Average Error: 13.3 → 1.0
Time: 2.2m
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.10407274317618 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 8.10407274317618 \cdot 10^{-09}:\\
\;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\

\end{array}
double f(double wj, double x) {
        double r49540945 = wj;
        double r49540946 = exp(r49540945);
        double r49540947 = r49540945 * r49540946;
        double r49540948 = x;
        double r49540949 = r49540947 - r49540948;
        double r49540950 = r49540946 + r49540947;
        double r49540951 = r49540949 / r49540950;
        double r49540952 = r49540945 - r49540951;
        return r49540952;
}
↓
double f(double wj, double x) {
        double r49540953 = wj;
        double r49540954 = 8.10407274317618e-09;
        bool r49540955 = r49540953 <= r49540954;
        double r49540956 = x;
        double r49540957 = -2.0;
        double r49540958 = fma(r49540956, r49540957, r49540953);
        double r49540959 = fma(r49540958, r49540953, r49540956);
        double r49540960 = exp(r49540953);
        double r49540961 = r49540956 / r49540960;
        double r49540962 = r49540953 - r49540961;
        double r49540963 = 1.0;
        double r49540964 = r49540953 + r49540963;
        double r49540965 = r49540962 / r49540964;
        double r49540966 = r49540953 - r49540965;
        double r49540967 = r49540955 ? r49540959 : r49540966;
        return r49540967;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.3
Target12.7
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 8.10407274317618e-09

    1. Initial program 13.0

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

    2. Taylor expanded around 0 0.9

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

    3. Simplified1.0

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

    if 8.10407274317618e-09 < wj

    1. Initial program 23.9

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied distribute-rgt1-in24.0

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

    4. Applied *-un-lft-identity24.0

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

    5. Applied times-frac23.9

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

    6. Applied add-cube-cbrt24.7

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

    7. Applied prod-diff24.7

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

    8. Simplified23.9

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

    9. Simplified2.8

      \[\leadsto \left(wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\right) + \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.10407274317618 \cdot 10^{-09}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

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

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