Lambert W₋₁ Calculator

What does this lambert w₋₁ calculator compute?

The secondary real Lambert W function W_{-1}(x) solves W e^W = x for −1/e ≤ x < 0. W_{-1}(−1/e)=−1, W_{-1}(−2/e²)=−2, and W_{-1}(x)→−∞ as x→0⁻.

The published formula or method is W_{-1}(x) e^{W_{-1}(x)} = x. The following references support the stated mathematics or method:

• NIST DLMF §4.13 — Lambert W-Function: The inverse relation W exp(W)=x and the real principal and secondary branches.

Which inputs and limits should you check?

The mathematical function can have a wider domain than this calculator supports. Choose the intended function and branch, then check its input domain and published precision limits.

The worked example uses x = −0.1 with W_{-1}(−0.1). Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.special v1.10.6 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical precision and input policy

• Real, dimensionless binary64 inputs; publishes12 significant digits, no arbitrary-precision/exact claim.
• Default mode erf and Bessel order0 apply only when absent; explicit empty/null rejects. All supplied numeric aliases must agree; known irrelevant numeric parameters are parsed then ignored.
• Gamma poles reject. Nonzero Gamma that underflows rejects NUMERICAL_RANGE_ERROR; positive overflow suggests lgamma. lgamma is computed directly for x>0, using a local Taylor expansion near its zeros1 and2.
• Bessel integer n0–8, |x|≤80, Y/K require x>0; subnormal error and zeros limit relative accuracy.
• Lambert W0 and W-1 explicit real branches; binary64 -1/e designated endpoint, no tolerance band absorbs adjacent inputs.
• Beta shapes positive and at most1e6, except exact B(x,1)=1/x or B(1,y)=1/y. Incomplete beta shapes at most1e6. CF capped10000 steps; failure NO_CONVERGENCE.
• erf/erfc and beta tails can round to zero only on binary64 underflow, not an absolute display cutoff. Regularized incomplete beta evaluated separately in logs.
• Normalized sinc uses sinpi reduction and returns zero at representable nonzero integer x. Other tiny nonzero results retain significance.
• Numeric references have relative1e-10, with J/Y near zeros absolute2e-14,4 minimum subnormals otherwise. No global error proof for all real inputs.
• Browser interaction computes locally; parameterized share pages, REST and MCP send inputs to server. Logs/retention/region are deployment concerns; no zero-log promise.

Discovery pages lock their advertised function; the hub selects all14 operations. A shared link replays with the current engine and identifies its calculation version; it does not preserve a historical runtime. CSV and JSON exports retain the original decimal inputs, complete function result, formula and version.

Definitions and methods: NIST DLMF Gamma, incomplete beta, and Lambert W. CVP verification is separate from expert review and production freshness.