Average Error: 13.5 → 1.1
Time: 4.5s
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 r266133 = wj;
        double r266134 = exp(r266133);
        double r266135 = r266133 * r266134;
        double r266136 = x;
        double r266137 = r266135 - r266136;
        double r266138 = r266134 + r266135;
        double r266139 = r266137 / r266138;
        double r266140 = r266133 - r266139;
        return r266140;
}
↓
double f(double wj, double x) {
        double r266141 = x;
        double r266142 = wj;
        double r266143 = 1.0;
        double r266144 = r266142 + r266143;
        double r266145 = r266141 / r266144;
        double r266146 = exp(r266142);
        double r266147 = r266145 / r266146;
        double r266148 = 4.0;
        double r266149 = pow(r266142, r266148);
        double r266150 = 2.0;
        double r266151 = pow(r266142, r266150);
        double r266152 = r266149 + r266151;
        double r266153 = 3.0;
        double r266154 = pow(r266142, r266153);
        double r266155 = r266152 - r266154;
        double r266156 = r266147 + r266155;
        return r266156;
}

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

Original13.5
Target12.8
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

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

  2. Simplified12.8

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

  4. Applied associate--l+6.6

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

  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 2020060 
(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))))))