Fix the accuracy of ERF and ERFC near zero and for large arguments

[marcin-kordas-hoc]

Summary

ERF and ERFC lost precision for arguments close to zero and for large arguments. Excel is accurate there, so the results differed from Excel's:

Formula	Before	Excel / exact	Now
ERF(1e-10)	1.128379612e-10 (relative error 3.9e-7)	1.12837916709551e-10	agrees
ERF(1e-300) (raw function)	about 1.1e-16	1.12837916709551e-300	agrees
ERFC(5)	1.5374368445e-12 (relative error 1.5e-5)	1.53745979442803e-12	agrees

The cause was erf returning 1 - exp(...) (every digit below about 1e-16 is lost) and erfc being 1 - erf. erf is now computed from its series below 2 (all terms positive, no cancellation) and from the continued fraction of erfc above; erfc uses the continued fraction from 1 upwards. Relative error against Python's math.erf and math.erfc over 46 arguments from 1e-300 to 27 (negative ones included) is at most 7e-16 for erf and 3e-15 for erfc (a prototype of the same formulas, measured before they went into the repository).

The change is in the vendored jstat.ts, so the normal and log-normal distributions that call erf and erfc (NORM.DIST, NORM.INV through erfcinv, LOGNORM.DIST, LOGNORM.INV) are more accurate as well. The existing tests for them still pass unchanged.

Behaviour check against Excel

71 ERF and ERFC formulas were evaluated in Excel Online (MS Graph live session) and in HyperFormula, from 1e-300 to 27, with negative arguments and the two-argument form of ERF: all agree within a relative 4e-11, which is the 11 significant digits HyperFormula returns by default against Excel's 15. Two of them need a note:

• ERF(1e-300) and ERFC(27) return NaN from the engine, not from erf: with the default smartRounding, any result below about 1e-290 becomes NaN, and so does a plain =1e-300*1. The raw functions return the right values (1.1283791670955126e-300 and 5.23705e-319). That is a separate defect of the number rounding and is not changed here, so the tests start at 1e-100.
• The uppercase exponent form 1E-10 does not parse on develop (it is fixed in the GESTEP pull request), so the cases use 1e-10.

Tests

Paired tests branch: fix/erf-accuracy-near-zero in the tests repository: 29 Excel-measured cases added to the ERF and ERFC specs, 12 or more of which fail on the previous implementation. Full local run: 502 suites, 6262 passed, 3 skipped.

🤖 Generated with Claude Code

---

Note

Medium Risk
Changes numeric outputs for ERF/ERFC and dependent statistical functions across many formulas; behavior is intentional and Excel-aligned but may shift edge-case results in existing sheets.

Overview
Improves numeric accuracy of ERF and ERFC so tiny and large arguments match Excel instead of losing precision from 1 - exp(...) and erfc = 1 - erf.

The vendored jstat.ts implementation is replaced with piecewise algorithms: a positive-term series for small |x|, a continued-fraction erfc for larger values, and explicit handling for NaN, sign, and underflow (e.g. erfc → 0 for very large x). Normal and log-normal distribution helpers that call these primitives inherit the same accuracy. The unreleased changelog documents the fix (#1795).

^{Reviewed by Cursor Bugbot for commit 6fc292b. Bugbot is set up for automated code reviews on this repo. Configure here.}

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Status	Name	Latest Commit	Preview URL	Updated (UTC)
✅ Deployment successful! View logs	hyperformula-docs	6fc292b	Commit Preview URL Branch Preview URL	Oct 07 2026, 06:25 AM

[github-actions]

Performance comparison of head (6fc292b) vs base (5abbbd9)

                                     testName |    base |    head | change
--------------------------------------------------------------------------
                                      Sheet A |  489.51 |  491.03 | +0.31%
                                      Sheet B |  158.88 |  162.37 | +2.20%
                                      Sheet T |   137.4 |  141.58 | +3.04%
                                Column ranges |  466.64 |  471.93 | +1.13%
                                Sorted lookup | 14145.4 | 14196.5 | +0.36%
Sheet A:  change value, add/remove row/column |   16.07 |   16.59 | +3.24%
 Sheet B: change value, add/remove row/column |  129.31 |  134.06 | +3.67%
                   Column ranges - add column |   148.3 |  152.59 | +2.89%
                Column ranges - without batch |  469.39 |  482.88 | +2.87%
                        Column ranges - batch |  113.49 |  118.42 | +4.34%

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 97.32%. Comparing base (5abbbd9) to head (6fc292b).

Additional details and impacted files

@@           Coverage Diff            @@
##           develop    #1795   +/-   ##
========================================
  Coverage    97.32%   97.32%           
========================================
  Files          195      195           
  Lines        15739    15753   +14     
  Branches      3461     3499   +38     
========================================
+ Hits         15318    15332   +14     
  Misses         413      413           
  Partials         8        8           

Files with missing lines	Coverage Δ
src/interpreter/plugin/3rdparty/jstat/jstat.ts	88.48% <100.00%> (+0.30%)	⬆️

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