Average Error: 13.9 → 1.2
Time: 21.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x}{e^{wj} + e^{wj} \cdot wj} + \left(\left(wj \cdot wj + \left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right)\right) - wj \cdot \left(wj \cdot wj\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x}{e^{wj} + e^{wj} \cdot wj} + \left(\left(wj \cdot wj + \left(wj \cdot wj\right) \cdot \left(wj \cdot wj\right)\right) - wj \cdot \left(wj \cdot wj\right)\right)
double f(double wj, double x) {
        double r4475223 = wj;
        double r4475224 = exp(r4475223);
        double r4475225 = r4475223 * r4475224;
        double r4475226 = x;
        double r4475227 = r4475225 - r4475226;
        double r4475228 = r4475224 + r4475225;
        double r4475229 = r4475227 / r4475228;
        double r4475230 = r4475223 - r4475229;
        return r4475230;
}
↓
double f(double wj, double x) {
        double r4475231 = x;
        double r4475232 = wj;
        double r4475233 = exp(r4475232);
        double r4475234 = r4475233 * r4475232;
        double r4475235 = r4475233 + r4475234;
        double r4475236 = r4475231 / r4475235;
        double r4475237 = r4475232 * r4475232;
        double r4475238 = r4475237 * r4475237;
        double r4475239 = r4475237 + r4475238;
        double r4475240 = r4475232 * r4475237;
        double r4475241 = r4475239 - r4475240;
        double r4475242 = r4475236 + r4475241;
        return r4475242;
}

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
Target13.2
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  3. Applied div-sub13.9

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

  4. Applied associate--r-7.6

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

  5. Taylor expanded around 0 1.2

    \[\leadsto \color{blue}{\left(\left({wj}^{2} + {wj}^{4}\right) - {wj}^{3}\right)} + \frac{x}{e^{wj} + wj \cdot e^{wj}}\]

  6. Simplified1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2019151 
(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))))))