Average Error: 13.7 → 0.9
Time: 19.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 6.728609179253453840938316012033545976401 \cdot 10^{-9}:\\ \;\;\;\;\left(x + wj \cdot wj\right) + \left(x \cdot wj\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}} - wj}{1 - wj \cdot wj} \cdot \left(1 - wj\right) + wj\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 6.728609179253453840938316012033545976401 \cdot 10^{-9}:\\
\;\;\;\;\left(x + wj \cdot wj\right) + \left(x \cdot wj\right) \cdot -2\\

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

\end{array}
double f(double wj, double x) {
        double r300628 = wj;
        double r300629 = exp(r300628);
        double r300630 = r300628 * r300629;
        double r300631 = x;
        double r300632 = r300630 - r300631;
        double r300633 = r300629 + r300630;
        double r300634 = r300632 / r300633;
        double r300635 = r300628 - r300634;
        return r300635;
}
↓
double f(double wj, double x) {
        double r300636 = wj;
        double r300637 = 6.728609179253454e-09;
        bool r300638 = r300636 <= r300637;
        double r300639 = x;
        double r300640 = r300636 * r300636;
        double r300641 = r300639 + r300640;
        double r300642 = r300639 * r300636;
        double r300643 = -2.0;
        double r300644 = r300642 * r300643;
        double r300645 = r300641 + r300644;
        double r300646 = exp(r300636);
        double r300647 = r300639 / r300646;
        double r300648 = r300647 - r300636;
        double r300649 = 1.0;
        double r300650 = r300649 - r300640;
        double r300651 = r300648 / r300650;
        double r300652 = r300649 - r300636;
        double r300653 = r300651 * r300652;
        double r300654 = r300653 + r300636;
        double r300655 = r300638 ? r300645 : r300654;
        return r300655;
}

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

Derivation

  1. Split input into 2 regimes

  2. if wj < 6.728609179253454e-09

    1. Initial program 13.4

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 0.8

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

    4. Simplified0.8

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

    if 6.728609179253454e-09 < wj

    1. Initial program 24.5

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

    2. Simplified2.9

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

    4. Applied flip-+2.9

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

    5. Applied associate-/r/2.9

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

    6. Simplified2.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 6.728609179253453840938316012033545976401 \cdot 10^{-9}:\\ \;\;\;\;\left(x + wj \cdot wj\right) + \left(x \cdot wj\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{wj}} - wj}{1 - wj \cdot wj} \cdot \left(1 - wj\right) + wj\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))))))