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Accuracy

Every row below is checked by a test. The tests compare against independent reference data in test/reference/. That folder is for development and is not published. The references are mpmath (arbitrary precision) for decimal-math, the special functions and the distributions, and the NIST StRD certified values for statistics and regression. Seeded generator scripts rebuild the data.

The guarantees come in three kinds:

• Exact. No rounding happens unless you ask for it: Decimal and Rational arithmetic, convertExact, exact finance, and integer functions.
• Correctly rounded. The result is the true value rounded once, in the mode you chose. The tests compare digit for digit, and test/reference/ERROR-ANALYSIS.md has the proof.
• Float bounds. These are the largest error seen over every reference point, tails included. They are measured, not proven. A change that exceeds one fails the build.

Decimal math (@arki/math/decimal-math)

Each case runs in all 8 rounding modes, at scales from 0 to 100. In unnecessary mode, an inexact value must throw.

• Near-tie cases sit within 10^-(scale+14) to 10^-(scale+90) of a rounding boundary. They force the Ziv loop to retry with 20, 40, 80 and 160 guard digits.
• Exact cases are roots that land exactly on a tie, and logarithms or powers with exact results.

Function	Guarantee	Evidence: cases (random / near-tie / exact)
sqrt	correctly rounded; exact or RoundingNecessaryError without options	120 / 12 / 12
cbrt	correctly rounded; negative input gives a negative root	80 / 12 / 12
nthRoot (n 4–12)	correctly rounded	80 / 12 / 12
exp	correctly rounded	120 / 32 / —
ln	correctly rounded	120 / 32 / —
log10, log2	correctly rounded; exact results detected	159 / 64 / 24
log (any base)	correctly rounded; exact results detected	60 / — / 24
pow (non-integer)	correctly rounded; exact results detected	114 / 32 / 30
PI, E	correctly rounded (half-even)	Machin and Taylor kernels audited in the error analysis

Special functions and distributions (@arki/math/distributions)

The error is relative: |got − want| / |want|. The noncentral t is the exception, measured in absolute error. One ulp is about 1.1e-16 to 2.2e-16. The reference values go down to about 1e-300. Quantiles are tested at p from 1e-300 up to 1 − 1e-15.

Function	Max error	Points	Range covered
erf	4e-16 (2 ulp)	180	|x| from 1e-300 to 6
erfc	5e-16 (3 ulp)	149	x from −6 to 26.5 (values down to 6e-296)
erfInv	5e-16 (2 ulp)	186	p from 1e-300 to 1 − 1e-15
gamma	3e-13	75	x from 1e-300 to 171.6, and negative non-integers
gammaLn	1e-13	180	x from 1e-300 to 1e300
beta, betaLn	2e-12, 5e-13	336	a, b from 1e-3 to 1e3
regularizedGammaP	1e-12	271	a from 0.01 to 1000, x from a/1000 to 10a
regularizedBeta	2e-11	800	a, b from 0.1 to 1000, x from 0.001 to 0.999
normal cdf, sf / pdf	3e-13 / 2e-13	570	z from −37 to 37
normal quantile	5e-16 (2 ulp)	68	p from 1e-300 to 1 − 1e-15
studentT cdf, sf / pdf	5e-12 / 1e-12	1 560	df from 1 to 1000, |t| up to 1e8
studentT quantile	3e-12	224	p from 1e-100 to 1 − 1e-12
chiSquare cdf, sf, pdf	2e-12	836	df from 1 to 1000, x from df/10⁴ to 50·df
chiSquare quantile	2e-12	224	p from 1e-100 to 1 − 1e-12
fisherF cdf, sf, pdf	3e-13	693	(df1, df2) from (1, 1) to (100, 100), x from 1e-4 to 1e4
fisherF quantile	3e-12	175	p from 1e-60 to 1 − 1e-10
noncentralT cdf	2e-14 absolute	174	df from 5 to 50, ncp from −2 to 10
exponential cdf, sf, pdf, quantile	correctly rounded at every point (bound 2.3e-16)	211	x from 1e-10 to 316; p down to 1e-300
binomial pmf, cdf, sf	2e-12	1 022	n up to 1000, p from 0.01 to 0.9, the whole support
poisson pmf, cdf, sf	8e-12	1 558	λ from 0.5 to 1000, out to λ + 40√λ + 40
binomial, poisson quantile	exact (the smallest k with cdf(k) ≥ p)	360	random p

The noncentral t (AS 243) is accurate in absolute terms. Where its cdf is at least 1e-3, the relative error is at most 1.2e-12. Far down the lower tail, the relative error grows: 4e-7 where the cdf is 1e-8. For comparison, scipy (Boost) is off by up to 1.7e-8 relative on the same points.

Statistics and regression (NIST StRD)

The figures are digits of agreement with the certified values, out of the 15 that NIST certifies. Higher is better. The measure is the log relative error, LRE = −log₁₀(relative error).

Dataset (difficulty)	Function	Digits on number	On Decimal
PiDigits, Lottery, Lew, NumAcc1 (lower)	mean, stddev	15, 15	equal to the certified value
Mavro, Michelso (lower)	mean / stddev	15 / ≥ 13	equal to the certified value
NumAcc2 (average)	mean / stddev	15 / ≥ 14.5	equal to the certified value
NumAcc3 (average), NumAcc4 (higher)	mean / stddev	15 / ≥ 9, ≥ 8	equal to the certified value
Norris (lower)	linearRegression: slope, intercept, their standard errors, residual SD, r²	≥ 12.5	—
Pontius (lower, quadratic)	polynomialRegression	≥ 11.5	—
Filip (higher, degree 10)	polynomialRegression	≥ 13.5	—
Wampler1–5 (higher, degree 5)	polynomialRegression	≥ 9, 12, 9, 9, 7.5	—
Longley (higher, 6 predictors); NoInt1, NoInt2 (average)	leastSquares	≥ 10.5; ≥ 14	—

On Decimal input, mean is exact and stddev is correctly rounded. Both reproduce every certified value digit for digit.

On number input, NumAcc3 and NumAcc4 are limited by the data, not by the algorithm: values such as 10000000.2 are not exact doubles. Against the exact standard deviation of the binary inputs, stddev keeps at least 14 digits on every dataset.

The polynomial and least-squares rows hold on both V8 and JavaScriptCore.