Error

Target

Derivation

1. Split input into 2 regimes

2. if wj < -8.76421065063732e-09

   1. Initial program 4.5

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt4.8

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\sqrt{e^{wj}} \cdot \sqrt{e^{wj}}} + wj \cdot e^{wj}}\]

   4. Applied fma-def4.8

      \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{(\left(\sqrt{e^{wj}}\right) \cdot \left(\sqrt{e^{wj}}\right) + \left(wj \cdot e^{wj}\right))_*}}\]

   if -8.76421065063732e-09 < wj

   1. Initial program 13.7

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]

   2. Taylor expanded around 0 1.4

      \[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)}\]

   3. Simplified1.5

      \[\leadsto \color{blue}{(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -8.76421065063732 \cdot 10^{-09}:\\ \;\;\;\;wj - \frac{e^{wj} \cdot wj - x}{(\left(\sqrt{e^{wj}}\right) \cdot \left(\sqrt{e^{wj}}\right) + \left(e^{wj} \cdot wj\right))_*}\\ \mathbf{else}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \end{array}\]

Runtime

Time bar (total: 36.5s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))