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

↓

\[\begin{array}{l} \mathbf{if}\;wj \le 6.82905740885949 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array}\]

double f(double wj, double x) {
        double r55895679 = wj;
        double r55895680 = exp(r55895679);
        double r55895681 = r55895679 * r55895680;
        double r55895682 = x;
        double r55895683 = r55895681 - r55895682;
        double r55895684 = r55895680 + r55895681;
        double r55895685 = r55895683 / r55895684;
        double r55895686 = r55895679 - r55895685;
        return r55895686;
}

↓

double f(double wj, double x) {
        double r55895687 = wj;
        double r55895688 = 6.82905740885949e-09;
        bool r55895689 = r55895687 <= r55895688;
        double r55895690 = x;
        double r55895691 = -2.0;
        double r55895692 = r55895690 * r55895691;
        double r55895693 = r55895692 + r55895687;
        double r55895694 = r55895687 * r55895693;
        double r55895695 = r55895690 + r55895694;
        double r55895696 = exp(r55895687);
        double r55895697 = r55895690 / r55895696;
        double r55895698 = r55895687 - r55895697;
        double r55895699 = 1.0;
        double r55895700 = r55895687 + r55895699;
        double r55895701 = r55895699 / r55895700;
        double r55895702 = r55895698 * r55895701;
        double r55895703 = r55895687 - r55895702;
        double r55895704 = r55895689 ? r55895695 : r55895703;
        return r55895704;
}

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

↓

\begin{array}{l}
\mathbf{if}\;wj \le 6.82905740885949 \cdot 10^{-09}:\\
\;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\

\mathbf{else}:\\
\;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\

\end{array}

Original	13.6
Target	13.0
Herbie	1.0

Derivation

Split input into 2 regimes
if wj < 6.82905740885949e-09
1. Initial program 13.3
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Taylor expanded around 0 0.9
  \[\leadsto \color{blue}{\left({wj}^{2} + x\right) - 2 \cdot \left(x \cdot wj\right)}\]
3. Simplified1.0
  \[\leadsto \color{blue}{x + \left(wj + -2 \cdot x\right) \cdot wj}\]
if 6.82905740885949e-09 < wj
1. Initial program 24.6
  \[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
2. Using strategy rm
3. Applied distribute-rgt1-in24.6
  \[\leadsto wj - \frac{wj \cdot e^{wj} - x}{\color{blue}{\left(wj + 1\right) \cdot e^{wj}}}\]
4. Applied *-un-lft-identity24.6
  \[\leadsto wj - \frac{\color{blue}{1 \cdot \left(wj \cdot e^{wj} - x\right)}}{\left(wj + 1\right) \cdot e^{wj}}\]
5. Applied times-frac24.5
  \[\leadsto wj - \color{blue}{\frac{1}{wj + 1} \cdot \frac{wj \cdot e^{wj} - x}{e^{wj}}}\]
6. Simplified2.4
  \[\leadsto wj - \frac{1}{wj + 1} \cdot \color{blue}{\left(wj - \frac{x}{e^{wj}}\right)}\]
Recombined 2 regimes into one program.
Final simplification1.0
\[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 6.82905740885949 \cdot 10^{-09}:\\ \;\;\;\;x + wj \cdot \left(x \cdot -2 + wj\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \left(wj - \frac{x}{e^{wj}}\right) \cdot \frac{1}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 
(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))))))

could not determine a ground truth for program body (more)

Error

Target

Derivation

`if wj < 6.82905740885949e-09`

`if 6.82905740885949e-09 < wj`

Reproduce