Average Error: 13.9 → 2.2
Time: 26.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj - x \cdot \left(wj + wj\right)\right) + x\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj - x \cdot \left(wj + wj\right)\right) + x
double f(double wj, double x) {
        double r11597383 = wj;
        double r11597384 = exp(r11597383);
        double r11597385 = r11597383 * r11597384;
        double r11597386 = x;
        double r11597387 = r11597385 - r11597386;
        double r11597388 = r11597384 + r11597385;
        double r11597389 = r11597387 / r11597388;
        double r11597390 = r11597383 - r11597389;
        return r11597390;
}
↓
double f(double wj, double x) {
        double r11597391 = wj;
        double r11597392 = r11597391 * r11597391;
        double r11597393 = x;
        double r11597394 = r11597391 + r11597391;
        double r11597395 = r11597393 * r11597394;
        double r11597396 = r11597392 - r11597395;
        double r11597397 = r11597396 + r11597393;
        return r11597397;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.3
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  2. Taylor expanded around 0 2.2

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

  3. Simplified2.2

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

  4. Final simplification2.2

    \[\leadsto \left(wj \cdot wj - x \cdot \left(wj + wj\right)\right) + x\]

Reproduce

herbie shell --seed 2019162 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

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