Average Error: 14.2 → 2.1
Time: 24.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2
double f(double wj, double x) {
        double r9392373 = wj;
        double r9392374 = exp(r9392373);
        double r9392375 = r9392373 * r9392374;
        double r9392376 = x;
        double r9392377 = r9392375 - r9392376;
        double r9392378 = r9392374 + r9392375;
        double r9392379 = r9392377 / r9392378;
        double r9392380 = r9392373 - r9392379;
        return r9392380;
}
↓
double f(double wj, double x) {
        double r9392381 = x;
        double r9392382 = wj;
        double r9392383 = r9392382 * r9392382;
        double r9392384 = r9392381 + r9392383;
        double r9392385 = r9392382 * r9392381;
        double r9392386 = -2.0;
        double r9392387 = r9392385 * r9392386;
        double r9392388 = r9392384 + r9392387;
        return r9392388;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.2
Target13.6
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.2

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

  2. Taylor expanded around 0 2.1

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

  3. Simplified2.1

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

  4. Final simplification2.1

    \[\leadsto \left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2\]

Reproduce

herbie shell --seed 2019171 
(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))))))