Average Error: 13.5 → 1.4
Time: 58.8s
Precision: 64
Internal Precision: 896
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{e^{wj}}}{1 + wj} + \left(wj \cdot wj - {wj}^{3}\right)\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.8
Herbie1.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  3. Applied div-sub13.5

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

  4. Applied simplify12.8

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

  5. Applied simplify12.8

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

  6. Taylor expanded around 0 14.1

    \[\leadsto wj - \left(\color{blue}{\left(\left({wj}^{3} + wj\right) - {wj}^{2}\right)} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)\]

  7. Applied simplify1.4

    \[\leadsto \color{blue}{\frac{\frac{x}{e^{wj}}}{1 + wj} + \left(wj \cdot wj - {wj}^{3}\right)}\]
  8. Removed slow pow expressions.

Runtime

Time bar (total: 58.8s)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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))))))