Average Error: 14.2 → 1.1
Time: 7.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.06252678575455972 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.06252678575455972 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r250087 = wj;
        double r250088 = exp(r250087);
        double r250089 = r250087 * r250088;
        double r250090 = x;
        double r250091 = r250089 - r250090;
        double r250092 = r250088 + r250089;
        double r250093 = r250091 / r250092;
        double r250094 = r250087 - r250093;
        return r250094;
}
↓
double f(double wj, double x) {
        double r250095 = wj;
        double r250096 = 9.06252678575456e-09;
        bool r250097 = r250095 <= r250096;
        double r250098 = x;
        double r250099 = fma(r250095, r250095, r250098);
        double r250100 = 2.0;
        double r250101 = r250095 * r250098;
        double r250102 = r250100 * r250101;
        double r250103 = r250099 - r250102;
        double r250104 = 1.0;
        double r250105 = r250095 + r250104;
        double r250106 = r250098 / r250105;
        double r250107 = exp(r250095);
        double r250108 = r250106 / r250107;
        double r250109 = r250108 + r250095;
        double r250110 = r250095 * r250095;
        double r250111 = r250110 - r250104;
        double r250112 = r250095 / r250111;
        double r250113 = r250095 - r250104;
        double r250114 = r250112 * r250113;
        double r250115 = r250109 - r250114;
        double r250116 = r250097 ? r250103 : r250115;
        return r250116;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.2
Target13.5
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 9.06252678575456e-09

    1. Initial program 13.9

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

    2. Simplified13.9

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

    3. Taylor expanded around 0 1.0

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

    4. Taylor expanded around 0 1.0

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

    5. Simplified1.0

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

    if 9.06252678575456e-09 < wj

    1. Initial program 26.3

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

    2. Simplified3.1

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

    4. Applied flip-+3.2

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

    5. Applied associate-/r/3.0

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

    6. Simplified3.0

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.06252678575455972 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))))))