Average Error: 13.8 → 0.9
Time: 5.6s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)
double f(double wj, double x) {
        double r253206 = wj;
        double r253207 = exp(r253206);
        double r253208 = r253206 * r253207;
        double r253209 = x;
        double r253210 = r253208 - r253209;
        double r253211 = r253207 + r253208;
        double r253212 = r253210 / r253211;
        double r253213 = r253206 - r253212;
        return r253213;
}
↓
double f(double wj, double x) {
        double r253214 = x;
        double r253215 = wj;
        double r253216 = 1.0;
        double r253217 = r253215 + r253216;
        double r253218 = r253214 / r253217;
        double r253219 = exp(r253215);
        double r253220 = r253218 / r253219;
        double r253221 = 4.0;
        double r253222 = pow(r253215, r253221);
        double r253223 = 3.0;
        double r253224 = pow(r253215, r253223);
        double r253225 = r253222 - r253224;
        double r253226 = fma(r253215, r253215, r253225);
        double r253227 = r253220 + r253226;
        return r253227;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.2
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.8

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

  2. Simplified13.2

    \[\leadsto \color{blue}{\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj + 1}}\]
  3. Using strategy rm

  4. Applied associate--l+6.9

    \[\leadsto \color{blue}{\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(wj - \frac{wj}{wj + 1}\right)}\]

  5. Taylor expanded around 0 0.9

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]

  6. Simplified0.9

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)}\]

  7. Final simplification0.9

    \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \mathsf{fma}\left(wj, wj, {wj}^{4} - {wj}^{3}\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))