Average Error: 13.7 → 1.5
Time: 39.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -6.07033611174586248575152911712005110445 \cdot 10^{-9}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;\left(x + wj \cdot wj\right) - x \cdot \left(wj + wj\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -6.07033611174586248575152911712005110445 \cdot 10^{-9}:\\
\;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\

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

\end{array}
double f(double wj, double x) {
        double r10711214 = wj;
        double r10711215 = exp(r10711214);
        double r10711216 = r10711214 * r10711215;
        double r10711217 = x;
        double r10711218 = r10711216 - r10711217;
        double r10711219 = r10711215 + r10711216;
        double r10711220 = r10711218 / r10711219;
        double r10711221 = r10711214 - r10711220;
        return r10711221;
}
↓
double f(double wj, double x) {
        double r10711222 = wj;
        double r10711223 = -6.0703361117458625e-09;
        bool r10711224 = r10711222 <= r10711223;
        double r10711225 = 1.0;
        double r10711226 = r10711225 + r10711222;
        double r10711227 = r10711222 / r10711226;
        double r10711228 = r10711222 - r10711227;
        double r10711229 = x;
        double r10711230 = exp(r10711222);
        double r10711231 = r10711229 / r10711230;
        double r10711232 = r10711231 / r10711226;
        double r10711233 = r10711228 + r10711232;
        double r10711234 = r10711222 * r10711222;
        double r10711235 = r10711229 + r10711234;
        double r10711236 = r10711222 + r10711222;
        double r10711237 = r10711229 * r10711236;
        double r10711238 = r10711235 - r10711237;
        double r10711239 = r10711224 ? r10711233 : r10711238;
        return r10711239;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.1
Herbie1.5
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -6.0703361117458625e-09

    1. Initial program 5.7

      \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity5.7

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

    4. Applied distribute-rgt-out5.7

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

    5. Applied associate-/r*5.5

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

    7. Applied div-sub5.5

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

    8. Applied div-sub5.5

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

    9. Applied associate--r-5.5

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

    10. Simplified5.5

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

    if -6.0703361117458625e-09 < wj

    1. Initial program 13.9

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

    2. Taylor expanded around 0 1.4

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

    3. Simplified1.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -6.07033611174586248575152911712005110445 \cdot 10^{-9}:\\ \;\;\;\;\left(wj - \frac{wj}{1 + wj}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\\ \mathbf{else}:\\ \;\;\;\;\left(x + wj \cdot wj\right) - x \cdot \left(wj + wj\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))