Average Error: 13.4 → 1.3
Time: 5.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 7.7872172205598176 \cdot 10^{-7}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 7.7872172205598176 \cdot 10^{-7}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r286361 = wj;
        double r286362 = exp(r286361);
        double r286363 = r286361 * r286362;
        double r286364 = x;
        double r286365 = r286363 - r286364;
        double r286366 = r286362 + r286363;
        double r286367 = r286365 / r286366;
        double r286368 = r286361 - r286367;
        return r286368;
}
↓
double f(double wj, double x) {
        double r286369 = wj;
        double r286370 = 7.787217220559818e-07;
        bool r286371 = r286369 <= r286370;
        double r286372 = x;
        double r286373 = 2.0;
        double r286374 = pow(r286369, r286373);
        double r286375 = r286372 + r286374;
        double r286376 = r286369 * r286372;
        double r286377 = r286373 * r286376;
        double r286378 = r286375 - r286377;
        double r286379 = 1.0;
        double r286380 = r286369 + r286379;
        double r286381 = r286372 / r286380;
        double r286382 = exp(r286369);
        double r286383 = r286381 / r286382;
        double r286384 = r286383 * r286383;
        double r286385 = r286369 * r286369;
        double r286386 = r286384 - r286385;
        double r286387 = r286386 * r286380;
        double r286388 = r286383 - r286369;
        double r286389 = r286388 * r286369;
        double r286390 = r286387 - r286389;
        double r286391 = r286388 * r286380;
        double r286392 = r286390 / r286391;
        double r286393 = r286371 ? r286378 : r286392;
        return r286393;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.7
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 < 7.787217220559818e-07

    1. Initial program 13.1

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

    2. Simplified13.0

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

    3. Taylor expanded around 0 1.0

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

    if 7.787217220559818e-07 < wj

    1. Initial program 25.3

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

    2. Simplified2.0

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

    4. Applied flip-+11.0

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

    5. Applied frac-sub11.2

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 7.7872172205598176 \cdot 10^{-7}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} \cdot \frac{\frac{x}{wj + 1}}{e^{wj}} - wj \cdot wj\right) \cdot \left(wj + 1\right) - \left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot wj}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} - wj\right) \cdot \left(wj + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(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))))))