Average Error: 14.3 → 2.2
Time: 4.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)
double f(double wj, double x) {
        double r187309 = wj;
        double r187310 = exp(r187309);
        double r187311 = r187309 * r187310;
        double r187312 = x;
        double r187313 = r187311 - r187312;
        double r187314 = r187310 + r187311;
        double r187315 = r187313 / r187314;
        double r187316 = r187309 - r187315;
        return r187316;
}
↓
double f(double wj, double x) {
        double r187317 = x;
        double r187318 = wj;
        double r187319 = 2.0;
        double r187320 = pow(r187318, r187319);
        double r187321 = r187317 + r187320;
        double r187322 = r187318 * r187317;
        double r187323 = r187319 * r187322;
        double r187324 = r187321 - r187323;
        return r187324;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.3
Target13.6
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.3

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

  2. Simplified13.6

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

  3. Taylor expanded around 0 2.2

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

  4. Final simplification2.2

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(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))))))