Average Error: 13.6 → 2.1
Time: 27.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2
double f(double wj, double x) {
        double r4612268 = wj;
        double r4612269 = exp(r4612268);
        double r4612270 = r4612268 * r4612269;
        double r4612271 = x;
        double r4612272 = r4612270 - r4612271;
        double r4612273 = r4612269 + r4612270;
        double r4612274 = r4612272 / r4612273;
        double r4612275 = r4612268 - r4612274;
        return r4612275;
}
↓
double f(double wj, double x) {
        double r4612276 = x;
        double r4612277 = wj;
        double r4612278 = r4612277 * r4612277;
        double r4612279 = r4612276 + r4612278;
        double r4612280 = r4612277 * r4612276;
        double r4612281 = -2.0;
        double r4612282 = r4612280 * r4612281;
        double r4612283 = r4612279 + r4612282;
        return r4612283;
}

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

Original13.6
Target13.0
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.6

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

  2. Taylor expanded around 0 2.1

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

  3. Simplified2.1

    \[\leadsto \color{blue}{-2 \cdot \left(x \cdot wj\right) + \left(x + wj \cdot wj\right)}\]

  4. Final simplification2.1

    \[\leadsto \left(x + wj \cdot wj\right) + \left(wj \cdot x\right) \cdot -2\]

Reproduce

herbie shell --seed 2019141 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

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