Average Error: 13.7 → 1.2
Time: 34.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{\frac{x}{1 + wj \cdot \left(wj \cdot wj\right)}}{e^{wj + \left(wj + wj\right)}} \cdot \left(\left(\left(wj \cdot e^{wj}\right) \cdot \left(wj \cdot e^{wj}\right) - e^{wj} \cdot \left(wj \cdot e^{wj}\right)\right) + e^{wj} \cdot e^{wj}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{\frac{x}{1 + wj \cdot \left(wj \cdot wj\right)}}{e^{wj + \left(wj + wj\right)}} \cdot \left(\left(\left(wj \cdot e^{wj}\right) \cdot \left(wj \cdot e^{wj}\right) - e^{wj} \cdot \left(wj \cdot e^{wj}\right)\right) + e^{wj} \cdot e^{wj}\right)
double f(double wj, double x) {
        double r7961488 = wj;
        double r7961489 = exp(r7961488);
        double r7961490 = r7961488 * r7961489;
        double r7961491 = x;
        double r7961492 = r7961490 - r7961491;
        double r7961493 = r7961489 + r7961490;
        double r7961494 = r7961492 / r7961493;
        double r7961495 = r7961488 - r7961494;
        return r7961495;
}
↓
double f(double wj, double x) {
        double r7961496 = wj;
        double r7961497 = r7961496 * r7961496;
        double r7961498 = r7961497 - r7961496;
        double r7961499 = r7961498 * r7961497;
        double r7961500 = r7961497 + r7961499;
        double r7961501 = x;
        double r7961502 = 1.0;
        double r7961503 = r7961496 * r7961497;
        double r7961504 = r7961502 + r7961503;
        double r7961505 = r7961501 / r7961504;
        double r7961506 = r7961496 + r7961496;
        double r7961507 = r7961496 + r7961506;
        double r7961508 = exp(r7961507);
        double r7961509 = r7961505 / r7961508;
        double r7961510 = exp(r7961496);
        double r7961511 = r7961496 * r7961510;
        double r7961512 = r7961511 * r7961511;
        double r7961513 = r7961510 * r7961511;
        double r7961514 = r7961512 - r7961513;
        double r7961515 = r7961510 * r7961510;
        double r7961516 = r7961514 + r7961515;
        double r7961517 = r7961509 * r7961516;
        double r7961518 = r7961500 + r7961517;
        return r7961518;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.1
Herbie1.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  3. Applied div-sub13.7

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

  4. Applied associate--r-7.4

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

  5. Taylor expanded around 0 1.0

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

  6. Simplified1.1

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

  8. Applied flip3-+1.2

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

  9. Applied associate-/r/1.2

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

  10. Simplified1.2

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

  11. Final simplification1.2

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

Reproduce

herbie shell --seed 2019133 
(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))))))