Average Error: 13.2 → 2.1
Time: 21.1s
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 r7691231 = wj;
        double r7691232 = exp(r7691231);
        double r7691233 = r7691231 * r7691232;
        double r7691234 = x;
        double r7691235 = r7691233 - r7691234;
        double r7691236 = r7691232 + r7691233;
        double r7691237 = r7691235 / r7691236;
        double r7691238 = r7691231 - r7691237;
        return r7691238;
}
↓
double f(double wj, double x) {
        double r7691239 = wj;
        double r7691240 = x;
        double r7691241 = fma(r7691239, r7691239, r7691240);
        double r7691242 = r7691239 + r7691239;
        double r7691243 = r7691242 * r7691240;
        double r7691244 = r7691241 - r7691243;
        return r7691244;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.2
Target12.5
Herbie2.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.2

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

  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

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

  4. Final simplification2.1

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

Reproduce

herbie shell --seed 2019149 +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))))))