Average Error: 13.3 → 2.2
Time: 41.7s
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[wj \cdot \left(wj + x \cdot -2\right) + x\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.3
Target12.7
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.3

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

  2. Taylor expanded around 0 2.2

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

  3. Simplified2.2

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

  4. Final simplification2.2

    \[\leadsto wj \cdot \left(wj + x \cdot -2\right) + x\]

Reproduce

herbie shell --seed 2019091 
(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))))))