Average Error: 13.6 → 0.5
Time: 16.1s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 2.7842014216165472 \cdot 10^{-14}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \left(x + \left(-2 + wj \cdot \frac{5}{2}\right) \cdot \left(x \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{wj - \frac{wj}{wj + 1}} \cdot \sqrt[3]{wj - \frac{wj}{wj + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{wj - \frac{wj}{wj + 1}} \cdot \left(\sqrt[3]{wj - \frac{wj}{wj + 1}} \cdot \sqrt[3]{wj - \frac{wj}{wj + 1}}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.6
Target13.0
Herbie0.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 2.7842014216165472e-14

    1. Initial program 18.0

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

    2. Initial simplification9.1

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

    3. Taylor expanded around 0 0.1

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

    4. Taylor expanded around 0 0.4

      \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(x + \frac{5}{2} \cdot \left(x \cdot {wj}^{2}\right)\right) - 2 \cdot \left(x \cdot wj\right)\right)}\]

    5. Simplified0.4

      \[\leadsto \left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right) + \color{blue}{\left(\left(wj \cdot \frac{5}{2} + -2\right) \cdot \left(x \cdot wj\right) + x\right)}\]

    if 2.7842014216165472e-14 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 2.4

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

    2. Initial simplification0.4

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

    4. Applied add-cube-cbrt0.5

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

    6. Applied add-cbrt-cube0.5

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 2.7842014216165472 \cdot 10^{-14}:\\ \;\;\;\;\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \left(x + \left(-2 + wj \cdot \frac{5}{2}\right) \cdot \left(x \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{wj - \frac{wj}{wj + 1}} \cdot \sqrt[3]{wj - \frac{wj}{wj + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{wj - \frac{wj}{wj + 1}} \cdot \left(\sqrt[3]{wj - \frac{wj}{wj + 1}} \cdot \sqrt[3]{wj - \frac{wj}{wj + 1}}\right)} + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Runtime

Time bar (total: 16.1s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes1.40.50.11.471.1%
herbie shell --seed 2018296 
(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))))))