Average Error: 13.6 → 0.4
Time: 21.8s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 0.38137679030448973:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj \cdot wj - 1} \cdot \left(wj - 1\right) + \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}} \cdot \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \left(\sqrt[3]{\frac{\frac{x}{e^{wj}}}{1 + wj}} \cdot \sqrt[3]{\frac{\frac{x}{e^{wj}}}{1 + wj}}\right) \cdot \sqrt[3]{\frac{\frac{x}{e^{wj}}}{1 + wj}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.0
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 < 0.38137679030448973

    1. Initial program 13.2

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

    2. Initial simplification6.9

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

    3. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
    4. Using strategy rm

    5. Applied flip-+0.4

      \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{\color{blue}{\frac{wj \cdot wj - 1 \cdot 1}{wj - 1}}}\]

    6. Applied associate-/r/0.4

      \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\frac{\frac{x}{e^{wj}}}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt0.4

      \[\leadsto \color{blue}{\sqrt{\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}} \cdot \sqrt{\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}}} + \frac{\frac{x}{e^{wj}}}{wj \cdot wj - 1 \cdot 1} \cdot \left(wj - 1\right)\]

    if 0.38137679030448973 < wj

    1. Initial program 42.5

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

    2. Initial simplification0.0

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

    4. Applied add-cube-cbrt0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 0.38137679030448973:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{wj \cdot wj - 1} \cdot \left(wj - 1\right) + \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}} \cdot \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \left(\sqrt[3]{\frac{\frac{x}{e^{wj}}}{1 + wj}} \cdot \sqrt[3]{\frac{\frac{x}{e^{wj}}}{1 + wj}}\right) \cdot \sqrt[3]{\frac{\frac{x}{e^{wj}}}{1 + wj}}\\ \end{array}\]

Runtime

Time bar (total: 21.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.10.40.11.068.8%
herbie shell --seed 2018286 
(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))))))