SPECIAL FUNCTIONS / INVERSE EXPONENTIAL PRODUCT
Solve w exp(w) = x on the principal real branch and expose the numerical residual.
METHOD / WORKED EXAMPLE
Lambert W reverses the combined operation w exp(w), not the exponential alone. The residual makes the numerical root auditable and the branch label prevents two real solutions from being silently conflated.
WORKED DEFAULT
For x = 1, the principal solution is approximately 0.5671432904. Substitution gives 0.5671432904 x exp(0.5671432904), which returns 1 within floating-point residual.
READ THE RESULT
A small residual supports numerical convergence for the entered equation. It does not prove another branch was evaluated or guarantee arbitrary-precision digits beyond JavaScript number accuracy.
ASSUMPTIONS AND LIMITS
Only W0 is returned. Inputs between -1/e and zero also have a second real W-1 solution that this page deliberately does not calculate.
COMMON QUESTIONS
The product w exp(w) reaches its real minimum at w = -1, where the value is -1/e. Below that input there is no real solution. At the branch point both real branches meet at -1. Between -1/e and zero, W0 and W-1 are different real solutions, so the branch must be stated explicitly.
No. This page returns only W0, the principal real branch. For x between -1/e and zero, the decreasing branch W-1 is also real and is at or below -1. Using W0 where a model requires W-1 can change the result substantially. Choose a branch from the underlying equation and domain, not from whichever value looks convenient.
The residual is the absolute difference between the entered x and W exp(W) reconstructed from the numerical answer. A value near floating-point scale indicates the equation was solved consistently. It is not an uncertainty estimate for measured inputs, and displayed decimal agreement does not create more precision than the source data or JavaScript arithmetic supports.
RELATED TOOLS
Use boundary
Use a branch-aware starting value and Halley's method to solve w exp(w) - x = 0, stopping when successive values converge at floating-point precision.
Find w such that w x exp(w) = x on the principal real branch.