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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le -7.928590425983746 \cdot 10^{-09} \lor \neg \left(wj \le 3.974142294647761 \cdot 10^{-09}\right):\\ \;\;\;\;\left(\frac{\frac{x}{e^{wj}}}{1 + wj} - \frac{wj}{1 + wj}\right) + wj\\ \mathbf{else}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)\\ \end{array}\]

Original	13.9
Target	13.2
Herbie	0.3

Derivation

Split input into 2 regimes
if wj < -7.928590425983746e-09 or 3.974142294647761e-09 < wj
1. Initial program 18.3
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Using strategy rm
3. Applied sub-neg18.3
  \[\leadsto \color{blue}{wj + \left(-\frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\right)}\]
4. Applied simplify3.8
  \[\leadsto wj + \color{blue}{\left(\frac{\frac{x}{e^{wj}}}{wj + 1} - \frac{wj}{wj + 1}\right)}\]
if -7.928590425983746e-09 < wj < 3.974142294647761e-09
1. Initial program 13.6
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Taylor expanded around 0 0.1
  \[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(wj \cdot x\right)}\]
Recombined 2 regimes into one program.
Applied simplify0.3
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;wj \le -7.928590425983746 \cdot 10^{-09} \lor \neg \left(wj \le 3.974142294647761 \cdot 10^{-09}\right):\\ \;\;\;\;\left(\frac{\frac{x}{e^{wj}}}{1 + wj} - \frac{wj}{1 + wj}\right) + wj\\ \mathbf{else}:\\ \;\;\;\;\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)\\ \end{array}}\]

Runtime

herbie shell --seed 2018201 
(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

Try it out

Target

Derivation

`if wj < -7.928590425983746e-09 or 3.974142294647761e-09 < wj`

`if -7.928590425983746e-09 < wj < 3.974142294647761e-09`

Runtime

In
Out