Average Error: 13.7 → 1.1
Time: 26.0s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\mathsf{fma}\left(wj, wj, \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\mathsf{fma}\left(wj, wj, \left(wj \cdot wj - wj\right) \cdot \left(wj \cdot wj\right)\right) + \frac{x}{wj \cdot e^{wj} + e^{wj}}
double f(double wj, double x) {
        double r7002937 = wj;
        double r7002938 = exp(r7002937);
        double r7002939 = r7002937 * r7002938;
        double r7002940 = x;
        double r7002941 = r7002939 - r7002940;
        double r7002942 = r7002938 + r7002939;
        double r7002943 = r7002941 / r7002942;
        double r7002944 = r7002937 - r7002943;
        return r7002944;
}
↓
double f(double wj, double x) {
        double r7002945 = wj;
        double r7002946 = r7002945 * r7002945;
        double r7002947 = r7002946 - r7002945;
        double r7002948 = r7002947 * r7002946;
        double r7002949 = fma(r7002945, r7002945, r7002948);
        double r7002950 = x;
        double r7002951 = exp(r7002945);
        double r7002952 = r7002945 * r7002951;
        double r7002953 = r7002952 + r7002951;
        double r7002954 = r7002950 / r7002953;
        double r7002955 = r7002949 + r7002954;
        return r7002955;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.7
Target13.1
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Initial program 13.7

    \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
  2. Using strategy rm

  3. Applied div-sub13.7

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

  4. Applied associate--r-7.7

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

  5. Taylor expanded around 0 1.1

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

  6. Simplified1.1

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

  8. Applied fma-def1.1

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

  9. Final simplification1.1

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

Reproduce

herbie shell --seed 2019152 +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))))))