Math calculator

Lambert W₋₁ Calculator

Q: Is this a second special-functions engine?

No. It is mode=lambert_wm1 on math.special, the same engine as /calc/math/special-functions.

Q: Does this compute incomplete beta?

No. This seed is the secondary real branch W_{-1} for −1/e ≤ x < 0. Incomplete beta is on /calc/math/incomplete-beta. Not CAS.

Q: Where does this run?

Locally in the browser by default. REST and MCP call the same special-functions engine.

Real W_{-1}(x) for −1/e ≤ x < 0. Same engine as Special Functions. Not incomplete beta. Not CAS. Runs locally.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes

Input interpretation: Enter values to calculate.
Result: —
Verified scope: all published modes
Assurance: Core
Declared partition coverage: PASS · 15/15 declared partitions (erf, erfc, gamma, lgamma, bessel-j, bessel-y, bessel-i, bessel-k, lambert-w, lambert-wm1, beta, incomplete-beta, sinc, normalized-sinc, invalid-domain) · Matrix
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: Secondary real Lambert W_{-1}(x) for −1/e ≤ x < 0, with an engine-linked Live graph of y = W₋₁(x).
Scope: The Result Card includes an engine-linked Live graph of y = W₋₁(x) using the same Overview / Focus template as erf (Overview window −1/e ≤ x ≤ −0.02 so the branch does not run to −∞). Same Halley solver as REST/MCP. W_0 is on /lambert-w. Not CAS.
Verification: Engine tested · Specification checked · v1.10.2 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions: Calculation 1.10.2 · CVP protocol 1.0.0-proposed · Evidence 2026-10-08.math-response-consistency
Verification revision: 2026-10-08.math-response-consistency · 1/1 property · digest 0b2836762d99
Legacy regression: 58/58 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference: O1 model · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Special response contract also checks all 14 mode formulas and unrounded numeric summaries, including erfc/K tails and a subnormal erf input; UI is shared presentation/SSR, not a live browser session.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 482/482 golden · 8/8 CVP boundary · 10/10 invalid · 1/1 property · 3/3 metamorphic · 24/24 cross-interface · 1/1 cross-calculator · 6/6 CVP contract · Manifest
Specification basis: ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence: 13 legacy golden · 8 legacy boundary · legacy regression suite · 482/482 oracle-backed golden · 10/10 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

DefinitionW_{-1}(x) e^{W_{-1}(x)} = x

Domain−1/e ≤ x < 0

iSame engine as /special-functions. Not a second engine. Not /wm1. Incomplete beta and normalized sinc later on this engine.

How to use

Enter x

Finite real x with −1/e ≤ x < 0. Default x=−0.1.

Read W_{-1}(x)

W_{-1}(−0.1) ≈ −3.57715206396. W_{-1}(−2/e²)=−2. x=0 is not real. The Live graph shows y = W₋₁(x) on −1/e ≤ x < 0.

Example calculations

Common configurations with formula and result.

W_{-1}(−0.1)

x = −0.1

W_{-1}(−0.1)

−3.57715206396

W_{-1}(−2/e²)

x = −2/e²

W_{-1}(−2/e²) = −2

−2