Average Error: 13.4 → 2.3
Time: 27.1s
Precision: 64
Internal Precision: 128
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[(\left((-2 \cdot x + wj)_*\right) \cdot wj + x)_*\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.4
Target12.7
Herbie2.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Taylor expanded around 0 2.3

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

  3. Simplified2.3

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

  4. Final simplification2.3

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

Reproduce

herbie shell --seed 2019089 +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))))))