Average Error: 13.7 → 2.1
Time: 21.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj \cdot x, -2, \mathsf{fma}\left(wj, wj, x\right)\right)
double f(double wj, double x) {
        double r6562208 = wj;
        double r6562209 = exp(r6562208);
        double r6562210 = r6562208 * r6562209;
        double r6562211 = x;
        double r6562212 = r6562210 - r6562211;
        double r6562213 = r6562209 + r6562210;
        double r6562214 = r6562212 / r6562213;
        double r6562215 = r6562208 - r6562214;
        return r6562215;
}
↓
double f(double wj, double x) {
        double r6562216 = wj;
        double r6562217 = x;
        double r6562218 = r6562216 * r6562217;
        double r6562219 = -2.0;
        double r6562220 = fma(r6562216, r6562216, r6562217);
        double r6562221 = fma(r6562218, r6562219, r6562220);
        return r6562221;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

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

  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

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019179 +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))))))