Average Error: 13.5 → 1.1
Time: 22.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{1}{e^{wj}} \cdot \frac{x}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj \cdot wj + \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{1}{e^{wj}} \cdot \frac{x}{1 + wj}
double f(double wj, double x) {
        double r7787106 = wj;
        double r7787107 = exp(r7787106);
        double r7787108 = r7787106 * r7787107;
        double r7787109 = x;
        double r7787110 = r7787108 - r7787109;
        double r7787111 = r7787107 + r7787108;
        double r7787112 = r7787110 / r7787111;
        double r7787113 = r7787106 - r7787112;
        return r7787113;
}
↓
double f(double wj, double x) {
        double r7787114 = wj;
        double r7787115 = r7787114 * r7787114;
        double r7787116 = r7787115 - r7787114;
        double r7787117 = r7787116 * r7787115;
        double r7787118 = r7787115 + r7787117;
        double r7787119 = 1.0;
        double r7787120 = exp(r7787114);
        double r7787121 = r7787119 / r7787120;
        double r7787122 = x;
        double r7787123 = r7787119 + r7787114;
        double r7787124 = r7787122 / r7787123;
        double r7787125 = r7787121 * r7787124;
        double r7787126 = r7787118 + r7787125;
        return r7787126;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target12.9
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. Using strategy rm

  3. Applied div-sub13.5

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

  4. Applied associate--r-7.5

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

  5. Taylor expanded around 0 1.1

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

  6. Simplified1.1

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

  8. Applied *-un-lft-identity1.1

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

  9. Applied distribute-rgt-out1.1

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

  10. Applied *-un-lft-identity1.1

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

  11. Applied times-frac1.1

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

  12. Final simplification1.1

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

Reproduce

herbie shell --seed 2019172 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))