Average Error: 13.8 → 2.1
Time: 16.9s
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 r3379226 = wj;
        double r3379227 = exp(r3379226);
        double r3379228 = r3379226 * r3379227;
        double r3379229 = x;
        double r3379230 = r3379228 - r3379229;
        double r3379231 = r3379227 + r3379228;
        double r3379232 = r3379230 / r3379231;
        double r3379233 = r3379226 - r3379232;
        return r3379233;
}
↓
double f(double wj, double x) {
        double r3379234 = wj;
        double r3379235 = x;
        double r3379236 = fma(r3379234, r3379234, r3379235);
        double r3379237 = r3379234 + r3379234;
        double r3379238 = r3379237 * r3379235;
        double r3379239 = r3379236 - r3379238;
        return r3379239;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.1
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  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 2019156 +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))))))