Average Error: 13.7 → 1.1
Time: 5.9s
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 r416758 = wj;
        double r416759 = exp(r416758);
        double r416760 = r416758 * r416759;
        double r416761 = x;
        double r416762 = r416760 - r416761;
        double r416763 = r416759 + r416760;
        double r416764 = r416762 / r416763;
        double r416765 = r416758 - r416764;
        return r416765;
}
↓
double f(double wj, double x) {
        double r416766 = x;
        double r416767 = wj;
        double r416768 = 1.0;
        double r416769 = r416767 + r416768;
        double r416770 = r416766 / r416769;
        double r416771 = exp(r416767);
        double r416772 = r416770 / r416771;
        double r416773 = 4.0;
        double r416774 = pow(r416767, r416773);
        double r416775 = 2.0;
        double r416776 = pow(r416767, r416775);
        double r416777 = r416774 + r416776;
        double r416778 = 3.0;
        double r416779 = pow(r416767, r416778);
        double r416780 = r416777 - r416779;
        double r416781 = r416772 + r416780;
        return r416781;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

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

Derivation

  1. Initial program 13.7

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

  2. 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+7.1

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

  5. Taylor expanded around 0 1.1

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

  6. Final simplification1.1

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

Reproduce

herbie shell --seed 2020033 
(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))))))