Average Error: 13.7 → 0.2
Time: 2.8m
Precision: 64
Internal Precision: 832
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 1.5297742957850337 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}}}{1 + wj} + \left({wj}^{4} + \left(1 - wj\right) \cdot \left(wj \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{wj \cdot wj - \frac{wj}{1 + wj} \cdot \frac{wj}{1 + wj}}{wj + \frac{wj}{1 + wj}} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\\ \end{array}\]

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.0
Herbie0.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 1.5297742957850337e-09

    1. Initial program 17.8

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

    3. Applied div-sub17.8

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

    4. Applied associate--r-9.6

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

    5. Applied simplify9.6

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

    6. Taylor expanded around 0 0.2

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

    7. Applied simplify0.2

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

    if 1.5297742957850337e-09 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (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 associate--r-2.6

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

    5. Applied simplify0.2

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

    7. Applied flip--0.2

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

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1072743783 989954326 4239155542 3782239461 3602631542 1719177920)' 
(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))))))