Average Error: 13.7 → 0.8
Time: 20.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 7.377123706691067175607606942237539690999 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 7.377123706691067175607606942237539690999 \cdot 10^{-20}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\

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

\end{array}
double f(double wj, double x) {
        double r119943 = wj;
        double r119944 = exp(r119943);
        double r119945 = r119943 * r119944;
        double r119946 = x;
        double r119947 = r119945 - r119946;
        double r119948 = r119944 + r119945;
        double r119949 = r119947 / r119948;
        double r119950 = r119943 - r119949;
        return r119950;
}
↓
double f(double wj, double x) {
        double r119951 = wj;
        double r119952 = 7.377123706691067e-20;
        bool r119953 = r119951 <= r119952;
        double r119954 = 4.0;
        double r119955 = pow(r119951, r119954);
        double r119956 = fma(r119951, r119951, r119955);
        double r119957 = sqrt(r119956);
        double r119958 = 3.0;
        double r119959 = pow(r119951, r119958);
        double r119960 = -r119959;
        double r119961 = fma(r119957, r119957, r119960);
        double r119962 = x;
        double r119963 = exp(r119951);
        double r119964 = r119962 / r119963;
        double r119965 = 1.0;
        double r119966 = r119965 + r119951;
        double r119967 = r119964 / r119966;
        double r119968 = r119961 + r119967;
        double r119969 = r119951 / r119966;
        double r119970 = r119951 - r119969;
        double r119971 = r119951 + r119965;
        double r119972 = r119971 * r119963;
        double r119973 = r119962 / r119972;
        double r119974 = r119970 + r119973;
        double r119975 = r119953 ? r119968 : r119974;
        return r119975;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.0
Herbie0.8
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 7.377123706691067e-20

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied div-sub13.2

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

    5. Applied associate--r-6.7

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

    6. Taylor expanded around 0 0.3

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

    7. Simplified0.3

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

    9. Applied add-sqr-sqrt0.3

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

    10. Applied fma-neg0.3

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

    if 7.377123706691067e-20 < wj

    1. Initial program 23.8

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

    2. Simplified10.1

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

    4. Applied div-sub10.1

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

    5. Applied associate--r-10.1

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

    7. Applied div-inv10.1

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

    8. Applied associate-/l*10.1

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

    9. Simplified10.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 7.377123706691067175607606942237539690999 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{x}{\left(wj + 1\right) \cdot e^{wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +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))))))