Average Error: 13.9 → 1.0
Time: 24.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{1 + wj}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\

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

\end{array}
double f(double wj, double x) {
        double r126075 = wj;
        double r126076 = exp(r126075);
        double r126077 = r126075 * r126076;
        double r126078 = x;
        double r126079 = r126077 - r126078;
        double r126080 = r126076 + r126077;
        double r126081 = r126079 / r126080;
        double r126082 = r126075 - r126081;
        return r126082;
}
↓
double f(double wj, double x) {
        double r126083 = wj;
        double r126084 = 8.133938014656618e-09;
        bool r126085 = r126083 <= r126084;
        double r126086 = x;
        double r126087 = r126083 * r126086;
        double r126088 = -2.0;
        double r126089 = fma(r126083, r126083, r126086);
        double r126090 = fma(r126087, r126088, r126089);
        double r126091 = exp(r126083);
        double r126092 = sqrt(r126091);
        double r126093 = r126086 / r126092;
        double r126094 = r126093 / r126092;
        double r126095 = r126083 - r126094;
        double r126096 = 1.0;
        double r126097 = r126096 + r126083;
        double r126098 = r126095 / r126097;
        double r126099 = r126083 - r126098;
        double r126100 = r126085 ? r126090 : r126099;
        return r126100;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.2
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 < 8.133938014656618e-09

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 1.0

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

    4. Simplified1.0

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

    if 8.133938014656618e-09 < wj

    1. Initial program 27.6

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

    2. Simplified2.3

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

    4. Applied add-sqr-sqrt2.4

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

    5. Applied associate-/r*2.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 8.133938014656618390032425086177275685984 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{wj - \frac{\frac{x}{\sqrt{e^{wj}}}}{\sqrt{e^{wj}}}}{1 + wj}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))))))