + } + } +} + +// loggam is numpy's random_loggam (SPECFUN's log-gamma). +func loggam(x float64) float64 { + a := [10]float64{8.333333333333333e-02, -2.777777777777778e-03, + 7.936507936507937e-04, -5.952380952380952e-04, + 8.417508417508418e-04, -1.917526917526918e-03, + 6.410256410256410e-03, -2.955065359477124e-02, + 1.796443723688307e-01, -1.39243221690590e+00} + if x == 1.0 || x == 2.0 { + return 0.0 + } + var n int64 + if x < 7.0 { + n = int64(7 - x) + } + x0 := x + float64(n) + x2 := (1.0 / x0) * (1.0 / x0) + const lg2pi = 1.8378770664093453e+00 + gl0 := a[9] + for k := 8; k >= 0; k-- { + gl0 = float64(gl0*x2) + a[k] + } + gl := gl0/x0 + float64(0.5*lg2pi) + float64((x0-0.5)*log(x0)) - x0 + if x < 7.0 { + for k := int64(1); k <= n; k++ { + gl -= log(x0 - 1.0) + x0 -= 1.0 + } + } + return gl +} + +func normal(bg BitGenerator, loc, scale float64) float64 { + return loc + float64(scale*standardNormal(bg)) +} + +func uniform(bg BitGenerator, lower, rng float64) float64 { + return lower + float64(rng*bg.Float64()) +} + +const betaTinyThreshold = 3e-103 + +func beta(bg BitGenerator, a, b float64) float64 { + if a <= 1.0 && b <= 1.0 { + if a < betaTinyThreshold && b < betaTinyThreshold { + u := bg.Float64() + if (a+b)*u < a { + return 1 + } + return 0 + } + for { + u := bg.Float64() + v := bg.Float64() + x := pow(u, 1.0/a) + y := pow(v, 1.0/b) + xpy := x + y + if xpy <= 1.0 && u+v > 0.0 { + if x > 0 && y > 0 { + return x / xpy + } + logX := log(u) / a + logY := log(v) / b + delta := logX - logY + if delta > 0 { + return exp(-log1p(exp(-delta))) + } + return exp(delta - log1p(exp(delta))) + } + } + } + ga := standardGamma(bg, a) + gb := standardGamma(bg, b) + return ga / (ga + gb) +} + +func chisquare(bg BitGenerator, df float64) float64 { + return float64(2.0 * standardGamma(bg, df/2.0)) +} + +func fDist(bg BitGenerator, dfnum, dfden float64) float64 { + s1 := float64(chisquare(bg, dfnum) * dfden) + s2 := float64(chisquare(bg, dfden) * dfnum) + return s1 / s2 +} + +func standardCauchy(bg BitGenerator) float64 { + a := standardNormal(bg) + return a / standardNormal(bg) +} + +func pareto(bg BitGenerator, a float64) float64 { + return expm1(standardExponential(bg) / a) +} + +func weibull(bg BitGenerator, a float64) float64 { + if a == 0.0 { + return 0.0 + } + return pow(standardExponential(bg), 1./a) +} + +func power(bg BitGenerator, a float64) float64 { + return pow(-expm1(-standardExponential(bg)), 1./a) +} + +func laplace(bg BitGenerator, loc, scale float64) float64 { + for { + u := bg.Float64() + if u > 0.0 { + lo := u + u + hi := 2.0 - u - u + s := 0 + if u >= 0.5 { + s = 1 + } + return loc + float64(signDouble[s]*scale*log(min(lo, hi))) + } + } +} + +func gumbel(bg BitGenerator, loc, scale float64) float64 { + for { + u := 1.0 - bg.Float64() + if u < 1.0 { + return loc - float64(scale*log(-log(u))) + } + } +} + +func logistic(bg BitGenerator, loc, scale float64) float64 { + for { + u := bg.Float64() + if u > 0.0 { + return loc + float64(scale*log(u/(1.0-u))) + } + } +} + +func lognormal(bg BitGenerator, mean, sigma float64) float64 { + return exp(normal(bg, mean, sigma)) +} + +func rayleigh(bg BitGenerator, mode float64) float64 { + return float64(mode * math.Sqrt(2.0*standardExponential(bg))) +} + +func standardT(bg BitGenerator, df float64) float64 { + num := standardNormal(bg) + denom := standardGamma(bg, df/2) + return float64(math.Sqrt(df/2)*num) / math.Sqrt(denom) +} + +func poissonMult(bg BitGenerator, lam float64) int64 { + enlam := exp(-lam) + var x int64 + prod := 1.0 + for { + prod *= bg.Float64() + if prod > enlam { + x++ + } else { + return x + } + } +} + +const ( + ls2pi = 0.91893853320467267 + twelfth = 0.083333333333333333333333 +) + +func poissonPtrs(bg BitGenerator, lam float64) int64 { + slam := math.Sqrt(lam) + loglam := log(lam) + b := 0.931 + float64(2.53*slam) + a := -0.059 + float64(0.02483*b) + invalpha := 1.1239 + 1.1328/(b-3.4) + vr := 0.9277 - 3.6224/(b-2) + for { + u := bg.Float64() - 0.5 + v := bg.Float64() + us := 0.5 - math.Abs(u) + k := int64(math.Floor(float64((2*a/us+b)*u) + lam + 0.43)) + if us >= 0.07 && v <= vr { + return k + } + if k < 0 || (us < 0.013 && v > us) { + continue + } + if log(v)+log(invalpha)-log(a/(us*us)+b) <= + -lam+float64(float64(k)*loglam)-loggam(float64(k)+1) { + return k + } + } +} + +func poisson(bg BitGenerator, lam float64) int64 { + if lam >= 10 { + return poissonPtrs(bg, lam) + } + if lam == 0 { + return 0 + } + return poissonMult(bg, lam) +} + +func negativeBinomial(bg BitGenerator, n, p float64) int64 { + y := float64(standardGamma(bg, n) * ((1 - p) / p)) + return poisson(bg, y) +} + +// binomialCache is numpy's binomial_t: the setup of the last (n, p), kept +// between calls by a Generator. +type binomialCache struct { + has bool + nsave int64 + psave float64 + m int64 + r, q, fm, p1, xm, xl, xr, c, laml, lamr, p2, p3, p4 float64 +} + +// binomialBTPE is numpy's random_binomial_btpe; legacy selects the NumPy +// 1.16 version RandomState keeps (13680 for 13860 and all four error terms +// added). +func binomialBTPE(bg BitGenerator, n int64, p float64, bc *binomialCache, legacy bool) int64 { + if !bc.has || bc.nsave != n || bc.psave != p { + bc.nsave, bc.psave, bc.has = n, p, true + bc.r = min(p, 1.0-p) + bc.q = 1.0 - bc.r + bc.fm = float64(float64(n)*bc.r) + bc.r + bc.m = int64(math.Floor(bc.fm)) + bc.p1 = math.Floor(float64(2.195*math.Sqrt(float64(n)*bc.r*bc.q))-float64(4.6*bc.q)) + 0.5 + bc.xm = float64(bc.m) + 0.5 + bc.xl = bc.xm - bc.p1 + bc.xr = bc.xm + bc.p1 + bc.c = 0.134 + 20.5/(15.3+float64(bc.m)) + a := (bc.fm - bc.xl) / (bc.fm - float64(bc.xl*bc.r)) + bc.laml = float64(a * (1.0 + a/2.0)) + a = (bc.xr - bc.fm) / (bc.xr * bc.q) + bc.lamr = float64(a * (1.0 + a/2.0)) + bc.p2 = float64(bc.p1 * (1.0 + float64(2.0*bc.c))) + bc.p3 = bc.p2 + bc.c/bc.laml + bc.p4 = bc.p3 + bc.c/bc.lamr + } + r, q, m, p1, xm, xl, xr, c := bc.r, bc.q, bc.m, bc.p1, bc.xm, bc.xl, bc.xr, bc.c + laml, lamr, p2, p3, p4 := bc.laml, bc.lamr, bc.p2, bc.p3, bc.p4 + nf := float64(n) + var y int64 + for { + nrq := float64(nf * r * q) + u := float64(bg.Float64() * p4) + v := bg.Float64() + if u <= p1 { + y = int64(math.Floor(xm - float64(p1*v) + u)) + break + } + if u <= p2 { + x := xl + (u-p1)/c + v = float64(v*c) + 1.0 - math.Abs(float64(m)-x+0.5)/p1 + if v > 1.0 { + continue + } + y = int64(math.Floor(x)) + } else if u <= p3 { + y = int64(math.Floor(xl + log(v)/laml)) + if y < 0 || v == 0.0 { + continue + } + v = float64(v * (u - p2) * laml) + } else { + y = int64(math.Floor(xr - log(v)/lamr)) + if y > n || v == 0.0 { + continue + } + v = float64(v * (u - p3) * lamr) + } + k := y - m + if k < 0 { + k = -k + } + kf := float64(k) + if k > 20 && kf < nrq/2.0-1 { + rho := float64((kf / nrq) * ((float64(kf*(kf/3.0+0.625))+0.16666666666666666)/nrq + 0.5)) + t := float64(-kf*kf) / float64(2*nrq) + A := log(v) + if A < t-rho { + break + } + if A > t+rho { + continue + } + x1 := float64(y) + 1 + f1 := float64(m) + 1 + z := nf + 1 - float64(m) + w := nf - float64(y) + 1 + if A > btpeBound(xm, nf, m, y, r, q, x1, f1, z, w, legacy) { + continue + } + break + } + s := r / q + a := float64(s * (nf + 1)) + F := 1.0 + if m < y { + for i := m + 1; i <= y; i++ { + F = float64(F * (a/float64(i) - s)) + } + } else if m > y { + for i := y + 1; i <= m; i++ { + F /= a/float64(i) - s + } + } + if v > F { + continue + } + break + } + // numpy flips y when p > 0.5 here; binomial only passes p <= 0.5. + return y +} + +// btpeBound is the right-hand side of BTPE's final squeeze, with the +// Stirling error terms of f1, z, x1 and w. +func btpeBound(xm, nf float64, m, y int64, r, q, x1, f1, z, w float64, legacy bool) float64 { + c0 := 13860. + if legacy { + c0 = 13680. + } + term := func(v float64) float64 { + v2 := v * v + return (c0 - (462.-(132.-(99.-140./v2)/v2)/v2)/v2) / v / 166320. + } + s := float64(xm*log(f1/x1)) + float64((nf-float64(m)+0.5)*log(z/w)) + + float64(float64(y-m)*log(w*r/(x1*q))) + term(f1) + term(z) + if legacy { + return s + term(x1) + term(w) + } + return s - term(x1) - term(w) +} + +// binomialInversion is numpy's random_binomial_inversion; legacy computes +// q**n as exp(n*log(q)) where the current code uses exp(n*log1p(-p)). +func binomialInversion(bg BitGenerator, n int64, p float64, bc *binomialCache, legacy bool) int64 { + nf := float64(n) + if !bc.has || bc.nsave != n || bc.psave != p { + bc.nsave, bc.psave, bc.has = n, p, true + bc.q = 1.0 - p + if legacy { + bc.r = exp(nf * log(bc.q)) + } else { + bc.r = exp(nf * log1p(-p)) + } + bc.c = float64(nf * p) + bc.m = int64(min(nf, bc.c+float64(10.0*math.Sqrt(float64(bc.c*bc.q)+1)))) + } + q, qn, bound := bc.q, bc.r, bc.m + var x int64 + px := qn + u := bg.Float64() + for u > px { + x++ + if x > bound { + x = 0 + px = qn + u = bg.Float64() + } else { + u -= px + px = float64(float64(n-x+1)*p*px) / float64(float64(x)*q) + } + } + return x +} + +func binomial(bg BitGenerator, p float64, n int64, bc *binomialCache, legacy bool) int64 { + if !legacy && (n == 0 || p == 0.0) { + return 0 + } + if p <= 0.5 { + if p*float64(n) <= 30.0 { + return binomialInversion(bg, n, p, bc, legacy) + } + return binomialBTPE(bg, n, p, bc, legacy) + } + q := 1.0 - p + if q*float64(n) <= 30.0 { + return n - binomialInversion(bg, n, q, bc, legacy) + } + return n - binomialBTPE(bg, n, q, bc, legacy) +} + +func noncentralChisquare(bg BitGenerator, df, nonc float64) float64 { + if math.IsNaN(nonc) { + return math.NaN() + } + if nonc == 0 { + return chisquare(bg, df) + } + if 1 < df { + chi2 := chisquare(bg, df-1) + n := standardNormal(bg) + math.Sqrt(nonc) + return chi2 + float64(n*n) + } + i := poisson(bg, nonc/2.0) + return chisquare(bg, df+float64(2*i)) +} + +func noncentralF(bg BitGenerator, dfnum, dfden, nonc float64) float64 { + t := noncentralChisquare(bg, dfnum, nonc) * dfden + return t / (chisquare(bg, dfden) * dfnum) +} + +func wald(bg BitGenerator, mean, scale float64) float64 { + y := standardNormal(bg) + y = float64(mean * y * y) + d := 1 + math.Sqrt(1+float64(4*scale)/y) + x := float64(mean * (1 - 2/d)) + u := bg.Float64() + if u <= mean/(mean+x) { + return x + } + return float64(mean*mean) / x +} + +func vonmises(bg BitGenerator, mu, kappa float64) float64 { + if math.IsNaN(kappa) { + return math.NaN() + } + if kappa < 1e-8 { + return math.Pi * (2*bg.Float64() - 1) + } + var s float64 + if kappa < 1e-5 { + s = 1./kappa + kappa + } else if kappa <= 1e6 { + r := 1 + math.Sqrt(1+float64(4*kappa*kappa)) + rho := (r - math.Sqrt(2*r)) / (2 * kappa) + s = (1 + float64(rho*rho)) / (2 * rho) + } else { + result := mu + float64(math.Sqrt(1./kappa)*standardNormal(bg)) + if result < -math.Pi { + result += 2 * math.Pi + } + if result > math.Pi { + result -= 2 * math.Pi + } + return result + } + var w float64 + for { + u := bg.Float64() + z := cos(math.Pi * u) + w = (1 + float64(s*z)) / (s + z) + y := float64(kappa * (s - w)) + v := bg.Float64() + if float64(y*(2-y))-v >= 0 || log(y/v)+1-y >= 0 { + break + } + } + u := bg.Float64() + result := acos(w) + if u < 0.5 { + result = -result + } + result += mu + neg := result < 0 + mod := math.Abs(result) + mod = math.Mod(mod+math.Pi, 2*math.Pi) - math.Pi + if neg { + mod *= -1 + } + return mod +} + +func logseries(bg BitGenerator, p float64) int64 { + r := log1p(-p) + for { + v := bg.Float64() + if v >= p { + return 1 + } + u := bg.Float64() + q := -expm1(float64(r * u)) + if v <= q*q { + result := int64(math.Floor(1 + log(v)/log(q))) + if result < 1 || v == 0.0 { + continue + } + return result + } + if v >= q { + return 1 + } + return 2 + } +} + +func geometricSearch(bg BitGenerator, p float64) int64 { + x := int64(1) + sum, prod := p, p + q := 1.0 - p + u := bg.Float64() + for u > sum { + prod = float64(prod * q) + sum += prod + x++ + } + return x +} + +func geometricInversion(bg BitGenerator, p float64) int64 { + z := math.Ceil(-standardExponential(bg) / log1p(-p)) + if z >= 9.223372036854776e+18 { + return math.MaxInt64 + } + return int64(z) +} + +func geometric(bg BitGenerator, p float64) int64 { + if p >= 0.333333333333333333333333 { + return geometricSearch(bg, p) + } + return geometricInversion(bg, p) +} + +func zipf(bg BitGenerator, a float64) int64 { + if a >= 1025 { + return 1 + } + am1 := a - 1.0 + b := pow(2.0, am1) + umin := pow(float64(math.MaxInt64), -am1) + for { + u01 := bg.Float64() + u := float64(u01*umin) + (1 - u01) + v := bg.Float64() + x := math.Floor(pow(u, -1.0/am1)) + // numpy rejects x beyond int64 or below 1 (neither can occur + // short of pow rounding past 2^63) and tests the rest. + if x <= float64(math.MaxInt64) && x >= 1.0 { + t := pow(1.0+1.0/x, am1) + if v*x*(t-1.0)/(b-1.0) <= t/b { + return int64(x) + } + } + } +} + +func triangular(bg BitGenerator, left, mode, right float64) float64 { + base := right - left + leftbase := mode - left + ratio := leftbase / base + leftprod := float64(leftbase * base) + rightprod := float64((right - mode) * base) + u := bg.Float64() + if u <= ratio { + return left + math.Sqrt(float64(u*leftprod)) + } + return right - math.Sqrt(float64((1.0-u)*rightprod)) +} + +// multinomial is numpy's random_multinomial: d-1 conditional binomials. +func multinomial(bg BitGenerator, n int64, mnix []int64, pix []float64, bc *binomialCache, legacy bool) { + remaining := 1.0 + dn := n + d := len(pix) + for j := 0; j < d-1; j++ { + mnix[j] = binomial(bg, pix[j]/remaining, dn, bc, legacy) + dn -= mnix[j] + if dn <= 0 { + break + } + remaining -= pix[j] + } + if dn > 0 { + mnix[d-1] = dn + } +} diff --git a/random/doc.go b/random/doc.go new file mode 100644 index 0000000..255f0b5 --- /dev/null +++ b/random/doc.go @@ -0,0 +1,52 @@ +// Package random is numpy.random for go-ndarray: the same numbers as NumPy +// for the same seed. +// +// A notebook that does +// +// rng = np.random.default_rng(42) +// x = rng.standard_normal((3, 4)) +// +// becomes +// +// rng := random.DefaultRNG(42) +// x, _ := rng.StandardNormal(3, 4) +// +// and x holds NumPy's twelve values, bit for bit. Old code that calls +// np.random.seed(0) and np.random.rand(...) has random.Seed(0) and +// random.Rand(...), over the legacy RandomState, with the same numbers too. +// +// # What is here +// +// - SeedSequence, NumPy's entropy mixer: pool, GenerateState, Spawn. +// - The bit generators PCG64 (the default), PCG64DXSM, MT19937, Philox and +// SFC64, with Jumped and Advance where NumPy has them, and RandomRaw. +// - Generator (DefaultRNG, NewGenerator): Random, Integers, Uniform, +// StandardNormal/Normal (ziggurat), StandardExponential/Exponential, +// StandardGamma/Gamma, Beta, ChiSquare, F, Binomial, Poisson, Geometric, +// Lognormal, Laplace, Logistic, Gumbel, Weibull, Pareto, Power, Rayleigh, +// Wald, Triangular, VonMises, StandardT, StandardCauchy, Zipf, Logseries, +// NegativeBinomial, the noncentral chi-square and F, Choice, Shuffle, +// Permutation, Permuted, Bytes, Dirichlet and Multinomial. +// - RandomState, the legacy generator (MT19937, the polar Gaussian, NumPy +// 1.16's samplers), and the module functions Seed, Rand, Randn, Randint, +// RandomSample, Normal, Uniform, Choice, Shuffle, Permutation, Binomial +// and Poisson over a shared one. +// +// Results are arrays with NumPy's dtypes and shapes: the size arguments come +// last (numpy's size), and no size gives a 0-d array holding what NumPy +// returns as a scalar. Parameters are scalars (NumPy also broadcasts array +// parameters; that is not implemented here). +// +// # Exactness +// +// Integers, raw bit-generator output and every sampler that uses only +// arithmetic match NumPy to the bit. Samplers that call log, exp, pow ... +// match it except where NumPy's C library rounds those functions +// incorrectly: this package rounds them correctly, the C libraries almost +// always do, and the remainder is a fraction of a percent of values, one +// ULP apart. docs/random.md has the measurements, including the ones where +// NumPy's own aarch64 builds disagree with its x86-64 builds. +// +// Generators are not safe for concurrent use; give each goroutine its own +// (Generator.Spawn). +package random diff --git a/random/example_test.go b/random/example_test.go new file mode 100644 index 0000000..780ea48 --- /dev/null +++ b/random/example_test.go @@ -0,0 +1,56 @@ +package random_test + +import ( + "fmt" + + "github.com/go-ndarray/ndarray" + "github.com/go-ndarray/ndarray/random" +) + +// The numbers below are NumPy's, for the same seed: +// +// rng = np.random.default_rng(42) +// rng.random(3); rng.integers(0, 10, 5); rng.standard_normal(2) +func Example() { + rng := random.DefaultRNG(42) + u, _ := rng.Random(3) + i, _ := rng.Integers(0, 10, 5) + z, _ := rng.StandardNormal(2) + fu, _ := ndarray.Data[float64](u) + fi, _ := ndarray.Data[int64](i) + fz, _ := ndarray.Data[float64](z) + fmt.Println(fu) + fmt.Println(fi) + fmt.Println(fz) + // Output: + // [0.7739560485559633 0.4388784397520523 0.8585979199113825] + // [0 6 2 0 5] + // [0.12784040316728537 -0.3162425923435822] +} + +// The legacy API: np.random.seed(0); np.random.rand(3); np.random.randn(2); +// np.random.randint(0, 10, 5). +func ExampleSeed() { + random.Seed(0) + u, _ := random.Rand(3) + z, _ := random.Randn(2) + i, _ := random.Randint(0, 10, 5) + fu, _ := ndarray.Data[float64](u) + fz, _ := ndarray.Data[float64](z) + fi, _ := ndarray.Data[int64](i) + fmt.Println(fu) + fmt.Println(fz) + fmt.Println(fi) + // Output: + // [0.5488135039273248 0.7151893663724195 0.6027633760716439] + // [-2.268328201180374 1.3335453816218967] + // [5 2 4 7 6] +} + +// rng.choice(5, 3, replace=False) +func ExampleGenerator_ChoiceN() { + idx, _ := random.DefaultRNG(42).ChoiceN(5, random.ChoiceOptions{NoReplace: true}, 3) + d, _ := ndarray.Data[int64](idx) + fmt.Println(d) + // Output: [4 0 3] +} diff --git a/random/generator.go b/random/generator.go new file mode 100644 index 0000000..3f81da4 --- /dev/null +++ b/random/generator.go @@ -0,0 +1,950 @@ +package random + +import ( + "encoding/binary" + "math" + "reflect" + "sort" + + "github.com/go-ndarray/ndarray" +) + +// Generator draws from distributions, numpy.random.Generator. Every method +// takes the output shape last (numpy's size): no shape gives a 0-d array, +// which holds what numpy returns as a scalar for size=None and consumes the +// same draws. +// +// A Generator is not safe for concurrent use; give each goroutine its own, +// from Spawn. +type Generator struct { + bg BitGenerator + binom binomialCache +} + +// NewGenerator returns numpy's Generator(bit_generator). +func NewGenerator(bg BitGenerator) *Generator { return &Generator{bg: bg} } + +// DefaultRNG returns numpy's default_rng(seed): a Generator over +// PCG64(SeedSequence(seed)). One value is numpy's integer seed, several its +// list seed (default_rng([1, 2, 3])); no value seeds from the OS. +func DefaultRNG(seed ...uint64) *Generator { return NewGenerator(NewPCG64(seed...)) } + +// BitGenerator returns the generator's bit generator. +func (g *Generator) BitGenerator() BitGenerator { return g.bg } + +// Spawn returns n independent Generators, numpy's Generator.spawn: over +// bit generators of the same kind seeded from the SeedSequence's children. +func (g *Generator) Spawn(n int) ([]*Generator, error) { + sp, ok := g.bg.(Spawner) + if !ok { + return nil, ErrNoSeedSequence + } + bgs, err := sp.Spawn(n) + if err != nil { + return nil, err + } + out := make([]*Generator, len(bgs)) + for i, b := range bgs { + out[i] = NewGenerator(b) + } + return out, nil +} + +// fill64 makes a float64 array of the given shape from n draws of f. +func fill64(shape []int, f func() float64) (*ndarray.Array, error) { + a, d, err := newArray[float64](ndarray.Float64, shape) + if err != nil { + return nil, err + } + for i := range d { + d[i] = f() + } + return a, nil +} + +func fill32(shape []int, f func() float32) (*ndarray.Array, error) { + a, d, err := newArray[float32](ndarray.Float32, shape) + if err != nil { + return nil, err + } + for i := range d { + d[i] = f() + } + return a, nil +} + +func fillInt(shape []int, f func() int64) (*ndarray.Array, error) { + a, d, err := newArray[int64](ndarray.Int64, shape) + if err != nil { + return nil, err + } + for i := range d { + d[i] = f() + } + return a, nil +} + +func floatOnly(dt ndarray.DType, what string) error { + if dt != ndarray.Float64 && dt != ndarray.Float32 { + return valueErr("Unsupported dtype %v for %s", dt, what) + } + return nil +} + +// Random returns float64 values in [0, 1), numpy's random(size). +func (g *Generator) Random(size ...int) (*ndarray.Array, error) { + a, d, err := newArray[float64](ndarray.Float64, size) + if err != nil { + return nil, err + } + if p, ok := g.bg.(*PCG64); ok { // the default, called directly + for i := range d { + d[i] = uint64ToDouble(p.Uint64()) + } + return a, nil + } + for i := range d { + d[i] = g.bg.Float64() + } + return a, nil +} + +// RandomOf returns values in [0, 1) of dtype Float64 or Float32, numpy's +// random(size, dtype). Float32 values have 24 random bits, from 32-bit draws. +func (g *Generator) RandomOf(dt ndarray.DType, size ...int) (*ndarray.Array, error) { + if err := floatOnly(dt, "random"); err != nil { + return nil, err + } + if dt == ndarray.Float64 { + return g.Random(size...) + } + if p, ok := g.bg.(*PCG64); ok { + a, d, err := newArray[float32](ndarray.Float32, size) + if err != nil { + return nil, err + } + for i := range d { + d[i] = float32(p.Uint32()>>8) * (1.0 / 16777216.0) + } + return a, nil + } + return fill32(size, func() float32 { return nextFloat(g.bg) }) +} + +// StandardNormal returns float64 standard normal values (ziggurat), numpy's +// standard_normal(size). +func (g *Generator) StandardNormal(size ...int) (*ndarray.Array, error) { + a, d, err := newArray[float64](ndarray.Float64, size) + if err != nil { + return nil, err + } + if p, ok := g.bg.(*PCG64); ok { + for i := range d { + d[i] = standardNormalPCG(p) + } + return a, nil + } + for i := range d { + d[i] = standardNormal(g.bg) + } + return a, nil +} + +// StandardNormalOf is numpy's standard_normal(size, dtype) for Float64 or +// Float32 (a separate 32-bit ziggurat). +func (g *Generator) StandardNormalOf(dt ndarray.DType, size ...int) (*ndarray.Array, error) { + if err := floatOnly(dt, "standard_normal"); err != nil { + return nil, err + } + if dt == ndarray.Float32 { + return fill32(size, func() float32 { return standardNormalF(g.bg) }) + } + return g.StandardNormal(size...) +} + +// Normal returns loc + scale * standard normal, numpy's normal(loc, scale, size). +func (g *Generator) Normal(loc, scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return normal(g.bg, loc, scale) }) +} + +// StandardExponential returns float64 values of the unit exponential +// (ziggurat), numpy's standard_exponential(size). +func (g *Generator) StandardExponential(size ...int) (*ndarray.Array, error) { + return g.StandardExponentialOf(ndarray.Float64, "zig", size...) +} + +// StandardExponentialOf is numpy's standard_exponential(size, dtype, method) +// with method "zig" (ziggurat) or "inv" (inversion, -log1p(-U)). +func (g *Generator) StandardExponentialOf(dt ndarray.DType, method string, size ...int) (*ndarray.Array, error) { + if err := floatOnly(dt, "standard_exponential"); err != nil { + return nil, err + } + if method != "zig" && method != "inv" { + return nil, valueErr("method must be 'zig' or 'inv'") + } + inv := method == "inv" + if dt == ndarray.Float32 { + return fill32(size, func() float32 { + if inv { + return float32(-log1p(-float64(nextFloat(g.bg)))) + } + return standardExponentialF(g.bg) + }) + } + return fill64(size, func() float64 { + if inv { + return -log1p(-g.bg.Float64()) + } + return standardExponential(g.bg) + }) +} + +// Exponential returns scale * the unit exponential, numpy's +// exponential(scale, size). +func (g *Generator) Exponential(scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return scale * standardExponential(g.bg) }) +} + +// Uniform returns values in [low, high), numpy's uniform(low, high, size): +// low + (high-low)*U. +func (g *Generator) Uniform(low, high float64, size ...int) (*ndarray.Array, error) { + rng := high - low + if math.IsInf(rng, 0) || math.IsNaN(rng) { + return nil, valueErr("high - low range exceeds valid bounds") + } + if err := nonNegative(rng, "high - low"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return uniform(g.bg, low, rng) }) +} + +// StandardGamma is numpy's standard_gamma(shape, size) (Marsaglia-Tsang). +func (g *Generator) StandardGamma(shape float64, size ...int) (*ndarray.Array, error) { + return g.StandardGammaOf(ndarray.Float64, shape, size...) +} + +// StandardGammaOf is numpy's standard_gamma(shape, size, dtype). +func (g *Generator) StandardGammaOf(dt ndarray.DType, shape float64, size ...int) (*ndarray.Array, error) { + if err := floatOnly(dt, "standard_gamma"); err != nil { + return nil, err + } + if err := nonNegative(shape, "shape"); err != nil { + return nil, err + } + if dt == ndarray.Float32 { + s := float32(shape) + return fill32(size, func() float32 { return standardGammaF(g.bg, s) }) + } + return fill64(size, func() float64 { return standardGamma(g.bg, shape) }) +} + +// Gamma is numpy's gamma(shape, scale, size). +func (g *Generator) Gamma(shape, scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(shape, "shape"); err != nil { + return nil, err + } + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return scale * standardGamma(g.bg, shape) }) +} + +// Beta is numpy's beta(a, b, size). +func (g *Generator) Beta(a, b float64, size ...int) (*ndarray.Array, error) { + if err := positive(a, "a"); err != nil { + return nil, err + } + if err := positive(b, "b"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return beta(g.bg, a, b) }) +} + +// ChiSquare is numpy's chisquare(df, size). +func (g *Generator) ChiSquare(df float64, size ...int) (*ndarray.Array, error) { + if err := positive(df, "df"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return chisquare(g.bg, df) }) +} + +// F is numpy's f(dfnum, dfden, size). +func (g *Generator) F(dfnum, dfden float64, size ...int) (*ndarray.Array, error) { + if err := positive(dfnum, "dfnum"); err != nil { + return nil, err + } + if err := positive(dfden, "dfden"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return fDist(g.bg, dfnum, dfden) }) +} + +// NoncentralChiSquare is numpy's noncentral_chisquare(df, nonc, size). +func (g *Generator) NoncentralChiSquare(df, nonc float64, size ...int) (*ndarray.Array, error) { + if err := positive(df, "df"); err != nil { + return nil, err + } + if err := nonNegative(nonc, "nonc"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return noncentralChisquare(g.bg, df, nonc) }) +} + +// NoncentralF is numpy's noncentral_f(dfnum, dfden, nonc, size). +func (g *Generator) NoncentralF(dfnum, dfden, nonc float64, size ...int) (*ndarray.Array, error) { + if err := positive(dfnum, "dfnum"); err != nil { + return nil, err + } + if err := positive(dfden, "dfden"); err != nil { + return nil, err + } + if err := nonNegative(nonc, "nonc"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return noncentralF(g.bg, dfnum, dfden, nonc) }) +} + +// StandardCauchy is numpy's standard_cauchy(size). +func (g *Generator) StandardCauchy(size ...int) (*ndarray.Array, error) { + return fill64(size, func() float64 { return standardCauchy(g.bg) }) +} + +// StandardT is numpy's standard_t(df, size). +func (g *Generator) StandardT(df float64, size ...int) (*ndarray.Array, error) { + if err := positive(df, "df"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return standardT(g.bg, df) }) +} + +// VonMises is numpy's vonmises(mu, kappa, size). +func (g *Generator) VonMises(mu, kappa float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(kappa, "kappa"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return vonmises(g.bg, mu, kappa) }) +} + +// Pareto is numpy's pareto(a, size). +func (g *Generator) Pareto(a float64, size ...int) (*ndarray.Array, error) { + if err := positive(a, "a"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return pareto(g.bg, a) }) +} + +// Weibull is numpy's weibull(a, size). +func (g *Generator) Weibull(a float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(a, "a"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return weibull(g.bg, a) }) +} + +// Power is numpy's power(a, size). +func (g *Generator) Power(a float64, size ...int) (*ndarray.Array, error) { + if err := positive(a, "a"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return power(g.bg, a) }) +} + +// Laplace is numpy's laplace(loc, scale, size). +func (g *Generator) Laplace(loc, scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return laplace(g.bg, loc, scale) }) +} + +// Gumbel is numpy's gumbel(loc, scale, size). +func (g *Generator) Gumbel(loc, scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return gumbel(g.bg, loc, scale) }) +} + +// Logistic is numpy's logistic(loc, scale, size). +func (g *Generator) Logistic(loc, scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return logistic(g.bg, loc, scale) }) +} + +// Lognormal is numpy's lognormal(mean, sigma, size). +func (g *Generator) Lognormal(mean, sigma float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(sigma, "sigma"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return lognormal(g.bg, mean, sigma) }) +} + +// Rayleigh is numpy's rayleigh(scale, size). +func (g *Generator) Rayleigh(scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return rayleigh(g.bg, scale) }) +} + +// Wald is numpy's wald(mean, scale, size). +func (g *Generator) Wald(mean, scale float64, size ...int) (*ndarray.Array, error) { + if err := positive(mean, "mean"); err != nil { + return nil, err + } + if err := positive(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return wald(g.bg, mean, scale) }) +} + +// Triangular is numpy's triangular(left, mode, right, size). +func (g *Generator) Triangular(left, mode, right float64, size ...int) (*ndarray.Array, error) { + switch { + case left > mode: + return nil, valueErr("left > mode") + case mode > right: + return nil, valueErr("mode > right") + case left == right: + return nil, valueErr("left == right") + } + return fill64(size, func() float64 { return triangular(g.bg, left, mode, right) }) +} + +// Binomial is numpy's binomial(n, p, size), int64 values. +func (g *Generator) Binomial(n int64, p float64, size ...int) (*ndarray.Array, error) { + if err := bounded01(p, "p"); err != nil { + return nil, err + } + if n < 0 { + return nil, valueErr("n < 0") + } + return fillInt(size, func() int64 { return binomial(g.bg, p, n, &g.binom, false) }) +} + +// NegativeBinomial is numpy's negative_binomial(n, p, size), int64 values. +func (g *Generator) NegativeBinomial(n, p float64, size ...int) (*ndarray.Array, error) { + if math.IsNaN(n) { + return nil, valueErr("n must not be NaN") + } + if err := positive(n, "n"); err != nil { + return nil, err + } + if err := boundedGT01(p, "p"); err != nil { + return nil, err + } + if (1-p)/p*(n+float64(10*math.Sqrt(n))) > poissonLamMax { + return nil, valueErr("n too large or p too small, see Generator.negative_binomial Notes") + } + return fillInt(size, func() int64 { return negativeBinomial(g.bg, n, p) }) +} + +// poissonLamMax is numpy's POISSON_LAM_MAX. +var poissonLamMax = float64(math.MaxInt64) - float64(math.Sqrt(float64(math.MaxInt64))*10) + +// Poisson is numpy's poisson(lam, size), int64 values. +func (g *Generator) Poisson(lam float64, size ...int) (*ndarray.Array, error) { + if err := poissonLam(lam); err != nil { + return nil, err + } + return fillInt(size, func() int64 { return poisson(g.bg, lam) }) +} + +// Zipf is numpy's zipf(a, size), int64 values. +func (g *Generator) Zipf(a float64, size ...int) (*ndarray.Array, error) { + if !(a > 1) { + return nil, valueErr("a <= 1 or a is NaN") + } + return fillInt(size, func() int64 { return zipf(g.bg, a) }) +} + +// Geometric is numpy's geometric(p, size), int64 values. +func (g *Generator) Geometric(p float64, size ...int) (*ndarray.Array, error) { + if err := boundedGT01(p, "p"); err != nil { + return nil, err + } + return fillInt(size, func() int64 { return geometric(g.bg, p) }) +} + +// Logseries is numpy's logseries(p, size), int64 values. +func (g *Generator) Logseries(p float64, size ...int) (*ndarray.Array, error) { + if !(p >= 0) || !(p < 1) { + return nil, valueErr("p < 0, p >= 1 or p is NaN") + } + return fillInt(size, func() int64 { return logseries(g.bg, p) }) +} + +// Bytes returns n random bytes, numpy's bytes(n): little-endian uint32 draws, +// the same bytes on every platform. +func (g *Generator) Bytes(n int) []byte { + words := (n + 3) / 4 + b := make([]byte, 4*words) + for i := range words { + binary.LittleEndian.PutUint32(b[4*i:], g.bg.Uint32()) + } + return b[:n] +} + +// Dirichlet is numpy's dirichlet(alpha, size): shape size + (len(alpha),). +func (g *Generator) Dirichlet(alpha []float64, size ...int) (*ndarray.Array, error) { + k := len(alpha) + amax := math.Inf(-1) + for _, a := range alpha { + if a < 0 { + return nil, valueErr("alpha < 0") + } + amax = max(amax, a) + } + shape := append(append([]int(nil), size...), k) + out, d, err := newArray[float64](ndarray.Float64, shape) + if err != nil { + return nil, err + } + if k > 0 && amax < 0.1 { + csum := make([]float64, k) + acc := 0.0 + for j := k - 1; j >= 0; j-- { + acc += alpha[j] + csum[j] = acc + } + if acc > 0 { + for i := 0; i < len(d); i += k { + acc := 1. + for j := 0; j < k-1; j++ { + v := beta(g.bg, alpha[j], csum[j+1]) + d[i+j] = acc * v + acc *= 1. - v + if csum[j+1] == 0 { + break + } + } + d[i+k-1] = acc + } + } + return out, nil + } + for i := 0; i < len(d); i += k { + acc := 0. + for j := range k { + d[i+j] = standardGamma(g.bg, alpha[j]) + acc += d[i+j] + } + inv := 1. / acc + for j := range k { + d[i+j] *= inv + } + } + return out, nil +} + +// Multinomial is numpy's multinomial(n, pvals, size): int64 counts of shape +// size + (len(pvals),). +func (g *Generator) Multinomial(n int64, pvals []float64, size ...int) (*ndarray.Array, error) { + return multinomialArray(g.bg, &g.binom, false, n, pvals, size) +} + +func multinomialArray(bg BitGenerator, bc *binomialCache, legacy bool, n int64, pvals []float64, size []int) (*ndarray.Array, error) { + d := len(pvals) + if d == 0 { + return nil, valueErr("pvals must have at least 1 dimension and the last dimension of pvals must be greater than 0.") + } + for _, p := range pvals { + if err := bounded01(p, "pvals"); err != nil { + return nil, err + } + } + if kahanSum(pvals[:d-1]) > 1.0+1e-12 { + return nil, valueErr("sum(pvals[:-1]) > 1.0") + } + if n < 0 { + return nil, valueErr("n < 0") + } + shape := append(append([]int(nil), size...), d) + out, m, err := newArray[int64](ndarray.Int64, shape) + if err != nil { + return nil, err + } + for off := 0; off < len(m); off += d { + multinomial(bg, n, m[off:off+d], pvals, bc, legacy) + } + return out, nil +} + +// kahanSum is numpy's kahan_sum. +func kahanSum(x []float64) float64 { + if len(x) == 0 { + return 0 + } + sum, c := x[0], 0.0 + for _, v := range x[1:] { + y := v - c + t := sum + y + c = (t - sum) - y + sum = t + } + return sum +} + +func nonNegative(v float64, name string) error { + if !math.IsNaN(v) && math.Signbit(v) { + return valueErr("%s < 0", name) + } + return nil +} + +func positive(v float64, name string) error { + if v <= 0 { + return valueErr("%s <= 0", name) + } + return nil +} + +func bounded01(v float64, name string) error { + if !(v >= 0) || !(v <= 1) { + return valueErr("%s < 0, %s > 1 or %s is NaN", name, name, name) + } + return nil +} + +func boundedGT01(v float64, name string) error { + if !(v > 0) || !(v <= 1) { + return valueErr("%s <= 0, %s > 1 or %s contains NaNs", name, name, name) + } + return nil +} + +func poissonLam(lam float64) error { + if !(lam >= 0) { + return valueErr("lam < 0 or lam is NaN") + } + if !(lam <= poissonLamMax) { + return valueErr("lam value too large") + } + return nil +} + +// Shuffle permutes a in place along axis, numpy's shuffle(x, axis): the +// Fisher-Yates swaps of slabs along the axis, from the last to the second. +// a must own contiguous storage (not be a view). +func (g *Generator) Shuffle(a *ndarray.Array, axis int) error { + if a.Ndim() == 0 { + return valueErr("x must be an array of at least one dimension") + } + ax, err := normAxis(axis, a.Ndim()) + if err != nil { + return err + } + s, err := ownStorage(a) + if err != nil || a.Size() == 0 { + return err + } + outer, n, inner := layout(a.Shape(), ax) + swap := reflect.Swapper(s.Interface()) + for i := n - 1; i >= 1; i-- { + j := int(interval(g.bg, uint64(i))) + if i != j { + swapSlabs(swap, outer, n, inner, i, j) + } + } + return nil +} + +// Permutation returns a shuffled arange(n) of int64, numpy's permutation(n). +func (g *Generator) Permutation(n int) (*ndarray.Array, error) { + a, d, err := newArray[int64](ndarray.Int64, []int{n}) + if err != nil { + return nil, err + } + for i := range d { + d[i] = int64(i) + } + return a, g.Shuffle(a, 0) +} + +// PermutationOf returns a copy of a with its slices along axis shuffled, +// numpy's permutation(x, axis). +func (g *Generator) PermutationOf(a *ndarray.Array, axis int) (*ndarray.Array, error) { + if a.Ndim() == 0 { + return nil, valueErr("axis %d is out of bounds for array of dimension 0", axis) + } + ax, err := normAxis(axis, a.Ndim()) + if err != nil { + return nil, err + } + if a.Ndim() == 1 { + c := a.Copy() + return c, g.Shuffle(c, 0) + } + idx, _ := g.Permutation(a.Shape()[ax]) + d, _ := ndarray.Data[int64](idx) + return takeAxis(a, d, []int{len(d)}, ax) +} + +// Permuted returns a copy of a with its elements shuffled as one flat array, +// numpy's permuted(x) (axis=None). +func (g *Generator) Permuted(a *ndarray.Array) (*ndarray.Array, error) { + c := a.Copy() + s := reflect.ValueOf(storage(c)) + swap := reflect.Swapper(s.Interface()) + for i := s.Len() - 1; i >= 1; i-- { + swap(i, int(interval(g.bg, uint64(i)))) + } + return c, nil +} + +// PermutedAxis returns a copy of a with each 1-d slice along axis shuffled +// independently, numpy's permuted(x, axis=axis); slices are taken in C order. +func (g *Generator) PermutedAxis(a *ndarray.Array, axis int) (*ndarray.Array, error) { + ax, err := normAxis(axis, a.Ndim()) + if err != nil { + return nil, err + } + c := a.Copy() + s := reflect.ValueOf(storage(c)) + swap := reflect.Swapper(s.Interface()) + outer, n, inner := layout(c.Shape(), ax) + for o := range outer { + for in := range inner { + base := o*n*inner + in + for i := n - 1; i >= 0; i-- { + j := int(interval(g.bg, uint64(i))) + swap(base+i*inner, base+j*inner) + } + } + } + return c, nil +} + +// ChoiceOptions are numpy's choice keywords. Their zero values are numpy's +// defaults: with replacement, uniform, axis 0, shuffled. +type ChoiceOptions struct { + // NoReplace samples without replacement (numpy's replace=False). + NoReplace bool + // P are the probabilities of the entries; nil is uniform. + P []float64 + // Axis is the axis of a to sample along. + Axis int + // NoShuffle leaves a sample without replacement in selection order + // (numpy's shuffle=False). + NoShuffle bool +} + +// ChoiceN returns int64 indices sampled from arange(n), numpy's +// choice(n, size, replace, p, shuffle=...). +func (g *Generator) ChoiceN(n int64, o ChoiceOptions, size ...int) (*ndarray.Array, error) { + if n <= 0 && ndarrayProd(size) != 0 { + return nil, valueErr("a must be a positive integer unless no samples are taken") + } + idx, err := g.choiceIndices(n, o, size) + if err != nil { + return nil, err + } + return ndarray.FromSlice(idx, size...) +} + +// Choice returns entries of a sampled along o.Axis, numpy's +// choice(a, size, replace, p, axis, shuffle): a's shape with that axis +// replaced by size. +func (g *Generator) Choice(a *ndarray.Array, o ChoiceOptions, size ...int) (*ndarray.Array, error) { + if a.Ndim() == 0 { + return nil, valueErr("a must be a sequence or an integer, not a 0-d array") + } + ax, err := normAxis(o.Axis, a.Ndim()) + if err != nil { + return nil, err + } + n := int64(a.Shape()[ax]) + if n == 0 && ndarrayProd(size) != 0 { + return nil, valueErr("a cannot be empty unless no samples are taken") + } + idx, err := g.choiceIndices(n, o, size) + if err != nil { + return nil, err + } + return takeAxis(a, idx, size, ax) +} + +func ndarrayProd(size []int) int { + p := 1 + for _, d := range size { + p *= d + } + return p +} + +// checkP validates choice's probabilities as numpy does. +func checkP(p []float64, n int64) error { + if int64(len(p)) != n { + return valueErr("a and p must have same size") + } + s := kahanSum(p) + if math.IsNaN(s) { + return valueErr("Probabilities contain NaN") + } + for _, v := range p { + if v < 0 { + return valueErr("Probabilities are not non-negative") + } + } + if math.Abs(s-1.) > math.Sqrt(0x1p-52) { + return valueErr("Probabilities do not sum to 1. See Notes section of docstring for more information.") + } + return nil +} + +// cdfOf is numpy's cdf = p.cumsum(); cdf /= cdf[-1]. +func cdfOf(p []float64) []float64 { + cdf := make([]float64, len(p)) + acc := 0.0 + for i, v := range p { + acc += v + cdf[i] = acc + } + last := cdf[len(cdf)-1] + for i := range cdf { + cdf[i] /= last + } + return cdf +} + +// searchRight is numpy's searchsorted(side='right'). +func searchRight(cdf []float64, u float64) int64 { + return int64(sort.Search(len(cdf), func(i int) bool { return cdf[i] > u })) +} + +func (g *Generator) choiceIndices(n int64, o ChoiceOptions, size []int) ([]int64, error) { + if o.P != nil { + if err := checkP(o.P, n); err != nil { + return nil, err + } + } + count := ndarrayProd(size) + if count < 0 { + return nil, valueErr("negative dimensions are not allowed") + } + idx := make([]int64, count) + if !o.NoReplace { + if o.P != nil { + cdf := cdfOf(o.P) + for i := range idx { + idx[i] = searchRight(cdf, g.bg.Float64()) + } + return idx, nil + } + u := make([]uint64, count) + fillUint64(g.bg, 0, uint64(n-1), false, u) + for i := range u { + idx[i] = int64(u[i]) + } + return idx, nil + } + if int64(count) > n { + return nil, valueErr("Cannot take a larger sample than population when replace is False") + } + if o.P != nil { + return g.choiceWeighted(o.P, idx) + } + g.choiceFloyd(n, int64(count), !o.NoShuffle, idx) + return idx, nil +} + +// choiceWeighted is numpy's weighted sampling without replacement: draw, +// keep the first occurrence of each new index, zero the found weights, +// repeat. +func (g *Generator) choiceWeighted(p []float64, found []int64) ([]int64, error) { + nz := 0 + for _, v := range p { + if v > 0 { + nz++ + } + } + if nz < len(found) { + return nil, valueErr("Fewer non-zero entries in p than size") + } + p = append([]float64(nil), p...) + nUniq := 0 + for nUniq < len(found) { + x := make([]float64, len(found)-nUniq) + for i := range x { + x[i] = g.bg.Float64() + } + for _, f := range found[:nUniq] { + p[f] = 0 + } + cdf := cdfOf(p) + seen := map[int64]bool{} + for _, u := range x { + k := searchRight(cdf, u) + if !seen[k] { + seen[k] = true + found[nUniq] = k + nUniq++ + } + } + } + return found, nil +} + +// choiceFloyd is numpy's unweighted sampling without replacement: a partial +// Fisher-Yates over arange(n) when the sample is a large part of a large +// population, otherwise Floyd's algorithm with a hash set, then (if shuffle) +// a Fisher-Yates over the sample. +func (g *Generator) choiceFloyd(n, size int64, shuffle bool, idx []int64) { + cutoff := int64(20) + if shuffle { + cutoff = 50 + } + if n > 10000 && size > n/cutoff { + all := make([]int64, n) + for i := range all { + all[i] = int64(i) + } + shuffleInt(g.bg, all, max(n-size, 1)) + copy(idx, all[n-size:]) + return + } + setSize := uint64(1.2 * float64(size)) + mask := genMask(setSize) + hash := make([]uint64, mask+1) + for i := range hash { + hash[i] = ^uint64(0) + } + for j := n - size; j < n; j++ { + val := boundedUint64(g.bg, 0, uint64(j), false) + loc := val & mask + for hash[loc] != ^uint64(0) && hash[loc] != val { + loc = (loc + 1) & mask + } + if hash[loc] == ^uint64(0) { + hash[loc] = val + idx[j-n+size] = int64(val) + } else { + loc = uint64(j) & mask + for hash[loc] != ^uint64(0) { + loc = (loc + 1) & mask + } + hash[loc] = uint64(j) + idx[j-n+size] = j + } + } + if shuffle { + shuffleInt(g.bg, idx, 1) + } +} + +// shuffleInt is numpy's _shuffle_int: Fisher-Yates from the end down to +// first, with Lemire bounded draws. +func shuffleInt(bg BitGenerator, d []int64, first int64) { + for i := int64(len(d)) - 1; i >= first; i-- { + j := boundedUint64(bg, 0, uint64(i), false) + d[i], d[j] = d[j], d[i] + } +} diff --git a/random/integers.go b/random/integers.go new file mode 100644 index 0000000..49056a8 --- /dev/null +++ b/random/integers.go @@ -0,0 +1,129 @@ +package random + +import ( + "math" + + "github.com/go-ndarray/ndarray" +) + +// intBounds are the bounds of numpy's integer dtypes, as int64 (uint64's +// upper bound does not fit and is handled by IntegersUint64). +var intBounds = map[ndarray.DType][2]int64{ + ndarray.Bool: {0, 1}, + ndarray.Int8: {math.MinInt8, math.MaxInt8}, + ndarray.Int16: {math.MinInt16, math.MaxInt16}, + ndarray.Int32: {math.MinInt32, math.MaxInt32}, + ndarray.Int64: {math.MinInt64, math.MaxInt64}, + ndarray.Uint8: {0, math.MaxUint8}, + ndarray.Uint16: {0, math.MaxUint16}, + ndarray.Uint32: {0, math.MaxUint32}, + ndarray.Uint64: {0, math.MaxInt64}, +} + +// boundsErr is numpy's format_bounds_error. +func boundsErr(closed bool, low int64) error { + switch { + case low == 0 && closed: + return valueErr("high < 0") + case low == 0: + return valueErr("high <= 0") + case closed: + return valueErr("low > high") + } + return valueErr("low >= high") +} + +// Integers returns int64 values in [low, high), numpy's +// integers(low, high, size). +func (g *Generator) Integers(low, high int64, size ...int) (*ndarray.Array, error) { + return g.IntegersOf(ndarray.Int64, low, high, false, size...) +} + +// IntegersOf is numpy's integers(low, high, size, dtype, endpoint): values in +// [low, high), or [low, high] with endpoint, of an integer or Bool dtype, +// drawn with Lemire's method. +func (g *Generator) IntegersOf(dt ndarray.DType, low, high int64, endpoint bool, size ...int) (*ndarray.Array, error) { + return integers(g.bg, false, dt, low, high, endpoint, size) +} + +// IntegersUint64 is numpy's integers(low, high, size, dtype=np.uint64, +// endpoint) for bounds beyond int64. +func (g *Generator) IntegersUint64(low, high uint64, endpoint bool, size ...int) (*ndarray.Array, error) { + return integersUint64(g.bg, false, low, high, endpoint, size) +} + +func integersUint64(bg BitGenerator, masked bool, low, high uint64, endpoint bool, size []int) (*ndarray.Array, error) { + if !endpoint { + if high == 0 { + return nil, boundsErr(false, int64(min(low, 1))) + } + high-- + } + if low > high { + return nil, boundsErr(true, int64(min(low, 1))) + } + a, d, err := newArray[uint64](ndarray.Uint64, size) + if err != nil { + return nil, err + } + fillUint64(bg, low, high-low, masked, d) + return a, nil +} + +func integers(bg BitGenerator, masked bool, dt ndarray.DType, low, high int64, endpoint bool, size []int) (*ndarray.Array, error) { + lim, ok := intBounds[dt] + if !ok { + return nil, valueErr("Unsupported dtype %v for integers", dt) + } + if !endpoint { + if high == math.MinInt64 { + return nil, boundsErr(false, low) + } + high-- + } + if low < lim[0] { + return nil, valueErr("low is out of bounds for %v", dt) + } + if high > lim[1] { + return nil, valueErr("high is out of bounds for %v", dt) + } + if low > high { + return nil, boundsErr(endpoint, low) + } + if dt == ndarray.Uint64 { + return integersUint64(bg, masked, uint64(low), uint64(high), true, size) + } + rng := uint64(high - low) + off := uint64(low) + a, err := ndarray.ZerosOf(dt, size...) + if err != nil { + return nil, err + } + switch dt { + case ndarray.Bool: + d, _ := ndarray.Data[bool](a) + fillBool(bg, off != 0, rng != 0, d) + case ndarray.Int8: + d, _ := ndarray.Data[int8](a) + fillUint8(bg, uint8(off), uint8(rng), masked, asUint8(d)) + case ndarray.Uint8: + d, _ := ndarray.Data[uint8](a) + fillUint8(bg, uint8(off), uint8(rng), masked, d) + case ndarray.Int16: + d, _ := ndarray.Data[int16](a) + fillUint16(bg, uint16(off), uint16(rng), masked, asUint16(d)) + case ndarray.Uint16: + d, _ := ndarray.Data[uint16](a) + fillUint16(bg, uint16(off), uint16(rng), masked, d) + case ndarray.Int32: + d, _ := ndarray.Data[int32](a) + fillUint32(bg, uint32(off), uint32(rng), masked, asUint32(d)) + case ndarray.Uint32: + d, _ := ndarray.Data[uint32](a) + fillUint32(bg, uint32(off), uint32(rng), masked, d) + default: // Int64 + d, _ := ndarray.Data[int64](a) + fillUint64(bg, off, rng, masked, asUint64(d)) + } + return a, nil +} diff --git a/random/legacy.go b/random/legacy.go new file mode 100644 index 0000000..696c282 --- /dev/null +++ b/random/legacy.go @@ -0,0 +1,511 @@ +package random + +import ( + "encoding/binary" + "math" + "reflect" + "sync" + + "github.com/go-ndarray/ndarray" +) + +// RandomState is numpy's legacy numpy.random.RandomState: MT19937 with +// NumPy 1.16's samplers, frozen so that old seeds keep their streams. It is +// what np.random.seed(0) followed by np.random.rand(...) uses, and it +// reproduces those numbers bit for bit. +// +// A RandomState is not safe for concurrent use; the package-level functions +// (Seed, Rand, Randn ...) share one behind a mutex, as numpy's module-level +// functions share one behind a lock. +type RandomState struct { + bg BitGenerator + hasGauss bool + gauss float64 + binom binomialCache +} + +// NewRandomState returns numpy's RandomState(seed) for an integer seed. +func NewRandomState(seed uint32) *RandomState { + return &RandomState{bg: NewMT19937Legacy(seed)} +} + +// NewRandomStateArray returns numpy's RandomState(seed) for an array seed +// (init_by_array). +func NewRandomStateArray(seed []uint32) (*RandomState, error) { + m := &MT19937{} + if err := m.SeedLegacyArray(seed); err != nil { + return nil, err + } + return &RandomState{bg: m}, nil +} + +// NewRandomStateFrom returns numpy's RandomState(bit_generator): the legacy +// samplers over any bit generator. +func NewRandomStateFrom(bg BitGenerator) *RandomState { return &RandomState{bg: bg} } + +// NewRandomStateEntropy returns numpy's RandomState() (seed=None): MT19937 +// seeded from a fresh SeedSequence. +func NewRandomStateEntropy() *RandomState { return &RandomState{bg: NewMT19937()} } + +// BitGenerator returns the underlying bit generator. +func (r *RandomState) BitGenerator() BitGenerator { return r.bg } + +func (r *RandomState) mt() (*MT19937, error) { + m, ok := r.bg.(*MT19937) + if !ok { + return nil, valueErr("can only re-seed a MT19937 BitGenerator") + } + return m, nil +} + +// Seed reseeds as np.random.seed(seed) with an integer seed and drops a +// cached Gaussian. +func (r *RandomState) Seed(seed uint32) error { + m, err := r.mt() + if err != nil { + return err + } + m.SeedLegacy(seed) + r.hasGauss, r.gauss = false, 0 + return nil +} + +// SeedArray reseeds as np.random.seed(seed) with an array seed. +func (r *RandomState) SeedArray(seed []uint32) error { + m, err := r.mt() + if err != nil { + return err + } + if err := m.SeedLegacyArray(seed); err != nil { + return err + } + r.hasGauss, r.gauss = false, 0 + return nil +} + +// LegacyState is numpy's legacy get_state() tuple: the MT19937 key and +// position and the cached Gaussian. +type LegacyState struct { + MT19937State + HasGauss bool + Gauss float64 +} + +// GetState returns the state, numpy's get_state() for an MT19937. +func (r *RandomState) GetState() (LegacyState, error) { + m, err := r.mt() + if err != nil { + return LegacyState{}, err + } + return LegacyState{m.State(), r.hasGauss, r.gauss}, nil +} + +// SetState restores a state from GetState, numpy's set_state. +func (r *RandomState) SetState(s LegacyState) error { + m, err := r.mt() + if err != nil { + return err + } + m.SetState(s.MT19937State) + r.hasGauss, r.gauss = s.HasGauss, s.Gauss + return nil +} + +// gaussian is numpy's legacy_gauss: the polar method, which makes two values +// and keeps the second for the next call. +func (r *RandomState) gaussian() float64 { + if r.hasGauss { + t := r.gauss + r.hasGauss, r.gauss = false, 0 + return t + } + var x1, x2, r2 float64 + for { + x1 = 2.0*r.bg.Float64() - 1.0 + x2 = 2.0*r.bg.Float64() - 1.0 + r2 = float64(x1*x1) + float64(x2*x2) + if r2 < 1.0 && r2 != 0.0 { + break + } + } + f := math.Sqrt(-2.0 * log(r2) / r2) + r.gauss = f * x1 + r.hasGauss = true + return f * x2 +} + +func (r *RandomState) standardExponential() float64 { return -log(1.0 - r.bg.Float64()) } + +// standardGamma is numpy's legacy_standard_gamma: Generator's algorithm over +// the legacy Gaussian and exponential. +func (r *RandomState) standardGamma(shape float64) float64 { + switch { + case shape == 1.0: + return r.standardExponential() + case shape == 0.0: + return 0.0 + case shape < 1.0: + for { + u := r.bg.Float64() + v := r.standardExponential() + if u <= 1.0-shape { + x := pow(u, 1./shape) + if x <= v { + return x + } + } else { + y := -log((1 - u) / shape) + x := pow(1.0-shape+float64(shape*y), 1./shape) + if x <= v+y { + return x + } + } + } + } + b := shape - 1./3. + c := 1. / math.Sqrt(9*b) + for { + var x, v float64 + for { + x = r.gaussian() + v = 1.0 + float64(c*x) + if v > 0.0 { + break + } + } + v = v * v * v + u := r.bg.Float64() + if u < 1.0-float64(0.0331*(x*x)*(x*x)) { + return b * v + } + if log(u) < float64(0.5*x*x)+float64(b*(1.-v+log(v))) { + return b * v + } + } +} + +// RandomSample returns float64 values in [0, 1), numpy's random_sample(size). +func (r *RandomState) RandomSample(size ...int) (*ndarray.Array, error) { + return fill64(size, r.bg.Float64) +} + +// Rand is numpy's rand(d0, d1, ...): RandomSample with the dimensions as +// arguments. +func (r *RandomState) Rand(dims ...int) (*ndarray.Array, error) { return r.RandomSample(dims...) } + +// Randn is numpy's randn(d0, d1, ...): standard normal values from the +// legacy polar method. +func (r *RandomState) Randn(dims ...int) (*ndarray.Array, error) { return r.StandardNormal(dims...) } + +// StandardNormal is numpy's legacy standard_normal(size). +func (r *RandomState) StandardNormal(size ...int) (*ndarray.Array, error) { + return fill64(size, r.gaussian) +} + +// Normal is numpy's legacy normal(loc, scale, size). +func (r *RandomState) Normal(loc, scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return loc + float64(scale*r.gaussian()) }) +} + +// Lognormal is numpy's legacy lognormal(mean, sigma, size). +func (r *RandomState) Lognormal(mean, sigma float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(sigma, "sigma"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return exp(mean + float64(sigma*r.gaussian())) }) +} + +// Uniform is numpy's legacy uniform(low, high, size); unlike Generator's it +// accepts high < low. +func (r *RandomState) Uniform(low, high float64, size ...int) (*ndarray.Array, error) { + rng := high - low + if math.IsInf(rng, 0) || math.IsNaN(rng) { + return nil, valueErr("Range exceeds valid bounds") + } + return fill64(size, func() float64 { return uniform(r.bg, low, rng) }) +} + +// StandardExponential is numpy's legacy standard_exponential(size): -log(1-U). +func (r *RandomState) StandardExponential(size ...int) (*ndarray.Array, error) { + return fill64(size, r.standardExponential) +} + +// Exponential is numpy's legacy exponential(scale, size). +func (r *RandomState) Exponential(scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return scale * r.standardExponential() }) +} + +// StandardGamma is numpy's legacy standard_gamma(shape, size). +func (r *RandomState) StandardGamma(shape float64, size ...int) (*ndarray.Array, error) { + return r.Gamma(shape, 1, size...) +} + +// Gamma is numpy's legacy gamma(shape, scale, size). +func (r *RandomState) Gamma(shape, scale float64, size ...int) (*ndarray.Array, error) { + if err := nonNegative(shape, "shape"); err != nil { + return nil, err + } + if err := nonNegative(scale, "scale"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return scale * r.standardGamma(shape) }) +} + +// ChiSquare is numpy's legacy chisquare(df, size). +func (r *RandomState) ChiSquare(df float64, size ...int) (*ndarray.Array, error) { + if err := positive(df, "df"); err != nil { + return nil, err + } + return fill64(size, func() float64 { return 2.0 * r.standardGamma(df/2.0) }) +} + +// Randint returns int64 values in [low, high), numpy's randint(low, high, +// size) with its default dtype (C long, which is int64 on Linux and macOS +// but int32 on Windows; this always uses int64), by masked rejection. +func (r *RandomState) Randint(low, high int64, size ...int) (*ndarray.Array, error) { + return r.RandintOf(ndarray.Int64, low, high, size...) +} + +// RandintOf is numpy's randint(low, high, size, dtype). +func (r *RandomState) RandintOf(dt ndarray.DType, low, high int64, size ...int) (*ndarray.Array, error) { + return integers(r.bg, true, dt, low, high, false, size) +} + +// Bytes returns n random bytes, numpy's legacy bytes(n). +func (r *RandomState) Bytes(n int) []byte { + words := (n + 3) / 4 + b := make([]byte, 4*words) + for i := range words { + binary.LittleEndian.PutUint32(b[4*i:], r.bg.Uint32()) + } + return b[:n] +} + +// Binomial is numpy's legacy binomial(n, p, size), int64 values. +func (r *RandomState) Binomial(n int64, p float64, size ...int) (*ndarray.Array, error) { + if err := bounded01(p, "p"); err != nil { + return nil, err + } + if n < 0 { + return nil, valueErr("n < 0") + } + return fillInt(size, func() int64 { return binomial(r.bg, p, n, &r.binom, true) }) +} + +// Poisson is numpy's legacy poisson(lam, size), int64 values. +func (r *RandomState) Poisson(lam float64, size ...int) (*ndarray.Array, error) { + if err := poissonLam(lam); err != nil { + return nil, err + } + return fillInt(size, func() int64 { return poisson(r.bg, lam) }) +} + +// Multinomial is numpy's legacy multinomial(n, pvals, size). +func (r *RandomState) Multinomial(n int64, pvals []float64, size ...int) (*ndarray.Array, error) { + return multinomialArray(r.bg, &r.binom, true, n, pvals, size) +} + +// Shuffle permutes a in place along its first axis, numpy's legacy +// shuffle(x). a must own contiguous storage. +func (r *RandomState) Shuffle(a *ndarray.Array) error { + if a.Ndim() == 0 { + return valueErr("len() of unsized object") + } + s, err := ownStorage(a) + if err != nil || a.Size() == 0 { + return err + } + outer, n, inner := layout(a.Shape(), 0) + swap := reflect.Swapper(s.Interface()) + for i := n - 1; i >= 1; i-- { + j := int(interval(r.bg, uint64(i))) + if i != j { + swapSlabs(swap, outer, n, inner, i, j) + } + } + return nil +} + +// Permutation returns a shuffled arange(n) of int64, numpy's legacy +// permutation(n). +func (r *RandomState) Permutation(n int) (*ndarray.Array, error) { + a, d, err := newArray[int64](ndarray.Int64, []int{n}) + if err != nil { + return nil, err + } + for i := range d { + d[i] = int64(i) + } + return a, r.Shuffle(a) +} + +// PermutationOf returns a copy of a with its first axis shuffled, numpy's +// legacy permutation(x). +func (r *RandomState) PermutationOf(a *ndarray.Array) (*ndarray.Array, error) { + if a.Ndim() == 0 { + return nil, valueErr("x must be an integer or at least 1-dimensional") + } + if a.Ndim() == 1 { + c := a.Copy() + return c, r.Shuffle(c) + } + idx, _ := r.Permutation(a.Shape()[0]) + d, _ := ndarray.Data[int64](idx) + return takeAxis(a, d, []int{len(d)}, 0) +} + +// ChoiceN is numpy's legacy choice(n, size, replace, p); o.Axis and +// o.NoShuffle do not apply. +func (r *RandomState) ChoiceN(n int64, o ChoiceOptions, size ...int) (*ndarray.Array, error) { + if n <= 0 && ndarrayProd(size) != 0 { + return nil, valueErr("a must be greater than 0 unless no samples are taken") + } + idx, err := r.choiceIndices(n, o, size) + if err != nil { + return nil, err + } + return ndarray.FromSlice(idx, size...) +} + +// Choice is numpy's legacy choice(a, size, replace, p) for a 1-d a. +func (r *RandomState) Choice(a *ndarray.Array, o ChoiceOptions, size ...int) (*ndarray.Array, error) { + if a.Ndim() != 1 { + return nil, valueErr("a must be 1-dimensional") + } + n := int64(a.Shape()[0]) + if n == 0 && ndarrayProd(size) != 0 { + return nil, valueErr("'a' cannot be empty unless no samples are taken") + } + idx, err := r.choiceIndices(n, o, size) + if err != nil { + return nil, err + } + return takeAxis(a, idx, size, 0) +} + +func (r *RandomState) choiceIndices(n int64, o ChoiceOptions, size []int) ([]int64, error) { + if o.P != nil { + if err := checkP(o.P, n); err != nil { + return nil, err + } + } + count := ndarrayProd(size) + idx := make([]int64, count) + // The weighted paths draw exactly as Generator's do (random_sample and + // searchsorted); only the unweighted ones differ. + g := &Generator{bg: r.bg} + switch { + case !o.NoReplace && o.P != nil: + return g.choiceIndices(n, o, size) + case !o.NoReplace: + fillUint64(r.bg, 0, uint64(n-1), true, asUint64(idx)) + return idx, nil + case int64(count) > n: + return nil, valueErr("Cannot take a larger sample than population when 'replace=False'") + case o.P != nil: + return g.choiceWeighted(o.P, idx) + } + // Beyond int (32-bit platforms) the allocation fails as too big. + perm, err := r.Permutation(int(min(n, int64(math.MaxInt)))) + if err != nil { + return nil, err + } + d, _ := ndarray.Data[int64](perm) + copy(idx, d) + return idx, nil +} + +// global is the RandomState behind the package-level functions, numpy's +// np.random._rand, seeded from the OS until Seed is called. +var ( + globalMu sync.Mutex + global *RandomState +) + +func globalState() *RandomState { + if global == nil { + global = NewRandomStateEntropy() + } + return global +} + +func withGlobal[T any](f func(r *RandomState) T) T { + globalMu.Lock() + defer globalMu.Unlock() + return f(globalState()) +} + +// Seed is np.random.seed(seed): it reseeds the shared RandomState. +func Seed(seed uint32) { + withGlobal(func(r *RandomState) error { return r.Seed(seed) }) +} + +// Rand is np.random.rand(d0, d1, ...). +func Rand(dims ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Rand(dims...)) }).unpack() +} + +// Randn is np.random.randn(d0, d1, ...). +func Randn(dims ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Randn(dims...)) }).unpack() +} + +// RandomSample is np.random.random_sample(size) (also np.random.random). +func RandomSample(size ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.RandomSample(size...)) }).unpack() +} + +// Randint is np.random.randint(low, high, size). +func Randint(low, high int64, size ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Randint(low, high, size...)) }).unpack() +} + +// Normal is np.random.normal(loc, scale, size). +func Normal(loc, scale float64, size ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Normal(loc, scale, size...)) }).unpack() +} + +// Uniform is np.random.uniform(low, high, size). +func Uniform(low, high float64, size ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Uniform(low, high, size...)) }).unpack() +} + +// Choice is np.random.choice(a, size, replace, p). +func Choice(a *ndarray.Array, o ChoiceOptions, size ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Choice(a, o, size...)) }).unpack() +} + +// Shuffle is np.random.shuffle(x). +func Shuffle(a *ndarray.Array) error { + return withGlobal(func(r *RandomState) error { return r.Shuffle(a) }) +} + +// Permutation is np.random.permutation(n). +func Permutation(n int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Permutation(n)) }).unpack() +} + +// Binomial is np.random.binomial(n, p, size). +func Binomial(n int64, p float64, size ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Binomial(n, p, size...)) }).unpack() +} + +// Poisson is np.random.poisson(lam, size). +func Poisson(lam float64, size ...int) (*ndarray.Array, error) { + return withGlobal(func(r *RandomState) result { return wrap(r.Poisson(lam, size...)) }).unpack() +} + +type result struct { + a *ndarray.Array + err error +} + +func wrap(a *ndarray.Array, err error) result { return result{a, err} } + +func (r result) unpack() (*ndarray.Array, error) { return r.a, r.err } diff --git a/random/libm.go b/random/libm.go new file mode 100644 index 0000000..e488d48 --- /dev/null +++ b/random/libm.go @@ -0,0 +1,344 @@ +package random + +import "math" + +// The elementary functions numpy takes from the C library. +// +// log, log1p, exp, expm1 and pow are correctly rounded here: each is +// evaluated in double-double arithmetic (about 100 good bits, from exact +// FMA-based products) and rounded once. The C libraries numpy is built +// against (glibc, Apple's libm, the Windows CRT) round these within about +// half an ULP, so they too return the correctly rounded value almost always +// — measured on 20,000 arguments per function, Apple's x86-64 libm does in +// all but 0.06% (log) to 0.3% (exp) of them, where Go's math package +// differs from it in 6.5% (log) to 71% (pow). The result is a stream that +// matches numpy's to the bit in all but those rare cases, and that is the +// same on every Go architecture (math.Log, math.Exp ... use assembly on some +// and pure Go on others). +// +// The double-double result is within about 2^-100 of the true value, so it +// rounds correctly unless the true value lies within that distance of a +// midpoint between two doubles: about one argument in 2^47. +// +// cos and acos (only vonmises uses them) come from the math package; the C +// libraries are not correctly rounded there either (Apple's differs from the +// correctly rounded cos for 17% of arguments in [0, pi]). + +// Go may fuse a product with a later sum into one FMA, even across +// statements and inlined calls, which breaks these error-free +// transformations; every product below is converted explicitly, which +// forbids it. + +// twoSum returns s, e with s = fl(a+b) and s+e = a+b exactly. +func twoSum(a, b float64) (float64, float64) { + s := a + b + bb := s - a + return s, (a - (s - bb)) + (b - bb) +} + +// fastTwoSum is twoSum for |a| >= |b|. +func fastTwoSum(a, b float64) (float64, float64) { + s := a + b + return s, b - (s - a) +} + +// twoProd returns p, e with p = fl(a*b) and p+e = a*b exactly. +func twoProd(a, b float64) (float64, float64) { + p := float64(a * b) // the conversion forbids fusing p into a later sum + return p, math.FMA(a, b, -p) +} + +// dd is a double-double: the unevaluated sum hi + lo, |lo| <= ulp(hi)/2. +type dd struct{ hi, lo float64 } + +func (a dd) add(b dd) dd { + s, e := twoSum(a.hi, b.hi) + e += a.lo + b.lo + h, l := fastTwoSum(s, e) + return dd{h, l} +} + +func (a dd) mul(b dd) dd { + p, e := twoProd(a.hi, b.hi) + e += float64(a.hi*b.lo) + float64(a.lo*b.hi) + h, l := fastTwoSum(p, e) + return dd{h, l} +} + +func (a dd) mulF(b float64) dd { + p, e := twoProd(a.hi, b) + e += float64(a.lo * b) + h, l := fastTwoSum(p, e) + return dd{h, l} +} + +func (a dd) div(b dd) dd { + q1 := a.hi / b.hi + r := a.add(b.mulF(-q1)) + q2 := r.hi / b.hi + r = r.add(b.mulF(-q2)) + q3 := r.hi / b.hi + h, l := fastTwoSum(q1, q2) + return dd{h, l}.add(dd{q3, 0}) +} + +func (a dd) round() float64 { return a.hi + a.lo } + +// atanh2 returns 2*atanh(s) = log((1+s)/(1-s)) for |s| <= 2^-6, to about +// 2^-104 relative: 2s(1 + s²/3 + s⁴/5 + ...), the leading terms in +// double-double. +func atanh2(s dd) dd { + z := s.mul(s) + zf := z.hi + tail := horner(zf, 1.0/11, 1.0/13, 1.0/15, 1.0/17, 1.0/19) + p := invOdd[9].add(dd{float64(zf * tail), 0}) + for _, c := range [...]int{7, 5, 3} { + p = z.mul(p).add(invOdd[c]) + } + p = z.mul(p).add(dd{1, 0}) + return s.mul(p).mulF(2) +} + +// ddRecip returns 1/c as a double-double. +func ddRecip(c float64) dd { return dd{1, 0}.div(dd{c, 0}) } + +// invOdd[k] is 1/k for the odd k of atanh2; invFact[k] is 1/k!. +var ( + invOdd = map[int]dd{3: ddRecip(3), 5: ddRecip(5), 7: ddRecip(7), 9: ddRecip(9)} + invFact = [7]dd{{1, 0}, {1, 0}, {0.5, 0}, ddRecip(6), ddRecip(24), ddRecip(120), ddRecip(720)} +) + +// horner evaluates c[0] + x*(c[1] + x*(...)) in plain double arithmetic, +// each product rounded on its own (no FMA), so that it is the same on +// every architecture. +func horner(x float64, c ...float64) float64 { + r := c[len(c)-1] + for i := len(c) - 2; i >= 0; i-- { + r = c[i] + float64(x*r) + } + return r +} + +// logDD returns log(x) for a positive finite x as a double-double. +func logDD(x float64) dd { + m, e := math.Frexp(x) // x = m * 2^e, m in [0.5, 1) + m *= 2 + e-- + if m >= math.Sqrt2 { + m /= 2 + e++ + } + j := int(math.Floor(m*64 + 0.5)) + c := float64(j) / 64 + num := m - c // exact: c/2 <= m <= 2c + den := dd{m, 0}.add(dd{c, 0}) + s := dd{num, 0}.div(den) + r := atanh2(s).add(dd{logTab[j-45][0], logTab[j-45][1]}) + if e != 0 { + ef := float64(e) + h, l := twoProd(ef, ln2Hi) + r = dd{h, l + float64(ef*ln2Lo)}.add(r) + } + return r +} + +// logFast returns log(x) for a positive finite x as hi + lo, to about +// 2^-70 relative, or ok = false for a subnormal x. Callers apply Ziv's +// rounding test and fall back on the slower logDD when it fails. +func logFast(x float64) (hi, lo float64, ok bool) { + b := math.Float64bits(x) + if b < 1<<52 { + return 0, 0, false // subnormal: logDD normalises it + } + e := int(b>>52) - 1023 + m := math.Float64frombits(b&(1<<52-1) | 1023<<52) // in [1, 2) + if m >= math.Sqrt2 { + m *= 0.5 + e++ + } + j := int(m*64 + 0.5) + c := float64(j) / 64 + num := m - c // exact + dh, dl := twoSum(m, c) + sh := num / dh + sl := (math.FMA(-sh, dh, num) - float64(sh*dl)) / dh + z := float64(sh * sh) + t := float64(sh * float64(z*horner(z, 2.0/3, 2.0/5, 2.0/7, 2.0/9))) + ef := float64(e) + ah, al := twoProd(ef, ln2Hi) + h1, l1 := twoSum(ah, logTab[j-45][0]) + h2, l2 := twoSum(h1, 2*sh) + low := l1 + l2 + al + float64(ef*ln2Lo) + logTab[j-45][1] + 2*sl + t + hi, lo = fastTwoSum(h2, low) + return hi, lo, true +} + +// rounds reports whether hi+lo, known to within eps*|hi|, rounds to the +// same double at both ends of that interval, and returns that double. +func rounds(hi, lo, eps float64) (float64, bool) { + d := math.Abs(hi) * eps + r := hi + (lo - d) + return r, r == hi+(lo+d) +} + +func log(x float64) float64 { + switch { + case x > 0 && !math.IsInf(x, 1): + if h, l, ok := logFast(x); ok { + if r, ok := rounds(h, l, 0x1p-68); ok { + return r + } + } + return logDD(x).round() + case x == 0: + return math.Inf(-1) + } + return math.Log(x) // NaN, negative, +Inf +} + +// log1pDD returns log(1+u) for u > -1 as a double-double. +func log1pDD(u float64) dd { + if math.Abs(u) < 0x1p-5 { + // s = u/(2+u) <= 2^-6 and log1p(u) = 2 atanh(s). + return atanh2(dd{u, 0}.div(dd{2, 0}.add(dd{u, 0}))) + } + h, l := twoSum(1, u) + return logDD(h).add(dd{l / h, 0}) +} + +func log1p(u float64) float64 { + switch { + case u > -1 && !math.IsInf(u, 1): + if u == 0 { + return u // keeps -0 + } + return log1pDD(u).round() + case u == -1: + return math.Inf(-1) + } + return math.Log1p(u) // NaN, < -1, +Inf +} + +// expDD returns exp(a) for a double-double argument within the finite range, +// as a double-double and a power of two to scale it by. +func expDD(a dd) (dd, int) { + k := math.Floor(float64(a.hi*(64/math.Ln2)) + 0.5) + // r = a - k*ln2/64, with ln2/64 in three parts. + p, pe := twoProd(k, ln2by64A) + r := a.add(dd{-p, -pe}).add(dd{float64(-k * ln2by64B), float64(-k * ln2by64C)}) + // exp(r), |r| <= ln2/128: Taylor to degree 11, the first six terms + // in double-double. + tail := horner(r.hi, 1.0/5040, 1.0/40320, 1.0/362880, 1.0/3628800, 1.0/39916800) + q := dd{tail, 0} + for c := 6; c >= 0; c-- { + q = r.mul(q).add(invFact[c]) + } + ki := int(k) + j := ki & 63 + q = q.mul(dd{exp2Tab[j][0], exp2Tab[j][1]}) + return q, (ki - j) / 64 +} + +func exp(x float64) float64 { + switch { + case math.IsNaN(x): + return x + case x > 709.8: + return math.Inf(1) + case x < -745.2: + return 0 + } + if h, l, e := expFast(x, 0); e > -1000 { + if r, ok := rounds(h, l, 0x1p-66); ok { + return math.Ldexp(r, e) + } + } + q, e := expDD(dd{x, 0}) + return scale(q, e) +} + +// expFast returns exp(xh+xl) as (hi+lo) * 2^e, to about 2^-67 relative, +// for |xh| <= 745.2 and |xl| <= 2^-40 |xh|. Callers apply Ziv's rounding +// test (and scale only normal results) and fall back on expDD. +func expFast(xh, xl float64) (hi, lo float64, e int) { + k := math.Floor(float64(xh*(64/math.Ln2)) + 0.5) + ph, pl := twoProd(k, ln2by64A) + rh, rl := twoSum(xh, -ph) + rh, rl = fastTwoSum(rh, rl+(xl-pl-float64(k*ln2by64B))) + // exp(rh+rl) = 1 + rh + rh²(1/2 + rh/6 + ...) + rl(1 + rh), |rh| <= 2^-7.5. + p := float64(float64(rh*rh) * horner(rh, 1.0/2, 1.0/6, 1.0/24, 1.0/120, 1.0/720, 1.0/5040)) + s1, e1 := fastTwoSum(1, rh) + eh, el := fastTwoSum(s1, e1+rl+float64(rl*rh)+p) + ki := int(k) + j := ki & 63 + th, tl := exp2Tab[j][0], exp2Tab[j][1] + hi, lo = twoProd(th, eh) + lo += float64(th*el) + float64(tl*eh) + hi, lo = fastTwoSum(hi, lo) + return hi, lo, (ki - j) / 64 +} + +// scale returns q * 2^e rounded once when the result is normal; a subnormal +// result is rounded twice (to 53 bits, then to its precision). +func scale(q dd, e int) float64 { + return math.Ldexp(q.round(), e) +} + +func expm1(x float64) float64 { + switch { + case math.IsNaN(x) || x == 0: + return x + case x > 709.8: + return math.Inf(1) + case x < -40: + return -1 // exp(x) < 2^-57: -1 + exp(x) rounds to -1 + case x > 700: + return exp(x) // exp(x) > 2^1009: the -1 is far below an ULP + case math.Abs(x) < 0x1p-30: + // x + x²/2 + x³/6 + x⁴/24, the square exact. + h, l := twoProd(x, x) + t := dd{h / 2, l / 2}.add(dd{float64(x*(h/6)) + float64(h*h/24), 0}) + return dd{x, 0}.add(t).round() + } + q, e := expDD(dd{x, 0}) + q = q.mul(dd{math.Ldexp(1, e), 0}) + return q.add(dd{-1, 0}).round() +} + +func pow(x, y float64) float64 { + if !(x > 0) || math.IsInf(x, 0) || math.IsNaN(y) || math.IsInf(y, 0) || y == 0 || x == 1 { + return math.Pow(x, y) // the C99 special cases, and x <= 0 + } + if lh, ll, ok := logFast(x); ok { + ph, pl := twoProd(lh, y) + pl += float64(ll * y) + if math.Abs(ph) <= 709 { + // The argument is good to 709*2^-70 absolute, so the result to + // about 2^-60 relative. + if h, l, e := expFast(ph, pl); e > -1000 { + if r, ok := rounds(h, l, 0x1p-59); ok { + return math.Ldexp(r, e) + } + } + } + } + l := logDD(x) + p, pe := twoProd(l.hi, y) + a := dd{p, pe + float64(l.lo*y)} + switch { + case a.hi > 709.8: + return math.Inf(1) + case a.hi < -745.2: + return 0 + } + q, e := expDD(a) + return scale(q, e) +} + +func cos(x float64) float64 { return math.Cos(x) } +func acos(x float64) float64 { return math.Acos(x) } + +func logf(x float32) float32 { return float32(log(float64(x))) } +func expf(x float32) float32 { return float32(exp(float64(x))) } +func log1pf(x float32) float32 { return float32(log1p(float64(x))) } +func powf(x, y float32) float32 { return float32(pow(float64(x), float64(y))) } diff --git a/random/libm_test.go b/random/libm_test.go new file mode 100644 index 0000000..1058846 --- /dev/null +++ b/random/libm_test.go @@ -0,0 +1,101 @@ +package random + +import ( + _ "embed" + "encoding/json" + "math" + "math/rand/v2" + "strconv" + "testing" +) + +// libm.json holds correctly rounded values of log, log1p, exp, expm1 and pow +// (mpmath at 300 bits) on random and edge arguments. +// +//go:embed testdata/libm.json +var libmJSON []byte + +func hexF(t *testing.T, s string) float64 { + t.Helper() + f, err := strconv.ParseFloat(s, 64) + if err != nil { + t.Fatal(err) + } + return f +} + +func TestCorrectlyRounded(t *testing.T) { + var c map[string][][]string + if err := json.Unmarshal(libmJSON, &c); err != nil { + t.Fatal(err) + } + one := map[string]func(float64) float64{"log": log, "log1p": log1p, "exp": exp, "expm1": expm1} + for name, rows := range c { + bad := 0 + for _, r := range rows { + var got, want float64 + if name == "pow" { + got, want = pow(hexF(t, r[0]), hexF(t, r[1])), hexF(t, r[2]) + } else { + got, want = one[name](hexF(t, r[0])), hexF(t, r[1]) + } + if got != want { + if bad < 3 { + t.Errorf("%s(%v) = %v, correctly rounded %v", name, r[:len(r)-1], got, want) + } + bad++ + } + } + t.Logf("%s: %d arguments, %d not correctly rounded", name, len(rows), bad) + } +} + +// TestFastPathsAgreeWithDoubleDouble checks the fast paths of log, exp and +// pow (about 2^-68, used when Ziv's rounding test passes) against the +// double-double evaluations, which libm.json checks against mpmath. +func TestFastPathsAgreeWithDoubleDouble(t *testing.T) { + r := rand.New(rand.NewPCG(1, 2)) + n := 400_000 + if testing.Short() { + n = 40_000 + } + var slow [3]int + for i := range n { + var x float64 + switch i % 4 { + case 0: + x = r.Float64() + case 1: + x = math.Float64frombits(r.Uint64() >> 1) // any positive double + case 2: + x = 1 + (r.Float64()-0.5)*1e-6 + default: + x = r.Float64() * 100 + } + if x > 0 && !math.IsInf(x, 0) && log(x) != logDD(x).round() { + t.Fatalf("log(%v) = %v, double-double %v", x, log(x), logDD(x).round()) + } + if _, _, ok := logFast(x); !ok { + slow[0]++ + } + y := (r.Float64() - 0.5) * 1400 + if i%3 == 0 { + y = (r.Float64() - 0.5) * 1e-3 + } + q, e := expDD(dd{y, 0}) + if want := scale(q, e); math.Abs(y) < 709 && exp(y) != want { + t.Fatalf("exp(%v) = %v, double-double %v", y, exp(y), want) + } + px, py := r.Float64()*10, (r.Float64()-0.5)*60 + l := logDD(px) + p, pe := twoProd(l.hi, py) + q, e = expDD(dd{p, pe + l.lo*py}) + if want := scale(q, e); pow(px, py) != want { + t.Fatalf("pow(%v, %v) = %v, double-double %v", px, py, pow(px, py), want) + } + } + // The slow paths: subnormal arguments and results. + if log(0x1p-1060) != logDD(0x1p-1060).round() || exp(-720) == 0 || pow(0x1p-1060, 0.5) != 0x1p-530 { + t.Error("subnormal paths") + } +} diff --git a/random/mt19937.go b/random/mt19937.go new file mode 100644 index 0000000..43ad5a2 --- /dev/null +++ b/random/mt19937.go @@ -0,0 +1,235 @@ +package random + +import "errors" + +const ( + mtN = 624 + mtM = 397 + mtMatrixA = 0x9908b0df + mtUpperMask = 0x80000000 + mtLowerMask = 0x7fffffff +) + +// MT19937State is the state of an MT19937, numpy's state dict: the 624-word +// key and the position in it. +type MT19937State struct { + Key [mtN]uint32 + Pos int +} + +// MT19937 is the Mersenne Twister, numpy's MT19937 and the generator behind +// the legacy RandomState. +type MT19937 struct { + key [mtN]uint32 + pos int + seed *SeedSequence +} + +// NewMT19937 returns numpy's MT19937(seed): seeded from +// NewSeedSequence(seed...). This is not np.random.seed(seed); for that use +// NewMT19937Legacy or RandomState. +func NewMT19937(seed ...uint64) *MT19937 { + return NewMT19937FromSeedSequence(NewSeedSequence(seed...)) +} + +// NewMT19937FromSeedSequence returns numpy's MT19937(seed_seq). +func NewMT19937FromSeedSequence(ss *SeedSequence) *MT19937 { + m := &MT19937{seed: ss} + m.seedSequence(ss) + return m +} + +// seedSequence is MT19937.__init__: key[0] is 0x80000000, the other words +// come from generate_state(624), and pos is left at 623 (the loop variable +// of numpy's Python loop), so the first output is word 623 untouched. +func (m *MT19937) seedSequence(ss *SeedSequence) { + v := ss.GenerateState(mtN) + m.key[0] = 0x80000000 + copy(m.key[1:], v[1:]) + m.pos = mtN - 1 +} + +// NewMT19937Legacy returns an MT19937 seeded as np.random.seed(seed) seeds +// one: Knuth's initialisation of the reference implementation. +func NewMT19937Legacy(seed uint32) *MT19937 { + m := &MT19937{} + m.SeedLegacy(seed) + return m +} + +// SeedLegacy reseeds as np.random.seed(seed) with an integer seed. +func (m *MT19937) SeedLegacy(seed uint32) { + for pos := range m.key { + m.key[pos] = seed + seed = 1812433253*(seed^(seed>>30)) + uint32(pos) + 1 + } + m.pos = mtN + m.seed = nil +} + +// SeedLegacyArray reseeds as np.random.seed(seed) with an array seed: the +// reference init_by_array. +func (m *MT19937) SeedLegacyArray(key []uint32) error { + if len(key) == 0 { + return errors.New("random: Seed must be non-empty") + } + mt := &m.key + mt[0] = 19650218 + for i := 1; i < mtN; i++ { + mt[i] = 1812433253*(mt[i-1]^(mt[i-1]>>30)) + uint32(i) + } + i, j := 1, 0 + for k := max(mtN, len(key)); k > 0; k-- { + mt[i] = (mt[i] ^ ((mt[i-1] ^ (mt[i-1] >> 30)) * 1664525)) + key[j] + uint32(j) + i++ + j++ + if i >= mtN { + mt[0] = mt[mtN-1] + i = 1 + } + if j >= len(key) { + j = 0 + } + } + for k := mtN - 1; k > 0; k-- { + mt[i] = (mt[i] ^ ((mt[i-1] ^ (mt[i-1] >> 30)) * 1566083941)) - uint32(i) + i++ + if i >= mtN { + mt[0] = mt[mtN-1] + i = 1 + } + } + mt[0] = 0x80000000 + m.pos = mtN + m.seed = nil + return nil +} + +func (m *MT19937) gen() { + k := &m.key + var i int + for i = 0; i < mtN-mtM; i++ { + y := k[i]&mtUpperMask | k[i+1]&mtLowerMask + k[i] = k[i+mtM] ^ y>>1 ^ (-(y & 1) & mtMatrixA) + } + for ; i < mtN-1; i++ { + y := k[i]&mtUpperMask | k[i+1]&mtLowerMask + k[i] = k[i+mtM-mtN] ^ y>>1 ^ (-(y & 1) & mtMatrixA) + } + y := k[mtN-1]&mtUpperMask | k[0]&mtLowerMask + k[mtN-1] = k[mtM-1] ^ y>>1 ^ (-(y & 1) & mtMatrixA) + m.pos = 0 +} + +// Uint32 returns the next tempered output. +func (m *MT19937) Uint32() uint32 { + if m.pos == mtN { + m.gen() + } + y := m.key[m.pos] + m.pos++ + y ^= y >> 11 + y ^= (y << 7) & 0x9d2c5680 + y ^= (y << 15) & 0xefc60000 + y ^= y >> 18 + return y +} + +// Uint64 returns two outputs, the first in the high half. +func (m *MT19937) Uint64() uint64 { + hi := uint64(m.Uint32()) + return hi<<32 | uint64(m.Uint32()) +} + +// Float64 is MT19937's own next_double: 27 bits of one output and 26 of the +// next, (a*2**26 + b) / 2**53. +func (m *MT19937) Float64() float64 { + a := int32(m.Uint32() >> 5) + b := int32(m.Uint32() >> 6) + return (float64(a)*67108864.0 + float64(b)) / 9007199254740992.0 +} + +// Raw returns the next 32-bit output, widened. +func (m *MT19937) Raw() uint64 { return uint64(m.Uint32()) } + +// State returns the generator's state. +func (m *MT19937) State() MT19937State { return MT19937State{m.key, m.pos} } + +// SetState replaces the generator's state. +func (m *MT19937) SetState(s MT19937State) { + m.key = s.Key + m.pos = s.Pos +} + +// SeedSeq returns the SeedSequence the generator was made from, nil after +// legacy seeding. +func (m *MT19937) SeedSeq() *SeedSequence { return m.seed } + +// Spawn returns n children seeded from SeedSeq().Spawn(n), numpy's spawn. +func (m *MT19937) Spawn(n int) ([]BitGenerator, error) { + return spawnWith(m.seed, n, func(s *SeedSequence) BitGenerator { return NewMT19937FromSeedSequence(s) }) +} + +// Jumped returns a copy advanced by jumps * 2**128 outputs, numpy's +// jumped(jumps), by the reference jump-ahead polynomial. +func (m *MT19937) Jumped(jumps int) *MT19937 { + c := &MT19937{key: m.key, pos: m.pos} + for range jumps { + c.jump() + } + return c +} + +// genNext advances the state by one word, as the jump code's gen_next. +func (m *MT19937) genNext() { + k := &m.key + n := m.pos + switch { + case n < mtN-mtM: + y := k[n]&mtUpperMask | k[n+1]&mtLowerMask + k[n] = k[n+mtM] ^ y>>1 ^ (-(y & 1) & mtMatrixA) + m.pos++ + case n < mtN-1: + y := k[n]&mtUpperMask | k[n+1]&mtLowerMask + k[n] = k[n+mtM-mtN] ^ y>>1 ^ (-(y & 1) & mtMatrixA) + m.pos++ + default: + y := k[mtN-1]&mtUpperMask | k[0]&mtLowerMask + k[mtN-1] = k[mtM-1] ^ y>>1 ^ (-(y & 1) & mtMatrixA) + m.pos = 0 + } +} + +// addState xors b's state into m's, each read from its own position. +func (m *MT19937) addState(b *MT19937) { + p1, p2 := m.pos, b.pos + for i := range mtN { + m.key[(i+p1)%mtN] ^= b.key[(i+p2)%mtN] + } +} + +func mtCoef(deg int) bool { return mtJumpPoly[deg>>5]&(1<<(deg&31)) != 0 } + +// jump is numpy's mt19937_jump_state: Horner evaluation of the jump +// polynomial on the state. +func (m *MT19937) jump() { + if m.pos >= mtN { + m.pos = 0 + } + i := 19937 - 1 + for !mtCoef(i) { + i-- + } + t := &MT19937{key: m.key, pos: m.pos} + t.genNext() + for i--; i > 0; i-- { + if mtCoef(i) { + t.addState(m) + } + t.genNext() + } + if mtCoef(0) { + t.addState(m) + } + m.key, m.pos = t.key, t.pos +} diff --git a/random/numpy_test.go b/random/numpy_test.go new file mode 100644 index 0000000..2f6430f --- /dev/null +++ b/random/numpy_test.go @@ -0,0 +1,825 @@ +package random + +import ( + "bytes" + "crypto/sha256" + _ "embed" + "encoding/binary" + "encoding/hex" + "encoding/json" + "fmt" + "math" + "math/big" + "sort" + "strconv" + "testing" + + "github.com/go-ndarray/ndarray" +) + +// cases.json holds NumPy's output for every function on several seeds and +// bit generators, made by testdata/cases.py with an x86-64 NumPy. +// +//go:embed testdata/cases.json +var casesJSON []byte + +type encoded struct { + DType string `json:"dtype"` + Shape []int `json:"shape"` + V []string `json:"v"` +} + +type call struct { + Fn string `json:"fn"` + Kw map[string]any `json:"kw"` + Size any `json:"size"` + R encoded `json:"r"` +} + +type script struct { + API string `json:"api"` + BG string `json:"bg"` + Seed string `json:"seed"` + Calls []call `json:"calls"` +} + +type bitgenRec struct { + BG string `json:"bg"` + Seed string `json:"seed"` + Raw []string `json:"raw"` + State []string `json:"state"` + Jumped1 []string `json:"jumped1"` + Jumped3 []string `json:"jumped3"` + Advance []string `json:"advance"` + AdvanceNeg []string `json:"advance_neg"` + Spawn [][]string `json:"spawn"` +} + +type fixture struct { + NumPy string `json:"numpy"` + Machine string `json:"machine"` + Cases []script `json:"cases"` + Digests []struct { + API string `json:"api"` + Seed string `json:"seed"` + Fn string `json:"fn"` + Kw map[string]any `json:"kw"` + Size any `json:"size"` + D struct { + DType string `json:"dtype"` + Shape []int `json:"shape"` + SHA256 string `json:"sha256"` + } `json:"d"` + } `json:"digests"` + Bitgens []bitgenRec `json:"bitgens"` + PhiloxKey struct { + Raw []string `json:"raw"` + Carry []string `json:"carry"` + } `json:"philox_key"` + LegacyMT []struct { + Seed string `json:"seed"` + Raw []string `json:"raw"` + Jumped []string `json:"jumped"` + JumpedFresh []string `json:"jumped_fresh"` + } `json:"legacy_mt"` + SeedSeqs []struct { + Name string `json:"name"` + Pool []string `json:"pool"` + State32 []string `json:"state32"` + State64 []string `json:"state64"` + Kids []struct { + Key []string `json:"key"` + Pool []string `json:"pool"` + } `json:"kids"` + } `json:"seedseqs"` +} + +var fx = func() *fixture { + var f fixture + dec := json.NewDecoder(bytes.NewReader(casesJSON)) + dec.UseNumber() + if err := dec.Decode(&f); err != nil { + panic(err) + } + return &f +}() + +func bigOf(s string) *big.Int { + v, ok := new(big.Int).SetString(s, 10) + if !ok { + panic(s) + } + return v +} + +func u64(s string) uint64 { return bigOf(s).Uint64() } + +// seedSeqs are the fixture's seeds, as NumPy's SeedSequence(seed). +var seedSeqs = map[string]func() *SeedSequence{ + "0": func() *SeedSequence { return NewSeedSequence(0) }, + "1": func() *SeedSequence { return NewSeedSequence(1) }, + "42": func() *SeedSequence { return NewSeedSequence(42) }, + "u64max": func() *SeedSequence { return NewSeedSequence(math.MaxUint64) }, + "list": func() *SeedSequence { return NewSeedSequence(1, 2, 3) }, + "big": func() *SeedSequence { + ss, err := NewSeedSequenceBig(bigOf("340282366920938463463374607431768223801")) + if err != nil { + panic(err) + } + return ss + }, +} + +func newBG(kind, seed string) BitGenerator { + ss := seedSeqs[seed]() + switch kind { + case "PCG64": + return NewPCG64FromSeedSequence(ss) + case "PCG64DXSM": + return NewPCG64DXSMFromSeedSequence(ss) + case "MT19937": + return NewMT19937FromSeedSequence(ss) + case "Philox": + return NewPhiloxFromSeedSequence(ss) + } + return NewSFC64FromSeedSequence(ss) +} + +var legacySeeds = map[string]func() *RandomState{ + "0": func() *RandomState { return NewRandomState(0) }, + "1": func() *RandomState { return NewRandomState(1) }, + "42": func() *RandomState { return NewRandomState(42) }, + "u32max": func() *RandomState { return NewRandomState(math.MaxUint32) }, + "arr": func() *RandomState { r, _ := NewRandomStateArray([]uint32{1, 2, 3}); return r }, + "arr700": func() *RandomState { + k := make([]uint32, 700) + for i := range k { + k[i] = uint32(i) + } + r, _ := NewRandomStateArray(k) + return r + }, +} + +func wantStrings(t *testing.T, what string, got []uint64, want []string) { + t.Helper() + if len(got) != len(want) { + t.Errorf("%s: %d values, NumPy %d", what, len(got), len(want)) + return + } + for i := range want { + if got[i] != u64(want[i]) { + t.Errorf("%s[%d] = %d, NumPy %s", what, i, got[i], want[i]) + return + } + } +} + +func TestSeedSequenceMatchesNumPy(t *testing.T) { + mk := map[string]func() *SeedSequence{ + "0": seedSeqs["0"], "42": seedSeqs["42"], "u64max": seedSeqs["u64max"], "list": seedSeqs["list"], + "big": seedSeqs["big"], + "long": func() *SeedSequence { return NewSeedSequence(1, 2, 3, 4, 5, 6, 7, 8, 9, 10) }, + "empty": func() *SeedSequence { s, _ := NewSeedSequenceWith(SeedSequenceOptions{Words: []uint32{}}); return s }, + "pool8": func() *SeedSequence { + s, _ := NewSeedSequenceWith(SeedSequenceOptions{Entropy: []uint64{7}, PoolSize: 8}) + return s + }, + "spawnkey": func() *SeedSequence { + s, _ := NewSeedSequenceWith(SeedSequenceOptions{Entropy: []uint64{7}, SpawnKey: []uint64{3, 1 << 40}}) + return s + }, + "spawnkey_long": func() *SeedSequence { + s, _ := NewSeedSequenceWith(SeedSequenceOptions{Entropy: []uint64{0, 1, 2, 3, 4, 5}, SpawnKey: []uint64{1}}) + return s + }, + } + for _, c := range fx.SeedSeqs { + ss := mk[c.Name]() + wantStrings(t, c.Name+" pool", widen(ss.Pool()), c.Pool) + wantStrings(t, c.Name+" state32", widen(ss.GenerateState(9)), c.State32) + wantStrings(t, c.Name+" state64", ss.GenerateState64(5), c.State64) + kids := append(ss.Spawn(3), ss.Spawn(1)...) + for i, k := range kids { + wantStrings(t, fmt.Sprintf("%s kid %d key", c.Name, i), k.SpawnKey(), c.Kids[i].Key) + wantStrings(t, fmt.Sprintf("%s kid %d pool", c.Name, i), widen(k.Pool()), c.Kids[i].Pool) + } + } +} + +func widen(w []uint32) []uint64 { + out := make([]uint64, len(w)) + for i, v := range w { + out[i] = uint64(v) + } + return out +} + +func TestBitGeneratorsMatchNumPy(t *testing.T) { + for _, r := range fx.Bitgens { + name := r.BG + "/" + r.Seed + bg := newBG(r.BG, r.Seed) + var state []uint64 + switch b := bg.(type) { + case *PCG64: + s := b.State() + state = []uint64{s.StateHi, s.StateLo, s.IncHi, s.IncLo} + case *PCG64DXSM: + s := b.State() + state = []uint64{s.StateHi, s.StateLo, s.IncHi, s.IncLo} + case *MT19937: + s := b.State() + state = []uint64{uint64(s.Key[0]), uint64(s.Key[1]), uint64(s.Key[2]), uint64(s.Key[623]), uint64(s.Pos)} + case *Philox: + s := b.State() + state = s.Key[:] + case *SFC64: + s := b.State() + state = s.S[:] + } + if r.BG == "PCG64" || r.BG == "PCG64DXSM" { + st, inc := bigOf(r.State[0]), bigOf(r.State[1]) + want := []uint64{new(big.Int).Rsh(st, 64).Uint64(), st.Uint64(), new(big.Int).Rsh(inc, 64).Uint64(), inc.Uint64()} + if fmt.Sprint(state) != fmt.Sprint(want) { + t.Errorf("%s state %v, NumPy %v", name, state, want) + } + } else { + wantStrings(t, name+" state", state, r.State) + } + wantStrings(t, name+" raw", RandomRaw(bg, len(r.Raw)), r.Raw) + for j, want := range map[int][]string{1: r.Jumped1, 3: r.Jumped3} { + if want == nil { + continue + } + var jb BitGenerator + switch b := newBG(r.BG, r.Seed).(type) { + case *PCG64: + jb = b.Jumped(uint64(j)) + case *PCG64DXSM: + jb = b.Jumped(uint64(j)) + case *MT19937: + jb = b.Jumped(j) + case *Philox: + jb = b.Jumped(uint64(j)) + } + wantStrings(t, fmt.Sprintf("%s jumped(%d)", name, j), RandomRaw(jb, 5), want) + } + if r.Advance != nil { + adv := func(skip int, d *big.Int) BitGenerator { + b := newBG(r.BG, r.Seed) + RandomRaw(b, skip) + switch b := b.(type) { + case *PCG64: + return b.Advance(d) + case *PCG64DXSM: + return b.Advance(d) + } + return b.(*Philox).Advance(d) + } + d := new(big.Int).Lsh(big.NewInt(1), 100) + wantStrings(t, name+" advance", RandomRaw(adv(3, d.Add(d, big.NewInt(7))), 5), r.Advance) + wantStrings(t, name+" advance(-3)", RandomRaw(adv(7, big.NewInt(-3)), 5), r.AdvanceNeg) + } + kids, err := newBG(r.BG, r.Seed).(Spawner).Spawn(2) + if err != nil { + t.Fatal(err) + } + for i, k := range kids { + wantStrings(t, fmt.Sprintf("%s spawn %d", name, i), RandomRaw(k, 5), r.Spawn[i]) + } + } + ph := NewPhiloxKey([4]uint64{1, 2, 3, math.MaxUint64}, [2]uint64{5, 6}) + wantStrings(t, "Philox(counter, key)", RandomRaw(ph, 12), fx.PhiloxKey.Raw) + ph = NewPhiloxKey([4]uint64{math.MaxUint64, math.MaxUint64, 0, 0}, [2]uint64{1, 2}).Advance(big.NewInt(1)) + wantStrings(t, "Philox advance carry", RandomRaw(ph, 4), fx.PhiloxKey.Carry) + for _, r := range fx.LegacyMT { + m := legacySeeds[r.Seed]().BitGenerator().(*MT19937) + wantStrings(t, "legacy MT19937 "+r.Seed, RandomRaw(m, len(r.Raw)), r.Raw) + wantStrings(t, "legacy MT19937 jumped "+r.Seed, RandomRaw(m.Jumped(1), 5), r.Jumped) + fresh := legacySeeds[r.Seed]().BitGenerator().(*MT19937) + wantStrings(t, "legacy MT19937 jumped fresh "+r.Seed, RandomRaw(fresh.Jumped(1), 5), r.JumpedFresh) + } +} + +// libmULP bounds, per sampler, the distance in ULPs from NumPy's value of a +// result that passes through a C library function. This package's log, +// log1p, exp, expm1 and pow are correctly rounded (libm.go); NumPy's come +// from the C library, which is too in all but a fraction of a percent of +// arguments. The fixture holds a few such values (each checked against +// mpmath: NumPy's is the misrounded one); beta magnifies its one to 2 ULP. +// vonmises calls cos and acos, which neither side rounds correctly (Apple's +// cos misrounds 17% of arguments), and acos near ±1 magnifies an ULP. +// +// Every other sampler, every integer-valued one, and every value not listed +// here must match to the bit: a libm difference that flipped an +// accept/reject decision would change the stream and fail. +var libmULP = map[string]int64{ + "standard_exponential": 1, // the "inv" method is -log1p(-U) + "standard_gamma": 2, // pow (shape < 1); log in the rejection test + "beta": 2, + "lognormal": 1, + "randn": 1, // RandomState's polar method takes a log per pair + "normal": 1, // RandomState's, likewise + "vonmises": 64, // in ULPs of pi: see absULP +} + +// absULP lists the samplers whose differences are measured in ULPs of a +// fixed scale rather than of the value: vonmises wraps its angle into +// [-pi, pi], so a value near 0 carries the absolute error of one near pi. +var absULP = map[string]float64{"vonmises": math.Pi} + +func sizeOf(v any) []int { + switch s := v.(type) { + case nil: + return nil + case json.Number: + return []int{int(numF(s))} + } + var out []int + for _, d := range v.([]any) { + out = append(out, int(numF(d))) + } + return out +} + +func kwF(kw map[string]any, k string, def float64) float64 { + if v, ok := kw[k]; ok { + return numF(v) + } + return def +} + +func kwI(kw map[string]any, k string) int64 { + return bigOf(string(kw[k].(json.Number))).Int64() +} + +func kwDType(kw map[string]any, def ndarray.DType) ndarray.DType { + if v, ok := kw["dtype"]; ok { + dt, err := ndarray.ParseDType(v.(string)) + if err != nil { + panic(err) + } + return dt + } + return def +} + +func kwFloats(kw map[string]any, k string) []float64 { + v, ok := kw[k].([]any) + if !ok { + return nil + } + out := make([]float64, len(v)) + for i, x := range v { + out[i] = numF(x) + } + return out +} + +func namedArray(name string) *ndarray.Array { + switch name { + case "arange10": + a, _ := ndarray.FromSlice([]int64{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 10) + return a + case "arange12": + d := make([]int64, 12) + for i := range d { + d[i] = int64(i) + } + a, _ := ndarray.FromSlice(d, 3, 4) + return a + } + a, _ := ndarray.FromSlice([]float64{0.5, 1.5, 2.5, 3.5, 4.5, 5.5}, 6) + return a +} + +func choiceOpts(kw map[string]any) ChoiceOptions { + o := ChoiceOptions{P: kwFloats(kw, "p")} + if r, ok := kw["replace"]; ok && !r.(bool) { + o.NoReplace = true + } + if s, ok := kw["shuffle"]; ok && !s.(bool) { + o.NoShuffle = true + } + if a, ok := kw["axis"]; ok { + o.Axis = int(numF(a)) + } + return o +} + +func bytesArray(b []byte) *ndarray.Array { + a, _ := ndarray.FromSlice(b, len(b)) + return a +} + +// bigBounds handles integers bounds beyond int64 (uint64 dtype). +func bigBounds(kw map[string]any) (lo, hi *big.Int) { + return bigOf(numString(kw["low"])), bigOf(numString(kw["high"])) +} + +func numString(v any) string { return string(v.(json.Number)) } + +func numF(v any) float64 { + f, err := v.(json.Number).Float64() + if err != nil { + panic(err) + } + return f +} + +func runGenerator(g *Generator, c call) (*ndarray.Array, error) { + kw, size := c.Kw, sizeOf(c.Size) + switch c.Fn { + case "random": + return g.RandomOf(kwDType(kw, ndarray.Float64), size...) + case "integers": + dt := kwDType(kw, ndarray.Int64) + ep, _ := kw["endpoint"].(bool) + if dt == ndarray.Uint64 { + lo, hi := bigBounds(kw) + return g.IntegersUint64(lo.Uint64(), hi.Uint64(), ep, size...) + } + lo, hi := bigBounds(kw) + return g.IntegersOf(dt, lo.Int64(), hi.Int64(), ep, size...) + case "standard_normal": + return g.StandardNormalOf(kwDType(kw, ndarray.Float64), size...) + case "standard_exponential": + m, ok := kw["method"].(string) + if !ok { + m = "zig" + } + return g.StandardExponentialOf(kwDType(kw, ndarray.Float64), m, size...) + case "standard_gamma": + return g.StandardGammaOf(kwDType(kw, ndarray.Float64), kwF(kw, "shape", 0), size...) + case "normal": + return g.Normal(kwF(kw, "loc", 0), kwF(kw, "scale", 1), size...) + case "exponential": + return g.Exponential(kwF(kw, "scale", 1), size...) + case "uniform": + return g.Uniform(kwF(kw, "low", 0), kwF(kw, "high", 1), size...) + case "gamma": + return g.Gamma(kwF(kw, "shape", 0), kwF(kw, "scale", 1), size...) + case "beta": + return g.Beta(kwF(kw, "a", 0), kwF(kw, "b", 0), size...) + case "chisquare": + return g.ChiSquare(kwF(kw, "df", 0), size...) + case "f": + return g.F(kwF(kw, "dfnum", 0), kwF(kw, "dfden", 0), size...) + case "noncentral_chisquare": + return g.NoncentralChiSquare(kwF(kw, "df", 0), kwF(kw, "nonc", 0), size...) + case "noncentral_f": + return g.NoncentralF(kwF(kw, "dfnum", 0), kwF(kw, "dfden", 0), kwF(kw, "nonc", 0), size...) + case "standard_cauchy": + return g.StandardCauchy(size...) + case "standard_t": + return g.StandardT(kwF(kw, "df", 0), size...) + case "vonmises": + return g.VonMises(kwF(kw, "mu", 0), kwF(kw, "kappa", 0), size...) + case "pareto": + return g.Pareto(kwF(kw, "a", 0), size...) + case "weibull": + return g.Weibull(kwF(kw, "a", 0), size...) + case "power": + return g.Power(kwF(kw, "a", 0), size...) + case "laplace": + return g.Laplace(kwF(kw, "loc", 0), kwF(kw, "scale", 1), size...) + case "gumbel": + return g.Gumbel(kwF(kw, "loc", 0), kwF(kw, "scale", 1), size...) + case "logistic": + return g.Logistic(kwF(kw, "loc", 0), kwF(kw, "scale", 1), size...) + case "lognormal": + return g.Lognormal(kwF(kw, "mean", 0), kwF(kw, "sigma", 1), size...) + case "rayleigh": + return g.Rayleigh(kwF(kw, "scale", 1), size...) + case "wald": + return g.Wald(kwF(kw, "mean", 0), kwF(kw, "scale", 0), size...) + case "triangular": + return g.Triangular(kwF(kw, "left", 0), kwF(kw, "mode", 0), kwF(kw, "right", 0), size...) + case "binomial": + return g.Binomial(kwI(kw, "n"), kwF(kw, "p", 0), size...) + case "negative_binomial": + return g.NegativeBinomial(kwF(kw, "n", 0), kwF(kw, "p", 0), size...) + case "poisson": + return g.Poisson(kwF(kw, "lam", 1), size...) + case "zipf": + return g.Zipf(kwF(kw, "a", 0), size...) + case "geometric": + return g.Geometric(kwF(kw, "p", 0), size...) + case "logseries": + return g.Logseries(kwF(kw, "p", 0), size...) + case "choice": + if name, ok := kw["a"].(string); ok { + return g.Choice(namedArray(name), choiceOpts(kw), size...) + } + return g.ChoiceN(kwI(kw, "a"), choiceOpts(kw), size...) + case "shuffle": + a := namedArray(kw["x"].(string)) + axis := 0 + if v, ok := kw["axis"]; ok { + axis = int(numF(v)) + } + return a, g.Shuffle(a, axis) + case "permutation": + if name, ok := kw["x"].(string); ok { + axis := 0 + if v, ok := kw["axis"]; ok { + axis = int(numF(v)) + } + return g.PermutationOf(namedArray(name), axis) + } + return g.Permutation(int(kwI(kw, "x"))) + case "permuted": + a := namedArray(kw["x"].(string)) + if v, ok := kw["axis"]; ok { + return g.PermutedAxis(a, int(numF(v))) + } + return g.Permuted(a) + case "bytes": + return bytesArray(g.Bytes(int(kwI(kw, "length")))), nil + case "dirichlet": + return g.Dirichlet(kwFloats(kw, "alpha"), size...) + case "multinomial": + return g.Multinomial(kwI(kw, "n"), kwFloats(kw, "pvals"), size...) + } + return nil, fmt.Errorf("unknown function %s", c.Fn) +} + +func runLegacy(r *RandomState, c call) (*ndarray.Array, error) { + kw, size := c.Kw, sizeOf(c.Size) + switch c.Fn { + case "rand": + return r.Rand(size...) + case "randn": + return r.Randn(size...) + case "random_sample": + return r.RandomSample(size...) + case "randint": + dt := kwDType(kw, ndarray.Int64) + lo, hi := bigBounds(kw) + if dt == ndarray.Uint64 { + return integersUint64(r.bg, true, lo.Uint64(), hi.Uint64(), false, size) + } + return r.RandintOf(dt, lo.Int64(), hi.Int64(), size...) + case "normal": + return r.Normal(kwF(kw, "loc", 0), kwF(kw, "scale", 1), size...) + case "uniform": + return r.Uniform(kwF(kw, "low", 0), kwF(kw, "high", 1), size...) + case "standard_normal": + return r.StandardNormal(size...) + case "choice": + if name, ok := kw["a"].(string); ok { + return r.Choice(namedArray(name), choiceOpts(kw), size...) + } + return r.ChoiceN(kwI(kw, "a"), choiceOpts(kw), size...) + case "shuffle": + a := namedArray(kw["x"].(string)) + return a, r.Shuffle(a) + case "permutation": + if name, ok := kw["x"].(string); ok { + return r.PermutationOf(namedArray(name)) + } + return r.Permutation(int(kwI(kw, "x"))) + case "binomial": + return r.Binomial(kwI(kw, "n"), kwF(kw, "p", 0), size...) + case "poisson": + return r.Poisson(kwF(kw, "lam", 1), size...) + case "standard_exponential": + return r.StandardExponential(size...) + case "exponential": + return r.Exponential(kwF(kw, "scale", 1), size...) + case "standard_gamma": + return r.StandardGamma(kwF(kw, "shape", 0), size...) + case "gamma": + return r.Gamma(kwF(kw, "shape", 0), kwF(kw, "scale", 1), size...) + case "chisquare": + return r.ChiSquare(kwF(kw, "df", 0), size...) + case "lognormal": + return r.Lognormal(kwF(kw, "mean", 0), kwF(kw, "sigma", 1), size...) + case "multinomial": + return r.Multinomial(kwI(kw, "n"), kwFloats(kw, "pvals"), size...) + case "bytes": + return bytesArray(r.Bytes(int(kwI(kw, "length")))), nil + } + return nil, fmt.Errorf("unknown function %s", c.Fn) +} + +// compare returns how a result differs from NumPy's: a structural +// difference (dtype, shape, or a differing integer) as an error message, +// and for floats the number of differing values and the largest ULP +// distance. +func compare(got *ndarray.Array, want encoded, scale float64) (msg string, diff int, maxULP int64) { + if got.DType().String() != want.DType { + return fmt.Sprintf("dtype %v, NumPy %s", got.DType(), want.DType), 0, 0 + } + if fmt.Sprint(got.Shape()) != fmt.Sprint(want.Shape) { + return fmt.Sprintf("shape %v, NumPy %v", got.Shape(), want.Shape), 0, 0 + } + vals := flatStrings(got) + for i, w := range want.V { + if vals[i] == w { + continue + } + if !got.DType().IsFloat() { + return fmt.Sprintf("[%d] = %s, NumPy %s", i, vals[i], w), 0, 0 + } + gf, _ := strconv.ParseFloat(vals[i], 64) + wf, _ := strconv.ParseFloat(w, 64) + diff++ + u := ulps(gf, wf, got.DType() == ndarray.Float32) + if scale > 0 { + u = int64(math.Ceil(math.Abs(gf-wf) / (scale * 0x1p-52))) + } + maxULP = max(maxULP, u) + } + return "", diff, maxULP +} + +func ulps(a, b float64, single bool) int64 { + if single { + d := int64(math.Float32bits(float32(a))) - int64(math.Float32bits(float32(b))) + return max(d, -d) + } + d := int64(math.Float64bits(a)) - int64(math.Float64bits(b)) + return max(d, -d) +} + +func flatStrings(a *ndarray.Array) []string { + var out []string + switch a.DType() { + case ndarray.Float64: + d, _ := ndarray.Data[float64](a) + for _, v := range d { + out = append(out, strconv.FormatFloat(v, 'g', -1, 64)) + } + case ndarray.Float32: + d, _ := ndarray.Data[float32](a) + for _, v := range d { + out = append(out, strconv.FormatFloat(float64(v), 'g', -1, 64)) + } + case ndarray.Bool: + d, _ := ndarray.Data[bool](a) + for _, v := range d { + out = append(out, map[bool]string{true: "1", false: "0"}[v]) + } + case ndarray.Uint64: + d, _ := ndarray.Data[uint64](a) + for _, v := range d { + out = append(out, strconv.FormatUint(v, 10)) + } + default: + d, _ := ndarray.Data[int64](a.AsType(ndarray.Int64)) + for _, v := range d { + out = append(out, strconv.FormatInt(v, 10)) + } + } + return out +} + +func normFloat(s string) string { + f, _ := strconv.ParseFloat(s, 64) + return strconv.FormatFloat(f, 'g', -1, 64) +} + +type libmStat struct { + calls, diffCalls, values, diffValues int + maxULP int64 +} + +// replay runs every script of the fixture and returns, per sampler, how +// many float values differ from NumPy's; it fails the test on anything +// else that differs. +func replay(t *testing.T) map[string]*libmStat { + stats := map[string]*libmStat{} + for i, s := range fx.Cases { + var g *Generator + var r *RandomState + if s.API == "Generator" { + g = NewGenerator(newBG(s.BG, s.Seed)) + } else { + r = legacySeeds[s.Seed]() + } + for j, c := range s.Calls { + if c.R.DType == "float64" || c.R.DType == "float32" { + for k := range c.R.V { + c.R.V[k] = normFloat(c.R.V[k]) + } + } + var got *ndarray.Array + var err error + if g != nil { + got, err = runGenerator(g, c) + } else { + got, err = runLegacy(r, c) + } + where := fmt.Sprintf("case %d call %d %s/%s/%s %s %v size %v", i, j, s.API, s.BG, s.Seed, c.Fn, c.Kw, c.Size) + if err != nil { + t.Errorf("%s: %v", where, err) + break + } + msg, diff, u := compare(got, c.R, absULP[c.Fn]) + if msg != "" { + t.Errorf("%s: %s", where, msg) + break + } + st := stats[c.Fn] + if st == nil { + st = &libmStat{} + stats[c.Fn] = st + } + st.calls++ + st.values += len(c.R.V) + if diff == 0 { + continue + } + if testing.Verbose() { + t.Logf("%s: %d values differ, up to %d ULP", where, diff, u) + } + st.diffCalls++ + st.diffValues += diff + st.maxULP = max(st.maxULP, u) + if bound, ok := libmULP[c.Fn]; !ok || u > bound { + t.Errorf("%s: %d values differ, up to %d ULP (bound %d)", where, diff, u, bound) + break + } + } + } + return stats +} + +// TestMatchesNumPy replays the fixture: everything to the bit, except the +// documented ULP-level differences of the C library functions. +func TestMatchesNumPy(t *testing.T) { + stats := replay(t) + names := make([]string, 0, len(stats)) + for k := range stats { + names = append(names, k) + } + sort.Strings(names) + for _, k := range names { + st := stats[k] + if st.diffValues > 0 { + t.Logf("%-22s %4d/%4d calls, %5d/%6d values differ, max %d ULP", k, st.diffCalls, st.calls, st.diffValues, st.values, st.maxULP) + } + } +} + +// leBytes returns the array's elements as little-endian bytes, as numpy's +// tobytes() of a little-endian array. +func leBytes(a *ndarray.Array) []byte { + var b []byte + switch a.DType() { + case ndarray.Float64: + d, _ := ndarray.Data[float64](a) + for _, v := range d { + b = binary.LittleEndian.AppendUint64(b, math.Float64bits(v)) + } + case ndarray.Float32: + d, _ := ndarray.Data[float32](a) + for _, v := range d { + b = binary.LittleEndian.AppendUint32(b, math.Float32bits(v)) + } + case ndarray.Int64: + d, _ := ndarray.Data[int64](a) + for _, v := range d { + b = binary.LittleEndian.AppendUint64(b, uint64(v)) + } + case ndarray.Int8: + d, _ := ndarray.Data[int8](a) + for _, v := range d { + b = append(b, byte(v)) + } + case ndarray.Uint16: + d, _ := ndarray.Data[uint16](a) + for _, v := range d { + b = binary.LittleEndian.AppendUint16(b, v) + } + case ndarray.Bool: + d, _ := ndarray.Data[bool](a) + for _, v := range d { + b = append(b, map[bool]byte{true: 1}[v]) + } + } + return b +} + +// TestLongRunsMatchNumPy compares long runs, which reach the rare paths +// (ziggurat tails and wedges, rejection loops), by digest. +func TestLongRunsMatchNumPy(t *testing.T) { + for _, d := range fx.Digests { + c := call{Fn: d.Fn, Kw: d.Kw, Size: d.Size} + var got *ndarray.Array + var err error + if d.API == "Generator" { + got, err = runGenerator(NewGenerator(newBG("PCG64", d.Seed)), c) + } else { + got, err = runLegacy(legacySeeds[d.Seed](), c) + } + if err != nil { + t.Fatal(err) + } + sum := sha256.Sum256(leBytes(got)) + if got.DType().String() != d.D.DType || hex.EncodeToString(sum[:]) != d.D.SHA256 { + t.Errorf("%s/%s %s %v size %v: %v digest differs from NumPy's", d.API, d.Seed, d.Fn, d.Kw, d.Size, got.DType()) + } + } +} diff --git a/random/pcg64.go b/random/pcg64.go new file mode 100644 index 0000000..977a928 --- /dev/null +++ b/random/pcg64.go @@ -0,0 +1,230 @@ +package random + +import ( + "errors" + "math/big" + "math/bits" +) + +// PCG constants, from numpy/random/src/pcg64/pcg64.h. +var ( + pcgMultiplier = uint128{2549297995355413924, 4865540595714422341} + // pcgJumpStep is the step of PCG64.jumped: 0x9e3779b97f4a7c15f39cc0605cedc835. + pcgJumpStep = uint128{0x9e3779b97f4a7c15, 0xf39cc0605cedc835} +) + +const pcgCheapMultiplier = 0xda942042e4dd58b5 + +// ErrNoSeedSequence is returned by Spawn on a bit generator that was not made +// from a SeedSequence (one seeded by legacy seeding or a raw state). +var ErrNoSeedSequence = errors.New("random: the bit generator was not made from a SeedSequence") + +// PCG64State is the state of a PCG64 or PCG64DXSM, numpy's state dict: the +// 128-bit LCG state and increment, and the buffered upper half of the last +// 64-bit output. +type PCG64State struct { + StateHi, StateLo uint64 + IncHi, IncLo uint64 + HasUint32 bool + Uinteger uint32 +} + +type pcgCore struct { + state, inc uint128 + half half32 + seed *SeedSequence +} + +// PCG64 is numpy's default bit generator: a 128-bit LCG with the XSL-RR +// output function. +type PCG64 struct{ pcgCore } + +// PCG64DXSM is numpy's PCG64DXSM: PCG64 with the cheap 64-bit multiplier and +// the stronger DXSM output function. +type PCG64DXSM struct{ pcgCore } + +// NewPCG64 returns numpy's PCG64(seed): seeded from +// NewSeedSequence(seed...). No seed draws fresh OS entropy. +func NewPCG64(seed ...uint64) *PCG64 { return NewPCG64FromSeedSequence(NewSeedSequence(seed...)) } + +// NewPCG64FromSeedSequence returns numpy's PCG64(seed_seq). +func NewPCG64FromSeedSequence(ss *SeedSequence) *PCG64 { + p := &PCG64{} + p.seedFrom(ss, p.step) + return p +} + +// NewPCG64DXSM returns numpy's PCG64DXSM(seed). +func NewPCG64DXSM(seed ...uint64) *PCG64DXSM { + return NewPCG64DXSMFromSeedSequence(NewSeedSequence(seed...)) +} + +// NewPCG64DXSMFromSeedSequence returns numpy's PCG64DXSM(seed_seq). +func NewPCG64DXSMFromSeedSequence(ss *SeedSequence) *PCG64DXSM { + // numpy seeds both with pcg64_set_seed, whose srandom steps with the + // full 128-bit multiplier even for DXSM. + p := &PCG64DXSM{} + q := &PCG64{} + q.seedFrom(ss, q.step) + p.pcgCore = q.pcgCore + return p +} + +// seedFrom is numpy's pcg64_set_seed after generate_state(4, uint64): the +// words are (state hi, state lo, increment hi, increment lo). +func (c *pcgCore) seedFrom(ss *SeedSequence, step func()) { + v := ss.GenerateState64(4) + c.seed = ss + initState := uint128{v[0], v[1]} + initSeq := uint128{v[2], v[3]} + c.state = uint128{} + c.inc = uint128{initSeq.hi<<1 | initSeq.lo>>63, initSeq.lo<<1 | 1} + step() + c.state = c.state.add(initState) + step() +} + +func (p *PCG64) step() { p.state = p.state.mul(pcgMultiplier).add(p.inc) } + +func (p *PCG64DXSM) step() { + hi, lo := bits.Mul64(p.state.lo, pcgCheapMultiplier) + hi += p.state.hi * pcgCheapMultiplier + p.state = uint128{hi, lo}.add(p.inc) +} + +// Uint64 returns the next 64-bit output. +func (p *PCG64) Uint64() uint64 { + p.step() + return bits.RotateLeft64(p.state.hi^p.state.lo, -int(p.state.hi>>58)) +} + +// Uint64 returns the next 64-bit output: DXSM of the state, then a step. +func (p *PCG64DXSM) Uint64() uint64 { + hi := p.state.hi + lo := p.state.lo | 1 + hi ^= hi >> 32 + hi *= pcgCheapMultiplier + hi ^= hi >> 48 + hi *= lo + p.step() + return hi +} + +// Uint32 returns the low half of a 64-bit output, or the high half kept from +// the previous call. +func (p *PCG64) Uint32() uint32 { + if p.half.has { + p.half.has = false + return p.half.uint + } + return p.half.split(p.Uint64()) +} + +// Uint32 returns the low half of a 64-bit output, or the high half kept from +// the previous call. +func (p *PCG64DXSM) Uint32() uint32 { + if p.half.has { + p.half.has = false + return p.half.uint + } + return p.half.split(p.Uint64()) +} + +// Float64 returns uint64ToDouble of the next output. +func (p *PCG64) Float64() float64 { return uint64ToDouble(p.Uint64()) } + +// Float64 returns uint64ToDouble of the next output. +func (p *PCG64DXSM) Float64() float64 { return uint64ToDouble(p.Uint64()) } + +// Raw returns the next 64-bit output. +func (p *PCG64) Raw() uint64 { return p.Uint64() } + +// Raw returns the next 64-bit output. +func (p *PCG64DXSM) Raw() uint64 { return p.Uint64() } + +// State returns the generator's state. +func (c *pcgCore) State() PCG64State { + return PCG64State{c.state.hi, c.state.lo, c.inc.hi, c.inc.lo, c.half.has, c.half.uint} +} + +// SetState replaces the generator's state. +func (c *pcgCore) SetState(s PCG64State) { + c.state = uint128{s.StateHi, s.StateLo} + c.inc = uint128{s.IncHi, s.IncLo} + c.half = half32{s.HasUint32, s.Uinteger} +} + +// SeedSeq returns the SeedSequence the generator was made from. +func (c *pcgCore) SeedSeq() *SeedSequence { return c.seed } + +// wrap128 reduces a non-negative or negative integer modulo 2**128, numpy's +// wrap_int(delta, 128). +func wrap128(delta *big.Int) uint128 { + m := new(big.Int).Lsh(big.NewInt(1), 128) + d := new(big.Int).Mod(delta, m) + lo := new(big.Int).And(d, new(big.Int).SetUint64(^uint64(0))) + return uint128{new(big.Int).Rsh(d, 64).Uint64(), lo.Uint64()} +} + +// Advance advances the state as if delta outputs had been drawn, numpy's +// advance(delta); delta is taken modulo 2**128 and may be negative. It drops +// a buffered 32-bit half. +func (p *PCG64) Advance(delta *big.Int) *PCG64 { + p.state = advanceLCG128(p.state, wrap128(delta), pcgMultiplier, p.inc) + p.half = half32{} + return p +} + +// Advance advances the state as if delta outputs had been drawn, numpy's +// advance(delta). +func (p *PCG64DXSM) Advance(delta *big.Int) *PCG64DXSM { + p.state = advanceLCG128(p.state, wrap128(delta), uint128{0, pcgCheapMultiplier}, p.inc) + p.half = half32{} + return p +} + +func jumpDelta(jumps uint64) *big.Int { + step := new(big.Int).SetUint64(pcgJumpStep.hi) + step.Lsh(step, 64).Or(step, new(big.Int).SetUint64(pcgJumpStep.lo)) + return step.Mul(step, new(big.Int).SetUint64(jumps)) +} + +// Jumped returns a copy advanced by jumps * 0x9e3779b97f4a7c15f39cc0605cedc835 +// steps, numpy's jumped(jumps). The copy has no SeedSequence, as in numpy +// (whose copy gets a fresh one that is never used). +func (p *PCG64) Jumped(jumps uint64) *PCG64 { + q := &PCG64{pcgCore{state: p.state, inc: p.inc, half: p.half}} + return q.Advance(jumpDelta(jumps)) +} + +// Jumped returns a copy advanced by jumps * 0x9e3779b97f4a7c15f39cc0605cedc835 +// steps, numpy's jumped(jumps). +func (p *PCG64DXSM) Jumped(jumps uint64) *PCG64DXSM { + q := &PCG64DXSM{pcgCore{state: p.state, inc: p.inc, half: p.half}} + return q.Advance(jumpDelta(jumps)) +} + +// Spawn returns n children seeded from SeedSeq().Spawn(n), numpy's spawn. +func (p *PCG64) Spawn(n int) ([]BitGenerator, error) { + return spawnWith(p.seed, n, func(s *SeedSequence) BitGenerator { return NewPCG64FromSeedSequence(s) }) +} + +// Spawn returns n children seeded from SeedSeq().Spawn(n), numpy's spawn. +func (p *PCG64DXSM) Spawn(n int) ([]BitGenerator, error) { + return spawnWith(p.seed, n, func(s *SeedSequence) BitGenerator { return NewPCG64DXSMFromSeedSequence(s) }) +} + +func spawnWith(ss *SeedSequence, n int, mk func(*SeedSequence) BitGenerator) ([]BitGenerator, error) { + if ss == nil { + return nil, ErrNoSeedSequence + } + if n < 0 { + return nil, errors.New("random: n_children must be non-negative") + } + kids := ss.Spawn(n) + out := make([]BitGenerator, n) + for i, k := range kids { + out[i] = mk(k) + } + return out, nil +} diff --git a/random/philox.go b/random/philox.go new file mode 100644 index 0000000..97ec4f4 --- /dev/null +++ b/random/philox.go @@ -0,0 +1,229 @@ +package random + +import ( + "math/big" + "math/bits" +) + +// PhiloxState is the state of a Philox, numpy's state dict. +type PhiloxState struct { + Counter [4]uint64 + Key [2]uint64 + Buffer [4]uint64 + BufferPos int + HasUint32 bool + Uinteger uint32 +} + +// Philox is numpy's Philox: the counter-based Philox4x64-10 of Salmon et al., +// four 64-bit outputs per counter value. +type Philox struct { + ctr [4]uint64 + key [2]uint64 + buf [4]uint64 + bufPos int + half half32 + seed *SeedSequence +} + +// NewPhilox returns numpy's Philox(seed): the key from +// NewSeedSequence(seed...).generate_state(2, uint64), the counter zero. +func NewPhilox(seed ...uint64) *Philox { + return NewPhiloxFromSeedSequence(NewSeedSequence(seed...)) +} + +// NewPhiloxFromSeedSequence returns numpy's Philox(seed_seq). +func NewPhiloxFromSeedSequence(ss *SeedSequence) *Philox { + k := ss.GenerateState64(2) + p := &Philox{key: [2]uint64{k[0], k[1]}, seed: ss} + p.reset() + return p +} + +// NewPhiloxKey returns numpy's Philox(counter=counter, key=key): counter and +// key as little-endian 64-bit words. +func NewPhiloxKey(counter [4]uint64, key [2]uint64) *Philox { + p := &Philox{ctr: counter, key: key} + p.reset() + return p +} + +func (p *Philox) reset() { + p.half = half32{} + p.bufPos = 4 + p.buf = [4]uint64{} +} + +func philoxRound(ctr [4]uint64, key [2]uint64) [4]uint64 { + hi0, lo0 := bits.Mul64(0xD2E7470EE14C6C93, ctr[0]) + hi1, lo1 := bits.Mul64(0xCA5A826395121157, ctr[2]) + return [4]uint64{hi1 ^ ctr[1] ^ key[0], lo1, hi0 ^ ctr[3] ^ key[1], lo0} +} + +func philox4x64(ctr [4]uint64, key [2]uint64) [4]uint64 { + ctr = philoxRound(ctr, key) + for range 9 { + key[0] += 0x9E3779B97F4A7C15 + key[1] += 0xBB67AE8584CAA73B + ctr = philoxRound(ctr, key) + } + return ctr +} + +// Uint64 returns the next output: the next of the four buffered ones, or the +// first of four new ones after incrementing the counter. +func (p *Philox) Uint64() uint64 { + if p.bufPos < 4 { + v := p.buf[p.bufPos] + p.bufPos++ + return v + } + for i := range p.ctr { + p.ctr[i]++ + if p.ctr[i] != 0 { + break + } + } + p.buf = philox4x64(p.ctr, p.key) + p.bufPos = 1 + return p.buf[0] +} + +// Uint32 returns the low half of a 64-bit output, or the high half kept from +// the previous call. +func (p *Philox) Uint32() uint32 { + if p.half.has { + p.half.has = false + return p.half.uint + } + return p.half.split(p.Uint64()) +} + +// Float64 returns uint64ToDouble of the next output. +func (p *Philox) Float64() float64 { return uint64ToDouble(p.Uint64()) } + +// Raw returns the next 64-bit output. +func (p *Philox) Raw() uint64 { return p.Uint64() } + +// State returns the generator's state. +func (p *Philox) State() PhiloxState { + return PhiloxState{p.ctr, p.key, p.buf, p.bufPos, p.half.has, p.half.uint} +} + +// SetState replaces the generator's state. +func (p *Philox) SetState(s PhiloxState) { + p.ctr, p.key, p.buf, p.bufPos = s.Counter, s.Key, s.Buffer, s.BufferPos + p.half = half32{s.HasUint32, s.Uinteger} +} + +// SeedSeq returns the SeedSequence the generator was made from. +func (p *Philox) SeedSeq() *SeedSequence { return p.seed } + +// Spawn returns n children seeded from SeedSeq().Spawn(n), numpy's spawn. +func (p *Philox) Spawn(n int) ([]BitGenerator, error) { + return spawnWith(p.seed, n, func(s *SeedSequence) BitGenerator { return NewPhiloxFromSeedSequence(s) }) +} + +// Advance adds delta (taken modulo 2**256, may be negative) to the counter, +// numpy's advance(delta), and drops the buffered outputs. +func (p *Philox) Advance(delta *big.Int) *Philox { + m := new(big.Int).Lsh(big.NewInt(1), 256) + d := new(big.Int).Mod(delta, m) + var step [4]uint64 + mask := new(big.Int).SetUint64(^uint64(0)) + for i := range step { + step[i] = new(big.Int).And(d, mask).Uint64() + d.Rsh(d, 64) + } + carry := false + for i := range p.ctr { + if carry { + p.ctr[i]++ + carry = p.ctr[i] == 0 + } + orig := p.ctr[i] + p.ctr[i] += step[i] + if p.ctr[i] < orig && !carry { + carry = true + } + } + p.reset() + return p +} + +// Jumped returns a copy advanced by jumps * 2**128 counter steps, numpy's +// jumped(jumps). +func (p *Philox) Jumped(jumps uint64) *Philox { + q := &Philox{ctr: p.ctr, key: p.key, buf: p.buf, bufPos: p.bufPos, half: p.half} + d := new(big.Int).Lsh(new(big.Int).SetUint64(jumps), 128) + return q.Advance(d) +} + +// SFC64State is the state of an SFC64, numpy's state dict. +type SFC64State struct { + S [4]uint64 + HasUint32 bool + Uinteger uint32 +} + +// SFC64 is numpy's SFC64, Chris Doty-Humphrey's Small Fast Chaotic generator. +type SFC64 struct { + s [4]uint64 + half half32 + seed *SeedSequence +} + +// NewSFC64 returns numpy's SFC64(seed). +func NewSFC64(seed ...uint64) *SFC64 { return NewSFC64FromSeedSequence(NewSeedSequence(seed...)) } + +// NewSFC64FromSeedSequence returns numpy's SFC64(seed_seq): three words from +// generate_state(3, uint64), a counter of 1, then twelve discarded outputs. +func NewSFC64FromSeedSequence(ss *SeedSequence) *SFC64 { + v := ss.GenerateState64(3) + g := &SFC64{s: [4]uint64{v[0], v[1], v[2], 1}, seed: ss} + for range 12 { + g.Uint64() + } + return g +} + +// Uint64 returns the next output. +func (g *SFC64) Uint64() uint64 { + s := &g.s + tmp := s[0] + s[1] + s[3] + s[3]++ + s[0] = s[1] ^ (s[1] >> 11) + s[1] = s[2] + (s[2] << 3) + s[2] = bits.RotateLeft64(s[2], 24) + tmp + return tmp +} + +// Uint32 returns the low half of a 64-bit output, or the high half kept from +// the previous call. +func (g *SFC64) Uint32() uint32 { + if g.half.has { + g.half.has = false + return g.half.uint + } + return g.half.split(g.Uint64()) +} + +// Float64 returns uint64ToDouble of the next output. +func (g *SFC64) Float64() float64 { return uint64ToDouble(g.Uint64()) } + +// Raw returns the next 64-bit output. +func (g *SFC64) Raw() uint64 { return g.Uint64() } + +// State returns the generator's state. +func (g *SFC64) State() SFC64State { return SFC64State{g.s, g.half.has, g.half.uint} } + +// SetState replaces the generator's state. +func (g *SFC64) SetState(s SFC64State) { g.s, g.half = s.S, half32{s.HasUint32, s.Uinteger} } + +// SeedSeq returns the SeedSequence the generator was made from. +func (g *SFC64) SeedSeq() *SeedSequence { return g.seed } + +// Spawn returns n children seeded from SeedSeq().Spawn(n), numpy's spawn. +func (g *SFC64) Spawn(n int) ([]BitGenerator, error) { + return spawnWith(g.seed, n, func(s *SeedSequence) BitGenerator { return NewSFC64FromSeedSequence(s) }) +} diff --git a/random/quant_test.go b/random/quant_test.go new file mode 100644 index 0000000..33970c6 --- /dev/null +++ b/random/quant_test.go @@ -0,0 +1,76 @@ +package random + +import ( + "encoding/binary" + "math" + "os" + "path/filepath" + "testing" + + "github.com/go-ndarray/ndarray" +) + +// TestQuantifyLibm measures, on 10^6 draws per sampler, how many values +// differ from NumPy's and by how much. It needs NumPy's output, written by +// testdata/quant.py (not embedded: 180 MB), and runs only when +// RANDOM_QUANT_DIR names its directory; docs/random.md quotes the result. +func TestQuantifyLibm(t *testing.T) { + dir := os.Getenv("RANDOM_QUANT_DIR") + if dir == "" { + t.Skip("RANDOM_QUANT_DIR not set") + } + const n = 1_000_000 + g := func() *Generator { return DefaultRNG(0) } + l := func() *RandomState { return NewRandomState(0) } + asF := func(a *ndarray.Array, err error) []float64 { + if err != nil { + t.Fatal(err) + } + d, _ := ndarray.Data[float64](a.AsType(ndarray.Float64)) + return d + } + jobs := []struct { + name string + f func() []float64 + }{ + {"legacy_randn", func() []float64 { return asF(l().Randn(n)) }}, + {"legacy_normal", func() []float64 { return asF(l().Normal(1, 3, n)) }}, + {"legacy_gamma3", func() []float64 { return asF(l().StandardGamma(3, n)) }}, + {"legacy_exp", func() []float64 { return asF(l().StandardExponential(n)) }}, + {"gamma05", func() []float64 { return asF(g().StandardGamma(0.5, n)) }}, + {"gamma3", func() []float64 { return asF(g().StandardGamma(3, n)) }}, + {"beta23", func() []float64 { return asF(g().Beta(2, 3, n)) }}, + {"beta05", func() []float64 { return asF(g().Beta(0.5, 0.5, n)) }}, + {"lognormal", func() []float64 { return asF(g().Lognormal(0.5, 0.25, n)) }}, + {"laplace", func() []float64 { return asF(g().Laplace(0, 1, n)) }}, + {"logistic", func() []float64 { return asF(g().Logistic(0, 1, n)) }}, + {"gumbel", func() []float64 { return asF(g().Gumbel(0, 1, n)) }}, + {"exp_inv", func() []float64 { return asF(g().StandardExponentialOf(ndarray.Float64, "inv", n)) }}, + {"pareto", func() []float64 { return asF(g().Pareto(3, n)) }}, + {"weibull", func() []float64 { return asF(g().Weibull(2, n)) }}, + {"vonmises", func() []float64 { return asF(g().VonMises(0.5, 2, n)) }}, + {"chisquare", func() []float64 { return asF(g().ChiSquare(3, n)) }}, + {"standard_t", func() []float64 { return asF(g().StandardT(4, n)) }}, + {"normal", func() []float64 { return asF(g().StandardNormal(n)) }}, + {"binomial", func() []float64 { return asF(g().Binomial(1000, 0.4, n)) }}, + {"poisson", func() []float64 { return asF(g().Poisson(50, n)) }}, + {"poisson3", func() []float64 { return asF(g().Poisson(3, n)) }}, + {"geometric", func() []float64 { return asF(g().Geometric(0.05, n)) }}, + } + for _, j := range jobs { + raw, err := os.ReadFile(filepath.Join(dir, j.name+".bin")) + if err != nil { + t.Fatal(err) + } + got := j.f() + diff, maxU := 0, int64(0) + for i := range got { + w := math.Float64frombits(binary.LittleEndian.Uint64(raw[8*i:])) + if got[i] != w { + diff++ + maxU = max(maxU, ulps(got[i], w, false)) + } + } + t.Logf("%-14s %7d of %d differ, max %d ULP", j.name, diff, len(got), maxU) + } +} diff --git a/random/seedseq.go b/random/seedseq.go new file mode 100644 index 0000000..e053435 --- /dev/null +++ b/random/seedseq.go @@ -0,0 +1,307 @@ +package random + +import ( + "crypto/rand" + "encoding/binary" + "fmt" + "math/big" +) + +// SeedSequence constants, from numpy/random/bit_generator.pyx (after +// Melissa O'Neill's seed_seq_fe). +const ( + defaultPoolSize = 4 + initA = 0x43b0d7e5 + multA = 0x931e8875 + initB = 0x8b51f9dd + multB = 0x58f38ded + mixMultL = 0xca01f9dd + mixMultR = 0x4973f715 + xshift = 16 +) + +// SeedSequence mixes user entropy into a well-distributed pool and derives +// bit-generator states from it, exactly as numpy.random.SeedSequence: the +// same entropy, spawn key and pool size give the same pool, the same +// GenerateState words and the same children. +// +// A SeedSequence is not safe for concurrent use (Spawn mutates it). +type SeedSequence struct { + entropy []uint32 // the entropy as uint32 words, lowest first + spawnKey []uint32 // the spawn key as uint32 words + spawnKeyInts []uint64 // the spawn key as given + poolSize int + nChildrenSpawned uint64 + pool []uint32 +} + +// NewSeedSequence returns the SeedSequence of numpy's SeedSequence(entropy). +// One value is numpy's integer entropy (SeedSequence(42)); several are its +// sequence form (SeedSequence([1, 2, 3])), where each value is split into +// uint32 words, lowest first. With no value the entropy is 128 fresh bits +// from the operating system, as SeedSequence() draws them. +// +// For an integer wider than 64 bits use NewSeedSequenceBig. Python's entropy +// 2**64 + 5, for instance, is the sequence [5, 0, 1] of words and not +// expressible as uint64s: [5, 2**32] would be [5, 0, 1] as well — numpy +// itself cannot tell those apart, and neither can this. +func NewSeedSequence(entropy ...uint64) *SeedSequence { + return newSeedSequence(entropyWords(entropy, defaultPoolSize), nil, defaultPoolSize, 0) +} + +// NewSeedSequenceBig returns the SeedSequence of numpy's +// SeedSequence(entropy) for an arbitrary non-negative integer. +func NewSeedSequenceBig(entropy *big.Int) (*SeedSequence, error) { + words, err := bigWords(entropy) + if err != nil { + return nil, err + } + return NewSeedSequenceWith(SeedSequenceOptions{Words: words}) +} + +// SeedSequenceOptions are the arguments of numpy's SeedSequence constructor. +type SeedSequenceOptions struct { + // Entropy is numpy's entropy as integers (see NewSeedSequence). + Entropy []uint64 + // Words, when not nil, is the entropy as uint32 words, lowest first, + // and Entropy must be empty. It expresses any Python integer or list. + Words []uint32 + // SpawnKey is numpy's spawn_key. + SpawnKey []uint64 + // PoolSize is numpy's pool_size; zero means the default, 4. + PoolSize int + // NChildrenSpawned is numpy's n_children_spawned. + NChildrenSpawned uint64 +} + +// NewSeedSequenceWith returns the SeedSequence of numpy's +// SeedSequence(entropy, spawn_key=..., pool_size=..., n_children_spawned=...). +// Entropy and Words both empty draw fresh entropy from the OS. +func NewSeedSequenceWith(o SeedSequenceOptions) (*SeedSequence, error) { + pool := o.PoolSize + if pool == 0 { + pool = defaultPoolSize + } + if pool < defaultPoolSize { + return nil, fmt.Errorf("random: the size of the entropy pool should be at least %d", defaultPoolSize) + } + if len(o.Words) > 0 && len(o.Entropy) > 0 { + return nil, fmt.Errorf("random: Entropy and Words are exclusive") + } + var words []uint32 + if o.Words != nil { + words = make([]uint32, len(o.Words)) + copy(words, o.Words) + } else { + words = entropyWords(o.Entropy, pool) + } + return newSeedSequence(words, o.SpawnKey, pool, o.NChildrenSpawned), nil +} + +// entropyWords is the uint32 words of numpy's integer (one value) or +// sequence entropy, or fresh OS entropy for none. +func entropyWords(entropy []uint64, pool int) []uint32 { + switch len(entropy) { + case 0: + return osEntropy(pool) + case 1: + return intWords(entropy[0]) + } + return sequenceWords(entropy) +} + +func newSeedSequence(words []uint32, spawnKey []uint64, pool int, spawned uint64) *SeedSequence { + ss := &SeedSequence{ + entropy: words, + spawnKey: sequenceWords(spawnKey), + spawnKeyInts: append([]uint64(nil), spawnKey...), + poolSize: pool, + nChildrenSpawned: spawned, + pool: make([]uint32, pool), + } + ss.mixEntropy(ss.assembledEntropy()) + return ss +} + +// readOSEntropy fills b from the operating system; a variable so a test can +// observe the call. +var readOSEntropy = rand.Read + +// osEntropy draws pool*32 bits, as numpy's randbits(pool_size * 32), and +// returns them as the words of that integer (leading zero words dropped). +func osEntropy(pool int) []uint32 { + b := make([]byte, 4*pool) + _, _ = readOSEntropy(b) // crypto/rand.Read never fails (Go 1.24+) + w := make([]uint32, pool) + for i := range w { + w[i] = binary.LittleEndian.Uint32(b[4*i:]) + } + return trimWords(w) +} + +// intWords splits a non-negative integer into uint32 words, lowest first; +// zero is the single word 0 (numpy's _int_to_uint32_array). +func intWords(v uint64) []uint32 { + if v == 0 { + return []uint32{0} + } + var w []uint32 + for v > 0 { + w = append(w, uint32(v)) + v >>= 32 + } + return w +} + +// sequenceWords concatenates the words of each integer, as numpy's +// _coerce_to_uint32_array does for a sequence. +func sequenceWords(vs []uint64) []uint32 { + var w []uint32 + for _, v := range vs { + w = append(w, intWords(v)...) + } + return w +} + +// trimWords drops high zero words, keeping at least one: the words of the +// integer the slice spells. +func trimWords(w []uint32) []uint32 { + n := len(w) + for n > 1 && w[n-1] == 0 { + n-- + } + return w[:n] +} + +// bigWords returns the uint32 words of a non-negative big integer. +func bigWords(v *big.Int) ([]uint32, error) { + if v.Sign() < 0 { + return nil, fmt.Errorf("random: expected non-negative integer") + } + b := v.Bytes() // big-endian + w := make([]uint32, (len(b)+3)/4) + for i := range b { + pos := len(b) - 1 - i // little-endian byte index + w[pos/4] |= uint32(b[i]) << (8 * (pos % 4)) + } + if len(w) == 0 { + w = []uint32{0} + } + return w, nil +} + +func hashmix(value uint32, hashConst *uint32) uint32 { + value ^= *hashConst + *hashConst *= multA + value *= *hashConst + value ^= value >> xshift + return value +} + +func mix(x, y uint32) uint32 { + r := mixMultL*x - mixMultR*y + r ^= r >> xshift + return r +} + +// assembledEntropy is numpy's get_assembled_entropy: the run entropy, padded +// with zeros to the pool size when there is a spawn key, then the spawn key. +func (s *SeedSequence) assembledEntropy() []uint32 { + run := append([]uint32(nil), s.entropy...) + if len(s.spawnKey) > 0 && len(run) < s.poolSize { + run = append(run, make([]uint32, s.poolSize-len(run))...) + } + return append(run, s.spawnKey...) +} + +func (s *SeedSequence) mixEntropy(e []uint32) { + h := uint32(initA) + m := s.pool + for i := range m { + if i < len(e) { + m[i] = hashmix(e[i], &h) + } else { + m[i] = hashmix(0, &h) + } + } + for src := range m { + for dst := range m { + if src != dst { + m[dst] = mix(m[dst], hashmix(m[src], &h)) + } + } + } + for src := len(m); src < len(e); src++ { + for dst := range m { + m[dst] = mix(m[dst], hashmix(e[src], &h)) + } + } +} + +// Entropy returns the entropy as uint32 words, lowest first. +func (s *SeedSequence) Entropy() []uint32 { return append([]uint32(nil), s.entropy...) } + +// EntropyInt returns the entropy as the integer numpy shows for an integer +// entropy (the words read as one little-endian number). +func (s *SeedSequence) EntropyInt() *big.Int { + v := new(big.Int) + for i := len(s.entropy) - 1; i >= 0; i-- { + v.Lsh(v, 32) + v.Or(v, big.NewInt(int64(s.entropy[i]))) + } + return v +} + +// SpawnKey returns the spawn key. +func (s *SeedSequence) SpawnKey() []uint64 { return append([]uint64(nil), s.spawnKeyInts...) } + +// PoolSize returns the pool size in uint32 words. +func (s *SeedSequence) PoolSize() int { return s.poolSize } + +// NChildrenSpawned returns how many children Spawn has made so far. +func (s *SeedSequence) NChildrenSpawned() uint64 { return s.nChildrenSpawned } + +// Pool returns a copy of the mixed entropy pool. +func (s *SeedSequence) Pool() []uint32 { return append([]uint32(nil), s.pool...) } + +// GenerateState returns n uint32 words of state, numpy's +// generate_state(n, np.uint32). +func (s *SeedSequence) GenerateState(n int) []uint32 { + h := uint32(initB) + out := make([]uint32, n) + for i := range out { + v := s.pool[i%len(s.pool)] + v ^= h + h *= multB + v *= h + v ^= v >> xshift + out[i] = v + } + return out +} + +// GenerateState64 returns n uint64 words of state, numpy's +// generate_state(n, np.uint64): pairs of uint32 words, low word first. +func (s *SeedSequence) GenerateState64(n int) []uint64 { + w := s.GenerateState(2 * n) + out := make([]uint64, n) + for i := range out { + out[i] = uint64(w[2*i]) | uint64(w[2*i+1])<<32 + } + return out +} + +// Spawn returns n child SeedSequences, numpy's spawn(n): each has this +// entropy and pool size, and this spawn key extended by its index. +func (s *SeedSequence) Spawn(n int) []*SeedSequence { + if n < 0 { + panic("random: n_children must be non-negative") + } + out := make([]*SeedSequence, n) + for i := range out { + key := append(append([]uint64(nil), s.spawnKeyInts...), s.nChildrenSpawned+uint64(i)) + out[i] = newSeedSequence(s.entropy, key, s.poolSize, 0) + } + s.nChildrenSpawned += uint64(n) + return out +} diff --git a/random/testdata/cases.json b/random/testdata/cases.json new file mode 100644 index 0000000..94ec1cb --- /dev/null +++ b/random/testdata/cases.json @@ -0,0 +1 @@ +{"numpy": "2.5.3", "machine": "x86_64", "cases": [{"api": "Generator", "bg": "PCG64", "seed": "0", "calls": [{"fn": "random", "kw": {}, "size": 7, "r": {"dtype": "float64", "shape": [7], "v": ["0.6369616873214543", "0.2697867137638703", "0.04097352393619469", "0.016527635528529094", "0.8132702392002724", "0.9127555772777217", "0.6066357757671799"]}}]}, {"api": "Generator", "bg": "PCG64", "seed": "0", "calls": [{"fn": "random", "kw": {"dtype": "float32"}, "size": [2, 3], "r": {"dtype": "float32", "shape": [2, 3], "v": ["0.8506242036819458", "0.6369616389274597", "0.5111364722251892", "0.26978665590286255", "0.3078293800354004", "0.0409734845161438"]}}]}, {"api": "Generator", "bg": "PCG64", "seed": "0", "calls": [{"fn": "random", "kw": {}, "size": null, "r": {"dtype": "float64", "shape": [], "v": ["0.6369616873214543"]}}]}, {"api": "Generator", "bg": 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"state32": ["2968811710", "3677149159", "745650761", "2884920346", "2642120001", "549907821", "574372308", "742431198", "2597619198"], "state64": ["15793235383387715774", "12390638538380655177", "2361836109651742017", "3188717715514472916", "648184599915300350"], "kids": [{"key": ["0"], "pool": ["2415978763", "195748244", "4024957757", "2824017155"]}, {"key": ["1"], "pool": ["443930042", "1047521080", "1138512698", "1433149623"]}, {"key": ["2"], "pool": ["2759570840", "133463972", "3748601666", "1494986599"]}, {"key": ["3"], "pool": ["2185221192", "331189864", "2705061464", "3070475053"]}]}, {"name": "pool8", "pool": ["2916486286", "1055748940", "1992122540", "3812097620", "4091149860", "2005385835", "2695311824", "2101776118"], "state32": ["4073991759", "3860319326", "1691695900", "522132654", "3069485573", "2982616557", "3936096425", "3846322761", "1529687222"], "state64": ["16579945261360754255", "2242542674795379484", "12810240571892605445", "16519830472291520681", "9819212013839920310"], "kids": [{"key": ["0"], "pool": ["3094069420", "866908893", "3785055402", "3161365574", "2909817204", "3298726200", "3637029617", "2000012359"]}, {"key": ["1"], "pool": ["758496164", "3774962083", "4013662995", "982972829", "337360604", "2868122493", "246892715", "1892033551"]}, {"key": ["2"], "pool": ["2396896678", "1711547145", "331925276", "4116159375", "1723908885", "3203440696", "4087288475", "597878569"]}, {"key": ["3"], "pool": ["509053414", "1172691786", "2603386081", "2499502971", "3553037285", "1580280710", "1893112511", "1023741433"]}]}, {"name": "spawnkey", "pool": ["2330321549", "2989780581", "1502215597", "4041235760"], "state32": ["3537215493", "1608948106", "3193448784", "1139554609", "2827003808", "2084514691", "596767305", "356359648", "3760911194"], "state64": ["6910379499768356869", "4894349780854516048", "8952922428703549344", "1530553034370839113", "7055575628257096538"], "kids": [{"key": ["3", "1099511627776", "0"], "pool": ["923873267", "1556642480", "688490278", "2332621719"]}, {"key": ["3", "1099511627776", "1"], "pool": ["915315904", "3533824334", "3084052531", "1836895589"]}, {"key": ["3", "1099511627776", "2"], "pool": ["868066103", "486117525", "4069129723", "3715513033"]}, {"key": ["3", "1099511627776", "3"], "pool": ["1662235061", "4005877860", "3046318805", "3815904727"]}]}, {"name": "spawnkey_long", "pool": ["2339234176", "1802292265", "3659146643", "404625381"], "state32": ["4123051093", "2876466179", "1303656962", "442040333", "3012560549", "514027427", "2367882341", "2297165455", "2789144043"], "state64": ["12354328170978133077", "1898548775051606530", "2207730991224587941", "9866250505093842021", "8088995520921400811"], "kids": [{"key": ["1", "0"], "pool": ["2959424460", "459843524", "3200397", "44149057"]}, {"key": ["1", "1"], "pool": ["2950935286", "2436979554", "2398685590", "3843374671"]}, {"key": ["1", "2"], "pool": ["2903682627", "3684366650", "3383819952", "1427142160"]}, {"key": ["1", "3"], "pool": ["3697842437", "2909036976", "2361012846", "1527540458"]}]}]} diff --git a/random/testdata/cases.py b/random/testdata/cases.py new file mode 100644 index 0000000..0a9b69e --- /dev/null +++ b/random/testdata/cases.py @@ -0,0 +1,379 @@ +# Generates random/testdata/cases.json, NumPy's output for every function of +# the package on several seeds and bit generators: +# +# uv run --python cpython-3.12-macos-x86_64 --with numpy==2.5.3 \ +# python random/testdata/cases.py > random/testdata/cases.json +# +# The reference is an x86-64 build of NumPy (on Apple silicon it runs under +# Rosetta). NumPy's aarch64 builds let the C compiler fuse a*b+c into one FMA +# in some samplers (uniform, for one), so their last bits differ from its own +# x86-64 builds; the x86-64 output is the one that follows the C source +# operation by operation, and the one this package reproduces. See +# docs/random.md. +# +# Every case is a script: a fresh generator, then calls in order, each with +# its result. Integers are decimal strings, floats repr strings (exact). +import json, platform +import numpy as np +from numpy.random import (SeedSequence, PCG64, PCG64DXSM, MT19937, Philox, + SFC64, Generator, RandomState) + +SEEDS = { + "0": 0, "1": 1, "42": 42, "u64max": 2**64 - 1, + "list": [1, 2, 3], "big": 2**128 + 12345, +} +BG = {"PCG64": PCG64, "PCG64DXSM": PCG64DXSM, "MT19937": MT19937, + "Philox": Philox, "SFC64": SFC64} + + +def enc(v): + v = np.asarray(v) + out = [] + for z in v.ravel(): + if v.dtype.kind == 'f': + out.append(repr(float(z))) + elif v.dtype.kind == 'b': + out.append("1" if z else "0") + else: + out.append(str(int(z))) + return {"dtype": str(v.dtype), "shape": list(v.shape), "v": out} + + +ARANGE12 = np.arange(12).reshape(3, 4) +FLOATS6 = np.array([0.5, 1.5, 2.5, 3.5, 4.5, 5.5]) + + +def gen_calls(): + c = [] + a = c.append + a(("random", {}, 7)); a(("random", {"dtype": "float32"}, [2, 3])); a(("random", {}, None)) + for dt, bounds in [ + ("int8", [(0, 10, False), (-128, 127, True), (-5, 5, False), (3, 4, False), (0, 64, False)]), + ("int16", [(0, 10, False), (-32768, 32767, True), (-300, 300, False), (0, 256, False)]), + ("int32", [(0, 10, False), (-2**31, 2**31 - 1, True), (-70000, 70000, False), (0, 2**16, False)]), + ("int64", [(0, 10, False), (-2**63, 2**63 - 1, True), (0, 2**32, False), (0, 2**32, True), + (-2**40, 2**40, False), (5, 2**31 + 5, False), (7, 8, False), (0, 3 * 2**30, False)]), + ("uint8", [(0, 10, False), (0, 255, True), (200, 256, False)]), + ("uint16", [(0, 10, False), (0, 65535, True), (1000, 1001, False)]), + ("uint32", [(0, 10, False), (0, 2**32 - 1, True), (2**31, 2**32, False)]), + ("uint64", [(0, 10, False), (0, 2**64 - 1, True), (2**63, 2**64 - 1, False), (0, 2**33, False), + (0, 3 * 2**62, False)]), + ("bool", [(0, 2, False), (0, 1, True), (1, 2, False), (0, 1, False)]), + ]: + for lo, hi, ep in bounds: + a(("integers", {"low": lo, "high": hi, "dtype": dt, "endpoint": ep}, 9)) + a(("integers", {"low": 0, "high": 100}, None)) + for dt in ["float64", "float32"]: + a(("standard_normal", {"dtype": dt}, 60)) + a(("standard_exponential", {"dtype": dt}, 60)) + a(("standard_exponential", {"dtype": dt, "method": "inv"}, 10)) + for sh in [0.5, 1.0, 3.0, 0.0, 0.05]: + a(("standard_gamma", {"shape": sh, "dtype": dt}, 20)) + a(("normal", {"loc": 1.5, "scale": 2.0}, 20)) + a(("exponential", {"scale": 2.0}, 10)) + a(("uniform", {"low": -1.0, "high": 3.0}, 20)) + a(("gamma", {"shape": 2.5, "scale": 1.5}, 10)) + for ab in [(0.5, 0.5), (2.0, 3.0), (1e-104, 1e-104), (0.01, 0.01), (1e-5, 1e-5)]: + a(("beta", {"a": ab[0], "b": ab[1]}, 10)) + a(("chisquare", {"df": 3.0}, 10)) + a(("f", {"dfnum": 3.0, "dfden": 5.0}, 10)) + for df, nonc in [(3.0, 2.0), (0.5, 2.0), (3.0, 0.0)]: + a(("noncentral_chisquare", {"df": df, "nonc": nonc}, 10)) + a(("noncentral_f", {"dfnum": 3.0, "dfden": 5.0, "nonc": 2.0}, 10)) + a(("standard_cauchy", {}, 10)) + a(("standard_t", {"df": 4.0}, 10)) + for mu, kappa in [(0.5, 2.0), (0.0, 1e-9), (0.0, 1e-6), (0.3, 1e7), (3.1415, 1e7), (-3.1415, 1e7)]: + a(("vonmises", {"mu": mu, "kappa": kappa}, 10)) + a(("pareto", {"a": 3.0}, 10)) + a(("weibull", {"a": 2.0}, 10)); a(("weibull", {"a": 0.0}, 3)) + a(("power", {"a": 2.0}, 10)) + a(("laplace", {"loc": 0.5, "scale": 2.0}, 10)) + a(("gumbel", {"loc": 1.0, "scale": 2.0}, 10)) + a(("logistic", {"loc": 1.0, "scale": 2.0}, 10)) + a(("lognormal", {"mean": 0.5, "sigma": 0.25}, 10)) + a(("rayleigh", {"scale": 2.0}, 10)) + a(("wald", {"mean": 1.0, "scale": 2.0}, 10)) + a(("triangular", {"left": 0.0, "mode": 1.0, "right": 3.0}, 10)) + for n, p in [(10, 0.3), (1000, 0.4), (1000, 0.7), (5, 0.9), (0, 0.5), (10, 0.0), (100000, 0.5), (3, 1.0)]: + a(("binomial", {"n": n, "p": p}, 20)) + a(("negative_binomial", {"n": 5.0, "p": 0.3}, 10)) + for lam in [3.0, 50.0, 0.0, 1e6, 10.0]: + a(("poisson", {"lam": lam}, 20)) + a(("zipf", {"a": 2.0}, 10)); a(("zipf", {"a": 1.5}, 10)); a(("zipf", {"a": 2000.0}, 2)) + a(("geometric", {"p": 0.5}, 10)); a(("geometric", {"p": 0.05}, 10)); a(("geometric", {"p": 1e-300}, 2)) + a(("logseries", {"p": 0.5}, 10)); a(("logseries", {"p": 0.99}, 10)) + a(("choice", {"a": 10}, 5)) + a(("choice", {"a": 10}, None)) + a(("choice", {"a": 10, "replace": False}, 5)) + a(("choice", {"a": 10, "replace": False, "shuffle": False}, 5)) + a(("choice", {"a": 10000, "replace": False}, 5)) + a(("choice", {"a": 20000, "replace": False}, 500)) + a(("choice", {"a": 20000, "replace": False, "shuffle": False}, 1500)) + a(("choice", {"a": 5, "p": [0.1, 0.2, 0.3, 0.15, 0.25]}, 6)) + a(("choice", {"a": 5, "p": [0.1, 0.2, 0.3, 0.15, 0.25], "replace": False}, 4)) + a(("choice", {"a": "arange12", "axis": 1}, 2)) + a(("choice", {"a": "arange12"}, None)) + a(("choice", {"a": "floats6"}, [2, 2])) + a(("shuffle", {"x": "arange10"}, None)) + a(("shuffle", {"x": "arange12", "axis": 0}, None)) + a(("shuffle", {"x": "arange12", "axis": 1}, None)) + a(("permutation", {"x": 10}, None)) + a(("permutation", {"x": "arange12", "axis": 1}, None)) + a(("permutation", {"x": "floats6"}, None)) + a(("permuted", {"x": "arange12"}, None)) + a(("permuted", {"x": "arange12", "axis": 1}, None)) + a(("permuted", {"x": "arange12", "axis": 0}, None)) + a(("bytes", {"length": 10}, None)) + a(("bytes", {"length": 0}, None)) + a(("dirichlet", {"alpha": [0.5, 1.0, 2.0]}, 3)) + a(("dirichlet", {"alpha": [0.05, 0.02, 0.01]}, 2)) + a(("dirichlet", {"alpha": [0.05, 0.0, 0.0]}, 2)) + a(("multinomial", {"n": 20, "pvals": [0.2, 0.3, 0.5]}, 3)) + a(("multinomial", {"n": 1000, "pvals": [0.1, 0.2, 0.7]}, None)) + a(("multinomial", {"n": 10, "pvals": [1.0, 0.0, 0.0]}, 2)) + a(("multinomial", {"n": 0, "pvals": [0.5, 0.5]}, 2)) + return c + + +def arg_array(v): + return {"arange10": np.arange(10), "arange12": ARANGE12.copy(), "floats6": FLOATS6.copy()}.get(v, v) \ + if isinstance(v, str) else v + + +def run_gen(g, fn, kw, size): + kw = {k: arg_array(v) for k, v in kw.items()} + if fn in ("shuffle",): + x = kw.pop("x") + g.shuffle(x, **kw) + return x + if fn in ("permutation", "permuted"): + x = kw.pop("x") + return getattr(g, fn)(x, **kw) + if fn == "bytes": + return np.frombuffer(g.bytes(kw["length"]), dtype=np.uint8) + if fn == "dirichlet": + return g.dirichlet(kw["alpha"], size=size) + if fn == "multinomial": + return g.multinomial(kw["n"], kw["pvals"], size=size) + if fn == "choice": + a = kw.pop("a") + return g.choice(a, size=size, **kw) + return getattr(g, fn)(size=size, **kw) + + +def make_bg(kind, seed): + return BG[kind](SEEDS[seed]) + + +cases = [] + + +def script(api, kind, seed, calls): + if api == "Generator": + g = Generator(make_bg(kind, seed)) + run = run_gen + else: + g = RandomState(seed) + run = run_legacy + out = [] + for fn, kw, size in calls: + r = run(g, fn, dict(kw), size) + out.append({"fn": fn, "kw": kw, "size": size, "r": enc(r)}) + cases.append({"api": api, "bg": kind, "seed": seed, "calls": out}) + + +GEN_CALLS = gen_calls() +for seed in SEEDS: + for call in GEN_CALLS: + script("Generator", "PCG64", seed, [call]) +SHORT = [c for c in GEN_CALLS if c[0] in ("random", "integers", "standard_normal", "standard_exponential", + "standard_gamma", "binomial", "choice", "shuffle", "bytes")] +for kind in ["PCG64DXSM", "MT19937", "Philox", "SFC64"]: + for seed in ["0", "42", "big"]: + for call in SHORT: + script("Generator", kind, seed, [call]) +# The buffered upper half of a 64-bit draw outlives a call: interleave. +MIXED = [("integers", {"low": 0, "high": 10, "dtype": "int32"}, 3), ("random", {}, 2), + ("integers", {"low": 0, "high": 10, "dtype": "int16"}, 1), + ("standard_normal", {"dtype": "float32"}, 3), ("random", {"dtype": "float32"}, 1), + ("integers", {"low": 0, "high": 10, "dtype": "uint8"}, 5), ("random", {}, 1), + ("bytes", {"length": 3}, None), ("standard_normal", {}, 2)] +for kind in BG: + for seed in SEEDS: + script("Generator", kind, seed, MIXED) + +# Legacy RandomState. +LSEEDS = {"0": 0, "1": 1, "42": 42, "u32max": 2**32 - 1, "arr": [1, 2, 3], "arr700": list(range(700))} + + +def run_legacy(r, fn, kw, size): + kw = {k: arg_array(v) for k, v in kw.items()} + if fn == "shuffle": + x = kw.pop("x") + r.shuffle(x) + return x + if fn == "permutation": + return r.permutation(kw["x"]) + if fn == "bytes": + return np.frombuffer(r.bytes(kw["length"]), dtype=np.uint8) + if fn in ("rand", "randn"): + return getattr(r, fn)(*(size or [])) + if fn == "choice": + a = kw.pop("a") + return r.choice(a, size=size, **kw) + if fn == "multinomial": + return r.multinomial(kw["n"], kw["pvals"], size=size) + return getattr(r, fn)(size=size, **kw) + + +LEG_CALLS = [("rand", {}, [3, 2]), ("rand", {}, None), ("randn", {}, [7]), ("random_sample", {}, 5), + ("randint", {"low": 0, "high": 10}, 8), ("randint", {"low": 0, "high": 2**40}, 5), + ("randint", {"low": -2**63, "high": 2**63 - 1}, 5)] +for dt, lo, hi in [("int8", -5, 5), ("int16", -300, 300), ("int32", 0, 2**31 - 1), ("uint8", 0, 256), + ("uint16", 7, 8), ("uint32", 0, 2**32), ("uint64", 0, 2**64 - 1), ("bool", 0, 2), + ("int64", 0, 2**32)]: + LEG_CALLS.append(("randint", {"low": lo, "high": hi, "dtype": dt}, 8)) +LEG_CALLS += [ + ("normal", {"loc": 1.0, "scale": 2.0}, 6), ("uniform", {"low": -1.0, "high": 3.0}, 6), + ("uniform", {"low": 3.0, "high": -1.0}, 4), ("standard_normal", {}, 5), + ("choice", {"a": 10}, 5), ("choice", {"a": 10, "replace": False}, 5), + ("choice", {"a": 5, "p": [0.1, 0.2, 0.3, 0.15, 0.25]}, 4), + ("choice", {"a": 5, "p": [0.1, 0.2, 0.3, 0.15, 0.25], "replace": False}, 3), + ("choice", {"a": "floats6"}, [2, 2]), ("choice", {"a": 10}, None), + ("shuffle", {"x": "arange10"}, None), ("shuffle", {"x": "arange12"}, None), + ("permutation", {"x": 10}, None), ("permutation", {"x": "arange12"}, None), + ("permutation", {"x": "floats6"}, None), + ("binomial", {"n": 10, "p": 0.3}, 10), ("binomial", {"n": 1000, "p": 0.4}, 10), + ("binomial", {"n": 1000, "p": 0.7}, 10), ("binomial", {"n": 0, "p": 0.5}, 3), + ("binomial", {"n": 100000, "p": 0.5}, 10), + ("poisson", {"lam": 3.0}, 10), ("poisson", {"lam": 50.0}, 10), ("poisson", {"lam": 0.0}, 2), + ("standard_exponential", {}, 5), ("exponential", {"scale": 2.0}, 5), + ("standard_gamma", {"shape": 0.5}, 5), ("standard_gamma", {"shape": 3.0}, 5), + ("standard_gamma", {"shape": 1.0}, 3), ("standard_gamma", {"shape": 0.0}, 2), + ("gamma", {"shape": 2.0, "scale": 2.0}, 5), ("chisquare", {"df": 3.0}, 5), + ("lognormal", {"mean": 0.5, "sigma": 0.25}, 5), + ("multinomial", {"n": 20, "pvals": [0.2, 0.3, 0.5]}, 3), + ("bytes", {"length": 10}, None), +] +for seed in LSEEDS: + for call in LEG_CALLS: + r = RandomState(LSEEDS[seed]) + out = [{"fn": call[0], "kw": call[1], "size": call[2], "r": enc(run_legacy(r, call[0], dict(call[1]), call[2]))}] + cases.append({"api": "RandomState", "bg": "MT19937", "seed": seed, "calls": out}) + # The polar method caches its second value across calls. + r = RandomState(LSEEDS[seed]) + calls = [("randn", {}, [3]), ("rand", {}, [1]), ("randn", {}, [1]), ("normal", {"loc": 0.0, "scale": 1.0}, 2), + ("standard_gamma", {"shape": 3.0}, 3), ("randn", {}, [1])] + cases.append({"api": "RandomState", "bg": "MT19937", "seed": seed, + "calls": [{"fn": f, "kw": k, "size": s, "r": enc(run_legacy(r, f, dict(k), s))} for f, k, s in calls]}) + +# Samplers that call the C library on every draw, long enough to meet its +# rare misroundings (see libmULP in numpy_test.go). +for seed in ["0", "42"]: + for fn, kw, size in [("randn", {}, [3000]), ("normal", {"loc": 1.0, "scale": 3.0}, 2000), + ("standard_gamma", {"shape": 3.0}, 2000)]: + r = RandomState(LSEEDS[seed]) + cases.append({"api": "RandomState", "bg": "MT19937", "seed": seed, + "calls": [{"fn": fn, "kw": kw, "size": size, "r": enc(run_legacy(r, fn, dict(kw), size))}]}) + script("Generator", "PCG64", seed, [("standard_gamma", {"shape": 0.5}, 3000)]) + +# Bit generators: raw streams, seeding, jumps. +bitgens = [] +for kind, cls in BG.items(): + for seed, s in SEEDS.items(): + bg = cls(s) + st = bg.state + rec = {"bg": kind, "seed": seed, "raw": [str(int(v)) for v in bg.random_raw(700 if kind == "MT19937" else 100)]} + if kind in ("PCG64", "PCG64DXSM"): + rec["state"] = [str(st["state"]["state"]), str(st["state"]["inc"])] + elif kind == "MT19937": + rec["state"] = [str(int(st["state"]["key"][i])) for i in (0, 1, 2, 623)] + [str(st["state"]["pos"])] + elif kind == "Philox": + rec["state"] = [str(int(v)) for v in st["state"]["key"]] + else: + rec["state"] = [str(int(v)) for v in st["state"]["state"]] + for j in ((1, 3) if kind != "SFC64" else ()): + rec[f"jumped{j}"] = [str(int(v)) for v in cls(s).jumped(j).random_raw(5)] + if kind in ("PCG64", "PCG64DXSM", "Philox"): + b = cls(s) + b.random_raw(3) + rec["advance"] = [str(int(v)) for v in b.advance(2**100 + 7).random_raw(5)] + b = cls(s) + b.random_raw(7) + rec["advance_neg"] = [str(int(v)) for v in b.advance(-3).random_raw(5)] + kids = cls(s).spawn(2) + rec["spawn"] = [[str(int(v)) for v in k.random_raw(5)] for k in kids] + bitgens.append(rec) + +ph = Philox(counter=np.array([1, 2, 3, 2**64 - 1], dtype=np.uint64), key=np.array([5, 6], dtype=np.uint64)) +philox_key = {"raw": [str(int(v)) for v in ph.random_raw(12)]} +ph = Philox(counter=np.array([2**64 - 1, 2**64 - 1, 0, 0], dtype=np.uint64), key=np.array([1, 2], dtype=np.uint64)) +philox_key["carry"] = [str(int(v)) for v in ph.advance(1).random_raw(4)] +legacy_mt = [] +for seed, s in LSEEDS.items(): + m = MT19937() + m._legacy_seeding(s) + legacy_mt.append({"seed": seed, "raw": [str(int(v)) for v in m.random_raw(700)], + "jumped": [str(int(v)) for v in m.jumped(1).random_raw(5)]}) + m = MT19937() + m._legacy_seeding(s) + legacy_mt[-1]["jumped_fresh"] = [str(int(v)) for v in m.jumped(1).random_raw(5)] + +seedseqs = [] +for name, kw in [("0", {"entropy": 0}), ("42", {"entropy": 42}), ("u64max", {"entropy": 2**64 - 1}), + ("list", {"entropy": [1, 2, 3]}), ("big", {"entropy": 2**128 + 12345}), + ("long", {"entropy": list(range(1, 11))}), ("empty", {"entropy": []}), + ("pool8", {"entropy": 7, "pool_size": 8}), + ("spawnkey", {"entropy": 7, "spawn_key": (3, 2**40)}), + ("spawnkey_long", {"entropy": list(range(6)), "spawn_key": (1,)})]: + ss = SeedSequence(**kw) + kids = ss.spawn(3) + more = ss.spawn(1) + seedseqs.append({ + "name": name, "pool": [str(int(v)) for v in ss.pool], + "state32": [str(int(v)) for v in ss.generate_state(9)], + "state64": [str(int(v)) for v in ss.generate_state(5, np.uint64)], + "kids": [{"key": [str(k) for k in c.spawn_key], "pool": [str(int(v)) for v in c.pool]} for c in kids + more], + }) + +# Long runs, as SHA-256 digests of the little-endian bytes: they reach the +# rare paths (ziggurat tails and wedges, rejection loops) that short runs miss. +import hashlib +DIGESTS = [ + ("random", {}, 100000), ("random", {"dtype": "float32"}, 100000), + ("standard_normal", {}, 200000), ("standard_normal", {"dtype": "float32"}, 200000), + ("standard_exponential", {}, 200000), ("standard_exponential", {"dtype": "float32"}, 200000), + ("integers", {"low": 0, "high": 1000}, 100000), ("integers", {"low": -100, "high": 100, "dtype": "int8"}, 100000), + ("integers", {"low": 0, "high": 40000, "dtype": "uint16"}, 100000), ("integers", {"low": 0, "high": 2, "dtype": "bool"}, 100000), + ("standard_gamma", {"shape": 3.0}, 50000), + ("standard_gamma", {"shape": 3.0, "dtype": "float32"}, 50000), + ("binomial", {"n": 1000, "p": 0.4}, 50000), ("binomial", {"n": 20, "p": 0.3}, 50000), + ("poisson", {"lam": 50.0}, 50000), ("poisson", {"lam": 3.0}, 50000), + ("normal", {"loc": 1.0, "scale": 3.0}, 100000), ("uniform", {"low": -2.0, "high": 5.0}, 100000), + ("choice", {"a": 1000}, 50000), ("choice", {"a": 100000, "replace": False}, 5000), +] +LDIGESTS = [("rand", {}, [100000]), ("randint", {"low": 0, "high": 1000}, 100000), + ("binomial", {"n": 1000, "p": 0.4}, 20000), ("poisson", {"lam": 50.0}, 20000), + ("uniform", {"low": -1.0, "high": 2.0}, 50000)] + + +def digest(v): + v = np.ascontiguousarray(v) + b = v.astype(v.dtype.newbyteorder("<")).tobytes() + return {"dtype": str(v.dtype), "shape": list(v.shape), "sha256": hashlib.sha256(b).hexdigest()} + + +digests = [] +for seed in ["0", "42"]: + for fn, kw, size in DIGESTS: + g = Generator(PCG64(SEEDS[seed])) + digests.append({"api": "Generator", "seed": seed, "fn": fn, "kw": kw, "size": size, + "d": digest(run_gen(g, fn, dict(kw), size))}) + for fn, kw, size in LDIGESTS: + r = RandomState(LSEEDS[seed]) + digests.append({"api": "RandomState", "seed": seed, "fn": fn, "kw": kw, "size": size, + "d": digest(run_legacy(r, fn, dict(kw), size))}) + +print(json.dumps({"numpy": np.__version__, "machine": platform.machine(), "cases": cases, "digests": digests, + "bitgens": bitgens, "philox_key": philox_key, "legacy_mt": legacy_mt, + "seedseqs": seedseqs})) diff --git a/random/testdata/libm.json b/random/testdata/libm.json new file mode 100644 index 0000000..1e73cd4 --- /dev/null +++ b/random/testdata/libm.json @@ -0,0 +1 @@ +{"log": [["0x1.4b9ad0f953a6ep-2", "-0x1.20a5adb6d4b80p+0"], ["0x1.822c83be57f0cp+0", "0x1.a4fa1971b2544p-2"], ["0x1.be636ed5f203ep-998", "-0x1.599a3951519b1p+9"], ["0x1.fffffff8a78e1p-1", "-0x1.d61c7c035f4cap-31"], ["0x1.c7408892f902bp-831", "-0x1.1fb6fe3871df5p+9"], ["0x1.276f358a6a63cp-3", "-0x1.efa8fa1c40d82p+0"], ["0x1.2d8c5236b13fdp+0", "0x1.4f5d4f377322ep-3"], ["0x1.a717a6def2d13p-999", "-0x1.59f9ceb1438f2p+9"], ["0x1.00000002b72b5p+0", "0x1.5b95a7fe2810fp-31"], ["0x1.0edf12e44158bp-812", "-0x1.1963b7b65cb29p+9"], ["0x1.7579da5cb9b30p-1", "-0x1.430ace7951367p-2"], ["0x1.708f72616ef60p+1", "0x1.0ebd2e65dea3dp+0"], ["0x1.5015c5edc44e9p-997", "-0x1.5965d4a798bc3p+9"], ["0x1.fffffff9703c5p-1", "-0x1.a3f0ec02b0de8p-31"], ["0x1.d7a6a6b0a18e8p-686", "-0x1.dae34daded43ap+8"], ["0x1.4b547924fd122p-1", "-0x1.bda7a675be4a0p-2"], ["0x1.3dca6b5b3fde6p+3", "0x1.25d8176c66d82p+1"], ["0x1.19d2e6d409630p-997", "-0x1.597c5e8326bdbp+9"], ["0x1.fffffffc4ca42p-1", "-0x1.d9adf001b63a2p-32"], ["0x1.a894462c33a4fp+793", "0x1.1315f8252b697p+9"], ["0x1.6f61b55763780p-2", "-0x1.066b0577359cap+0"], ["0x1.1af1138f37b80p+3", "0x1.16fa04f01b995p+1"], ["0x1.4863b7214f73cp-997", "-0x1.5968cbafbf2d2p+9"], ["0x1.fffffffa00bbep-1", "-0x1.7fd108023f732p-31"], ["0x1.dbede2d1c9af0p-714", "-0x1.ee497883fd38ap+8"], ["0x1.91cbc6b2f3f20p-2", "-0x1.defd6f66bf193p-1"], ["0x1.feb1623f0f27dp+1", "0x1.623caa4b4ce1ep+0"], ["0x1.1c023841b4c5dp-1000", "-0x1.5a858c994c06dp+9"], ["0x1.00000001274e5p+0", "0x1.274e4fff55ad3p-32"], ["0x1.5e4be0fef7928p-633", "-0x1.b672d507afb23p+8"], ["0x1.dd2c573d7d962p-2", "-0x1.86f590bacc6cep-1"], ["0x1.35a77858f2bc4p+2", "0x1.939a4a7d1c38ap+0"], ["0x1.d72c854b3cee2p-1001", "-0x1.5a9d79bf401a4p+9"], ["0x1.fffffff92a671p-1", "-0x1.b5663c02eb563p-31"], ["0x1.3d43257b6fb7ep+493", "0x1.55efa4ff78dfbp+8"], ["0x1.5193b35176edcp-2", "-0x1.1c13e0a43eed2p+0"], ["0x1.1d7e4ba21dd3ep+1", "0x1.9ab8718f19b64p-1"], ["0x1.1640d27700c75p-997", "-0x1.597e00534ede3p+9"], ["0x1.000000042a5d2p+0", "0x1.0a9747fdd4c29p-30"], ["0x1.a8b83da45e18ap-624", "-0x1.b00480b1a7f75p+8"], ["0x1.92daae5675f68p-3", "-0x1.a0446f1a9d788p+0"], ["0x1.0599104d397a9p+1", "0x1.6df76d72332c0p-1"], ["0x1.abf694f5950d0p-998", "-0x1.599f9e8fb4870p+9"], ["0x1.0000000370499p+0", "0x1.b824c7fd0b419p-31"], ["0x1.e881ad726c86bp-1020", "-0x1.612e95df3a6f2p+9"], ["0x1.042cb1070d710p-3", "-0x1.08191acb0911bp+1"], ["0x1.82f21f7d0f57ep+0", "0x1.a70590d0dff33p-2"], ["0x1.36421801cf31ep-997", "-0x1.597010c441f33p+9"], ["0x1.00000002a2014p+0", "0x1.51009ffe445d6p-31"], ["0x1.808f7256badf9p+229", "0x1.3e467720313d5p+7"], ["0x1.0996ecc5ad59ep-2", "-0x1.597a200b9175fp+0"], ["0x1.0c2b0442bc504p+2", "0x1.6ec7704b84befp+0"], ["0x1.678b10d755731p-1000", "-0x1.5a675ceee5db9p+9"], ["0x1.0000000385a31p+0", "0x1.c2d187fce61abp-31"], ["0x1.7f6fd5a9196f0p+815", "0x1.1aa8d357af733p+9"], ["0x1.1c62e09ee1113p-1", "-0x1.2d0d3276a33fap-1"], ["0x1.f5ef33ce1e846p+2", "0x1.07a08645e4c10p+1"], ["0x1.231091113fa11p-1000", "-0x1.5a826871ab7fap+9"], ["0x1.000000008497bp+0", "0x1.092f5fffbb534p-33"], ["0x1.cb82c3f9d52f9p-632", "-0x1.b57be997fbfd4p+8"], ["0x1.1e7897f922e59p-1", "-0x1.294fd0f907737p-1"], ["0x1.2dd86f781a81ap+3", "0x1.1f4143daed7e7p+1"], ["0x1.20057d0c74295p-997", "-0x1.597995b7e99c3p+9"], ["0x1.fffffff9c419bp-1", "-0x1.8ef994026dcd0p-31"], ["0x1.8fef21f229dd0p-220", "-0x1.3017af0b44476p+7"], ["0x1.4aaeb7b4d19ecp-3", "-0x1.d2ce4b8deece7p+0"], ["0x1.142c88df869b6p+2", "0x1.764f52ea04affp+0"], ["0x1.61958bd1608eap-998", "-0x1.59b80e7c9347ap+9"], ["0x1.fffffffd3c6cep-1", "-0x1.61c99000f476cp-32"], ["0x1.58dba321c5296p-293", "-0x1.95969d48ba3f3p+7"], ["0x1.b063666cbf4b2p-2", "-0x1.b96b706ec1a0bp-1"], ["0x1.23a7051309f85p+3", "0x1.1adb9b35a2173p+1"], ["0x1.18d0531e8ed49p-997", "-0x1.597cd42a37e76p+9"], ["0x1.fffffffbda59fp-1", "-0x1.09698401132bap-31"], ["0x1.f45b5263cfa5ep+75", "0x1.a53fdc8dfea82p+5"], ["0x1.1249d34f93fa8p-2", "-0x1.513a088a135d9p+0"], ["0x1.4ad0c3cdd368cp+0", "0x1.06895a77a76ffp-2"], ["0x1.69571425664dfp-998", "-0x1.59b5477ac167bp+9"], ["0x1.fffffffb81a15p-1", "-0x1.1f97ac0143157p-31"], ["0x1.f2a6e1c0502c6p+961", "0x1.4d63fd6bc6630p+9"], ["0x1.a34980299b562p-1", "-0x1.992056326bd09p-3"], ["0x1.1497f9c485502p+2", "0x1.76b2d747094dbp+0"], ["0x1.53747b423ec00p-998", "-0x1.59bd46c52d97fp+9"], ["0x1.00000002dfd2ep+0", "0x1.6fe96ffdef40ep-31"], ["0x1.3325764a149f5p+964", "0x1.4e3021b729062p+9"], ["0x1.20b31e4ca4c56p-2", "-0x1.441e228f53634p+0"], ["0x1.3608558ed4661p+1", "0x1.c4f075249bcdfp-1"], ["0x1.91f05602e0851p-999", "-0x1.5a005f62dbaaep+9"], ["0x1.ffffffff4dac1p-1", "-0x1.64a7e0003e1c7p-34"], ["0x1.fcb7f28541424p-473", "-0x1.472c0191b5fdcp+8"], ["0x1.5555f1936f230p-5", "-0x1.96ca3d31e68b5p+1"], ["0x1.ccae1cff18090p-3", "-0x1.7dee24f7dc877p+0"], ["0x1.a147e5dc9a008p-999", "-0x1.59fb93e8b4e9ap+9"], ["0x1.fffffffb68e25p-1", "-0x1.25c76c0151222p-31"], ["0x1.3dae195e8c93ep+938", "0x1.4531a773d206cp+9"], ["0x1.023cec2461f7dp-1", "-0x1.5e6f526a6f43ep-1"], ["0x1.0b2e16708426bp+3", "0x1.0fa3d125fff4cp+1"], ["0x1.13e8fa986c24ap-997", "-0x1.597f156de556fp+9"], ["0x1.00000002cdc80p+0", "0x1.66e3fffe08dd8p-31"], ["0x1.3c9eb95850e21p+611", "0x1.a7b9b73cfd74cp+8"], ["0x1.518fb3e1a9d29p-1", "-0x1.aa93433dadbf2p-2"], ["0x1.522d85dd440f0p-1", "-0x1.a8b4f2afae113p-2"], ["0x1.f94397006de93p-998", "-0x1.598a5da2b9a6ap+9"], ["0x1.fffffffbbe22cp-1", "-0x1.1077500121fdcp-31"], ["0x1.e1b59130f27b2p+709", "0x1.ec12d1d4fcacfp+8"], ["0x1.f0ef5df1750b0p-5", "-0x1.66b6d08b74667p+1"], ["0x1.580775bb99f7fp+1", "0x1.fa36600c78db7p-1"], ["0x1.ccd5e39d378d5p-998", "-0x1.59962599d29cep+9"], ["0x1.00000001a69eap+0", "0x1.a69e9ffea3285p-32"], ["0x1.bae8bacfb2d5ep-21", "-0x1.c040bf36cb607p+3"], ["0x1.2248efbf5b410p-3", "-0x1.f4295c27969cbp+0"], ["0x1.4f66ed24f1d90p+2", "0x1.a80ce75e59a0cp+0"], ["0x1.46ada08806831p-997", "-0x1.596976e43822dp+9"], ["0x1.fffffff9b02e1p-1", "-0x1.93f47c027d6bap-31"], ["0x1.a80ebd1f9bdfep+270", "0x1.774f1185033d5p+7"], ["0x1.da50f38c8705dp-1", "-0x1.39240ec092869p-4"], ["0x1.c855c7f0f00f0p+2", "0x1.f6df4460420e2p+0"], ["0x1.3521b24388b33p-997", "-0x1.597087f6dcc6ep+9"], ["0x1.fffffffc63acfp-1", "-0x1.ce298801a12cfp-32"], ["0x1.12d8c5f49f0fcp-890", "-0x1.346a3bf21b2d8p+9"], ["0x1.4304c6599a485p-1", "-0x1.d7ab57c5e00bep-2"], ["0x1.244baf5307702p+3", "0x1.1b23cb6f8cb95p+1"], ["0x1.428cb061e556cp-997", "-0x1.596b17a9c8abcp+9"], ["0x1.000000006c410p+0", "0x1.b103ffffa4723p-34"], ["0x1.05805b835e8a5p-858", "-0x1.2959797235186p+9"], ["0x1.56b267a4a915cp-3", "-0x1.c9ab776ab6ca5p+0"], ["0x1.9dd781980df99p+0", "0x1.ebd56fa9da676p-2"], ["0x1.1d1a86356d78ep-999", "-0x1.5a2c5576deaf1p+9"], ["0x1.000000037cb72p+0", "0x1.be5b8ffcf5bcdp-31"], ["0x1.310f77f405bc8p+104", "0x1.210cf14a85c63p+6"], ["0x1.fc643dfe19f50p-5", "-0x1.63cbf1c1106bap+1"], ["0x1.63388e7be28a2p+1", "0x1.054d3a5476e7ep+0"], ["0x1.4bcd766a49bd1p-997", "-0x1.596778dfc4f9fp+9"], ["0x1.fffffff992936p-1", "-0x1.9b5b280294fddp-31"], ["0x1.daae580de8b3ep+60", "0x1.51a67ec15db65p+5"], 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"0x1.151b66401edeep+812"], ["0x1.403ef0fb2555ap-1", "0x1.49522d49439c2p+6", "0x1.33b3f91de9114p-56"], ["0x1.c46b3e4cf498fp-1", "0x1.e3a7208a7f254p+3", "0x1.3bb03c5a2412fp-3"], ["0x1.5d2a35f5014c2p+2", "-0x1.b5e97d0957494p+4", "0x1.010c5659c99c3p-67"], ["0x1.0000000000000p+1", "0x1.b9357688069cfp+7", "0x1.8536cb689e872p+220"], ["0x1.4d2ce35b3fa06p-1", "0x1.925547df9e144p+3", "0x1.276688e76c2cap-8"], ["0x1.81cac25d3cd66p-1", "0x1.bc640e0cb2715p+3", "0x1.41b2dd7b5ffe4p-6"], ["0x1.a71ab7acbce36p+0", "0x1.880aeb7afa096p+4", "0x1.b1e8284da00dcp+17"], ["0x1.0000000000000p+1", "0x1.f31fa420a6058p+9", "0x1.2fd8907ec457fp+998"], ["0x1.50533f6e067f5p-1", "0x1.88ee9a0d57121p+6", "0x1.5beb134733e26p-60"], ["0x1.b6f7204baa778p-4", "0x1.99e92dc322dc3p+3", "0x1.a79eebaf346e1p-42"], ["0x1.9bd3f8afab181p+2", "-0x1.210e328c809f6p+4", "0x1.643a3796ffed3p-49"], ["0x1.0000000000000p+1", "0x1.39936cabec740p+9", "0x1.1c65336cba00dp+627"], ["0x1.19301ad900b8cp-1", "0x1.efa6ee1e4d35dp+5", "0x1.595456ad21b13p-54"], ["0x1.ad0e19b0fe2d0p-5", "0x1.0ad5493617cd0p+1", "0x1.181fc366994dep-9"], ["0x1.09354a17e5e98p+3", "0x1.f28c641d4de58p+3", "0x1.72766a5889829p+47"], ["0x1.0000000000000p+1", "0x1.8a0ea851731a6p+7", "0x1.0521715ddaa31p+197"], ["0x1.451f13122e020p-2", "0x1.12e59d7ab7b00p+3", "0x1.b7f7b885e5afdp-15"], ["0x1.313cb8536d5edp-1", "0x1.67b0b494327abp+1", "0x1.debbfda7e01bep-3"], ["0x1.ad82f2ebb3e24p+1", "-0x1.f7b4c577774d4p+2", "0x1.3143a0f2c29bep-14"], ["0x1.0000000000000p+1", "0x1.70e7a007271ddp+9", "0x1.c0b0512e339b0p+737"], ["0x1.90868095441a3p-1", "0x1.171790eaa74edp+6", "0x1.3775d83925edep-25"], ["0x1.a9c5436e9fd94p-2", "0x1.bb7edb05fcadep+1", "0x1.879f372ee3d1cp-5"], ["0x1.e963b5a8c5ed8p+2", "0x1.8193b8e399f4cp+3", "0x1.492c9f9ad78b1p+35"], ["0x1.0000000000000p+1", "0x1.19d0bff26f794p+9", "0x1.8c69899f61fd0p+563"], ["0x1.7dd44ae69d717p-1", "0x1.7bcc05d102561p+6", "0x1.c2ad60a8286b7p-41"], ["0x1.bbe2116c28338p-3", "0x1.54fe1fdeed970p+3", "0x1.68561697a1a65p-24"], ["0x1.735ee37e9d609p+0", "-0x1.3c00cc3ad5ed2p+4", "0x1.51bc863c333f8p-11"], ["0x1.0000000000000p+1", "0x1.22d2308aae7bep+8", "0x1.c446238ec9185p+290"], ["0x1.855a0336c2c40p-5", "0x1.cb479bbce9ed3p+5", "0x1.99e9a64afdebbp-253"], ["0x1.ee8164ccc151ap-1", "0x1.08b53a837e266p+3", "0x1.8008cf61353f3p-1"], ["0x1.3c2d0eb37dad6p+3", "0x1.f83c95cad3498p+3", "0x1.0d086b4f36960p+52"], ["0x1.0000000000000p+1", "0x1.15f48827a7ed1p+8", "0x1.f058980a5792fp+277"], ["0x1.502ee6fbad73ap-2", "0x1.a6a94bd040e4ap+5", "0x1.13054eeee0557p-85"], ["0x1.9e86918a4b23cp-2", "0x1.388258c7923c7p+2", "0x1.8bfb91ab81a5ep-7"], ["0x1.f5fcd88ff49f0p-4", "-0x1.1233c9786c278p+2", "0x1.f74f587ae05c1p+12"], ["0x1.0000000000000p+1", "0x1.2bea162e71cb7p+9", "0x1.c6b5adee817f4p+599"], ["0x1.bdb4fc9a34920p-5", "0x1.1112f5be7af32p+1", "0x1.072253feaf24dp-9"]]} diff --git a/random/testdata/libm.py b/random/testdata/libm.py new file mode 100644 index 0000000..bca3d66 --- /dev/null +++ b/random/testdata/libm.py @@ -0,0 +1,28 @@ +# Generates random/testdata/libm.json, correctly rounded log, log1p, exp, expm1 and pow +# (mpmath, 300 bits) for libm_test.go: +# uv run --with mpmath python random/testdata/libm.py > random/testdata/libm.json +import math, random, json, mpmath, struct +mpmath.mp.prec = 300 +random.seed(7) +def f2s(x): return float.hex(x) +cases = {"log": [], "log1p": [], "exp": [], "expm1": [], "pow": []} +def rnd_double(): + return struct.unpack(' 0 and math.isfinite(x): + cases["log"].append([f2s(x), f2s(float(mpmath.log(x)))]) + for u in [-random.random(), random.uniform(-1e-3, 1e-3), random.uniform(-1e-12, 1e-12), random.uniform(0, 100), -0.5 + random.uniform(-0.1, 0.1)]: + cases["log1p"].append([f2s(u), f2s(float(mpmath.log1p(u)))]) + for x in [random.uniform(-20, 5), random.uniform(-745, 709.7), random.uniform(-1e-6, 1e-6)]: + cases["exp"].append([f2s(x), f2s(float(mpmath.exp(x)))]) + for x in [random.uniform(-3, 3), random.uniform(-1e-12, 1e-12), random.uniform(-1e-35, 1e-35), random.uniform(-50, 60)]: + cases["expm1"].append([f2s(x), f2s(float(mpmath.expm1(x)))]) + for x, y in [(random.random(), random.uniform(0.5, 20)), (random.uniform(0, 10), random.uniform(-30, 30)), + (2.0, random.uniform(0, 1000)), (random.random(), random.uniform(1, 100))]: + v = mpmath.mpf(x) ** mpmath.mpf(y) + if v < mpmath.mpf(2)**-1022 or v > mpmath.mpf(2)**1023: + continue + cases["pow"].append([f2s(x), f2s(y), f2s(float(v))]) +print(json.dumps(cases)) diff --git a/random/testdata/libm_tables.py b/random/testdata/libm_tables.py new file mode 100644 index 0000000..9baef18 --- /dev/null +++ b/random/testdata/libm_tables.py @@ -0,0 +1,33 @@ +# Generates random/zz_libm_tables.go: the double-double constants of the +# correctly rounded log/exp in libm.go, from mpmath at 300 bits: +# uv run --with mpmath python random/testdata/libm_tables.py > random/zz_libm_tables.go +import mpmath +mpmath.mp.prec = 300 + +def dd(v): + hi = float(v) + lo = float(v - mpmath.mpf(hi)) + return hi, lo + +def lit(x): + return float.hex(x) + +out = ["// Code generated by testdata/libm_tables.py (mpmath, 300 bits). DO NOT EDIT.", "", "package random", ""] +h, l = dd(mpmath.log(2)) +out.append(f"const (\n\tln2Hi = {h!r}\n\tln2Lo = {l!r}\n)\n") +x = mpmath.log(2) / 64 +a = float(x); b = float(x - a); c = float(x - a - b) +out.append(f"// ln2/64 as three doubles.\nconst (\n\tln2by64A = {a!r}\n\tln2by64B = {b!r}\n\tln2by64C = {c!r}\n)\n") +out.append("// logTab[j-45] is log(j/64) for j in [45, 91], as hi, lo.") +out.append("var logTab = [47][2]float64{") +for j in range(45, 92): + h, l = dd(mpmath.log(mpmath.mpf(j) / 64)) + out.append(f"\t{{{h!r}, {l!r}}},") +out.append("}\n") +out.append("// exp2Tab[j] is 2**(j/64) for j in [0, 64), as hi, lo.") +out.append("var exp2Tab = [64][2]float64{") +for j in range(64): + h, l = dd(mpmath.power(2, mpmath.mpf(j) / 64)) + out.append(f"\t{{{h!r}, {l!r}}},") +out.append("}") +print("\n".join(out)) diff --git a/random/testdata/quant.py b/random/testdata/quant.py new file mode 100644 index 0000000..ed20f2b --- /dev/null +++ b/random/testdata/quant.py @@ -0,0 +1,37 @@ +# Writes NumPy's output for 10^6 draws of the samplers that call the C library, +# as little-endian float64 files, for TestQuantifyLibm (docs/random.md quotes it): +# mkdir /tmp/q && uv run --python cpython-3.12-macos-x86_64 --with numpy==2.5.3 \ +# python random/testdata/quant.py /tmp/q +# RANDOM_QUANT_DIR=/tmp/q go test -run TestQuantifyLibm -v ./random +import numpy as np, sys, json +from numpy.random import Generator, PCG64, RandomState +N=1_000_000 +d=sys.argv[1] +jobs=[] +def save(name, v): + np.ascontiguousarray(v, dtype='/numpy/random/src/distributions/ziggurat_constants.h random/zz_ziggurat.go +# The C literals are copied verbatim (suffixes dropped): Go rounds an exact +# decimal constant to float32/float64 the way a C compiler rounds the literal. +import re, sys + +src = open(sys.argv[1]).read() +gotype = {'uint64_t': 'uint64', 'double': 'float64', 'uint32_t': 'uint32', 'float': 'float32'} +scalars = {'ziggurat_nor_r': 'zigNorR', 'ziggurat_nor_inv_r': 'zigNorInvR', 'ziggurat_exp_r': 'zigExpR', + 'ziggurat_nor_r_f': 'zigNorRF', 'ziggurat_nor_inv_r_f': 'zigNorInvRF', 'ziggurat_exp_r_f': 'zigExpRF'} + +def goname(c): + a, b = c.split('_') + return a + b.capitalize() + +out = ["// Code generated from numpy/random/src/distributions/ziggurat_constants.h\n" + "// (NumPy, BSD-3-Clause; see the NumPy license) by testdata/zig2go.py. DO NOT EDIT.\n\npackage random\n"] +for m in re.finditer(r'static const (\w+) (\w+)\[\] = \{(.*?)\};', src, re.S): + t, name, body = m.groups() + vals = [v.strip() for v in body.replace('\n', ' ').split(',') if v.strip()] + vals = [re.sub(r'(ULL|UL|U)$', '', v) if t.startswith('uint') else re.sub(r'[fF]$', '', v) for v in vals] + out.append(f"var {goname(name)} = [{len(vals)}]{gotype[t]}{{") + for i in range(0, len(vals), 4): + out.append("\t" + ", ".join(vals[i:i + 4]) + ",") + out.append("}\n") +for m in re.finditer(r'static const (double|float) (\w+) =\s*([0-9.eE+-]+)f?;', src): + t, name, v = m.groups() + out.append(f"const {scalars[name]} {gotype[t]} = {v}") +open(sys.argv[2], 'w').write("\n".join(out) + "\n") diff --git a/random/zz_libm_tables.go b/random/zz_libm_tables.go new file mode 100644 index 0000000..6213bc8 --- /dev/null +++ b/random/zz_libm_tables.go @@ -0,0 +1,134 @@ +// Code generated by testdata/libm_tables.py (mpmath, 300 bits). DO NOT EDIT. + +package random + +const ( + ln2Hi = 0.6931471805599453 + ln2Lo = 2.3190468138462996e-17 +) + +// ln2/64 as three doubles. +const ( + ln2by64A = 0.010830424696249145 + ln2by64B = 3.623510646634843e-19 + ln2by64C = 8.918294435025331e-36 +) + +// logTab[j-45] is log(j/64) for j in [45, 91], as hi, lo. +var logTab = [47][2]float64{ + {-0.3522205935893521, -5.7233316949182485e-18}, + {-0.33024168687057687, 1.0828321637483858e-17}, + {-0.3087354816496133, 1.6199186085148102e-17}, + {-0.2876820724517809, -2.607160616442564e-17}, + {-0.26706278524904525, 7.32891532732017e-18}, + {-0.24686007793152578, -1.361743371748368e-17}, + {-0.22705745063534608, -9.551415762738488e-18}, + {-0.2076393647782445, -1.2053243216686129e-17}, + {-0.18859116980755003, 7.432164219196925e-18}, + {-0.16989903679539747, 4.868008764439071e-19}, + {-0.15154989812720093, -5.1669593684615594e-18}, + {-0.13353139262452263, 3.664457663660085e-18}, + {-0.1158318155251217, -4.338484369808096e-18}, + {-0.09844007281325252, 4.439009633675136e-18}, + {-0.0813456394539524, -5.07707635593117e-18}, + {-0.06453852113757118, 6.470486661692933e-18}, + {-0.048009219186360606, -1.4390903347292205e-18}, + {-0.0317486983145803, -3.0382263084680858e-18}, + {-0.015748356968139168, -1.0021578630528974e-18}, + {0.0, 0.0}, + {0.015504186535965254, -3.278321022892429e-19}, + {0.030771658666753687, 1.0431732029005968e-18}, + {0.0458095360312942, 1.902959866474257e-18}, + {0.06062462181643484, 2.6424025938726934e-18}, + {0.07522342123758753, -5.930604196293241e-18}, + {0.08961215868968714, -5.4268129336647135e-18}, + {0.10379679368164356, 5.47772415726659e-18}, + {0.11778303565638346, -1.1971685747593677e-18}, + {0.13157635778871926, 1.1123000879729588e-17}, + {0.1451820098444979, 8.242418783022475e-18}, + {0.15860503017663857, 1.1257003872182592e-17}, + {0.17185025692665923, -6.0224538210113705e-18}, + {0.184922338494012, 3.0236614153574064e-18}, + {0.19782574332991987, 1.2821194372980142e-17}, + {0.21056476910734964, -4.249405314729895e-18}, + {0.22314355131420976, -9.091270597324799e-18}, + {0.2355660713127669, -2.3943371495187355e-18}, + {0.24783616390458127, -1.2432209578702523e-17}, + {0.25995752443692605, 2.069806938978935e-17}, + {0.27193371548364176, 7.83319637697442e-19}, + {0.2837681731306446, -2.032665581126656e-17}, + {0.2954642128938359, -2.16461086040599e-17}, + {0.3070250352949119, -1.2319916200101964e-17}, + {0.3184537311185346, 2.7114779367326236e-17}, + {0.329753286372468, 2.122020616196946e-18}, + {0.3409265869705932, 1.7467136443544747e-17}, + {0.3519764231571782, -1.2953893030191963e-17}, +} + +// exp2Tab[j] is 2**(j/64) for j in [0, 64), as hi, lo. +var exp2Tab = [64][2]float64{ + {1.0, 0.0}, + {1.0108892860517005, -1.5234778603368577e-17}, + {1.0218971486541166, 5.109225028973444e-17}, + {1.0330248790212284, 7.600838874027088e-18}, + {1.0442737824274138, 8.551889705537965e-17}, + {1.0556451783605572, 1.759325738772092e-18}, + {1.0671404006768237, -7.899853966841582e-17}, + {1.0787607977571199, -6.656660436056593e-17}, + {1.0905077326652577, -3.046782079812471e-17}, + {1.102382583307841, 5.2660368715706944e-17}, + {1.1143867425958924, 1.0410278456845571e-16}, + {1.1265216186082418, 5.165856758795457e-17}, + {1.1387886347566916, 8.912812676025408e-17}, + {1.1511892299529827, 3.250710218863827e-17}, + {1.1637248587775775, 3.8292048369240935e-17}, + {1.1763969916502812, 5.554203254218079e-17}, + {1.189207115002721, 3.982015231465646e-17}, + {1.202156731452703, 6.644981499252301e-17}, + {1.215247359980469, -7.712630692681488e-17}, + {1.22848053610687, -1.89878163130253e-17}, + {1.241857812073484, 4.658027591836937e-17}, + {1.255380757024691, -6.7113898212968784e-18}, + {1.2690509571917332, 2.667932131342186e-18}, + {1.2828700160787783, 1.713594918243561e-17}, + {1.2968395546510096, 2.5382502794888315e-17}, + {1.3109612115247644, -7.181536135519454e-17}, + {1.3252366431597413, -2.8587312100388614e-17}, + {1.339667524053303, 8.927282594831732e-17}, + {1.3542555469368927, 7.70094837980299e-17}, + {1.3690024229745905, 9.593797919118849e-17}, + {1.383909881963832, -6.770511658794786e-17}, + {1.3989796725383112, -9.614213209051323e-17}, + {1.4142135623730951, -9.667293313452913e-17}, + {1.42961333839197, -1.2031642489053655e-17}, + {1.4451808069770467, -3.0237581349939873e-17}, + {1.460917794180647, -5.600377186075216e-17}, + {1.4768261459394993, -3.483994556892796e-17}, + {1.4929077282912648, 1.4192920154284036e-17}, + {1.5091644275934228, -1.016455327754295e-16}, + {1.5255981507445384, -1.1024941712342561e-16}, + {1.5422108254079407, 7.949834809697621e-17}, + {1.559004400237837, 3.7812070533575275e-17}, + {1.5759808451078865, -1.0136916471278304e-17}, + {1.593142151342267, -1.0094406542311964e-16}, + {1.6104903319492543, 2.4707192569797888e-17}, + {1.6280274218573478, -6.712955084707084e-17}, + {1.645755478153965, -1.0125679913674773e-16}, + {1.6636765803267364, 5.8909926967131e-17}, + {1.681792830507429, 8.199010020581497e-17}, + {1.7001063537185235, -8.0237193703977e-18}, + {1.718619298122478, -1.851380418263111e-17}, + {1.7373338352737062, 3.164389299292957e-17}, + {1.7562521603732995, 2.960140695448873e-17}, + {1.7753764925265212, 6.429731796556572e-17}, + {1.7947090750031072, 1.8227458427912087e-17}, + {1.8142521755003989, -9.969531538920349e-17}, + {1.8340080864093424, 3.283107224245627e-17}, + {1.8539791250833855, 9.761887490727594e-17}, + {1.8741676341103, -6.122763413004143e-17}, + {1.8945759815869656, 3.4034035352165297e-17}, + {1.9152065613971474, -1.0619946056195963e-16}, + {1.9360617934922943, 1.0332385960676326e-16}, + {1.9571441241754002, 8.960767791036668e-17}, + {1.978456026387951, 4.0388753109278167e-17}, +} diff --git a/random/zz_mtjump.go b/random/zz_mtjump.go new file mode 100644 index 0000000..ee2cabf --- /dev/null +++ b/random/zz_mtjump.go @@ -0,0 +1,111 @@ +// Code generated from numpy/random/src/mt19937/mt19937-jump.h (NumPy, BSD-3-Clause). DO NOT EDIT. + +package random + +// mtJumpPoly is the 2**128-step jump polynomial of MT19937, 32 coefficients per word. +var mtJumpPoly = [624]uint32{ + 1927166307, 3044056772, 2284297142, 2820929765, 651705945, 69149273, + 3892165397, 2337412983, 1219880790, 3207074517, 3836784057, 189286826, + 1049791363, 3916249550, 2942382547, 166392552, 861176918, 3246476411, + 2302311555, 4273801148, 29196903, 1363664063, 3802562022, 2600400244, + 3090369801, 4040416970, 1432485208, 3632558139, 4015816763, 3013316418, + 551532385, 3592224467, 3479125595, 1195467127, 2391032553, 2393493419, + 1482493632, 1625159565, 748389672, 4042774030, 2998615036, 3393119101, + 2177492569, 2265897321, 2507383006, 3461498961, 2003319700, 1942857197, + 1455226044, 4097545580, 529653268, 3204756480, 2486748289, 495294513, + 3396001954, 2643963605, 2655404568, 3881604377, 624710790, 3443737948, + 1941294296, 2139259604, 3368734020, 422436761, 3602810182, 1384691081, + 3035786407, 2551797119, 537227499, 65486120, 642436100, 2023822537, + 2515598203, 1122953367, 2882306242, 1743213032, 321965189, 336496623, + 2436602518, 3556266590, 1055117829, 463541647, 743234441, 527083645, + 2606668346, 2274046499, 2761475053, 2760669048, 2538258534, 487125077, + 3365962306, 3604906217, 2714700608, 680709708, 2217161159, 1614899374, + 3710119533, 3201300658, 3752620679, 2755041105, 3129723037, 1247297753, + 2812642690, 4114340845, 3485092247, 2752814364, 3586551747, 4073138437, + 3462966585, 2924318358, 4061374901, 3314086806, 2640385723, 744590670, + 3007586513, 3959120371, 997207767, 3420235506, 2092400998, 3190305685, + 60965738, 549507222, 3784354415, 3209279509, 1238863299, 2605037827, + 178570440, 1743491299, 4079686640, 2136795825, 3435430548, 1679732443, + 1835708342, 2159367000, 1924487218, 4059723674, 996192116, 2308091645, + 1336281586, 674600050, 1642572529, 1383973289, 2202960007, 3165481279, + 3385474038, 2501318550, 2671842890, 3084085109, 3475033915, 1551329147, + 4101397249, 1205851807, 3641536021, 3607635071, 1609126163, 2910426664, + 3324508658, 4244311266, 254034382, 1258304384, 1914048768, 1358592011, + 527610138, 3072108727, 4289413885, 1417001678, 2445445945, 896462712, + 339855811, 3699378285, 2529457297, 3049459401, 2723472429, 2838633181, + 2520397330, 3272339035, 1667003847, 3742634787, 942706520, 2301027215, + 1907791250, 2306299096, 1021173342, 1539334516, 2907834628, 3199959207, + 1556251860, 3642580275, 2355865416, 285806145, 867932457, 1177354172, + 3291107470, 4022765061, 1613380116, 588147929, 650574324, 1236855601, + 1371354511, 2085218212, 1203081931, 420526905, 1022192219, 2903287064, + 2470845899, 3649873273, 2502333582, 3972385637, 4246356763, 199084157, + 1567178788, 2107121836, 4293612856, 1902910177, 332397359, 83422598, + 3614961721, 456321943, 2277615967, 2302518510, 3258315116, 2521897172, + 3900282042, 4186973154, 3146532165, 2299685029, 3889120948, 1293301857, + 187455105, 3395849230, 913321567, 3093513909, 1440944571, 1923481911, + 338680924, 1204882963, 2739724491, 2886241328, 2408907774, 1299817192, + 2474012871, 45400213, 553186784, 134558656, 2180943666, 2870807589, + 76511085, 3053566760, 2516601415, 4172865902, 1751297915, 1251975234, + 2964780642, 1412975316, 2739978478, 2171013719, 637935041, 975972384, + 3044407449, 3111425639, 1938684970, 2860857400, 13419586, 2772079268, + 3484375614, 3184054178, 159924837, 1386213021, 2765617231, 2523689118, + 1283505218, 3510789588, 4125878259, 2990287597, 2152014833, 3084155970, + 2815101609, 1932985704, 114887365, 1712687646, 2550515629, 3299051916, + 2022747614, 2143630992, 2244188960, 3309469192, 3234358520, 800720365, + 3278176634, 554357439, 2415629802, 1620877315, 2389462898, 2229691332, + 1007748450, 1966873768, 2264971043, 1214524156, 346854700, 3471905342, + 3984889660, 4034246840, 216712649, 4027196762, 3754772604, 2121785562, + 2347070732, 7457687, 1443375102, 683948143, 2940226032, 3211475670, + 2836507357, 774899409, 1588968308, 780438009, 3278878781, 2217181540, + 2184194887, 1642129086, 69346830, 297114710, 3841068188, 2631265450, + 4167492314, 2613519651, 1388582503, 2171556668, 1201873758, 2698772382, + 207791958, 3936134563, 3725025702, 3306317801, 1055730422, 4069230694, + 1767821343, 4252407395, 2422583118, 3158834399, 3754582617, 1112422556, + 376187931, 3137549150, 712221089, 3300799453, 3868250200, 1165257666, + 2494837767, 131304831, 1619349427, 1958236644, 3678218946, 3651007751, + 2261987899, 1567368524, 2193599522, 3034394674, 2994602555, 3072727647, + 889094521, 1089692095, 1822324824, 3876999182, 1703361286, 902229515, + 4213728487, 3838170364, 672727494, 2240733828, 3858539469, 1149254245, + 4166055926, 4193525313, 1709921593, 2278290377, 3190784116, 2919588882, + 1012709717, 3640562031, 2931984863, 3515665246, 250577343, 1147230194, + 1183856202, 3734511989, 3243867808, 3499383067, 2985115159, 2036821626, + 3298159553, 2726542838, 1686910320, 1778823772, 965412224, 233509772, + 3843098861, 1312622954, 500855830, 2950562091, 1915683607, 3405781138, + 596073719, 2195150546, 3381728478, 546426436, 3527890868, 2324975353, + 2241074266, 3992514859, 2576108287, 4077653225, 2632319392, 3127212632, + 917000669, 2498161805, 3980835128, 2259526768, 1083920509, 1187452089, + 97018536, 3056075838, 2059706760, 2373335692, 182196406, 2136713111, + 1762080153, 1572125803, 1145919955, 1023966754, 3921694345, 1632005969, + 1418372326, 354407429, 2438288265, 1620072033, 1586320921, 1044153697, + 969324572, 613487980, 4230993062, 397726764, 2194259193, 735511759, + 2066049260, 88093248, 1562536153, 2114157419, 3630951546, 589238503, + 3120654384, 2521793793, 2746692127, 2557723425, 889897693, 2778878177, + 643269509, 3342389831, 19218890, 3442706236, 3314581273, 3503147052, + 1546343434, 1448529060, 529038801, 2748942264, 2213019208, 111314040, + 2488697563, 1180642808, 2605272289, 4207476668, 1502558669, 2972370981, + 4204339995, 1046225278, 992840610, 3847290298, 2387673094, 2221565747, + 1045901716, 3997739302, 1556952765, 1103336648, 279418400, 2711316466, + 2336215718, 2317900806, 974624729, 909575434, 1675610631, 1922393214, + 2054896570, 3197007361, 3932554569, 1008619802, 3349254938, 113511461, + 932630384, 2098759268, 3436837432, 3119972401, 1612590197, 2281609013, + 4174211248, 4016332246, 2097525539, 1398632760, 1543697535, 2419227174, + 1676465074, 2882923045, 23216933, 808195649, 3690720147, 484419260, + 2254772642, 2975434733, 288528113, 204598404, 589968818, 3021152400, + 2463155141, 1397846755, 157285579, 4230258857, 2469135246, 625357422, + 3435224647, 465239124, 1022535736, 2823317040, 274194469, 2214966446, + 3661001613, 518802547, 2293436304, 1335881988, 2247010176, 1856732584, + 1088028094, 1877563709, 1015352636, 1700817932, 2960695857, 1882229300, + 1666906557, 1838841022, 3983797810, 1667630361, 385998221, 241341791, + 403550441, 2629200403, 3552759102, 2029750442, 2247999048, 2726665298, + 2507798776, 2419064129, 1266444923, 526255242, 2384866697, 1886200981, + 3954956408, 2171436866, 2295200753, 1047315850, 1967809707, 2860382973, + 3918334466, 3057439479, 952682588, 1925559679, 3112119050, 3833190964, + 1430139895, 2089165610, 3009202424, 3989186157, 3395807230, 347600520, + 120428923, 3017004655, 1384933954, 303039929, 234010146, 2278760249, + 315514836, 3987659575, 1239335668, 2387869477, 3885908826, 1983922602, + 698609264, 3009002846, 1520611399, 809159940, 3089771783, 374838722, + 2789914419, 2500831937, 3751970335, 4279852547, 2362894437, 1588814060, + 1671213155, 434218829, 2126587176, 2002526422, 2756464095, 141700479, + 2965974322, 2211530172, 992085992, 1943691492, 2705131817, 2519208889, + 1938768395, 3949294294, 354046666, 2158272751, 602858583, 0, +} diff --git a/random/zz_ziggurat.go b/random/zz_ziggurat.go new file mode 100644 index 0000000..1c5ae73 --- /dev/null +++ b/random/zz_ziggurat.go @@ -0,0 +1,815 @@ +// Code generated from numpy/random/src/distributions/ziggurat_constants.h +// (NumPy, BSD-3-Clause; see the NumPy license) by testdata/zig2go.py. DO NOT EDIT. + +package random + +var kiDouble = [256]uint64{ + 0x000EF33D8025EF6A, 0x0000000000000000, 0x000C08BE98FBC6A8, 0x000DA354FABD8142, + 0x000E51F67EC1EEEA, 0x000EB255E9D3F77E, 0x000EEF4B817ECAB9, 0x000F19470AFA44AA, + 0x000F37ED61FFCB18, 0x000F4F469561255C, 0x000F61A5E41BA396, 0x000F707A755396A4, + 0x000F7CB2EC28449A, 0x000F86F10C6357D3, 0x000F8FA6578325DE, 0x000F9724C74DD0DA, + 0x000F9DA907DBF509, 0x000FA360F581FA74, 0x000FA86FDE5B4BF8, 0x000FACF160D354DC, + 0x000FB0FB6718B90F, 0x000FB49F8D5374C6, 0x000FB7EC2366FE77, 0x000FBAECE9A1E50E, + 0x000FBDAB9D040BED, 0x000FC03060FF6C57, 0x000FC2821037A248, 0x000FC4A67AE25BD1, + 0x000FC6A2977AEE31, 0x000FC87AA92896A4, 0x000FCA325E4BDE85, 0x000FCBCCE902231A, + 0x000FCD4D12F839C4, 0x000FCEB54D8FEC99, 0x000FD007BF1DC930, 0x000FD1464DD6C4E6, + 0x000FD272A8E2F450, 0x000FD38E4FF0C91E, 0x000FD49A9990B478, 0x000FD598B8920F53, + 0x000FD689C08E99EC, 0x000FD76EA9C8E832, 0x000FD848547B08E8, 0x000FD9178BAD2C8C, + 0x000FD9DD07A7ADD2, 0x000FDA9970105E8C, 0x000FDB4D5DC02E20, 0x000FDBF95C5BFCD0, + 0x000FDC9DEBB99A7D, 0x000FDD3B8118729D, 0x000FDDD288342F90, 0x000FDE6364369F64, + 0x000FDEEE708D514E, 0x000FDF7401A6B42E, 0x000FDFF46599ED40, 0x000FE06FE4BC24F2, + 0x000FE0E6C225A258, 0x000FE1593C28B84C, 0x000FE1C78CBC3F99, 0x000FE231E9DB1CAA, + 0x000FE29885DA1B91, 0x000FE2FB8FB54186, 0x000FE35B33558D4A, 0x000FE3B799D0002A, + 0x000FE410E99EAD7F, 0x000FE46746D47734, 0x000FE4BAD34C095C, 0x000FE50BAED29524, + 0x000FE559F74EBC78, 0x000FE5A5C8E41212, 0x000FE5EF3E138689, 0x000FE6366FD91078, + 0x000FE67B75C6D578, 0x000FE6BE661E11AA, 0x000FE6FF55E5F4F2, 0x000FE73E5900A702, + 0x000FE77B823E9E39, 0x000FE7B6E37070A2, 0x000FE7F08D774243, 0x000FE8289053F08C, + 0x000FE85EFB35173A, 0x000FE893DC840864, 0x000FE8C741F0CEBC, 0x000FE8F9387D4EF6, + 0x000FE929CC879B1D, 0x000FE95909D388EA, 0x000FE986FB939AA2, 0x000FE9B3AC714866, + 0x000FE9DF2694B6D5, 0x000FEA0973ABE67C, 0x000FEA329CF166A4, 0x000FEA5AAB32952C, + 0x000FEA81A6D5741A, 0x000FEAA797DE1CF0, 0x000FEACC85F3D920, 0x000FEAF07865E63C, + 0x000FEB13762FEC13, 0x000FEB3585FE2A4A, 0x000FEB56AE3162B4, 0x000FEB76F4E284FA, + 0x000FEB965FE62014, 0x000FEBB4F4CF9D7C, 0x000FEBD2B8F449D0, 0x000FEBEFB16E2E3E, + 0x000FEC0BE31EBDE8, 0x000FEC2752B15A15, 0x000FEC42049DAFD3, 0x000FEC5BFD29F196, + 0x000FEC75406CEEF4, 0x000FEC8DD2500CB4, 0x000FECA5B6911F12, 0x000FECBCF0C427FE, + 0x000FECD38454FB15, 0x000FECE97488C8B3, 0x000FECFEC47F91B7, 0x000FED1377358528, + 0x000FED278F844903, 0x000FED3B10242F4C, 0x000FED4DFBAD586E, 0x000FED605498C3DD, + 0x000FED721D414FE8, 0x000FED8357E4A982, 0x000FED9406A42CC8, 0x000FEDA42B85B704, + 0x000FEDB3C8746AB4, 0x000FEDC2DF416652, 0x000FEDD171A46E52, 0x000FEDDF813C8AD3, + 0x000FEDED0F909980, 0x000FEDFA1E0FD414, 0x000FEE06AE124BC4, 0x000FEE12C0D95A06, + 0x000FEE1E579006E0, 0x000FEE29734B6524, 0x000FEE34150AE4BC, 0x000FEE3E3DB89B3C, + 0x000FEE47EE2982F4, 0x000FEE51271DB086, 0x000FEE59E9407F41, 0x000FEE623528B42E, + 0x000FEE6A0B5897F1, 0x000FEE716C3E077A, 0x000FEE7858327B82, 0x000FEE7ECF7B06BA, + 0x000FEE84D2484AB2, 0x000FEE8A60B66343, 0x000FEE8F7ACCC851, 0x000FEE94207E25DA, + 0x000FEE9851A829EA, 0x000FEE9C0E13485C, 0x000FEE9F557273F4, 0x000FEEA22762CCAE, + 0x000FEEA4836B42AC, 0x000FEEA668FC2D71, 0x000FEEA7D76ED6FA, 0x000FEEA8CE04FA0A, + 0x000FEEA94BE8333B, 0x000FEEA950296410, 0x000FEEA8D9C0075E, 0x000FEEA7E7897654, + 0x000FEEA678481D24, 0x000FEEA48AA29E83, 0x000FEEA21D22E4DA, 0x000FEE9F2E352024, + 0x000FEE9BBC26AF2E, 0x000FEE97C524F2E4, 0x000FEE93473C0A3A, 0x000FEE8E40557516, + 0x000FEE88AE369C7A, 0x000FEE828E7F3DFD, 0x000FEE7BDEA7B888, 0x000FEE749BFF37FF, + 0x000FEE6CC3A9BD5E, 0x000FEE64529E007E, 0x000FEE5B45A32888, 0x000FEE51994E57B6, + 0x000FEE474A0006CF, 0x000FEE3C53E12C50, 0x000FEE30B2E02AD8, 0x000FEE2462AD8205, + 0x000FEE175EB83C5A, 0x000FEE09A22A1447, 0x000FEDFB27E349CC, 0x000FEDEBEA76216C, + 0x000FEDDBE422047E, 0x000FEDCB0ECE39D3, 0x000FEDB964042CF4, 0x000FEDA6DCE938C9, + 0x000FED937237E98D, 0x000FED7F1C38A836, 0x000FED69D2B9C02B, 0x000FED538D06AE00, + 0x000FED3C41DEA422, 0x000FED23E76A2FD8, 0x000FED0A732FE644, 0x000FECEFDA07FE34, + 0x000FECD4100EB7B8, 0x000FECB708956EB4, 0x000FEC98B61230C1, 0x000FEC790A0DA978, + 0x000FEC57F50F31FE, 0x000FEC356686C962, 0x000FEC114CB4B335, 0x000FEBEB948E6FD0, + 0x000FEBC429A0B692, 0x000FEB9AF5EE0CDC, 0x000FEB6FE1C98542, 0x000FEB42D3AD1F9E, + 0x000FEB13B00B2D4B, 0x000FEAE2591A02E9, 0x000FEAAEAE992257, 0x000FEA788D8EE326, + 0x000FEA3FCFFD73E5, 0x000FEA044C8DD9F6, 0x000FE9C5D62F563B, 0x000FE9843BA947A4, + 0x000FE93F471D4728, 0x000FE8F6BD76C5D6, 0x000FE8AA5DC4E8E6, 0x000FE859E07AB1EA, + 0x000FE804F690A940, 0x000FE7AB488233C0, 0x000FE74C751F6AA5, 0x000FE6E8102AA202, + 0x000FE67DA0B6ABD8, 0x000FE60C9F38307E, 0x000FE5947338F742, 0x000FE51470977280, + 0x000FE48BD436F458, 0x000FE3F9BFFD1E37, 0x000FE35D35EEB19C, 0x000FE2B5122FE4FE, + 0x000FE20003995557, 0x000FE13C82788314, 0x000FE068C4EE67B0, 0x000FDF82B02B71AA, + 0x000FDE87C57EFEAA, 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1.53343161060262855866e-01, 1.50809290681845675763e-01, + 1.48286642732574552861e-01, 1.45775208005994028060e-01, 1.43274978973513461566e-01, 1.40785949814444699690e-01, + 1.38308116448550733057e-01, 1.35841476571253755301e-01, 1.33386029691669155683e-01, 1.30941777173644358090e-01, + 1.28508722279999570981e-01, 1.26086870220185887081e-01, 1.23676228201596571932e-01, 1.21276805484790306533e-01, + 1.18888613442910059947e-01, 1.16511665625610869035e-01, 1.14145977827838487895e-01, 1.11791568163838089811e-01, + 1.09448457146811797824e-01, 1.07116667774683801961e-01, 1.04796225622487068629e-01, 1.02487158941935246892e-01, + 1.00189498768810017482e-01, 9.79032790388624646338e-02, 9.56285367130089991594e-02, 9.33653119126910124859e-02, + 9.11136480663737591268e-02, 8.88735920682758862021e-02, 8.66451944505580717859e-02, 8.44285095703534715916e-02, + 8.22235958132029043366e-02, 8.00305158146630696292e-02, 7.78493367020961224423e-02, 7.56801303589271778804e-02, + 7.35229737139813238622e-02, 7.13779490588904025339e-02, 6.92451443970067553879e-02, 6.71246538277884968737e-02, + 6.50165779712428976156e-02, 6.29210244377581412456e-02, 6.08381083495398780614e-02, 5.87679529209337372930e-02, + 5.67106901062029017391e-02, 5.46664613248889208474e-02, 5.26354182767921896513e-02, 5.06177238609477817000e-02, + 4.86135532158685421122e-02, 4.66230949019303814174e-02, 4.46465522512944634759e-02, 4.26841449164744590750e-02, + 4.07361106559409394401e-02, 3.88027074045261474722e-02, 3.68842156885673053135e-02, 3.49809414617161251737e-02, + 3.30932194585785779961e-02, 3.12214171919203004046e-02, 2.93659397581333588001e-02, 2.75272356696031131329e-02, + 2.57058040085489103443e-02, 2.39022033057958785407e-02, 2.21170627073088502113e-02, 2.03510962300445102935e-02, + 1.86051212757246224594e-02, 1.68800831525431419000e-02, 1.51770883079353092332e-02, 1.34974506017398673818e-02, + 1.18427578579078790488e-02, 1.02149714397014590439e-02, 8.61658276939872638800e-03, 7.05087547137322242369e-03, + 5.52240329925099155545e-03, 4.03797259336302356153e-03, 2.60907274610215926189e-03, 1.26028593049859797236e-03, +} + +var kiFloat = [256]uint32{ + 0x007799EC, 0x00000000, 0x006045F5, 0x006D1AA8, + 0x00728FB4, 0x007592AF, 0x00777A5C, 0x0078CA38, + 0x0079BF6B, 0x007A7A35, 0x007B0D2F, 0x007B83D4, + 0x007BE597, 0x007C3788, 0x007C7D33, 0x007CB926, + 0x007CED48, 0x007D1B08, 0x007D437F, 0x007D678B, + 0x007D87DB, 0x007DA4FC, 0x007DBF61, 0x007DD767, + 0x007DED5D, 0x007E0183, 0x007E1411, 0x007E2534, + 0x007E3515, 0x007E43D5, 0x007E5193, 0x007E5E67, + 0x007E6A69, 0x007E75AA, 0x007E803E, 0x007E8A32, + 0x007E9395, 0x007E9C72, 0x007EA4D5, 0x007EACC6, + 0x007EB44E, 0x007EBB75, 0x007EC243, 0x007EC8BC, + 0x007ECEE8, 0x007ED4CC, 0x007EDA6B, 0x007EDFCB, + 0x007EE4EF, 0x007EE9DC, 0x007EEE94, 0x007EF31B, + 0x007EF774, 0x007EFBA0, 0x007EFFA3, 0x007F037F, + 0x007F0736, 0x007F0ACA, 0x007F0E3C, 0x007F118F, + 0x007F14C4, 0x007F17DC, 0x007F1ADA, 0x007F1DBD, + 0x007F2087, 0x007F233A, 0x007F25D7, 0x007F285D, + 0x007F2AD0, 0x007F2D2E, 0x007F2F7A, 0x007F31B3, + 0x007F33DC, 0x007F35F3, 0x007F37FB, 0x007F39F3, + 0x007F3BDC, 0x007F3DB7, 0x007F3F84, 0x007F4145, + 0x007F42F8, 0x007F449F, 0x007F463A, 0x007F47CA, + 0x007F494E, 0x007F4AC8, 0x007F4C38, 0x007F4D9D, + 0x007F4EF9, 0x007F504C, 0x007F5195, 0x007F52D5, + 0x007F540D, 0x007F553D, 0x007F5664, 0x007F5784, + 0x007F589C, 0x007F59AC, 0x007F5AB5, 0x007F5BB8, + 0x007F5CB3, 0x007F5DA8, 0x007F5E96, 0x007F5F7E, + 0x007F605F, 0x007F613B, 0x007F6210, 0x007F62E0, + 0x007F63AA, 0x007F646F, 0x007F652E, 0x007F65E8, + 0x007F669C, 0x007F674C, 0x007F67F6, 0x007F689C, + 0x007F693C, 0x007F69D9, 0x007F6A70, 0x007F6B03, + 0x007F6B91, 0x007F6C1B, 0x007F6CA0, 0x007F6D21, + 0x007F6D9E, 0x007F6E17, 0x007F6E8C, 0x007F6EFC, + 0x007F6F68, 0x007F6FD1, 0x007F7035, 0x007F7096, + 0x007F70F3, 0x007F714C, 0x007F71A1, 0x007F71F2, + 0x007F723F, 0x007F7289, 0x007F72CF, 0x007F7312, + 0x007F7350, 0x007F738B, 0x007F73C3, 0x007F73F6, + 0x007F7427, 0x007F7453, 0x007F747C, 0x007F74A1, + 0x007F74C3, 0x007F74E0, 0x007F74FB, 0x007F7511, + 0x007F7524, 0x007F7533, 0x007F753F, 0x007F7546, + 0x007F754A, 0x007F754B, 0x007F7547, 0x007F753F, + 0x007F7534, 0x007F7524, 0x007F7511, 0x007F74F9, + 0x007F74DE, 0x007F74BE, 0x007F749A, 0x007F7472, + 0x007F7445, 0x007F7414, 0x007F73DF, 0x007F73A5, + 0x007F7366, 0x007F7323, 0x007F72DA, 0x007F728D, + 0x007F723A, 0x007F71E3, 0x007F7186, 0x007F7123, + 0x007F70BB, 0x007F704D, 0x007F6FD9, 0x007F6F5F, + 0x007F6EDF, 0x007F6E58, 0x007F6DCB, 0x007F6D37, + 0x007F6C9C, 0x007F6BF9, 0x007F6B4F, 0x007F6A9C, + 0x007F69E2, 0x007F691F, 0x007F6854, 0x007F677F, + 0x007F66A1, 0x007F65B8, 0x007F64C6, 0x007F63C8, + 0x007F62C0, 0x007F61AB, 0x007F608A, 0x007F5F5D, + 0x007F5E21, 0x007F5CD8, 0x007F5B7F, 0x007F5A17, + 0x007F589E, 0x007F5713, 0x007F5575, 0x007F53C4, + 0x007F51FE, 0x007F5022, 0x007F4E2F, 0x007F4C22, + 0x007F49FA, 0x007F47B6, 0x007F4553, 0x007F42CF, + 0x007F4028, 0x007F3D5A, 0x007F3A64, 0x007F3741, + 0x007F33ED, 0x007F3065, 0x007F2CA4, 0x007F28A4, + 0x007F245F, 0x007F1FCE, 0x007F1AEA, 0x007F15A9, + 0x007F1000, 0x007F09E4, 0x007F0346, 0x007EFC16, + 0x007EF43E, 0x007EEBA8, 0x007EE237, 0x007ED7C8, + 0x007ECC2F, 0x007EBF37, 0x007EB09D, 0x007EA00A, + 0x007E8D0D, 0x007E7710, 0x007E5D47, 0x007E3E93, + 0x007E1959, 0x007DEB2C, 0x007DB036, 0x007D6203, + 0x007CF4B9, 0x007C4FD2, 0x007B3630, 0x0078D2D2, +} + +var wiFloat = [256]float32{ + 4.66198677960027669255e-07, 2.56588335019207033255e-08, 3.41146697750176784592e-08, 4.00230311410932959821e-08, + 4.47179475877737745459e-08, 4.86837785973537366722e-08, 5.21562578925932412861e-08, 5.52695199001886257153e-08, + 5.81078488992733116465e-08, 6.07279932024587421409e-08, 6.31701613261172047795e-08, 6.54639842900233842742e-08, + 6.76319905583641815324e-08, 6.96917493470166688656e-08, 7.16572544283857476692e-08, 7.35398519048393832969e-08, + 7.53488822443557479279e-08, 7.70921367281667127885e-08, 7.87761895947956022626e-08, 8.04066446825615346857e-08, + 8.19883218760237408659e-08, 8.35254002936857088917e-08, 8.50215298165053411740e-08, 8.64799190652369040985e-08, + 8.79034055989140110861e-08, 8.92945125124233511541e-08, 9.06554945027956262312e-08, 9.19883756905278607229e-08, + 9.32949809202232869780e-08, 9.45769618559625849039e-08, 9.58358188855612866442e-08, 9.70729196232813152662e-08, + 9.82895146313061088986e-08, 9.94867508514382224721e-08, 1.00665683139461669691e-07, 1.01827284217853923044e-07, + 1.02972453302539369464e-07, 1.04102023612124921572e-07, 1.05216768930574060431e-07, 1.06317409364335657741e-07, + 1.07404616410877866490e-07, 1.08479017436113134283e-07, 1.09541199642370962438e-07, 1.10591713595628691212e-07, + 1.11631076370069356306e-07, 1.12659774359245895023e-07, 1.13678265795837113569e-07, 1.14686983015899673063e-07, + 1.15686334498432158725e-07, 1.16676706706789039179e-07, 1.17658465754873988919e-07, 1.18631958917986203582e-07, + 1.19597516005596215528e-07, 1.20555450611113917226e-07, 1.21506061251817163689e-07, 1.22449632410483948386e-07, + 1.23386435488872536840e-07, 1.24316729681986364321e-07, 1.25240762781015530062e-07, 1.26158771911939892267e-07, + 1.27070984215989333455e-07, 1.27977617477468922011e-07, 1.28878880703854958297e-07, 1.29774974662539874521e-07, + 1.30666092378141980504e-07, 1.31552419593887221722e-07, 1.32434135200211397569e-07, 1.33311411633413359243e-07, + 1.34184415246907777059e-07, 1.35053306657377859830e-07, 1.35918241067904315860e-07, 1.36779368569952053923e-07, + 1.37636834425917531047e-07, 1.38490779333783508675e-07, 1.39341339675287344817e-07, 1.40188647748881762555e-07, + 1.41032831988654882776e-07, 1.41874017170273235693e-07, 1.42712324604921442006e-07, 1.43547872322127921816e-07, + 1.44380775242292721080e-07, 1.45211145339665544509e-07, 1.46039091796461362146e-07, 1.46864721148745476208e-07, + 1.47688137424670065700e-07, 1.48509442275598857119e-07, 1.49328735100614641423e-07, 1.50146113164867617390e-07, + 1.50961671712187416111e-07, 1.51775504072350982845e-07, 1.52587701763369746341e-07, 1.53398354589133671168e-07, + 1.54207550732725568797e-07, 1.55015376845697999657e-07, 1.55821918133584372604e-07, 1.56627258437898192833e-07, + 1.57431480314857468671e-07, 1.58234665111056041043e-07, 1.59036893036289199880e-07, 1.59838243233728855017e-07, + 1.60638793847630850137e-07, 1.61438622088746393909e-07, 1.62237804297600106296e-07, 1.63036416005787357730e-07, + 1.63834531995435479082e-07, 1.64632226356965902954e-07, 1.65429572545287097020e-07, 1.66226643434541294491e-07, + 1.67023511371523209274e-07, 1.67820248227882200051e-07, 1.68616925451215588827e-07, 1.69413614115155757272e-07, + 1.70210384968549673733e-07, 1.71007308483826142122e-07, 1.71804454904642543391e-07, 1.72601894292900061024e-07, + 1.73399696575213681990e-07, 1.74197931588920988271e-07, 1.74996669127712165834e-07, 1.75795978986961275677e-07, + 1.76595931008838063924e-07, 1.77396595127278238022e-07, 1.78198041412889183130e-07, 1.79000340117867431104e-07, + 1.79803561721004406185e-07, 1.80607776972855859813e-07, 1.81413056941151359868e-07, 1.82219473056520464354e-07, + 1.83027097158612474240e-07, 1.83836001542687613069e-07, 1.84646259006759307383e-07, 1.85457942899367347876e-07, + 1.86271127168064649331e-07, 1.87085886408701333260e-07, 1.87902295915592424729e-07, 1.88720431732658022414e-07, + 1.89540370705627262627e-07, 1.90362190535400839128e-07, 1.91185969832669990437e-07, 1.92011788173893651535e-07, + 1.92839726158739913768e-07, 1.93669865469102145482e-07, 1.94502288929804890433e-07, 1.95337080571120616772e-07, + 1.96174325693223683314e-07, 1.97014110932714374919e-07, 1.97856524331352952716e-07, 1.98701655407150388211e-07, + 1.99549595227971635348e-07, 2.00400436487814600236e-07, 2.01254273585938820883e-07, 2.02111202709026498408e-07, + 2.02971321916571014951e-07, 2.03834731229698846698e-07, 2.04701532723644121196e-07, 2.05571830624108885378e-07, + 2.06445731407757185541e-07, 2.07323343907107312957e-07, 2.08204779420104330037e-07, 2.09090151824673600213e-07, + 2.09979577698577670508e-07, 2.10873176444920111011e-07, 2.11771070423665379388e-07, 2.12673385089569268965e-07, + 2.13580249136944118603e-07, 2.14491794651713402832e-07, 2.15408157271244625533e-07, 2.16329476352486921685e-07, + 2.17255895148978920488e-07, 2.18187560997337924713e-07, 2.19124625513888206785e-07, 2.20067244802139479285e-07, + 2.21015579671883851683e-07, 2.21969795870742159701e-07, 2.22930064329060010376e-07, 2.23896561419128954210e-07, + 2.24869469229791575583e-07, 2.25848975857580322189e-07, 2.26835275715640744118e-07, 2.27828569861799901001e-07, + 2.28829066347263833069e-07, 2.29836980587561823183e-07, 2.30852535757505260518e-07, 2.31875963212094114516e-07, + 2.32907502935486642699e-07, 2.33947404020352726160e-07, 2.34995925180156140289e-07, 2.36053335297164516378e-07, + 2.37119914009265667728e-07, 2.38195952338983970691e-07, 2.39281753368440712742e-07, 2.40377632964396957621e-07, + 2.41483920557958384709e-07, 2.42600959984018662258e-07, 2.43729110386077326413e-07, 2.44868747192698939290e-07, + 2.46020263172594533433e-07, 2.47184069576113545901e-07, 2.48360597371852893654e-07, 2.49550298588131851232e-07, + 2.50753647770270890721e-07, 2.51971143565970967140e-07, 2.53203310452642767375e-07, 2.54450700622322097890e-07, + 2.55713896041856770961e-07, 2.56993510708419870887e-07, 2.58290193123138874550e-07, 2.59604629008804833146e-07, + 2.60937544301314385690e-07, 2.62289708448800566945e-07, 2.63661938057441759882e-07, 2.65055100928844238758e-07, + 2.66470120540847889467e-07, 2.67907981031821866252e-07, 2.69369732758258246335e-07, 2.70856498507068313229e-07, + 2.72369480457841388042e-07, 2.73909968006952220135e-07, 2.75479346585437289399e-07, 2.77079107626811561009e-07, + 2.78710859870496796972e-07, 2.80376342222588603820e-07, 2.82077438439999912690e-07, 2.83816193958769527230e-07, + 2.85594835255375795814e-07, 2.87415792215003905739e-07, 2.89281724087851835900e-07, 2.91195549750371467233e-07, + 2.93160483161771875581e-07, 2.95180075129332912389e-07, 2.97258262785797916083e-07, 2.99399428561531794298e-07, + 3.01608470935804138388e-07, 3.03890889921758510417e-07, 3.06252891144972267537e-07, 3.08701513613258141075e-07, + 3.11244787989714509378e-07, 3.13891934589336184321e-07, 3.16653613755314681314e-07, 3.19542246256559459667e-07, + 3.22572428717978242099e-07, 3.25761480217458181578e-07, 3.29130173358915628534e-07, 3.32703730345002116955e-07, + 3.36513208964639108346e-07, 3.40597478255417943913e-07, 3.45006114675213401550e-07, 3.49803789521323211592e-07, + 3.55077180848341416206e-07, 3.60946392031859609868e-07, 3.67584959507244041831e-07, 3.75257645787954431030e-07, + 3.84399301057791926300e-07, 3.95804015855768440983e-07, 4.11186015434435801956e-07, 4.35608969373823260746e-07, +} + +var fiFloat = [256]float32{ + 1.00000000000000000000e+00, 9.77101701267671596263e-01, 9.59879091800106665211e-01, 9.45198953442299649730e-01, + 9.32060075959230460718e-01, 9.19991505039347012840e-01, 9.08726440052130879366e-01, 8.98095921898343418910e-01, + 8.87984660755833377088e-01, 8.78309655808917399966e-01, 8.69008688036857046555e-01, 8.60033621196331532488e-01, + 8.51346258458677951353e-01, 8.42915653112204177333e-01, 8.34716292986883434679e-01, 8.26726833946221373317e-01, + 8.18929191603702366642e-01, 8.11307874312656274185e-01, 8.03849483170964274059e-01, 7.96542330422958966274e-01, + 7.89376143566024590648e-01, 7.82341832654802504798e-01, 7.75431304981187174974e-01, 7.68637315798486264740e-01, + 7.61953346836795386565e-01, 7.55373506507096115214e-01, 7.48892447219156820459e-01, 7.42505296340151055290e-01, + 7.36207598126862650112e-01, 7.29995264561476231435e-01, 7.23864533468630222401e-01, 7.17811932630721960535e-01, + 7.11834248878248421200e-01, 7.05928501332754310127e-01, 7.00091918136511615067e-01, 6.94321916126116711609e-01, + 6.88616083004671808432e-01, 6.82972161644994857355e-01, 6.77388036218773526009e-01, 6.71861719897082099173e-01, + 6.66391343908750100056e-01, 6.60975147776663107813e-01, 6.55611470579697264149e-01, 6.50298743110816701574e-01, + 6.45035480820822293424e-01, 6.39820277453056585060e-01, 6.34651799287623608059e-01, 6.29528779924836690007e-01, + 6.24450015547026504592e-01, 6.19414360605834324325e-01, 6.14420723888913888899e-01, 6.09468064925773433949e-01, + 6.04555390697467776029e-01, 5.99681752619125263415e-01, 5.94846243767987448159e-01, 5.90047996332826008015e-01, + 5.85286179263371453274e-01, 5.80559996100790898232e-01, 5.75868682972353718164e-01, 5.71211506735253227163e-01, + 5.66587763256164445025e-01, 5.61996775814524340831e-01, 5.57437893618765945014e-01, 5.52910490425832290562e-01, + 5.48413963255265812791e-01, 5.43947731190026262382e-01, 5.39511234256952132426e-01, 5.35103932380457614215e-01, + 5.30725304403662057062e-01, 5.26374847171684479008e-01, 5.22052074672321841931e-01, 5.17756517229756352272e-01, + 5.13487720747326958914e-01, 5.09245245995747941592e-01, 5.05028667943468123624e-01, 5.00837575126148681903e-01, + 4.96671569052489714213e-01, 4.92530263643868537748e-01, 4.88413284705458028423e-01, 4.84320269426683325253e-01, + 4.80250865909046753544e-01, 4.76204732719505863248e-01, 4.72181538467730199660e-01, 4.68180961405693596422e-01, + 4.64202689048174355069e-01, 4.60246417812842867345e-01, 4.56311852678716434184e-01, 4.52398706861848520777e-01, + 4.48506701507203064949e-01, 4.44635565395739396077e-01, 4.40785034665803987508e-01, 4.36954852547985550526e-01, + 4.33144769112652261445e-01, 4.29354541029441427735e-01, 4.25583931338021970170e-01, 4.21832709229495894654e-01, + 4.18100649837848226120e-01, 4.14387534040891125642e-01, 4.10693148270188157500e-01, 4.07017284329473372217e-01, + 4.03359739221114510510e-01, 3.99720314980197222177e-01, 3.96098818515832451492e-01, 3.92495061459315619512e-01, + 3.88908860018788715696e-01, 3.85340034840077283462e-01, 3.81788410873393657674e-01, 3.78253817245619183840e-01, + 3.74736087137891138443e-01, 3.71235057668239498696e-01, 3.67750569779032587814e-01, 3.64282468129004055601e-01, + 3.60830600989648031529e-01, 3.57394820145780500731e-01, 3.53974980800076777232e-01, 3.50570941481406106455e-01, + 3.47182563956793643900e-01, 3.43809713146850715049e-01, 3.40452257044521866547e-01, 3.37110066637006045021e-01, + 3.33783015830718454708e-01, 3.30470981379163586400e-01, 3.27173842813601400970e-01, 3.23891482376391093290e-01, + 3.20623784956905355514e-01, 3.17370638029913609834e-01, 3.14131931596337177215e-01, 3.10907558126286509559e-01, + 3.07697412504292056035e-01, 3.04501391976649993243e-01, 3.01319396100803049698e-01, 2.98151326696685481377e-01, + 2.94997087799961810184e-01, 2.91856585617095209972e-01, 2.88729728482182923521e-01, 2.85616426815501756042e-01, + 2.82516593083707578948e-01, 2.79430141761637940157e-01, 2.76356989295668320494e-01, 2.73297054068577072172e-01, + 2.70250256365875463072e-01, 2.67216518343561471038e-01, 2.64195763997261190426e-01, 2.61187919132721213522e-01, + 2.58192911337619235290e-01, 2.55210669954661961700e-01, 2.52241126055942177508e-01, 2.49284212418528522415e-01, + 2.46339863501263828249e-01, 2.43408015422750312329e-01, 2.40488605940500588254e-01, 2.37581574431238090606e-01, + 2.34686861872330010392e-01, 2.31804410824338724684e-01, 2.28934165414680340644e-01, 2.26076071322380278694e-01, + 2.23230075763917484855e-01, 2.20396127480151998723e-01, 2.17574176724331130872e-01, 2.14764175251173583536e-01, + 2.11966076307030182324e-01, 2.09179834621125076977e-01, 2.06405406397880797353e-01, 2.03642749310334908452e-01, + 2.00891822494656591136e-01, 1.98152586545775138971e-01, 1.95425003514134304483e-01, 1.92709036903589175926e-01, + 1.90004651670464985713e-01, 1.87311814223800304768e-01, 1.84630492426799269756e-01, 1.81960655599522513892e-01, + 1.79302274522847582272e-01, 1.76655321443734858455e-01, 1.74019770081838553999e-01, 1.71395595637505754327e-01, + 1.68782774801211288285e-01, 1.66181285764481906364e-01, 1.63591108232365584074e-01, 1.61012223437511009516e-01, + 1.58444614155924284882e-01, 1.55888264724479197465e-01, 1.53343161060262855866e-01, 1.50809290681845675763e-01, + 1.48286642732574552861e-01, 1.45775208005994028060e-01, 1.43274978973513461566e-01, 1.40785949814444699690e-01, + 1.38308116448550733057e-01, 1.35841476571253755301e-01, 1.33386029691669155683e-01, 1.30941777173644358090e-01, + 1.28508722279999570981e-01, 1.26086870220185887081e-01, 1.23676228201596571932e-01, 1.21276805484790306533e-01, + 1.18888613442910059947e-01, 1.16511665625610869035e-01, 1.14145977827838487895e-01, 1.11791568163838089811e-01, + 1.09448457146811797824e-01, 1.07116667774683801961e-01, 1.04796225622487068629e-01, 1.02487158941935246892e-01, + 1.00189498768810017482e-01, 9.79032790388624646338e-02, 9.56285367130089991594e-02, 9.33653119126910124859e-02, + 9.11136480663737591268e-02, 8.88735920682758862021e-02, 8.66451944505580717859e-02, 8.44285095703534715916e-02, + 8.22235958132029043366e-02, 8.00305158146630696292e-02, 7.78493367020961224423e-02, 7.56801303589271778804e-02, + 7.35229737139813238622e-02, 7.13779490588904025339e-02, 6.92451443970067553879e-02, 6.71246538277884968737e-02, + 6.50165779712428976156e-02, 6.29210244377581412456e-02, 6.08381083495398780614e-02, 5.87679529209337372930e-02, + 5.67106901062029017391e-02, 5.46664613248889208474e-02, 5.26354182767921896513e-02, 5.06177238609477817000e-02, + 4.86135532158685421122e-02, 4.66230949019303814174e-02, 4.46465522512944634759e-02, 4.26841449164744590750e-02, + 4.07361106559409394401e-02, 3.88027074045261474722e-02, 3.68842156885673053135e-02, 3.49809414617161251737e-02, + 3.30932194585785779961e-02, 3.12214171919203004046e-02, 2.93659397581333588001e-02, 2.75272356696031131329e-02, + 2.57058040085489103443e-02, 2.39022033057958785407e-02, 2.21170627073088502113e-02, 2.03510962300445102935e-02, + 1.86051212757246224594e-02, 1.68800831525431419000e-02, 1.51770883079353092332e-02, 1.34974506017398673818e-02, + 1.18427578579078790488e-02, 1.02149714397014590439e-02, 8.61658276939872638800e-03, 7.05087547137322242369e-03, + 5.52240329925099155545e-03, 4.03797259336302356153e-03, 2.60907274610215926189e-03, 1.26028593049859797236e-03, +} + +var keDouble = [256]uint64{ + 0x001C5214272497C6, 0x0000000000000000, 0x00137D5BD79C317E, 0x00186EF58E3F3C10, + 0x001A9BB7320EB0AE, 0x001BD127F719447C, 0x001C951D0F88651A, 0x001D1BFE2D5C3972, + 0x001D7E5BD56B18B2, 0x001DC934DD172C70, 0x001E0409DFAC9DC8, 0x001E337B71D47836, + 0x001E5A8B177CB7A2, 0x001E7B42096F046C, 0x001E970DAF08AE3E, 0x001EAEF5B14EF09E, + 0x001EC3BD07B46556, 0x001ED5F6F08799CE, 0x001EE614AE6E5688, 0x001EF46ECA361CD0, + 0x001F014B76DDD4A4, 0x001F0CE313A796B6, 0x001F176369F1F77A, 0x001F20F20C452570, + 0x001F29AE1951A874, 0x001F31B18FB95532, 0x001F39125157C106, 0x001F3FE2EB6E694C, + 0x001F463332D788FA, 0x001F4C10BF1D3A0E, 0x001F51874C5C3322, 0x001F56A109C3ECC0, + 0x001F5B66D9099996, 0x001F5FE08210D08C, 0x001F6414DD445772, 0x001F6809F6859678, + 0x001F6BC52A2B02E6, 0x001F6F4B3D32E4F4, 0x001F72A07190F13A, 0x001F75C8974D09D6, + 0x001F78C71B045CC0, 0x001F7B9F12413FF4, 0x001F7E5346079F8A, 0x001F80E63BE21138, + 0x001F835A3DAD9162, 0x001F85B16056B912, 0x001F87ED89B24262, 0x001F8A10759374FA, + 0x001F8C1BBA3D39AC, 0x001F8E10CC45D04A, 0x001F8FF102013E16, 0x001F91BD968358E0, + 0x001F9377AC47AFD8, 0x001F95204F8B64DA, 0x001F96B878633892, 0x001F98410C968892, + 0x001F99BAE146BA80, 0x001F9B26BC697F00, 0x001F9C85561B717A, 0x001F9DD759CFD802, + 0x001F9F1D6761A1CE, 0x001FA058140936C0, 0x001FA187EB3A3338, 0x001FA2AD6F6BC4FC, + 0x001FA3C91ACE0682, 0x001FA4DB5FEE6AA2, 0x001FA5E4AA4D097C, 0x001FA6E55EE46782, + 0x001FA7DDDCA51EC4, 0x001FA8CE7CE6A874, 0x001FA9B793CE5FEE, 0x001FAA9970ADB858, + 0x001FAB745E588232, 0x001FAC48A3740584, 0x001FAD1682BF9FE8, 0x001FADDE3B5782C0, + 0x001FAEA008F21D6C, 0x001FAF5C2418B07E, 0x001FB012C25B7A12, 0x001FB0C41681DFF4, + 0x001FB17050B6F1FA, 0x001FB2179EB2963A, 0x001FB2BA2BDFA84A, 0x001FB358217F4E18, + 0x001FB3F1A6C9BE0C, 0x001FB486E10CACD6, 0x001FB517F3C793FC, 0x001FB5A500C5FDAA, + 0x001FB62E2837FE58, 0x001FB6B388C9010A, 0x001FB7353FB50798, 0x001FB7B368DC7DA8, + 0x001FB82E1ED6BA08, 0x001FB8A57B0347F6, 0x001FB919959A0F74, 0x001FB98A85BA7204, + 0x001FB9F861796F26, 0x001FBA633DEEE286, 0x001FBACB2F41EC16, 0x001FBB3048B49144, + 0x001FBB929CAEA4E2, 0x001FBBF23CC8029E, 0x001FBC4F39D22994, 0x001FBCA9A3E140D4, + 0x001FBD018A548F9E, 0x001FBD56FBDE729C, 0x001FBDAA068BD66A, 0x001FBDFAB7CB3F40, + 0x001FBE491C7364DE, 0x001FBE9540C9695E, 0x001FBEDF3086B128, 0x001FBF26F6DE6174, + 0x001FBF6C9E828AE2, 0x001FBFB031A904C4, 0x001FBFF1BA0FFDB0, 0x001FC03141024588, + 0x001FC06ECF5B54B2, 0x001FC0AA6D8B1426, 0x001FC0E42399698A, 0x001FC11BF9298A64, + 0x001FC151F57D1942, 0x001FC1861F770F4A, 0x001FC1B87D9E74B4, 0x001FC1E91620EA42, + 0x001FC217EED505DE, 0x001FC2450D3C83FE, 0x001FC27076864FC2, 0x001FC29A2F90630E, + 0x001FC2C23CE98046, 0x001FC2E8A2D2C6B4, 0x001FC30D654122EC, 0x001FC33087DE9C0E, + 0x001FC3520E0B7EC6, 0x001FC371FADF66F8, 0x001FC390512A2886, 0x001FC3AD137497FA, + 0x001FC3C844013348, 0x001FC3E1E4CCAB40, 0x001FC3F9F78E4DA8, 0x001FC4107DB85060, + 0x001FC4257877FD68, 0x001FC438E8B5BFC6, 0x001FC44ACF15112A, 0x001FC45B2BF447E8, + 0x001FC469FF6C4504, 0x001FC477495001B2, 0x001FC483092BFBB8, 0x001FC48D3E457FF6, + 0x001FC495E799D21A, 0x001FC49D03DD30B0, 0x001FC4A29179B432, 0x001FC4A68E8E07FC, + 0x001FC4A8F8EBFB8C, 0x001FC4A9CE16EA9E, 0x001FC4A90B41FA34, 0x001FC4A6AD4E28A0, + 0x001FC4A2B0C82E74, 0x001FC49D11E62DE2, 0x001FC495CC852DF4, 0x001FC48CDC265EC0, + 0x001FC4823BEC237A, 0x001FC475E696DEE6, 0x001FC467D6817E82, 0x001FC458059DC036, + 0x001FC4466D702E20, 0x001FC433070BCB98, 0x001FC41DCB0D6E0E, 0x001FC406B196BBF6, + 0x001FC3EDB248CB62, 0x001FC3D2C43E593C, 0x001FC3B5DE0591B4, 0x001FC396F599614C, + 0x001FC376005A4592, 0x001FC352F3069370, 0x001FC32DC1B22818, 0x001FC3065FBD7888, + 0x001FC2DCBFCBF262, 0x001FC2B0D3B99F9E, 0x001FC2828C8FFCF0, 0x001FC251DA79F164, + 0x001FC21EACB6D39E, 0x001FC1E8F18C6756, 0x001FC1B09637BB3C, 0x001FC17586DCCD10, + 0x001FC137AE74D6B6, 0x001FC0F6F6BB2414, 0x001FC0B348184DA4, 0x001FC06C898BAFF0, + 0x001FC022A092F364, 0x001FBFD5710F72B8, 0x001FBF84DD29488E, 0x001FBF30C52FC60A, + 0x001FBED907770CC6, 0x001FBE7D80327DDA, 0x001FBE1E094BA614, 0x001FBDBA7A354408, + 0x001FBD52A7B9F826, 0x001FBCE663C6201A, 0x001FBC757D2C4DE4, 0x001FBBFFBF63B7AA, + 0x001FBB84F23FE6A2, 0x001FBB04D9A0D18C, 0x001FBA7F351A70AC, 0x001FB9F3BF92B618, + 0x001FB9622ED4ABFC, 0x001FB8CA33174A16, 0x001FB82B76765B54, 0x001FB7859C5B895C, + 0x001FB6D840D55594, 0x001FB622F7D96942, 0x001FB5654C6F37E0, 0x001FB49EBFBF69D2, + 0x001FB3CEC803E746, 0x001FB2F4CF539C3E, 0x001FB21032442852, 0x001FB1203E5A9604, + 0x001FB0243042E1C2, 0x001FAF1B31C479A6, 0x001FAE045767E104, 0x001FACDE9DBF2D72, + 0x001FABA8E640060A, 0x001FAA61F399FF28, 0x001FA908656F66A2, 0x001FA79AB3508D3C, + 0x001FA61726D1F214, 0x001FA47BD48BEA00, 0x001FA2C693C5C094, 0x001FA0F4F47DF314, + 0x001F9F04336BBE0A, 0x001F9CF12B79F9BC, 0x001F9AB84415ABC4, 0x001F98555B782FB8, + 0x001F95C3ABD03F78, 0x001F92FDA9CEF1F2, 0x001F8FFCDA9AE41C, 0x001F8CB99E7385F8, + 0x001F892AEC479606, 0x001F8545F904DB8E, 0x001F80FDC336039A, 0x001F7C427839E926, + 0x001F7700A3582ACC, 0x001F71200F1A241C, 0x001F6A8234B7352A, 0x001F630000A8E266, + 0x001F5A66904FE3C4, 0x001F50724ECE1172, 0x001F44C7665C6FDA, 0x001F36E5A38A59A2, + 0x001F26143450340A, 0x001F113E047B0414, 0x001EF6AEFA57CBE6, 0x001ED38CA188151E, + 0x001EA2A61E122DB0, 0x001E5961C78B267C, 0x001DDDF62BAC0BB0, 0x001CDB4DD9E4E8C0, +} + +var weDouble = [256]float64{ + 9.655740063209182975e-16, 7.089014243955414331e-18, 1.163941249669122378e-17, 1.524391512353216015e-17, + 1.833284885723743916e-17, 2.108965109464486630e-17, 2.361128077843138196e-17, 2.595595772310893952e-17, + 2.816173554197752338e-17, 3.025504130321382330e-17, 3.225508254836375280e-17, 3.417632340185027033e-17, + 3.602996978734452488e-17, 3.782490776869649048e-17, 3.956832198097553231e-17, 4.126611778175946428e-17, + 4.292321808442525631e-17, 4.454377743282371417e-17, 4.613133981483185932e-17, 4.768895725264635940e-17, + 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1.030481601712577161e-01, 1.013229974259536315e-01, + 9.961058367063713170e-02, 9.791085331149219917e-02, 9.622374255043279756e-02, 9.454918937605585882e-02, + 9.288713355604354127e-02, 9.123751663104015530e-02, 8.960028191003285847e-02, 8.797537446727021759e-02, + 8.636274114075691288e-02, 8.476233053236811865e-02, 8.317409300963238272e-02, 8.159798070923741931e-02, + 8.003394754231990538e-02, 7.848194920160642130e-02, 7.694194317048050347e-02, 7.541388873405840965e-02, + 7.389774699236474620e-02, 7.239348087570873780e-02, 7.090105516237182881e-02, 6.942043649872875477e-02, + 6.795159342193660135e-02, 6.649449638533977414e-02, 6.504911778675374900e-02, 6.361543199980733421e-02, + 6.219341540854099459e-02, 6.078304644547963265e-02, 5.938430563342026597e-02, 5.799717563120065922e-02, + 5.662164128374287675e-02, 5.525768967669703741e-02, 5.390531019604608703e-02, 5.256449459307169225e-02, + 5.123523705512628146e-02, 4.991753428270637172e-02, 4.861138557337949667e-02, 4.731679291318154762e-02, + 4.603376107617516977e-02, 4.476229773294328196e-02, 4.350241356888818328e-02, 4.225412241331623353e-02, + 4.101744138041481941e-02, 3.979239102337412542e-02, 3.857899550307485742e-02, 3.737728277295936097e-02, + 3.618728478193142251e-02, 3.500903769739741045e-02, 3.384258215087432992e-02, 3.268796350895953468e-02, + 3.154523217289360859e-02, 3.041444391046660423e-02, 2.929566022463739317e-02, 2.818894876397863569e-02, + 2.709438378095579969e-02, 2.601204664513421735e-02, 2.494202641973178314e-02, 2.388442051155817078e-02, + 2.283933540638524023e-02, 2.180688750428358066e-02, 2.078720407257811723e-02, 1.978042433800974303e-02, + 1.878670074469603046e-02, 1.780620041091136169e-02, 1.683910682603994777e-02, 1.588562183997316302e-02, + 1.494596801169114850e-02, 1.402039140318193759e-02, 1.310916493125499106e-02, 1.221259242625538123e-02, + 1.133101359783459695e-02, 1.046481018102997894e-02, 9.614413642502209895e-03, 8.780314985808975251e-03, + 7.963077438017040002e-03, 7.163353183634983863e-03, 6.381905937319179087e-03, 5.619642207205483020e-03, + 4.877655983542392333e-03, 4.157295120833795314e-03, 3.460264777836904049e-03, 2.788798793574076128e-03, + 2.145967743718906265e-03, 1.536299780301572356e-03, 9.672692823271745359e-04, 4.541343538414967652e-04, +} + +var keFloat = [256]uint32{ + 0x00714851, 0x00000000, 0x004DF56F, 0x0061BBD6, + 0x006A6EDD, 0x006F44A0, 0x00725474, 0x00746FF9, + 0x0075F96F, 0x007724D3, 0x00781027, 0x0078CDEE, + 0x00796A2C, 0x0079ED08, 0x007A5C37, 0x007ABBD7, + 0x007B0EF4, 0x007B57DC, 0x007B9853, 0x007BD1BB, + 0x007C052E, 0x007C338C, 0x007C5D8E, 0x007C83C8, + 0x007CA6B8, 0x007CC6C6, 0x007CE449, 0x007CFF8C, + 0x007D18CD, 0x007D3043, 0x007D461D, 0x007D5A84, + 0x007D6D9B, 0x007D7F82, 0x007D9053, 0x007DA028, + 0x007DAF15, 0x007DBD2D, 0x007DCA82, 0x007DD722, + 0x007DE31C, 0x007DEE7C, 0x007DF94D, 0x007E0399, + 0x007E0D69, 0x007E16C6, 0x007E1FB6, 0x007E2842, + 0x007E306F, 0x007E3843, 0x007E3FC4, 0x007E46F6, + 0x007E4DDF, 0x007E5481, 0x007E5AE2, 0x007E6104, + 0x007E66EC, 0x007E6C9B, 0x007E7215, 0x007E775D, + 0x007E7C76, 0x007E8160, 0x007E8620, 0x007E8AB6, + 0x007E8F24, 0x007E936D, 0x007E9793, 0x007E9B95, + 0x007E9F77, 0x007EA33A, 0x007EA6DE, 0x007EAA66, + 0x007EADD1, 0x007EB123, 0x007EB45A, 0x007EB779, + 0x007EBA80, 0x007EBD71, 0x007EC04B, 0x007EC310, + 0x007EC5C1, 0x007EC85E, 0x007ECAE9, 0x007ECD61, + 0x007ECFC7, 0x007ED21C, 0x007ED460, 0x007ED694, + 0x007ED8B9, 0x007EDACE, 0x007EDCD5, 0x007EDECE, + 0x007EE0B8, 0x007EE296, 0x007EE466, 0x007EE62A, + 0x007EE7E2, 0x007EE98D, 0x007EEB2D, 0x007EECC1, + 0x007EEE4A, 0x007EEFC9, 0x007EF13D, 0x007EF2A7, + 0x007EF406, 0x007EF55C, 0x007EF6A8, 0x007EF7EB, + 0x007EF924, 0x007EFA55, 0x007EFB7D, 0x007EFC9C, + 0x007EFDB2, 0x007EFEC1, 0x007EFFC7, 0x007F00C5, + 0x007F01BB, 0x007F02AA, 0x007F0391, 0x007F0470, + 0x007F0548, 0x007F0618, 0x007F06E2, 0x007F07A4, + 0x007F0860, 0x007F0914, 0x007F09C2, 0x007F0A69, + 0x007F0B09, 0x007F0BA3, 0x007F0C36, 0x007F0CC2, + 0x007F0D48, 0x007F0DC8, 0x007F0E41, 0x007F0EB4, + 0x007F0F21, 0x007F0F88, 0x007F0FE8, 0x007F1042, + 0x007F1096, 0x007F10E4, 0x007F112B, 0x007F116D, + 0x007F11A8, 0x007F11DD, 0x007F120C, 0x007F1235, + 0x007F1258, 0x007F1274, 0x007F128A, 0x007F129A, + 0x007F12A4, 0x007F12A7, 0x007F12A4, 0x007F129B, + 0x007F128B, 0x007F1274, 0x007F1257, 0x007F1233, + 0x007F1209, 0x007F11D8, 0x007F119F, 0x007F1160, + 0x007F111A, 0x007F10CC, 0x007F1077, 0x007F101B, + 0x007F0FB7, 0x007F0F4B, 0x007F0ED7, 0x007F0E5C, + 0x007F0DD8, 0x007F0D4C, 0x007F0CB7, 0x007F0C19, + 0x007F0B73, 0x007F0AC3, 0x007F0A0A, 0x007F0947, + 0x007F087B, 0x007F07A4, 0x007F06C2, 0x007F05D6, + 0x007F04DF, 0x007F03DC, 0x007F02CD, 0x007F01B2, + 0x007F008B, 0x007EFF56, 0x007EFE13, 0x007EFCC3, + 0x007EFB64, 0x007EF9F6, 0x007EF878, 0x007EF6EA, + 0x007EF54B, 0x007EF39A, 0x007EF1D6, 0x007EEFFF, + 0x007EEE14, 0x007EEC13, 0x007EE9FD, 0x007EE7CF, + 0x007EE589, 0x007EE329, 0x007EE0AE, 0x007EDE16, + 0x007EDB61, 0x007ED88C, 0x007ED595, 0x007ED27B, + 0x007ECF3B, 0x007ECBD3, 0x007EC841, 0x007EC481, + 0x007EC091, 0x007EBC6D, 0x007EB811, 0x007EB37A, + 0x007EAEA4, 0x007EA988, 0x007EA422, 0x007E9E6B, + 0x007E985D, 0x007E91EF, 0x007E8B1A, 0x007E83D4, + 0x007E7C11, 0x007E73C5, 0x007E6AE1, 0x007E6155, + 0x007E570F, 0x007E4BF7, 0x007E3FF3, 0x007E32E6, + 0x007E24AC, 0x007E1518, 0x007E03F7, 0x007DF10A, + 0x007DDC03, 0x007DC480, 0x007DAA09, 0x007D8C00, + 0x007D699A, 0x007D41C9, 0x007D131E, 0x007CDB97, + 0x007C9851, 0x007C44F8, 0x007BDABC, 0x007B4E33, + 0x007A8A98, 0x00796587, 0x007777D9, 0x00736D37, +} + +var weFloat = [256]float32{ + 1.03677719e-06, 7.61177108e-09, 1.24977240e-08, 1.63680292e-08, + 1.96847466e-08, 2.26448404e-08, 2.53524197e-08, 2.78699974e-08, + 3.02384333e-08, 3.24861032e-08, 3.46336312e-08, 3.66965478e-08, + 3.86868855e-08, 4.06141855e-08, 4.24861622e-08, 4.43091566e-08, + 4.60884545e-08, 4.78285168e-08, 4.95331490e-08, 5.12056279e-08, + 5.28488000e-08, 5.44651557e-08, 5.60568899e-08, 5.76259484e-08, + 5.91740662e-08, 6.07027987e-08, 6.22135462e-08, 6.37075759e-08, + 6.51860386e-08, 6.66499836e-08, 6.81003709e-08, 6.95380822e-08, + 7.09639292e-08, 7.23786618e-08, 7.37829746e-08, 7.51775128e-08, + 7.65628768e-08, 7.79396272e-08, 7.93082883e-08, 8.06693516e-08, + 8.20232788e-08, 8.33705045e-08, 8.47114385e-08, 8.60464681e-08, + 8.73759596e-08, 8.87002606e-08, 9.00197010e-08, 9.13345948e-08, + 9.26452410e-08, 9.39519249e-08, 9.52549192e-08, 9.65544849e-08, + 9.78508719e-08, 9.91443202e-08, 1.00435060e-07, 1.01723315e-07, + 1.03009296e-07, 1.04293211e-07, 1.05575259e-07, 1.06855633e-07, + 1.08134518e-07, 1.09412096e-07, 1.10688542e-07, 1.11964025e-07, + 1.13238713e-07, 1.14512767e-07, 1.15786343e-07, 1.17059595e-07, + 1.18332673e-07, 1.19605723e-07, 1.20878890e-07, 1.22152313e-07, + 1.23426131e-07, 1.24700479e-07, 1.25975490e-07, 1.27251294e-07, + 1.28528022e-07, 1.29805799e-07, 1.31084751e-07, 1.32365001e-07, + 1.33646673e-07, 1.34929886e-07, 1.36214760e-07, 1.37501415e-07, + 1.38789966e-07, 1.40080532e-07, 1.41373228e-07, 1.42668169e-07, + 1.43965470e-07, 1.45265245e-07, 1.46567606e-07, 1.47872669e-07, + 1.49180545e-07, 1.50491348e-07, 1.51805191e-07, 1.53122186e-07, + 1.54442445e-07, 1.55766083e-07, 1.57093212e-07, 1.58423946e-07, + 1.59758399e-07, 1.61096684e-07, 1.62438917e-07, 1.63785214e-07, + 1.65135690e-07, 1.66490462e-07, 1.67849647e-07, 1.69213364e-07, + 1.70581733e-07, 1.71954874e-07, 1.73332908e-07, 1.74715958e-07, + 1.76104148e-07, 1.77497602e-07, 1.78896448e-07, 1.80300814e-07, + 1.81710828e-07, 1.83126623e-07, 1.84548331e-07, 1.85976086e-07, + 1.87410026e-07, 1.88850288e-07, 1.90297012e-07, 1.91750343e-07, + 1.93210424e-07, 1.94677403e-07, 1.96151428e-07, 1.97632653e-07, + 1.99121231e-07, 2.00617321e-07, 2.02121082e-07, 2.03632677e-07, + 2.05152273e-07, 2.06680040e-07, 2.08216149e-07, 2.09760777e-07, + 2.11314104e-07, 2.12876312e-07, 2.14447590e-07, 2.16028129e-07, + 2.17618123e-07, 2.19217773e-07, 2.20827283e-07, 2.22446862e-07, + 2.24076723e-07, 2.25717086e-07, 2.27368174e-07, 2.29030216e-07, + 2.30703448e-07, 2.32388110e-07, 2.34084450e-07, 2.35792720e-07, + 2.37513182e-07, 2.39246101e-07, 2.40991752e-07, 2.42750416e-07, + 2.44522382e-07, 2.46307948e-07, 2.48107418e-07, 2.49921109e-07, + 2.51749342e-07, 2.53592452e-07, 2.55450781e-07, 2.57324683e-07, + 2.59214522e-07, 2.61120673e-07, 2.63043524e-07, 2.64983476e-07, + 2.66940939e-07, 2.68916342e-07, 2.70910123e-07, 2.72922739e-07, + 2.74954660e-07, 2.77006373e-07, 2.79078382e-07, 2.81171210e-07, + 2.83285396e-07, 2.85421503e-07, 2.87580110e-07, 2.89761822e-07, + 2.91967265e-07, 2.94197089e-07, 2.96451969e-07, 2.98732610e-07, + 3.01039742e-07, 3.03374127e-07, 3.05736557e-07, 3.08127859e-07, + 3.10548894e-07, 3.13000563e-07, 3.15483804e-07, 3.17999599e-07, + 3.20548974e-07, 3.23133003e-07, 3.25752811e-07, 3.28409576e-07, + 3.31104534e-07, 3.33838984e-07, 3.36614287e-07, 3.39431878e-07, + 3.42293264e-07, 3.45200034e-07, 3.48153864e-07, 3.51156520e-07, + 3.54209871e-07, 3.57315892e-07, 3.60476673e-07, 3.63694431e-07, + 3.66971518e-07, 3.70310433e-07, 3.73713834e-07, 3.77184553e-07, + 3.80725611e-07, 3.84340234e-07, 3.88031877e-07, 3.91804239e-07, + 3.95661291e-07, 3.99607304e-07, 4.03646879e-07, 4.07784981e-07, + 4.12026980e-07, 4.16378695e-07, 4.20846449e-07, 4.25437124e-07, + 4.30158235e-07, 4.35018005e-07, 4.40025460e-07, 4.45190536e-07, + 4.50524210e-07, 4.56038644e-07, 4.61747369e-07, 4.67665494e-07, + 4.73809965e-07, 4.80199879e-07, 4.86856855e-07, 4.93805512e-07, + 5.01074042e-07, 5.08694944e-07, 5.16705952e-07, 5.25151216e-07, + 5.34082859e-07, 5.43563016e-07, 5.53666578e-07, 5.64484953e-07, + 5.76131313e-07, 5.88748108e-07, 6.02518140e-07, 6.17681418e-07, + 6.34561837e-07, 6.53611496e-07, 6.75488730e-07, 7.01206245e-07, + 7.32441505e-07, 7.72282898e-07, 8.27435688e-07, 9.17567905e-07, +} + +var feFloat = [256]float32{ + 1.00000000e+00, 9.38143681e-01, 9.00469930e-01, 8.71704332e-01, + 8.47785501e-01, 8.26993297e-01, 8.08421652e-01, 7.91527637e-01, + 7.75956852e-01, 7.61463389e-01, 7.47868622e-01, 7.35038092e-01, + 7.22867660e-01, 7.11274761e-01, 7.00192655e-01, 6.89566496e-01, + 6.79350572e-01, 6.69506317e-01, 6.60000841e-01, 6.50805833e-01, + 6.41896716e-01, 6.33251994e-01, 6.24852739e-01, 6.16682181e-01, + 6.08725382e-01, 6.00968966e-01, 5.93400902e-01, 5.86010318e-01, + 5.78787359e-01, 5.71723049e-01, 5.64809193e-01, 5.58038282e-01, + 5.51403417e-01, 5.44898238e-01, 5.38516872e-01, 5.32253880e-01, + 5.26104214e-01, 5.20063177e-01, 5.14126394e-01, 5.08289776e-01, + 5.02549502e-01, 4.96901987e-01, 4.91343870e-01, 4.85871987e-01, + 4.80483364e-01, 4.75175193e-01, 4.69944825e-01, 4.64789756e-01, + 4.59707616e-01, 4.54696157e-01, 4.49753251e-01, 4.44876873e-01, + 4.40065101e-01, 4.35316103e-01, 4.30628137e-01, 4.25999541e-01, + 4.21428729e-01, 4.16914186e-01, 4.12454466e-01, 4.08048183e-01, + 4.03694013e-01, 3.99390684e-01, 3.95136982e-01, 3.90931737e-01, + 3.86773829e-01, 3.82662181e-01, 3.78595759e-01, 3.74573568e-01, + 3.70594648e-01, 3.66658080e-01, 3.62762973e-01, 3.58908473e-01, + 3.55093753e-01, 3.51318016e-01, 3.47580495e-01, 3.43880445e-01, + 3.40217149e-01, 3.36589914e-01, 3.32998069e-01, 3.29440964e-01, + 3.25917972e-01, 3.22428485e-01, 3.18971913e-01, 3.15547685e-01, + 3.12155249e-01, 3.08794067e-01, 3.05463619e-01, 3.02163401e-01, + 2.98892921e-01, 2.95651704e-01, 2.92439288e-01, 2.89255223e-01, + 2.86099074e-01, 2.82970415e-01, 2.79868833e-01, 2.76793928e-01, + 2.73745310e-01, 2.70722597e-01, 2.67725420e-01, 2.64753419e-01, + 2.61806243e-01, 2.58883550e-01, 2.55985007e-01, 2.53110290e-01, + 2.50259082e-01, 2.47431076e-01, 2.44625969e-01, 2.41843469e-01, + 2.39083290e-01, 2.36345152e-01, 2.33628783e-01, 2.30933917e-01, + 2.28260294e-01, 2.25607660e-01, 2.22975768e-01, 2.20364376e-01, + 2.17773247e-01, 2.15202151e-01, 2.12650862e-01, 2.10119159e-01, + 2.07606828e-01, 2.05113656e-01, 2.02639439e-01, 2.00183975e-01, + 1.97747066e-01, 1.95328521e-01, 1.92928150e-01, 1.90545770e-01, + 1.88181199e-01, 1.85834263e-01, 1.83504787e-01, 1.81192603e-01, + 1.78897547e-01, 1.76619455e-01, 1.74358169e-01, 1.72113535e-01, + 1.69885401e-01, 1.67673619e-01, 1.65478042e-01, 1.63298529e-01, + 1.61134940e-01, 1.58987139e-01, 1.56854992e-01, 1.54738369e-01, + 1.52637142e-01, 1.50551185e-01, 1.48480376e-01, 1.46424594e-01, + 1.44383722e-01, 1.42357645e-01, 1.40346251e-01, 1.38349429e-01, + 1.36367071e-01, 1.34399072e-01, 1.32445328e-01, 1.30505738e-01, + 1.28580205e-01, 1.26668629e-01, 1.24770919e-01, 1.22886980e-01, + 1.21016722e-01, 1.19160057e-01, 1.17316899e-01, 1.15487164e-01, + 1.13670768e-01, 1.11867632e-01, 1.10077676e-01, 1.08300825e-01, + 1.06537004e-01, 1.04786139e-01, 1.03048160e-01, 1.01322997e-01, + 9.96105837e-02, 9.79108533e-02, 9.62237426e-02, 9.45491894e-02, + 9.28871336e-02, 9.12375166e-02, 8.96002819e-02, 8.79753745e-02, + 8.63627411e-02, 8.47623305e-02, 8.31740930e-02, 8.15979807e-02, + 8.00339475e-02, 7.84819492e-02, 7.69419432e-02, 7.54138887e-02, + 7.38977470e-02, 7.23934809e-02, 7.09010552e-02, 6.94204365e-02, + 6.79515934e-02, 6.64944964e-02, 6.50491178e-02, 6.36154320e-02, + 6.21934154e-02, 6.07830464e-02, 5.93843056e-02, 5.79971756e-02, + 5.66216413e-02, 5.52576897e-02, 5.39053102e-02, 5.25644946e-02, + 5.12352371e-02, 4.99175343e-02, 4.86113856e-02, 4.73167929e-02, + 4.60337611e-02, 4.47622977e-02, 4.35024136e-02, 4.22541224e-02, + 4.10174414e-02, 3.97923910e-02, 3.85789955e-02, 3.73772828e-02, + 3.61872848e-02, 3.50090377e-02, 3.38425822e-02, 3.26879635e-02, + 3.15452322e-02, 3.04144439e-02, 2.92956602e-02, 2.81889488e-02, + 2.70943838e-02, 2.60120466e-02, 2.49420264e-02, 2.38844205e-02, + 2.28393354e-02, 2.18068875e-02, 2.07872041e-02, 1.97804243e-02, + 1.87867007e-02, 1.78062004e-02, 1.68391068e-02, 1.58856218e-02, + 1.49459680e-02, 1.40203914e-02, 1.31091649e-02, 1.22125924e-02, + 1.13310136e-02, 1.04648102e-02, 9.61441364e-03, 8.78031499e-03, + 7.96307744e-03, 7.16335318e-03, 6.38190594e-03, 5.61964221e-03, + 4.87765598e-03, 4.15729512e-03, 3.46026478e-03, 2.78879879e-03, + 2.14596774e-03, 1.53629978e-03, 9.67269282e-04, 4.54134354e-04, +} + +const zigNorR float64 = 3.6541528853610087963519472518 +const zigNorInvR float64 = 0.27366123732975827203338247596 +const zigExpR float64 = 7.6971174701310497140446280481 +const zigNorRF float32 = 3.6541528853610087963519472518 +const zigNorInvRF float32 = 0.27366123732975827203338247596 +const zigExpRF float32 = 7.6971174701310497140446280481