Average Error: 13.9 → 1.1
Time: 19.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right) + \frac{\frac{x}{e^{wj}}}{1 + wj}
double f(double wj, double x) {
        double r184328 = wj;
        double r184329 = exp(r184328);
        double r184330 = r184328 * r184329;
        double r184331 = x;
        double r184332 = r184330 - r184331;
        double r184333 = r184329 + r184330;
        double r184334 = r184332 / r184333;
        double r184335 = r184328 - r184334;
        return r184335;
}
↓
double f(double wj, double x) {
        double r184336 = wj;
        double r184337 = 4.0;
        double r184338 = pow(r184336, r184337);
        double r184339 = fma(r184336, r184336, r184338);
        double r184340 = sqrt(r184339);
        double r184341 = 3.0;
        double r184342 = pow(r184336, r184341);
        double r184343 = -r184342;
        double r184344 = fma(r184340, r184340, r184343);
        double r184345 = x;
        double r184346 = exp(r184336);
        double r184347 = r184345 / r184346;
        double r184348 = 1.0;
        double r184349 = r184348 + r184336;
        double r184350 = r184347 / r184349;
        double r184351 = r184344 + r184350;
        return r184351;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.2
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  2. Simplified13.2

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

  4. Applied div-sub13.2

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

  5. Applied associate--r-6.9

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

  6. Taylor expanded around 0 1.1

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

  7. Simplified1.1

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

  9. Applied add-sqr-sqrt1.1

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

  10. Applied fma-neg1.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, \sqrt{\mathsf{fma}\left(wj, wj, {wj}^{4}\right)}, -{wj}^{3}\right)} + \frac{\frac{x}{e^{wj}}}{1 + wj}\]

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2020045 +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))))))