Average Error: 14.0 → 2.1
Time: 25.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, x\right) - \left(wj + wj\right) \cdot x\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, x\right) - \left(wj + wj\right) \cdot x
double f(double wj, double x) {
        double r3838314 = wj;
        double r3838315 = exp(r3838314);
        double r3838316 = r3838314 * r3838315;
        double r3838317 = x;
        double r3838318 = r3838316 - r3838317;
        double r3838319 = r3838315 + r3838316;
        double r3838320 = r3838318 / r3838319;
        double r3838321 = r3838314 - r3838320;
        return r3838321;
}
↓
double f(double wj, double x) {
        double r3838322 = wj;
        double r3838323 = x;
        double r3838324 = fma(r3838322, r3838322, r3838323);
        double r3838325 = r3838322 + r3838322;
        double r3838326 = r3838325 * r3838323;
        double r3838327 = r3838324 - r3838326;
        return r3838327;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.0
Target13.3
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.0

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

  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

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019153 +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))))))