Average Error: 13.9 → 1.0
Time: 23.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.038831895678682838728544720295621228701 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - 1, \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(-wj, wj, 1\right)}, wj\right) + \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(-wj, wj, 1\right)} \cdot \left(\left(wj - 1\right) + \left(1 - wj\right)\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.038831895678682838728544720295621228701 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\

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

\end{array}
double f(double wj, double x) {
        double r116243 = wj;
        double r116244 = exp(r116243);
        double r116245 = r116243 * r116244;
        double r116246 = x;
        double r116247 = r116245 - r116246;
        double r116248 = r116244 + r116245;
        double r116249 = r116247 / r116248;
        double r116250 = r116243 - r116249;
        return r116250;
}
↓
double f(double wj, double x) {
        double r116251 = wj;
        double r116252 = 9.038831895678683e-09;
        bool r116253 = r116251 <= r116252;
        double r116254 = x;
        double r116255 = r116251 * r116254;
        double r116256 = -2.0;
        double r116257 = fma(r116251, r116251, r116254);
        double r116258 = fma(r116255, r116256, r116257);
        double r116259 = 1.0;
        double r116260 = r116251 - r116259;
        double r116261 = exp(r116251);
        double r116262 = r116254 / r116261;
        double r116263 = r116251 - r116262;
        double r116264 = -r116251;
        double r116265 = fma(r116264, r116251, r116259);
        double r116266 = r116263 / r116265;
        double r116267 = fma(r116260, r116266, r116251);
        double r116268 = r116259 - r116251;
        double r116269 = r116260 + r116268;
        double r116270 = r116266 * r116269;
        double r116271 = r116267 + r116270;
        double r116272 = r116253 ? r116258 : r116271;
        return r116272;
}

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 < 9.038831895678683e-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 9.038831895678683e-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 flip-+2.4

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

    5. Applied associate-/r/2.3

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

    6. Applied add-sqr-sqrt2.7

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

    7. Applied prod-diff2.7

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

    8. Simplified2.3

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

    9. Simplified2.2

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.038831895678682838728544720295621228701 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(wj - 1, \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(-wj, wj, 1\right)}, wj\right) + \frac{wj - \frac{x}{e^{wj}}}{\mathsf{fma}\left(-wj, wj, 1\right)} \cdot \left(\left(wj - 1\right) + \left(1 - wj\right)\right)\\ \end{array}\]

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))))))