Average Error: 13.9 → 1.2
Time: 25.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{wj + 1}
double f(double wj, double x) {
        double r183972 = wj;
        double r183973 = exp(r183972);
        double r183974 = r183972 * r183973;
        double r183975 = x;
        double r183976 = r183974 - r183975;
        double r183977 = r183973 + r183974;
        double r183978 = r183976 / r183977;
        double r183979 = r183972 - r183978;
        return r183979;
}
↓
double f(double wj, double x) {
        double r183980 = wj;
        double r183981 = 4.0;
        double r183982 = pow(r183980, r183981);
        double r183983 = 2.0;
        double r183984 = pow(r183980, r183983);
        double r183985 = r183982 + r183984;
        double r183986 = 3.0;
        double r183987 = pow(r183980, r183986);
        double r183988 = r183985 - r183987;
        double r183989 = x;
        double r183990 = exp(r183980);
        double r183991 = r183989 / r183990;
        double r183992 = 1.0;
        double r183993 = r183980 + r183992;
        double r183994 = r183991 / r183993;
        double r183995 = r183988 + r183994;
        return r183995;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.9
Target13.3
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. Simplified13.3

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

  4. Applied div-sub13.3

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

  5. Applied associate--r-7.0

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

  6. Taylor expanded around 0 1.2

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

  7. Final simplification1.2

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

Reproduce

herbie shell --seed 2019304 
(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))))))