Average Error: 13.8 → 1.0
Time: 15.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x \cdot e^{-wj}}{wj + 1} - \left({wj}^{3} - \left({wj}^{4} + {wj}^{2}\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x \cdot e^{-wj}}{wj + 1} - \left({wj}^{3} - \left({wj}^{4} + {wj}^{2}\right)\right)
double f(double wj, double x) {
        double r308671 = wj;
        double r308672 = exp(r308671);
        double r308673 = r308671 * r308672;
        double r308674 = x;
        double r308675 = r308673 - r308674;
        double r308676 = r308672 + r308673;
        double r308677 = r308675 / r308676;
        double r308678 = r308671 - r308677;
        return r308678;
}
↓
double f(double wj, double x) {
        double r308679 = x;
        double r308680 = wj;
        double r308681 = -r308680;
        double r308682 = exp(r308681);
        double r308683 = r308679 * r308682;
        double r308684 = 1.0;
        double r308685 = r308680 + r308684;
        double r308686 = r308683 / r308685;
        double r308687 = 3.0;
        double r308688 = pow(r308680, r308687);
        double r308689 = 4.0;
        double r308690 = pow(r308680, r308689);
        double r308691 = 2.0;
        double r308692 = pow(r308680, r308691);
        double r308693 = r308690 + r308692;
        double r308694 = r308688 - r308693;
        double r308695 = r308686 - r308694;
        return r308695;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.3
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  2. Simplified13.3

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

  4. Applied div-sub13.3

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

  5. Applied associate-+l-6.9

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

  6. Taylor expanded around 0 1.0

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

  8. Applied div-inv1.0

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

  9. Simplified1.0

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

  10. Final simplification1.0

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

Reproduce

herbie shell --seed 2020047 
(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))))))