Average Error: 13.9 → 2.2
Time: 22.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(x \cdot -2, wj, \mathsf{fma}\left(wj, wj, x\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(x \cdot -2, wj, \mathsf{fma}\left(wj, wj, x\right)\right)
double f(double wj, double x) {
        double r8769266 = wj;
        double r8769267 = exp(r8769266);
        double r8769268 = r8769266 * r8769267;
        double r8769269 = x;
        double r8769270 = r8769268 - r8769269;
        double r8769271 = r8769267 + r8769268;
        double r8769272 = r8769270 / r8769271;
        double r8769273 = r8769266 - r8769272;
        return r8769273;
}
↓
double f(double wj, double x) {
        double r8769274 = x;
        double r8769275 = -2.0;
        double r8769276 = r8769274 * r8769275;
        double r8769277 = wj;
        double r8769278 = fma(r8769277, r8769277, r8769274);
        double r8769279 = fma(r8769276, r8769277, r8769278);
        return r8769279;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.9
Target13.3
Herbie2.2
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.9

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

  2. Taylor expanded around 0 2.2

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

  3. Simplified2.2

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

  4. Final simplification2.2

    \[\leadsto \mathsf{fma}\left(x \cdot -2, wj, \mathsf{fma}\left(wj, wj, x\right)\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(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))))))