Average Error: 13.7 → 0.9
Time: 5.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 4.160404689641216852166604763104892916736 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 4.160404689641216852166604763104892916736 \cdot 10^{-9}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r166618 = wj;
        double r166619 = exp(r166618);
        double r166620 = r166618 * r166619;
        double r166621 = x;
        double r166622 = r166620 - r166621;
        double r166623 = r166619 + r166620;
        double r166624 = r166622 / r166623;
        double r166625 = r166618 - r166624;
        return r166625;
}
↓
double f(double wj, double x) {
        double r166626 = wj;
        double r166627 = 4.160404689641217e-09;
        bool r166628 = r166626 <= r166627;
        double r166629 = x;
        double r166630 = 2.0;
        double r166631 = pow(r166626, r166630);
        double r166632 = r166629 + r166631;
        double r166633 = r166626 * r166629;
        double r166634 = r166630 * r166633;
        double r166635 = r166632 - r166634;
        double r166636 = exp(r166626);
        double r166637 = 1.0;
        double r166638 = r166626 + r166637;
        double r166639 = r166636 * r166638;
        double r166640 = r166629 / r166639;
        double r166641 = r166640 + r166626;
        double r166642 = r166626 / r166638;
        double r166643 = r166641 - r166642;
        double r166644 = r166628 ? r166635 : r166643;
        return r166644;
}

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 < 4.160404689641217e-09

    1. Initial program 13.4

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 0.8

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

    if 4.160404689641217e-09 < wj

    1. Initial program 26.7

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

    2. Simplified2.9

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

    4. Applied div-inv2.9

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

    5. Applied associate-/l*2.9

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

    6. Simplified2.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.160404689641216852166604763104892916736 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

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