Average Error: 13.5 → 0.3
Time: 22.0s
Precision: 64
Internal Precision: 896
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \left(\frac{wj}{wj + 1} - \frac{1}{\sqrt[3]{e^{wj}} \cdot \sqrt[3]{e^{wj}}} \cdot \frac{\frac{x}{1 + wj}}{\sqrt[3]{e^{wj}}}\right) \le 7.565976676171613 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \left(wj \cdot wj - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(\frac{wj}{wj + 1} - \frac{1}{\sqrt[3]{e^{wj}} \cdot \sqrt[3]{e^{wj}}} \cdot \frac{\frac{x}{1 + wj}}{\sqrt[3]{e^{wj}}}\right)\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.9
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (- (/ wj (+ wj 1)) (* (/ 1 (* (cbrt (exp wj)) (cbrt (exp wj)))) (/ (/ x (+ 1 wj)) (cbrt (exp wj)))))) < 7.565976676171613e-13

    1. Initial program 17.7

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

    3. Applied div-sub17.7

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

    4. Applied simplify17.7

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

    5. Applied simplify17.7

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

    6. Taylor expanded around 0 17.9

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

    7. Applied simplify0.2

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

    if 7.565976676171613e-13 < (- wj (- (/ wj (+ wj 1)) (* (/ 1 (* (cbrt (exp wj)) (cbrt (exp wj)))) (/ (/ x (+ 1 wj)) (cbrt (exp wj))))))

    1. Initial program 2.6

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

    3. Applied div-sub2.6

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

    4. Applied simplify0.4

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

    5. Applied simplify0.4

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

    7. Applied add-cube-cbrt0.5

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

    8. Applied *-un-lft-identity0.5

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

    9. Applied times-frac0.5

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

Runtime

Time bar (total: 22.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(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))))))