Average Error: 13.6 → 1.1
Time: 6.1s
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 r241365 = wj;
        double r241366 = exp(r241365);
        double r241367 = r241365 * r241366;
        double r241368 = x;
        double r241369 = r241367 - r241368;
        double r241370 = r241366 + r241367;
        double r241371 = r241369 / r241370;
        double r241372 = r241365 - r241371;
        return r241372;
}
↓
double f(double wj, double x) {
        double r241373 = x;
        double r241374 = wj;
        double r241375 = 1.0;
        double r241376 = r241374 + r241375;
        double r241377 = r241373 / r241376;
        double r241378 = exp(r241374);
        double r241379 = r241377 / r241378;
        double r241380 = 4.0;
        double r241381 = pow(r241374, r241380);
        double r241382 = 2.0;
        double r241383 = pow(r241374, r241382);
        double r241384 = r241381 + r241383;
        double r241385 = 3.0;
        double r241386 = pow(r241374, r241385);
        double r241387 = r241384 - r241386;
        double r241388 = r241379 + r241387;
        return r241388;
}

Error

Bits error versus wj

Bits error versus x

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Your Program's Arguments

Results

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Target

Original13.6
Target13.0
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.6

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

  2. Simplified13.0

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

  4. Applied associate--l+6.9

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

  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 2020057 
(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))))))