Average Error: 13.4 → 0.1
Time: 21.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(wj - \frac{wj}{{wj}^{3} + 1}\right) - \left(\left(wj \cdot wj - wj\right) \cdot \frac{wj}{{wj}^{3} + 1} - \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(wj - \frac{wj}{{wj}^{3} + 1}\right) - \left(\left(wj \cdot wj - wj\right) \cdot \frac{wj}{{wj}^{3} + 1} - \frac{x}{\left(1 + wj\right) \cdot e^{wj}}\right)
double f(double wj, double x) {
        double r237115 = wj;
        double r237116 = exp(r237115);
        double r237117 = r237115 * r237116;
        double r237118 = x;
        double r237119 = r237117 - r237118;
        double r237120 = r237116 + r237117;
        double r237121 = r237119 / r237120;
        double r237122 = r237115 - r237121;
        return r237122;
}
↓
double f(double wj, double x) {
        double r237123 = wj;
        double r237124 = 3.0;
        double r237125 = pow(r237123, r237124);
        double r237126 = 1.0;
        double r237127 = r237125 + r237126;
        double r237128 = r237123 / r237127;
        double r237129 = r237123 - r237128;
        double r237130 = r237123 * r237123;
        double r237131 = r237130 - r237123;
        double r237132 = r237131 * r237128;
        double r237133 = x;
        double r237134 = r237126 + r237123;
        double r237135 = exp(r237123);
        double r237136 = r237134 * r237135;
        double r237137 = r237133 / r237136;
        double r237138 = r237132 - r237137;
        double r237139 = r237129 - r237138;
        return r237139;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.4
Target12.8
Herbie0.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.4

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

  2. Simplified12.8

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

  4. Applied div-sub12.8

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

  5. Simplified12.8

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

  7. Applied flip3-+12.8

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

  8. Applied associate-/r/12.8

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

  9. Simplified12.8

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

  11. Applied distribute-lft-in12.8

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

  12. Applied associate--l+12.8

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

  13. Applied associate--r+0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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))))))