Average Error: 13.8 → 0.3
Time: 5.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 4.811043021165421 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 4.811043021165421 \cdot 10^{-8}:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\\

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

\end{array}
double f(double wj, double x) {
        double r151542 = wj;
        double r151543 = exp(r151542);
        double r151544 = r151542 * r151543;
        double r151545 = x;
        double r151546 = r151544 - r151545;
        double r151547 = r151543 + r151544;
        double r151548 = r151546 / r151547;
        double r151549 = r151542 - r151548;
        return r151549;
}
↓
double f(double wj, double x) {
        double r151550 = wj;
        double r151551 = 4.811043021165421e-08;
        bool r151552 = r151550 <= r151551;
        double r151553 = x;
        double r151554 = 1.0;
        double r151555 = r151550 + r151554;
        double r151556 = r151553 / r151555;
        double r151557 = exp(r151550);
        double r151558 = r151556 / r151557;
        double r151559 = 4.0;
        double r151560 = pow(r151550, r151559);
        double r151561 = 3.0;
        double r151562 = pow(r151550, r151561);
        double r151563 = r151560 - r151562;
        double r151564 = fma(r151550, r151550, r151563);
        double r151565 = r151558 + r151564;
        double r151566 = r151557 * r151555;
        double r151567 = r151553 / r151566;
        double r151568 = r151550 / r151555;
        double r151569 = r151550 - r151568;
        double r151570 = r151567 + r151569;
        double r151571 = r151552 ? r151565 : r151570;
        return r151571;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.1
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 4.811043021165421e-08

    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. Using strategy rm

    4. Applied associate--l+6.9

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

    5. Taylor expanded around 0 0.2

      \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]

    6. Simplified0.2

      \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)}\]

    if 4.811043021165421e-08 < wj

    1. Initial program 28.7

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

    2. Simplified2.3

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

    4. Applied associate--l+2.3

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

    6. Applied div-inv2.3

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

    7. Applied associate-/l*2.3

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

    8. Simplified2.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.811043021165421 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))))