Average Error: 14.1 → 2.3
Time: 2.2m
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(\left(\mathsf{fma}\left(x, -2, wj\right)\right), wj, x\right)
double f(double wj, double x) {
        double r57385052 = wj;
        double r57385053 = exp(r57385052);
        double r57385054 = r57385052 * r57385053;
        double r57385055 = x;
        double r57385056 = r57385054 - r57385055;
        double r57385057 = r57385053 + r57385054;
        double r57385058 = r57385056 / r57385057;
        double r57385059 = r57385052 - r57385058;
        return r57385059;
}
↓
double f(double wj, double x) {
        double r57385060 = x;
        double r57385061 = -2.0;
        double r57385062 = wj;
        double r57385063 = fma(r57385060, r57385061, r57385062);
        double r57385064 = fma(r57385063, r57385062, r57385060);
        return r57385064;
}

Error

Bits error versus wj

Bits error versus x

Target

Original14.1
Target13.5
Herbie2.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 14.1

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

  2. Taylor expanded around 0 2.3

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

  3. Simplified2.3

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

  4. Final simplification2.3

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

Reproduce

herbie shell --seed 2019125 +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))))))