Average Error: 13.9 → 0.8
Time: 21.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 4.306602228527943150027854119990031549921 \cdot 10^{-16}:\\ \;\;\;\;x + \left({wj}^{2} - x \cdot \left(2 \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1} + wj\right) + \left(wj \cdot wj - wj\right) \cdot \frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 4.306602228527943150027854119990031549921 \cdot 10^{-16}:\\
\;\;\;\;x + \left({wj}^{2} - x \cdot \left(2 \cdot wj\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1} + wj\right) + \left(wj \cdot wj - wj\right) \cdot \frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1}\\

\end{array}
double f(double wj, double x) {
        double r121620 = wj;
        double r121621 = exp(r121620);
        double r121622 = r121620 * r121621;
        double r121623 = x;
        double r121624 = r121622 - r121623;
        double r121625 = r121621 + r121622;
        double r121626 = r121624 / r121625;
        double r121627 = r121620 - r121626;
        return r121627;
}
↓
double f(double wj, double x) {
        double r121628 = wj;
        double r121629 = 4.306602228527943e-16;
        bool r121630 = r121628 <= r121629;
        double r121631 = x;
        double r121632 = 2.0;
        double r121633 = pow(r121628, r121632);
        double r121634 = r121632 * r121628;
        double r121635 = r121631 * r121634;
        double r121636 = r121633 - r121635;
        double r121637 = r121631 + r121636;
        double r121638 = exp(r121628);
        double r121639 = r121631 / r121638;
        double r121640 = r121639 - r121628;
        double r121641 = 3.0;
        double r121642 = pow(r121628, r121641);
        double r121643 = 1.0;
        double r121644 = r121642 + r121643;
        double r121645 = r121640 / r121644;
        double r121646 = r121645 + r121628;
        double r121647 = r121628 * r121628;
        double r121648 = r121647 - r121628;
        double r121649 = r121648 * r121645;
        double r121650 = r121646 + r121649;
        double r121651 = r121630 ? r121637 : r121650;
        return r121651;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.2
Herbie0.8
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 4.306602228527943e-16

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 0.7

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

    4. Simplified0.7

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

    if 4.306602228527943e-16 < wj

    1. Initial program 22.5

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

    2. Simplified6.0

      \[\leadsto \color{blue}{wj + \frac{\frac{x}{e^{wj}} - \frac{wj}{1}}{1 + wj}}\]
    3. Using strategy rm

    4. Applied flip3-+6.0

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

    5. Applied associate-/r/6.0

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

    6. Simplified6.0

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

    8. Applied distribute-lft-in5.9

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

    9. Applied associate-+r+1.8

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

    10. Simplified1.8

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.306602228527943150027854119990031549921 \cdot 10^{-16}:\\ \;\;\;\;x + \left({wj}^{2} - x \cdot \left(2 \cdot wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1} + wj\right) + \left(wj \cdot wj - wj\right) \cdot \frac{\frac{x}{e^{wj}} - wj}{{wj}^{3} + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))