Distributions
Verification
Scope
Densities, masses, CDFs, survival functions, their logarithms, and quantiles of every family at saved points.
References
- Reference
- mpmath 1.4.1 at 60 digits, each value rounded once
- Versions
- mpmath 1.4.1, R 4.6.1 (2026-06-24), SciPy 1.18.1, NumPy 2.5.4, PHP 8.5.10
Data
- Inputs
- Saved cases and evaluation points for every family and operation, with R values and 60-digit mpmath references.
Method
Each case is evaluated by phit, R, and SciPy and compared with mpmath references computed at 60 digits and rounded once to double. A case passes when its largest relative error meets the case's rule.
- Check
- Comparison with a reference implementation
- Recorded
- 2026-11-01, PHP 8.5.10, macOS 27.0.1 (arm64), commit 066e2e1
Tolerances
|x − r| ≤ 1e-12 × max(|r|, 1e-300)
Applies to: Probabilities
Densities, masses, CDFs, survival functions
|x − r| ≤ 1e-12 × max(|r|, 1)
Applies to: Logarithms and quantiles
x: package value; r: reference value. Relative error: |x − r| / max(|r|, 1e-300 or 1). Max errors are over the values a row compares. r, scipy: each implementation’s error against the same reference.
Results
553 comparisons in 79 cases: 553 pass, 0 fail, 0 skipped.
By case
| Case | Comparisons | Largest rel. error | Status |
|---|---|---|---|
| Normal (mean = 0, sd = 1) | 7 | 6.7e-16 | pass |
| Normal (mean = 3, sd = 2) | 7 | 1.8e-14 | pass |
| Normal (mean = 1000000, sd = 1000) | 7 | 8.3e-15 | pass |
| Normal (mean = -2, sd = 1e-08) | 7 | 2.9e-14 | pass |
| Student's t (df = 0.5) | 7 | 9.2e-14 | pass |
| Student's t (df = 1) | 7 | 8.4e-15 | pass |
| Student's t (df = 2) | 7 | 6.0e-14 | pass |
| Student's t (df = 3.5) | 7 | 1.2e-14 | pass |
| Student's t (df = 12) | 7 | 3.9e-14 | pass |
| Student's t (df = 30) | 7 | 1.2e-13 | pass |
| Student's t (df = 100) | 7 | 2.7e-14 | pass |
| Student's t (df = 10000) | 7 | 1.8e-13 | pass |
| Student's t (df = 1000000) | 7 | 4.8e-14 | pass |
| Chi-square (df = 0.5) | 7 | 1.4e-13 | pass |
| Chi-square (df = 1) | 7 | 7.9e-14 | pass |
| Chi-square (df = 2) | 7 | 3.4e-14 | pass |
| Chi-square (df = 4) | 7 | 2.5e-14 | pass |
| Chi-square (df = 5) | 7 | 9.2e-14 | pass |
| Chi-square (df = 30) | 7 | 2.2e-14 | pass |
| Chi-square (df = 1000) | 7 | 1.5e-13 | pass |
| Chi-square (df = 1000000) | 7 | 8.3e-13 | pass |
| F (df1 = 4, df2 = 40) | 7 | 2.9e-14 | pass |
| F (df1 = 1, df2 = 1) | 7 | 2.9e-14 | pass |
| F (df1 = 0.5, df2 = 3) | 7 | 3.9e-14 | pass |
| F (df1 = 10, df2 = 1000000) | 7 | 3.2e-14 | pass |
| F (df1 = 100, df2 = 100) | 7 | 3.5e-14 | pass |
| F (df1 = 5, df2 = 0.5) | 7 | 1.6e-13 | pass |
| F (df1 = 2, df2 = 2) | 7 | 2.2e-14 | pass |
| Beta (alpha = 2, beta = 5) | 7 | 6.9e-14 | pass |
| Beta (alpha = 0.5, beta = 0.5) | 7 | 2.7e-14 | pass |
| Beta (alpha = 1, beta = 1) | 7 | 2.4e-14 | pass |
| Beta (alpha = 0.1, beta = 10) | 7 | 6.2e-14 | pass |
| Beta (alpha = 50, beta = 200) | 7 | 3.4e-14 | pass |
| Beta (alpha = 1000, beta = 1000) | 7 | 1.5e-13 | pass |
| Beta (alpha = 3, beta = 1) | 7 | 8.9e-14 | pass |
| Gamma (shape = 2, scale = 3) | 7 | 3.0e-14 | pass |
| Gamma (shape = 0.5, scale = 1) | 7 | 1.9e-14 | pass |
| Gamma (shape = 1, scale = 1) | 7 | 2.4e-14 | pass |
| Gamma (shape = 0.01, scale = 1) | 7 | 4.9e-14 | pass |
| Gamma (shape = 100, scale = 0.01) | 7 | 1.5e-13 | pass |
| Gamma (shape = 10000, scale = 2) | 7 | 7.5e-13 | pass |
| Gamma (shape = 3, scale = 100000) | 7 | 4.4e-14 | pass |
| Exponential (rate = 1) | 7 | 5.6e-17 | pass |
| Exponential (rate = 0.5) | 7 | 1.1e-16 | pass |
| Exponential (rate = 1e-05) | 7 | 5.6e-14 | pass |
| Exponential (rate = 100000) | 7 | 1.5e-14 | pass |
| Log-normal (meanlog = 0, sdlog = 1) | 7 | 1.3e-14 | pass |
| Log-normal (meanlog = 2, sdlog = 0.5) | 7 | 6.4e-15 | pass |
| Log-normal (meanlog = -5, sdlog = 3) | 7 | 9.1e-14 | pass |
| Logistic (location = 0, scale = 1) | 7 | 1.5e-16 | pass |
| Logistic (location = 5, scale = 2) | 7 | 7.3e-16 | pass |
| Logistic (location = 0, scale = 1e-05) | 7 | 1.5e-15 | pass |
| Binomial (size = 10, prob = 0.5) | 7 | 1.2e-15 | pass |
| Binomial (size = 1, prob = 0.3) | 7 | 1.6e-16 | pass |
| Binomial (size = 100, prob = 0.01) | 7 | 3.3e-14 | pass |
| Binomial (size = 1000, prob = 0.999) | 7 | 2.2e-16 | pass |
| Binomial (size = 1000000, prob = 0.3) | 7 | 7.8e-16 | pass |
| Binomial (size = 50, prob = 0) | 7 | 0 | pass |
| Binomial (size = 50, prob = 1) | 7 | 0 | pass |
| Binomial (size = 0, prob = 0.4) | 7 | 0 | pass |
| Poisson (lambda = 3.2) | 7 | 1.1e-14 | pass |
| Poisson (lambda = 1e-10) | 7 | 4.6e-14 | pass |
| Poisson (lambda = 0.5) | 7 | 5.2e-14 | pass |
| Poisson (lambda = 12) | 7 | 5.2e-14 | pass |
| Poisson (lambda = 85) | 7 | 3.7e-14 | pass |
| Poisson (lambda = 10000) | 7 | 5.2e-14 | pass |
| Poisson (lambda = 1000000) | 7 | 3.7e-13 | pass |
| Poisson (lambda = 0) | 7 | 0 | pass |
| Negative binomial (size = 3, prob = 0.4) | 7 | 6.8e-14 | pass |
| Negative binomial (size = 0.5, prob = 0.1) | 7 | 2.4e-15 | pass |
| Negative binomial (size = 100, prob = 0.99) | 7 | 2.1e-14 | pass |
| Negative binomial (size = 100000, prob = 0.5) | 7 | 9.7e-16 | pass |
| Negative binomial (size = 2, prob = 1) | 7 | 0 | pass |
| Negative binomial (mu = 4.5, size = 2) | 7 | 4.2e-14 | pass |
| Negative binomial (mu = 10, size = 0.5) | 7 | 5.1e-14 | pass |
| Negative binomial (mu = 3, size = 1000000) | 7 | 2.0e-14 | pass |
| Negative binomial (mu = 0.001, size = 10) | 7 | 2.2e-14 | pass |
| Negative binomial (mu = 100000, size = 1) | 7 | 2.4e-15 | pass |
| Negative binomial (mu = 0, size = 3) | 7 | 0 | pass |
All 553 comparisons
| Case | Quantity | Criterion | Values | Max rel. error | r | scipy | Status |
|---|---|---|---|---|---|---|---|
| Normal (mean = 0, sd = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 1.6e-16 | 1.6e-16 | 2.4e-16 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 0 | 0 | 1.6e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.8e-16 | 2.8e-16 | 1.1e-13 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.5e-16 | 1.1e-16 | 3.0e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.8e-16 | 2.8e-16 | 1.1e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.5e-16 | 6.9e-18 | 3.0e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 6.7e-16 | 2.3e-16 | 2.2e-16 | pass | |
| Normal (mean = 3, sd = 2) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 1.8e-14 | 2.2e-16 | 2.4e-16 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.5e-16 | 1.5e-16 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.8e-16 | 2.8e-16 | 1.1e-13 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.5e-16 | 1.1e-16 | 3.0e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.8e-16 | 2.8e-16 | 1.1e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.5e-16 | 6.9e-18 | 3.0e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.3e-15 | 8.4e-16 | 6.0e-16 | pass | |
| Normal (mean = 1000000, sd = 1000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 8.3e-15 | 2.3e-16 | 3.5e-16 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.8e-16 | 1.8e-16 | 1.8e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.8e-16 | 2.8e-16 | 1.1e-13 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.5e-16 | 1.2e-16 | 3.0e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.8e-16 | 2.8e-16 | 1.1e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.5e-16 | 1.2e-16 | 3.0e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 0 | pass | |
| Normal (mean = -2, sd = 1e-08) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.8e-14 | 4.2e-9 | 4.3e-10 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 3.6e-16 | 3.6e-16 | 3.6e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.9e-14 | 2.9e-14 | 1.4e-13 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.6e-16 | 1.6e-16 | 3.3e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 28 | 2.9e-14 | 2.9e-14 | 1.4e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 28 | 1.6e-16 | 1.6e-16 | 3.3e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 0 | pass | |
| Student's t (df = 0.5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 1.2e-15 | 1.1e-15 | 5.1e-16 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.0e-16 | 2.0e-16 | 1.7e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 2.8e-16 | 2.8e-16 | 2.2e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.9e-16 | 1.9e-16 | 1.9e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 2.4e-16 | 2.4e-16 | 3.8e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.9e-16 | 1.5e-16 | 1.9e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 9.2e-14 | 2.6e+284 | 1.0e+0 | pass | |
| Student's t (df = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 1.5e-15 | 2.0e-15 | 1.5e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.9e-16 | 1.7e-16 | 1.9e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 8.9e-16 | 8.9e-16 | 3.1e-9 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 6.4e-16 | 8.0e-16 | 3.1e-9 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 8.9e-16 | 8.9e-16 | 3.1e-9 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 6.4e-16 | 8.0e-16 | 3.1e-9 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 8.4e-15 | 1.8e-16 | 1.5e-16 | pass | |
| Student's t (df = 2) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 4.1e-15 | 6.4e-16 | 4.1e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.1e-16 | 2.1e-16 | 2.1e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 3.5e-16 | 1.8e-16 | 2.1e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.8e-17 | 3.0e-16 | 1.6e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 2.7e-16 | 1.6e-16 | 2.0e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.1e-16 | 3.0e-16 | 1.4e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 6.0e-14 | 2.1e-16 | 2.1e-16 | pass | |
| Student's t (df = 3.5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 3.6e-15 | 2.5e-15 | 3.6e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.4e-16 | 1.9e-16 | 1.4e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 7.1e-15 | 7.1e-15 | 2.7e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.7e-16 | 2.7e-16 | 1.1e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 7.1e-15 | 7.1e-15 | 4.2e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.7e-16 | 1.8e-16 | 1.1e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.2e-14 | 4.4e-8 | Inf | pass | |
| Student's t (df = 12) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 3.9e-14 | 7.9e-15 | 3.9e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 5.6e-16 | 3.0e-16 | 7.8e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 5.6e-15 | 5.6e-15 | 1.0e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.5e-16 | 2.1e-16 | 1.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 5.6e-15 | 5.6e-15 | 1.1e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.5e-16 | 2.5e-16 | 2.0e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.4e-15 | 3.6e-12 | Inf | pass | |
| Student's t (df = 30) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 1.2e-13 | 5.2e-14 | 1.2e-13 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.7e-16 | 3.5e-16 | 2.2e-15 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 3.8e-14 | 2.7e-14 | 3.3e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.9e-16 | 2.9e-16 | 1.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 3.8e-14 | 2.7e-14 | 3.3e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 1.9e-16 | 2.3e-16 | 1.6e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 4.3e-15 | 3.7e-15 | 1.7e-16 | pass | |
| Student's t (df = 100) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 1.1e-14 | 1.1e-14 | 3.1e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 3.1e-16 | 2.0e-16 | 3.1e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 2.7e-14 | 2.5e-15 | 1.0e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 4.8e-16 | 6.1e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 2.7e-14 | 5.3e-15 | 4.4e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 4.8e-16 | 6.1e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 5.6e-16 | 2.9e-15 | 1.9e-16 | pass | |
| Student's t (df = 10000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 1.6e-14 | 2.8e-14 | 2.1e-12 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 6.7e-16 | 1.6e-16 | 2.1e-12 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 8.7e-16 | 9.9e-15 | 4.5e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 4.8e-16 | 6.0e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 1.8e-13 | 7.8e-14 | 1.0e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 4.8e-16 | 6.0e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 6.5e-16 | 5.0e-16 | 2.2e-16 | pass | |
| Student's t (df = 1000000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 1.2e-14 | 2.6e-14 | 1.2e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 6.7e-16 | 1.8e-16 | 3.3e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 8.8e-15 | 3.7e-15 | 5.0e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 2.7e-16 | 3.5e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 19 | 4.8e-14 | 8.8e-15 | 5.6e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 19 | 3.5e-16 | 1.2e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 3.5e-16 | 4.4e-16 | 1.9e-16 | pass | |
| Chi-square (df = 0.5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 1.4e-13 | 4.2e-16 | 3.5e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 2.2e-16 | 2.2e-16 | 1.9e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 8.6e-15 | 1.7e-16 | 9.7e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 1.4e-16 | 5.6e-17 | 3.9e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 2.2e-14 | 1.4e-15 | 2.3e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 5.8e-16 | 2.2e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.4e-14 | 1.2e-14 | 7.9e-15 | pass | |
| Chi-square (df = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 7.9e-14 | 2.8e-16 | 3.4e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 3.9e-16 | 2.1e-16 | 1.7e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 6.4e-16 | 1.3e-15 | 2.5e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 6.7e-16 | 1.3e-15 | 4.8e-15 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 5.7e-14 | 7.2e-15 | 2.6e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 6.7e-16 | 3.8e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.1e-14 | 8.7e-15 | 1.9e-14 | pass | |
| Chi-square (df = 2) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 23 | 1.8e-15 | 0 | 1.8e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 23 | 1.1e-16 | 1.1e-16 | 1.1e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 23 | 3.4e-14 | 3.4e-14 | 3.4e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 23 | 1.5e-16 | 1.5e-16 | 2.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 23 | 2.6e-14 | 1.2e-16 | 2.6e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 23 | 1.1e-16 | 1.4e-17 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.4e-14 | 3.1e-14 | 3.9e-15 | pass | |
| Chi-square (df = 4) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 2.4e-14 | 3.8e-16 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 2.1e-16 | 2.1e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 2.5e-14 | 2.1e-16 | 2.5e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 1.1e-16 | 2.2e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 4.0e-15 | 4.3e-16 | 4.0e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 1.2e-16 | 2.1e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.5e-14 | 1.3e-14 | 1.5e-14 | pass | |
| Chi-square (df = 5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 23 | 3.6e-14 | 2.8e-16 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 23 | 2.2e-16 | 2.0e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 23 | 9.2e-14 | 2.5e-16 | 9.0e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 23 | 2.7e-15 | 2.0e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 23 | 7.5e-15 | 3.7e-16 | 7.5e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 23 | 2.2e-15 | 2.0e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.6e-14 | 1.8e-14 | 1.6e-14 | pass | |
| Chi-square (df = 30) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 2.2e-14 | 1.2e-15 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 5.1e-16 | 2.0e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 2.0e-14 | 7.4e-16 | 2.7e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 7.1e-16 | 2.2e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 6.5e-15 | 1.3e-15 | 1.0e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 1.9e-16 | 2.2e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.5e-15 | 5.0e-15 | 1.8e-15 | pass | |
| Chi-square (df = 1000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 1.5e-13 | 2.7e-14 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 2.0e-15 | 1.3e-15 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 3.1e-14 | 8.2e-14 | 3.0e-13 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 6.2e-16 | 1.6e-15 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 22 | 1.8e-14 | 2.0e-14 | 2.7e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 22 | 1.2e-15 | 1.5e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 7.1e-16 | 5.2e-16 | 1.8e-16 | pass | |
| Chi-square (df = 1000000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 2.7e-15 | 1.5e-11 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 6.0e-16 | 7.3e-13 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 1.1e-13 | 2.4e-15 | 8.2e-9 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 7.4e-15 | 1.5e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 24 | 4.2e-13 | 1.5e-14 | 2.8e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 24 | 8.3e-13 | 2.7e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.3e-14 | 1.2e-16 | 4.1e-14 | pass | |
| F (df1 = 4, df2 = 40) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 2.9e-14 | 2.3e-14 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 5.1e-16 | 2.0e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 1.7e-15 | 1.3e-15 | 1.1e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 5.0e-16 | 5.0e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 2.4e-14 | 2.4e-14 | 1.0e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 6.7e-16 | 2.2e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.1e-14 | 1.0e+0 | Inf | pass | |
| F (df1 = 1, df2 = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 4.0e-15 | Inf | 2.4e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 2.5e-16 | Inf | 2.2e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 8.9e-16 | 8.9e-16 | 6.4e-11 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 8.9e-16 | 1.0e-15 | 2.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 8.9e-16 | 8.9e-16 | 2.6e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 8.9e-16 | 1.0e-15 | 2.2e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.9e-14 | 1.0e+0 | 1.8e-16 | pass | |
| F (df1 = 0.5, df2 = 3) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.9e-14 | Inf | 1.8e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 2.1e-16 | Inf | 2.7e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 2.6e-15 | 2.6e-15 | 5.2e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 2.2e-16 | 2.2e-16 | 2.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 4.3e-15 | 4.3e-15 | 6.5e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 4.7e-16 | 3.3e-16 | 3.0e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.5e-14 | 1.0e+0 | 9.9e-16 | pass | |
| F (df1 = 10, df2 = 1000000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.2e-14 | 3.3e-14 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 6.9e-16 | 4.1e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 1.1e-14 | 9.9e-15 | 2.1e-11 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 1.1e-14 | 3.6e-15 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.2e-14 | 8.1e-14 | 2.7e-11 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 1.4e-14 | 3.0e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 5.9e-15 | 4.4e-5 | Inf | pass | |
| F (df1 = 100, df2 = 100) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 2.4e-14 | 2.6e-14 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 1.6e-15 | 1.8e-15 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 6.8e-15 | 6.8e-15 | 7.4e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 4.4e-16 | 6.7e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.5e-14 | 3.5e-14 | 1.6e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 4.4e-16 | 6.7e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 3.3e-16 | 4.3e-10 | 3.4e-16 | pass | |
| F (df1 = 5, df2 = 0.5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 1.6e-13 | 1.5e-13 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 3.3e-16 | 1.7e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 2.1e-15 | 4.9e-15 | 7.0e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 4.4e-16 | 4.4e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.3e-16 | 3.3e-16 | 2.3e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 2.8e-16 | 3.3e-16 | 2.2e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.6e-14 | 1.0e+0 | 1.3e-15 | pass | |
| F (df1 = 2, df2 = 2) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 1.4e-15 | 1.4e-16 | 1.4e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 1.1e-16 | 2.8e-17 | 2.5e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 2.2e-14 | 2.2e-14 | 1.7e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 7.5e-16 | 7.5e-16 | 1.7e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.2e-14 | 1.0e+0 | 0 | pass | |
| Beta (alpha = 2, beta = 5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 6.9e-14 | 6.9e-14 | 1.6e-13 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 1.1e-15 | 7.1e-16 | 7.1e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 3.3e-14 | 3.3e-14 | 6.4e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 4.1e-16 | 3.3e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 1.2e-15 | 2.4e-15 | 5.3e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 3.3e-16 | 3.3e-16 | 5.6e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 3.8e-14 | 2.2e-14 | 1.6e+100 | pass | |
| Beta (alpha = 0.5, beta = 0.5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 2.7e-14 | 2.7e-14 | 2.1e-16 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 5.7e-16 | 5.7e-16 | 3.3e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 8.9e-16 | 8.9e-16 | 2.2e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 8.9e-16 | 1.0e-15 | 2.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 8.9e-16 | 8.9e-16 | 2.6e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 8.9e-16 | 1.0e-15 | 2.2e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 4.5e-15 | 3.7e-16 | 1.7e-16 | pass | |
| Beta (alpha = 1, beta = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 0 | 0 | 1.3e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 2.4e-14 | 2.4e-14 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 3.5e-18 | 3.5e-18 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 5.2e-16 | 5.2e-16 | 1.6e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 1.4e-17 | 1.4e-17 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.4e-14 | 0 | 0 | pass | |
| Beta (alpha = 0.1, beta = 10) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 6.2e-14 | 6.2e-14 | 1.7e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 4.4e-16 | 4.4e-16 | 1.3e-15 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 5.3e-15 | 5.3e-15 | 4.2e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 1.6e-16 | 5.6e-17 | 2.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 7.8e-15 | 7.8e-15 | 2.3e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 2.5e-16 | 1.7e-16 | 5.0e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.0e-14 | 1.1e-14 | 2.6e-15 | pass | |
| Beta (alpha = 50, beta = 200) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 3.4e-14 | 3.4e-14 | 1.3e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 4.6e-16 | 4.6e-16 | 8.5e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 9.4e-15 | 9.4e-15 | 4.8e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 1.7e-15 | 8.9e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 1.8e-14 | 3.8e-14 | 1.2e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 1.8e-15 | 8.9e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 6.2e-16 | 7.6e-16 | 3.8e-15 | pass | |
| Beta (alpha = 1000, beta = 1000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 1.5e-13 | 1.5e-13 | 1.2e-13 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 8.3e-16 | 8.3e-16 | 5.4e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 1.2e-13 | 1.2e-13 | 1.2e-13 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 6.6e-16 | 2.2e-15 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 4.1e-14 | 4.1e-14 | 3.4e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 4.2e-16 | 2.2e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 7.8e-16 | 2.1e-16 | 8.0e-16 | pass | |
| Beta (alpha = 3, beta = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 2.4e-14 | 2.4e-14 | 9.1e-16 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 1.3e-16 | 1.3e-16 | 1.3e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 8.9e-14 | 3.5e-14 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 2.1e-16 | 2.2e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 16 | 6.9e-16 | 6.9e-16 | 1.7e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 16 | 1.7e-16 | 1.7e-16 | 1.7e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.8e-14 | 8.6e-15 | 1.3e-14 | pass | |
| Gamma (shape = 2, scale = 3) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 3.0e-14 | 2.8e-16 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 2.0e-16 | 2.0e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 1.1e-14 | 2.1e-16 | 1.1e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 1.1e-16 | 1.7e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 6.7e-16 | 3.0e-16 | 1.2e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 1.1e-16 | 1.1e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.5e-14 | 1.3e-14 | 1.5e-14 | pass | |
| Gamma (shape = 0.5, scale = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 1.9e-14 | 1.9e-16 | 1.9e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 2.8e-16 | 1.6e-16 | 1.5e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 6.4e-16 | 1.3e-15 | 4.8e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 6.7e-16 | 1.3e-15 | 4.8e-15 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 2.7e-15 | 7.2e-15 | 2.6e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 6.7e-16 | 3.8e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.1e-14 | 8.7e-15 | 1.9e-14 | pass | |
| Gamma (shape = 1, scale = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 0 | 0 | 0 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 2.4e-14 | 2.4e-14 | 2.4e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 1.9e-16 | 1.9e-16 | 2.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 1.1e-15 | 1.2e-16 | 1.1e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 1.1e-16 | 1.4e-17 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.4e-14 | 3.1e-14 | 3.9e-15 | pass | |
| Gamma (shape = 0.01, scale = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 4.9e-14 | 6.8e-16 | 4.9e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 6.7e-16 | 5.6e-16 | 4.4e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 4.3e-16 | 1.3e-16 | 7.0e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 1.9e-16 | 2.8e-17 | 5.8e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 1.4e-15 | 1.6e-15 | 8.9e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 4.9e-16 | 2.9e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 4.1e-14 | 4.2e-14 | 4.6e-14 | pass | |
| Gamma (shape = 100, scale = 0.01) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 1.5e-13 | 1.9e-14 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 8.4e-16 | 2.1e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 1.1e-13 | 1.2e-14 | 2.4e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 3.1e-16 | 2.2e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 7.4e-14 | 3.9e-14 | 4.7e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 2.2e-16 | 3.3e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 9.2e-16 | 2.9e-15 | 3.4e-16 | pass | |
| Gamma (shape = 10000, scale = 2) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 1.9e-13 | 1.6e-16 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 3.5e-16 | 2.2e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 2.5e-14 | 4.7e-15 | 1.7e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 1.6e-15 | 2.3e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 7.5e-13 | 5.0e-16 | 1.9e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 5.5e-15 | 2.2e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 3.1e-15 | 2.3e-16 | 1.7e-16 | pass | |
| Gamma (shape = 3, scale = 100000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 2.1e-14 | 3.9e-15 | Inf | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 1.9e-16 | 1.9e-16 | Inf | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 4.4e-14 | 3.5e-16 | 4.6e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 2.0e-16 | 1.8e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 17 | 2.7e-15 | 3.9e-15 | 4.7e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 17 | 2.2e-16 | 2.4e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.7e-14 | 1.9e-14 | 1.7e-14 | pass | |
| Exponential (rate = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 0 | pass | |
| Exponential (rate = 0.5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.1e-16 | 1.1e-16 | 1.1e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 0 | pass | |
| Exponential (rate = 1e-05) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 6.1e-15 | 6.1e-15 | 6.1e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.7e-16 | 1.7e-16 | 1.7e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.4e-16 | 1.7e-16 | 1.7e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.6e-17 | 1.1e-16 | 1.1e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 5.6e-14 | 5.6e-14 | 5.6e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 0 | 1.1e-16 | 1.1e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.0e-16 | 2.1e-16 | 2.1e-16 | pass | |
| Exponential (rate = 100000) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.5e-14 | 1.4e-14 | 1.4e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 4.4e-16 | 4.4e-16 | 4.4e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 1.7e-16 | 1.7e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.1e-16 | 1.1e-16 | 1.1e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.5e-14 | 1.5e-14 | 1.5e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 0 | 1.5e-16 | 1.5e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.0e-16 | 2.1e-16 | 2.1e-16 | pass | |
| Log-normal (meanlog = 0, sdlog = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.7e-15 | 8.2e-16 | 1.7e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.8e-16 | 1.8e-16 | 1.8e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 3.9e-16 | 3.9e-16 | 7.3e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.2e-16 | 1.2e-16 | 1.9e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 7.7e-16 | 7.7e-16 | 6.2e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.2e-16 | 1.2e-16 | 1.9e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.3e-14 | 2.7e-15 | 2.3e-15 | pass | |
| Log-normal (meanlog = 2, sdlog = 0.5) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.3e-15 | 1.6e-15 | 1.3e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 2.0e-16 | 2.1e-16 | 1.7e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 5.8e-16 | 5.8e-16 | 7.5e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.2e-16 | 1.7e-16 | 2.3e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.7e-15 | 1.7e-15 | 8.7e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.9e-16 | 1.9e-16 | 1.9e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 6.4e-15 | 1.3e-15 | 1.1e-15 | pass | |
| Log-normal (meanlog = -5, sdlog = 3) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 9.1e-14 | 1.0e+0 | 2.2e-14 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 2.9e-16 | 2.0e-16 | 1.8e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.9e-16 | 1.9e-16 | 6.7e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.4e-16 | 1.2e-16 | 2.8e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 3.9e-16 | 3.9e-16 | 6.6e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.2e-16 | 1.2e-16 | 1.9e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 3.5e-14 | 9.6e-15 | 6.5e-15 | pass | |
| Logistic (location = 0, scale = 1) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.5e-16 | 1.5e-16 | 0 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 1.5e-16 | 0 | 0 | pass | |
| Logistic (location = 5, scale = 2) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.5e-16 | 1.5e-16 | 0 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.6e-17 | 5.6e-17 | 5.6e-17 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 7.3e-16 | 7.3e-16 | 7.3e-16 | pass | |
| Logistic (location = 0, scale = 1e-05) | Density | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.4e-15 | 1.4e-15 | 1.4e-15 | pass |
| Log density | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.4e-16 | 1.8e-16 | 1.4e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.5e-15 | 1.5e-15 | 1.5e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.8e-16 | 1.8e-16 | 1.8e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.5e-15 | 1.5e-15 | 1.5e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.8e-16 | 1.8e-16 | 1.8e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 2.0e-16 | 1.8e-16 | 1.8e-16 | pass | |
| Binomial (size = 10, prob = 0.5) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 11 | 1.2e-15 | 7.1e-16 | 8.9e-16 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 11 | 6.2e-16 | 2.1e-16 | 6.3e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 11 | 6.5e-16 | 6.5e-16 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 11 | 2.0e-16 | 2.0e-16 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 11 | 5.1e-16 | 5.1e-16 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 11 | 2.2e-16 | 2.2e-16 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 1.0e+300 | pass | |
| Binomial (size = 1, prob = 0.3) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 9 | 0 | 0 | 3.2e-16 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 9 | 1.1e-16 | 0 | 1.1e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 9 | 1.6e-16 | 1.6e-16 | 1.6e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 9 | 0 | 0 | 1.1e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 9 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 9 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 1.0e+300 | pass | |
| Binomial (size = 100, prob = 0.01) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 3.3e-14 | 2.5e-14 | 5.5e-15 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.7e-16 | 1.7e-16 | 1.9e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.5e-16 | 3.0e-16 | 1.2e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.2e-16 | 2.2e-16 | 1.5e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.3e-14 | 7.0e-14 | 2.2e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 3.3e-16 | 5.3e-16 | 8.8e-16 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 5.9e-2 | 1.0e+300 | pass | |
| Binomial (size = 1000, prob = 0.999) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.5e-16 | 6.0e-16 | 1.5e-16 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.2e-16 | 6.7e-16 | 9.9e-13 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.6e-16 | 1.3e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 1.0e+300 | pass | |
| Binomial (size = 1000000, prob = 0.3) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 3.7e-16 | 6.2e-16 | 8.0e-12 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 5.0e-16 | 3.3e-16 | 1.3e-11 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 6.7e-16 | 2.2e-16 | 3.6e-12 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 7.8e-16 | 2.2e-15 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 7.8e-16 | 2.2e-16 | 3.6e-12 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 7.8e-16 | 2.2e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 4.2e-4 | 1.0e+300 | pass | |
| Binomial (size = 50, prob = 0) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 5.0e+301 | 5.0e+301 | pass | |
| Binomial (size = 50, prob = 1) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 1.0e+300 | pass | |
| Binomial (size = 0, prob = 0.4) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 8 | 0 | 0 | 0 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 8 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 8 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 8 | 0 | 0 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 8 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 8 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 1.0e+300 | pass | |
| Poisson (lambda = 3.2) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.1e-14 | 7.0e-15 | 4.6e-14 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 4.1e-16 | 1.5e-16 | 3.5e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.8e-16 | 1.8e-16 | 2.9e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 2.2e-16 | 1.3e-16 | 3.3e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 3.3e-15 | 6.9e-15 | 3.6e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 3.9e-16 | 1.7e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 3.8e-2 | 1.0e+300 | pass | |
| Poisson (lambda = 1e-10) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 10 | 4.6e-14 | 4.8e-16 | 4.6e-14 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 10 | 1.7e-16 | 1.7e-16 | 1.5e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 10 | 0 | 1.1e-16 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 10 | 1.3e-25 | 1.3e-25 | 8.3e-18 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 10 | 1.7e-14 | 5.6e-16 | 2.8e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 10 | 1.7e-16 | 2.0e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 1.0e+300 | pass | |
| Poisson (lambda = 0.5) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.5e-14 | 2.3e-16 | 4.5e-15 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.0e-16 | 2.0e-16 | 2.1e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.1e-16 | 1.1e-16 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.1e-16 | 1.4e-17 | 2.8e-17 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 5.2e-14 | 3.6e-16 | 2.7e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.1e-16 | 2.1e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 7.1e-2 | 1.0e+300 | pass | |
| Poisson (lambda = 12) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 5.2e-14 | 3.0e-16 | 1.1e-13 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 2.3e-16 | 2.0e-16 | 8.1e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 2.5e-15 | 1.9e-16 | 6.2e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 2.6e-15 | 1.9e-16 | 5.6e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 3.7e-14 | 4.3e-16 | 1.1e-13 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 3.4e-15 | 4.8e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 4.0e-2 | 1.0e+300 | pass | |
| Poisson (lambda = 85) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 1.9e-14 | 1.9e-16 | 1.1e-13 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 2.0e-16 | 1.9e-16 | 1.4e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 5.5e-15 | 1.3e-15 | 5.9e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 1.9e-16 | 1.3e-15 | 1.9e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 13 | 3.7e-14 | 1.4e-15 | 9.7e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 13 | 3.4e-16 | 1.4e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 1.8e-2 | 1.0e+300 | pass | |
| Poisson (lambda = 10000) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.5e-15 | 1.6e-16 | 1.1e-11 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 2.2e-15 | 1.9e-16 | 2.0e-13 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 5.2e-14 | 1.0e-14 | 1.4e-14 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 1.2e-15 | 2.6e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.2e-14 | 0 | 8.3e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 3.2e-14 | 2.1e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 2.6e-3 | 1.0e+300 | pass | |
| Poisson (lambda = 1000000) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 2.8e-15 | 5.3e-16 | 2.4e-9 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 2.0e-16 | 1.2e-16 | 8.7e-11 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 8.8e-14 | 2.2e-14 | 1.9e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 4.0e-14 | 4.0e-16 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 14 | 3.1e-13 | 7.9e-15 | 2.6e-10 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 14 | 3.7e-13 | 4.3e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 2.7e-4 | 1.0e+300 | pass | |
| Poisson (lambda = 0) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 10 | 0 | 0 | 0 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 10 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 10 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 10 | 0 | 0 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 10 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 10 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | Inf | pass | |
| Negative binomial (size = 3, prob = 0.4) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 6.8e-14 | 4.3e-14 | 2.3e-14 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.3e-16 | 2.8e-16 | 4.6e-15 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 3.5e-16 | 1.7e-16 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 3.9e-16 | 1.9e-16 | 1.9e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.5e-14 | 2.5e-14 | 1.8e-16 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.2e-16 | 1.6e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 6.0e-2 | 1.0e+300 | pass | |
| Negative binomial (size = 0.5, prob = 0.1) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.7e-15 | 2.2e-15 | 1.3e-15 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.1e-16 | 2.1e-16 | 1.1e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.0e-16 | 3.2e-16 | 2.0e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.9e-16 | 1.7e-16 | 2.2e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.4e-15 | 1.3e-14 | 1.6e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 3.9e-16 | 3.9e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 6.6e-2 | 1.0e+300 | pass | |
| Negative binomial (size = 100, prob = 0.99) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 5.6e-15 | 9.8e-15 | 1.4e-14 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.1e-16 | 4.4e-16 | 2.0e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 4.5e-16 | 4.5e-16 | 1.5e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 5.6e-16 | 6.1e-16 | 5.6e-17 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.1e-14 | 2.1e-14 | 1.4e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.2e-15 | 1.3e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 5.6e-2 | 1.0e+300 | pass | |
| Negative binomial (size = 100000, prob = 0.5) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 9.7e-16 | 8.5e-16 | 1.2e-16 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.1e-16 | 2.2e-16 | 1.0e-10 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.2e-16 | 2.2e-16 | 2.2e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.1e-16 | 2.0e-15 | Inf | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.7e-16 | 2.1e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 1.2e-3 | 1.0e+300 | pass | |
| Negative binomial (size = 2, prob = 1) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | Inf | pass | |
| Negative binomial (mu = 4.5, size = 2) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 6.4e-15 | 8.6e-15 | 9.1e-15 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 4.0e-16 | 3.2e-16 | 1.9e-15 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.5e-16 | 2.5e-16 | 1.5e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.2e-16 | 2.2e-16 | 1.6e-22 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 4.2e-14 | 4.2e-14 | 1.2e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 4.0e-16 | 3.5e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 5.6e-2 | 1.0e+300 | pass | |
| Negative binomial (mu = 10, size = 0.5) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.1e-14 | 3.2e-14 | 2.6e-14 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 3.0e-16 | 3.0e-16 | 2.3e-14 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.2e-16 | 2.2e-16 | 1.7e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.1e-16 | 1.1e-16 | 4.4e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 5.1e-14 | 2.4e-14 | 3.2e-14 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.7e-16 | 1.9e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 6.5e-2 | 1.0e+300 | pass | |
| Negative binomial (mu = 3, size = 1000000) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.2e-14 | 2.5e-11 | 5.7e-10 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.5e-16 | 1.3e-11 | 7.2e-11 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 4.5e-15 | 1.4e-15 | 3.6e-11 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 4.5e-15 | 1.4e-15 | 1.9e-11 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.0e-14 | 6.3e-15 | 5.8e-10 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.2e-14 | 3.7e-15 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 3.8e-2 | 1.0e+300 | pass | |
| Negative binomial (mu = 0.001, size = 10) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.5e-14 | 5.3e-14 | 4.7e-11 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 2.5e-16 | 2.5e-16 | 1.4e-13 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 1.0e-15 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 8.7e-19 | 1.1e-18 | 9.4e-16 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.2e-14 | 6.8e-14 | 4.8e-11 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 3.2e-16 | 2.0e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | 1.0e+300 | pass | |
| Negative binomial (mu = 100000, size = 1) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 2.4e-15 | 1.9e-15 | 6.8e-16 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.8e-16 | 1.8e-16 | 1.8e-16 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.1e-15 | 1.1e-15 | 2.5e-16 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 1.9e-16 | 1.9e-16 | 3.2e-12 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 1.2e-15 | 1.6e-15 | 1.0e-15 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 5.5e-16 | 2.2e-16 | Inf | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 6.2e-2 | 1.0e+300 | pass | |
| Negative binomial (mu = 0, size = 3) | Probability mass | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass |
| Log probability mass | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Lower tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log lower tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Upper tail | |x − r| ≤ 1e-12 × max(|r|, 1e-300) | 12 | 0 | 0 | 0 | pass | |
| Log upper tail | |x − r| ≤ 1e-12 × max(|r|, 1) | 12 | 0 | 0 | 0 | pass | |
| Quantile | |x − r| ≤ 1e-12 × max(|r|, 1) | 15 | 0 | 0 | Inf | pass |
Limitations
Exclusions
- Points where mpmath has no reference are counted per case and not compared.
- Documented boundary conventions are asserted separately, in the test-suite record.
Known differences
- For a point mass, the discrete quantile at p = 1 is the mass location; R and SciPy may return the upper support bound.
- R's F density at +INF is NAN; phit returns 0.
- R F quantiles can round to zero for tiny probabilities and small numerator df.
- R's Student-t quantile at df = 0.5 has a small nonzero center and some far-tail inaccuracy.
- R's discrete-quantile adjustment near 1 − 2^-52 can shift the count; phit uses a relative 64·EPSILON allowance on the smaller tail.
- R's mean-form negative binomial mass differs from the high-precision value when size is much larger than the count.
- SciPy's large-df chi-square CDF has about 8e-9 relative error at the saved lower-tail point.
Reproduction
From the repository root
cargo build -p phit --release --locked
.context/notebooks-venv/bin/python docs/verifications/compare_distributions.py- Raw record
- site/verifications/distributions.json
- Full results
- docs/verifications/distributions-results.json
- Input data
- tests/fixtures/distributions-mpmath.json