Original	13.3
Target	12.7
Herbie	0.8

Derivation

Split input into 2 regimes
if wj < 4.663236460460694e-09
1. Initial program 13.0
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Taylor expanded around 0 0.8
  \[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)}\]
3. Simplified0.8
  \[\leadsto \color{blue}{(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*}\]
if 4.663236460460694e-09 < wj
1. Initial program 24.7
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Using strategy rm
3. Applied div-sub24.7
  \[\leadsto wj - \color{blue}{\left(\frac{wj \cdot e^{wj}}{e^{wj} + wj \cdot e^{wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
4. Simplified2.5
  \[\leadsto wj - \left(\color{blue}{\frac{wj}{1 + wj}} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]
5. Using strategy rm
6. Applied distribute-rgt1-in2.5
  \[\leadsto wj - \left(\frac{wj}{1 + wj} - \frac{x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\right)\]
7. Applied associate-/r*2.5
  \[\leadsto wj - \left(\frac{wj}{1 + wj} - \color{blue}{\frac{\frac{x}{wj + 1}}{e^{wj}}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.663236460460694 \cdot 10^{-09}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{wj + 1}}{e^{wj}}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018285 +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))))))

Error

Target

Derivation

`if wj < 4.663236460460694e-09`

`if 4.663236460460694e-09 < wj`

Runtime