Average Error: 13.8 → 1.0
Time: 12.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{e^{wj}}}{wj + 1} - \left({wj}^{3} - \left({wj}^{4} + {wj}^{2}\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{e^{wj}}}{wj + 1} - \left({wj}^{3} - \left({wj}^{4} + {wj}^{2}\right)\right)
double f(double wj, double x) {
        double r247415 = wj;
        double r247416 = exp(r247415);
        double r247417 = r247415 * r247416;
        double r247418 = x;
        double r247419 = r247417 - r247418;
        double r247420 = r247416 + r247417;
        double r247421 = r247419 / r247420;
        double r247422 = r247415 - r247421;
        return r247422;
}
↓
double f(double wj, double x) {
        double r247423 = x;
        double r247424 = wj;
        double r247425 = exp(r247424);
        double r247426 = r247423 / r247425;
        double r247427 = 1.0;
        double r247428 = r247424 + r247427;
        double r247429 = r247426 / r247428;
        double r247430 = 3.0;
        double r247431 = pow(r247424, r247430);
        double r247432 = 4.0;
        double r247433 = pow(r247424, r247432);
        double r247434 = 2.0;
        double r247435 = pow(r247424, r247434);
        double r247436 = r247433 + r247435;
        double r247437 = r247431 - r247436;
        double r247438 = r247429 - r247437;
        return r247438;
}

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

Original13.8
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. Final simplification1.0

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

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))))))