Average Error: 13.4 → 1.5
Time: 48.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\
\;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\

\mathbf{else}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\end{array}
double f(double wj, double x) {
        double r176111 = wj;
        double r176112 = exp(r176111);
        double r176113 = r176111 * r176112;
        double r176114 = x;
        double r176115 = r176113 - r176114;
        double r176116 = r176112 + r176113;
        double r176117 = r176115 / r176116;
        double r176118 = r176111 - r176117;
        return r176118;
}
↓
double f(double wj, double x) {
        double r176119 = wj;
        double r176120 = -5.00771993997508e-09;
        bool r176121 = r176119 <= r176120;
        double r176122 = exp(r176119);
        double r176123 = r176119 * r176122;
        double r176124 = x;
        double r176125 = r176123 - r176124;
        double r176126 = r176122 + r176123;
        double r176127 = r176125 / r176126;
        double r176128 = r176119 - r176127;
        double r176129 = 2.0;
        double r176130 = pow(r176119, r176129);
        double r176131 = r176124 + r176130;
        double r176132 = r176119 * r176124;
        double r176133 = r176129 * r176132;
        double r176134 = r176131 - r176133;
        double r176135 = r176121 ? r176128 : r176134;
        return r176135;
}

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
Target12.8
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 < -5.00771993997508e-09

    1. Initial program 5.8

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

    if -5.00771993997508e-09 < wj

    1. Initial program 13.6

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

    2. Simplified12.9

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

    3. Taylor expanded around 0 1.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -5.007719939975079832531441257617230156107 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))