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Enumerate all binomial outcomes at n=20 and p=0.05, then compare exact interval-coverage calculations with a fixed-seed simulation.

Provide a small bisection implementation, a valid cubic-root example, an invalid-bracket control, and a high-precision Newton comparison.

Enumerate every balanced assignment for two groups of four observations and compare the exact two-sided randomization p-value with a sampled estimate.

Resample a fixed twenty-value empirical distribution and compare bootstrap moments with their conditional analytical values.

Use rational arithmetic to compare composite quadrature errors for x^4 on [0,1] across five nested grid resolutions.

Investigate four step sizes for y'=-10y using the known amplification factor 1-10h, including stable, boundary, and growing discrete trajectories.

Compare unadjusted and Bonferroni thresholds on fixed-size families of synthetic uniform null p-values, with analytical familywise probabilities.

Sweep sixteen step sizes for forward and central differentiation of exp(x) at zero, exposing truncation and rounding behavior against the analytical derivative.

Measure loss of precision in sqrt(x+1)-sqrt(x) at x=10^16 and compare an algebraically rationalized expression with a high-precision reference.

Compare residuals near simple and double roots at the same candidate locations to illustrate why a small residual can have different interpretations.

Provide Dijkstra and Bellman–Ford implementations with hand-enumerated reference graphs, explicit negative-edge rejection, and a negative-cycle control.

Compare naive exponentiation with a shifted log-sum-exp calculation and a 70-digit Decimal reference on three fixed large inputs.