Average Error: 13.8 → 0.9
Time: 5.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)
double f(double wj, double x) {
        double r217350 = wj;
        double r217351 = exp(r217350);
        double r217352 = r217350 * r217351;
        double r217353 = x;
        double r217354 = r217352 - r217353;
        double r217355 = r217351 + r217352;
        double r217356 = r217354 / r217355;
        double r217357 = r217350 - r217356;
        return r217357;
}
↓
double f(double wj, double x) {
        double r217358 = x;
        double r217359 = wj;
        double r217360 = 1.0;
        double r217361 = r217359 + r217360;
        double r217362 = r217358 / r217361;
        double r217363 = exp(r217359);
        double r217364 = r217362 / r217363;
        double r217365 = 4.0;
        double r217366 = pow(r217359, r217365);
        double r217367 = 2.0;
        double r217368 = pow(r217359, r217367);
        double r217369 = r217366 + r217368;
        double r217370 = 3.0;
        double r217371 = pow(r217359, r217370);
        double r217372 = r217369 - r217371;
        double r217373 = r217364 + r217372;
        return r217373;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.2
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  2. Simplified13.2

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

  4. Applied associate--l+6.9

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

  5. Taylor expanded around 0 0.9

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]

  6. Final simplification0.9

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\]

Reproduce

herbie shell --seed 2020056 
(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))))))