Average Error: 14.2 → 0.4
Time: 16.6s
Precision: 64
Internal Precision: 896
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.6937520875282744 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \left(\sqrt[3]{wj \cdot wj - {wj}^{3}} \cdot \sqrt[3]{wj \cdot wj - {wj}^{3}}\right) \cdot \sqrt[3]{wj \cdot wj - {wj}^{3}}\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{\frac{x}{1 + wj}}{e^{wj}}\right)\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original14.2
Target13.6
Herbie0.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 1.6937520875282744e-14

    1. Initial program 18.5

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

    3. Applied div-sub18.5

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

    4. Applied simplify18.5

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

    5. Applied simplify18.5

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

    6. Taylor expanded around 0 18.6

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

    7. Applied simplify0.2

      \[\leadsto \color{blue}{\frac{\frac{x}{e^{wj}}}{1 + wj} + \left(wj \cdot wj - {wj}^{3}\right)}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.4

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

    if 1.6937520875282744e-14 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 2.5

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

    3. Applied div-sub2.5

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

    4. Applied simplify0.4

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

    5. Applied simplify0.4

      \[\leadsto wj - \left(\frac{wj}{wj + 1} - \color{blue}{\frac{\frac{x}{1 + wj}}{e^{wj}}}\right)\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 16.6s)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))))))