Original	13.3
Target	12.7
Herbie	1.0

Derivation

Split input into 2 regimes
if wj < 1.7395295982610156e-13
1. Initial program 12.9
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
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 1.7395295982610156e-13 < wj
1. Initial program 22.7
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Using strategy rm
3. Applied distribute-rgt1-in22.7
  \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\]
4. Applied *-un-lft-identity22.7
  \[\leadsto wj - \frac{\color{blue}{1 \cdot \left(wj \cdot e^{wj} - x\right)}}{\left(wj + 1\right) \cdot e^{wj}}\]
5. Applied times-frac22.6
  \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]
6. Simplified5.5
  \[\leadsto wj - \frac{1}{wj + 1} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\]
Recombined 2 regimes into one program.
Final simplification1.0
\[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.7395295982610156 \cdot 10^{-13}:\\ \;\;\;\;(wj \cdot \left((x \cdot -2 + wj)_*\right) + x)_*\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{1}{wj + 1} \cdot \left(wj - \frac{x}{e^{wj}}\right)\\ \end{array}\]

Runtime

herbie shell --seed 2018250 +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 < 1.7395295982610156e-13`

`if 1.7395295982610156e-13 < wj`

Runtime