Average Error: 13.6 → 1.1
Time: 1.2m
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*} + (\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4}\right))_*\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.6
Target12.9
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.6

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
  2. Using strategy rm

  3. Applied div-sub13.6

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

  4. Applied associate--r-7.1

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

  5. Applied simplify6.5

    \[\leadsto \color{blue}{\left(wj - \frac{wj}{wj + 1}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]

  6. Taylor expanded around 0 1.1

    \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]

  7. Applied simplify1.1

    \[\leadsto \color{blue}{\frac{x}{(\left(e^{wj}\right) \cdot wj + \left(e^{wj}\right))_*} + (\left(1 - wj\right) \cdot \left(wj \cdot wj\right) + \left({wj}^{4}\right))_*}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2019053 +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))))))