Average Error: 13.7 → 0.3
Time: 35.1s
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.336989694749254 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}} \cdot \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{1}{\frac{1 + wj}{\frac{x}{e^{wj}}}}\\ \end{array}\]

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

Original13.7
Target13.2
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 < 8.336989694749254e-05

    1. Initial program 13.4

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

    2. Initial simplification7.0

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

    3. Taylor expanded around 0 0.3

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

    5. Applied add-sqr-sqrt0.3

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

    if 8.336989694749254e-05 < wj

    1. Initial program 31.5

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

    2. Initial simplification0.9

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

    4. Applied clear-num0.9

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.336989694749254 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}} \cdot \sqrt{\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{1}{\frac{1 + wj}{\frac{x}{e^{wj}}}}\\ \end{array}\]

Runtime

Time bar (total: 35.1s)Debug logProfile

herbie shell --seed 2018220 
(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))))))