Average Error: 14.1 → 1.3
Time: 23.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.425827286753258358087311776993266754232 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{{wj}^{3} - {\left(\frac{x}{e^{wj}}\right)}^{3}}{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}} \cdot \left(wj + \frac{x}{e^{wj}}\right)\right) \cdot \left(1 + wj\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.425827286753258358087311776993266754232 \cdot 10^{-7}:\\
\;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \frac{{wj}^{3} - {\left(\frac{x}{e^{wj}}\right)}^{3}}{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}} \cdot \left(wj + \frac{x}{e^{wj}}\right)\right) \cdot \left(1 + wj\right)}\\

\end{array}
double f(double wj, double x) {
        double r150269 = wj;
        double r150270 = exp(r150269);
        double r150271 = r150269 * r150270;
        double r150272 = x;
        double r150273 = r150271 - r150272;
        double r150274 = r150270 + r150271;
        double r150275 = r150273 / r150274;
        double r150276 = r150269 - r150275;
        return r150276;
}
↓
double f(double wj, double x) {
        double r150277 = wj;
        double r150278 = 1.4258272867532584e-07;
        bool r150279 = r150277 <= r150278;
        double r150280 = x;
        double r150281 = r150277 * r150280;
        double r150282 = -2.0;
        double r150283 = fma(r150277, r150277, r150280);
        double r150284 = fma(r150281, r150282, r150283);
        double r150285 = 3.0;
        double r150286 = pow(r150277, r150285);
        double r150287 = exp(r150277);
        double r150288 = r150280 / r150287;
        double r150289 = pow(r150288, r150285);
        double r150290 = r150286 - r150289;
        double r150291 = r150277 + r150288;
        double r150292 = r150288 * r150291;
        double r150293 = fma(r150277, r150277, r150292);
        double r150294 = 1.0;
        double r150295 = r150294 + r150277;
        double r150296 = r150293 * r150295;
        double r150297 = r150290 / r150296;
        double r150298 = r150277 - r150297;
        double r150299 = r150279 ? r150284 : r150298;
        return r150299;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.1
Target13.6
Herbie1.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.4258272867532584e-07

    1. Initial program 13.9

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

    2. Simplified13.9

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

    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 1.4258272867532584e-07 < wj

    1. Initial program 24.7

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

    2. Simplified2.5

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

    4. Applied flip3--15.1

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

    5. Applied associate-/l/15.3

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

    6. Simplified15.3

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.425827286753258358087311776993266754232 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{{wj}^{3} - {\left(\frac{x}{e^{wj}}\right)}^{3}}{\mathsf{fma}\left(wj, wj, \frac{x}{e^{wj}} \cdot \left(wj + \frac{x}{e^{wj}}\right)\right) \cdot \left(1 + wj\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))))))