Average Error: 14.1 → 1.0
Time: 5.6s
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 r381912 = wj;
        double r381913 = exp(r381912);
        double r381914 = r381912 * r381913;
        double r381915 = x;
        double r381916 = r381914 - r381915;
        double r381917 = r381913 + r381914;
        double r381918 = r381916 / r381917;
        double r381919 = r381912 - r381918;
        return r381919;
}
↓
double f(double wj, double x) {
        double r381920 = x;
        double r381921 = wj;
        double r381922 = 1.0;
        double r381923 = r381921 + r381922;
        double r381924 = r381920 / r381923;
        double r381925 = exp(r381921);
        double r381926 = r381924 / r381925;
        double r381927 = 4.0;
        double r381928 = pow(r381921, r381927);
        double r381929 = 2.0;
        double r381930 = pow(r381921, r381929);
        double r381931 = r381928 + r381930;
        double r381932 = 3.0;
        double r381933 = pow(r381921, r381932);
        double r381934 = r381931 - r381933;
        double r381935 = r381926 + r381934;
        return r381935;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.1
Target13.5
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.1

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

  2. Simplified13.5

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

  4. Applied associate--l+7.4

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

  5. Taylor expanded around 0 1.0

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

  6. Final simplification1.0

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

Reproduce

herbie shell --seed 2020100 
(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))))))