Average Error: 13.6 → 1.1
Time: 35.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.034798015591916 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\ \mathbf{elif}\;wj \le 706.4071446376307:\\ \;\;\;\;wj - \frac{1}{\frac{e^{wj} + e^{wj} \cdot wj}{e^{wj} \cdot wj - x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot wj} + wj\right) - 1\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.034798015591916 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\

\mathbf{elif}\;wj \le 706.4071446376307:\\
\;\;\;\;wj - \frac{1}{\frac{e^{wj} + e^{wj} \cdot wj}{e^{wj} \cdot wj - x}}\\

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

\end{array}
double f(double wj, double x) {
        double r3638629 = wj;
        double r3638630 = exp(r3638629);
        double r3638631 = r3638629 * r3638630;
        double r3638632 = x;
        double r3638633 = r3638631 - r3638632;
        double r3638634 = r3638630 + r3638631;
        double r3638635 = r3638633 / r3638634;
        double r3638636 = r3638629 - r3638635;
        return r3638636;
}
↓
double f(double wj, double x) {
        double r3638637 = wj;
        double r3638638 = 9.034798015591916e-09;
        bool r3638639 = r3638637 <= r3638638;
        double r3638640 = x;
        double r3638641 = -2.0;
        double r3638642 = r3638640 * r3638641;
        double r3638643 = r3638642 + r3638637;
        double r3638644 = r3638637 * r3638643;
        double r3638645 = r3638640 + r3638644;
        double r3638646 = 706.4071446376307;
        bool r3638647 = r3638637 <= r3638646;
        double r3638648 = 1.0;
        double r3638649 = exp(r3638637);
        double r3638650 = r3638649 * r3638637;
        double r3638651 = r3638649 + r3638650;
        double r3638652 = r3638650 - r3638640;
        double r3638653 = r3638651 / r3638652;
        double r3638654 = r3638648 / r3638653;
        double r3638655 = r3638637 - r3638654;
        double r3638656 = r3638640 / r3638650;
        double r3638657 = r3638656 + r3638637;
        double r3638658 = r3638657 - r3638648;
        double r3638659 = r3638647 ? r3638655 : r3638658;
        double r3638660 = r3638639 ? r3638645 : r3638659;
        return r3638660;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target13.1
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if wj < 9.034798015591916e-09

    1. Initial program 13.4

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

    2. Taylor expanded around 0 0.9

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

    3. Simplified0.9

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

    if 9.034798015591916e-09 < wj < 706.4071446376307

    1. Initial program 4.5

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

    3. Applied clear-num4.4

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

    if 706.4071446376307 < wj

    1. Initial program 62.0

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

    2. Taylor expanded around inf 14.8

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.034798015591916 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\ \mathbf{elif}\;wj \le 706.4071446376307:\\ \;\;\;\;wj - \frac{1}{\frac{e^{wj} + e^{wj} \cdot wj}{e^{wj} \cdot wj - x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot wj} + wj\right) - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(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))))))