Average Error: 13.7 → 0.9
Time: 5.9s
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 r310581 = wj;
        double r310582 = exp(r310581);
        double r310583 = r310581 * r310582;
        double r310584 = x;
        double r310585 = r310583 - r310584;
        double r310586 = r310582 + r310583;
        double r310587 = r310585 / r310586;
        double r310588 = r310581 - r310587;
        return r310588;
}
↓
double f(double wj, double x) {
        double r310589 = wj;
        double r310590 = 4.160404689641217e-09;
        bool r310591 = r310589 <= r310590;
        double r310592 = x;
        double r310593 = 2.0;
        double r310594 = pow(r310589, r310593);
        double r310595 = r310592 + r310594;
        double r310596 = r310589 * r310592;
        double r310597 = r310593 * r310596;
        double r310598 = r310595 - r310597;
        double r310599 = exp(r310589);
        double r310600 = 1.0;
        double r310601 = r310589 + r310600;
        double r310602 = r310599 * r310601;
        double r310603 = r310592 / r310602;
        double r310604 = r310603 + r310589;
        double r310605 = r310589 / r310601;
        double r310606 = r310604 - r310605;
        double r310607 = r310591 ? r310598 : r310606;
        return r310607;
}

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 +o rules:numerics
(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))))))