Average Error: 13.5 → 1.1
Time: 24.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{x}{e^{wj} + e^{wj} \cdot wj} + \mathsf{fma}\left(wj \cdot wj, wj \cdot wj, \mathsf{fma}\left(wj, wj, wj \cdot \left(\left(-wj\right) \cdot wj\right)\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{x}{e^{wj} + e^{wj} \cdot wj} + \mathsf{fma}\left(wj \cdot wj, wj \cdot wj, \mathsf{fma}\left(wj, wj, wj \cdot \left(\left(-wj\right) \cdot wj\right)\right)\right)
double f(double wj, double x) {
        double r7744402 = wj;
        double r7744403 = exp(r7744402);
        double r7744404 = r7744402 * r7744403;
        double r7744405 = x;
        double r7744406 = r7744404 - r7744405;
        double r7744407 = r7744403 + r7744404;
        double r7744408 = r7744406 / r7744407;
        double r7744409 = r7744402 - r7744408;
        return r7744409;
}
↓
double f(double wj, double x) {
        double r7744410 = x;
        double r7744411 = wj;
        double r7744412 = exp(r7744411);
        double r7744413 = r7744412 * r7744411;
        double r7744414 = r7744412 + r7744413;
        double r7744415 = r7744410 / r7744414;
        double r7744416 = r7744411 * r7744411;
        double r7744417 = -r7744411;
        double r7744418 = r7744417 * r7744411;
        double r7744419 = r7744411 * r7744418;
        double r7744420 = fma(r7744411, r7744411, r7744419);
        double r7744421 = fma(r7744416, r7744416, r7744420);
        double r7744422 = r7744415 + r7744421;
        return r7744422;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.9
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.5

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
  2. Using strategy rm

  3. Applied div-sub13.5

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

  4. Applied associate--r-7.5

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

  5. Taylor expanded around 0 1.1

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

  6. Simplified1.1

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

  8. Applied fma-neg1.1

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

  9. Final simplification1.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

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