Average Error: 13.6 → 0.9
Time: 14.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 6.444094386194336532475590329253553600886 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot \left(wj - 2 \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 6.444094386194336532475590329253553600886 \cdot 10^{-9}:\\
\;\;\;\;wj \cdot \left(wj - 2 \cdot x\right) + x\\

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

\end{array}
double f(double wj, double x) {
        double r354201 = wj;
        double r354202 = exp(r354201);
        double r354203 = r354201 * r354202;
        double r354204 = x;
        double r354205 = r354203 - r354204;
        double r354206 = r354202 + r354203;
        double r354207 = r354205 / r354206;
        double r354208 = r354201 - r354207;
        return r354208;
}
↓
double f(double wj, double x) {
        double r354209 = wj;
        double r354210 = 6.4440943861943365e-09;
        bool r354211 = r354209 <= r354210;
        double r354212 = 2.0;
        double r354213 = x;
        double r354214 = r354212 * r354213;
        double r354215 = r354209 - r354214;
        double r354216 = r354209 * r354215;
        double r354217 = r354216 + r354213;
        double r354218 = 1.0;
        double r354219 = r354209 + r354218;
        double r354220 = r354209 / r354219;
        double r354221 = r354209 - r354220;
        double r354222 = exp(r354209);
        double r354223 = r354213 / r354222;
        double r354224 = r354223 / r354219;
        double r354225 = r354221 + r354224;
        double r354226 = r354211 ? r354217 : r354225;
        return r354226;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.6
Target12.9
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 6.4440943861943365e-09

    1. Initial program 13.2

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

    2. Simplified13.2

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

    3. Taylor expanded around 0 0.9

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

    4. Simplified0.9

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

    if 6.4440943861943365e-09 < wj

    1. Initial program 27.2

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

    2. Simplified2.7

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

    4. Applied div-sub2.7

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

    5. Applied associate--r-2.7

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 6.444094386194336532475590329253553600886 \cdot 10^{-9}:\\ \;\;\;\;wj \cdot \left(wj - 2 \cdot x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(wj - \frac{wj}{wj + 1}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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))))))