Average Error: 14.0 → 1.4
Time: 8.3m
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 6.460732948436636 \cdot 10^{-24}:\\ \;\;\;\;x + \left(wj \cdot wj - x \cdot \left(wj + wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{wj \cdot e^{wj} - x}{1 + wj}}{e^{wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 6.460732948436636 \cdot 10^{-24}:\\
\;\;\;\;x + \left(wj \cdot wj - x \cdot \left(wj + wj\right)\right)\\

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

\end{array}
double f(double wj, double x) {
        double r18049043 = wj;
        double r18049044 = exp(r18049043);
        double r18049045 = r18049043 * r18049044;
        double r18049046 = x;
        double r18049047 = r18049045 - r18049046;
        double r18049048 = r18049044 + r18049045;
        double r18049049 = r18049047 / r18049048;
        double r18049050 = r18049043 - r18049049;
        return r18049050;
}
↓
double f(double wj, double x) {
        double r18049051 = wj;
        double r18049052 = exp(r18049051);
        double r18049053 = r18049051 * r18049052;
        double r18049054 = x;
        double r18049055 = r18049053 - r18049054;
        double r18049056 = r18049052 + r18049053;
        double r18049057 = r18049055 / r18049056;
        double r18049058 = r18049051 - r18049057;
        double r18049059 = 6.460732948436636e-24;
        bool r18049060 = r18049058 <= r18049059;
        double r18049061 = r18049051 * r18049051;
        double r18049062 = r18049051 + r18049051;
        double r18049063 = r18049054 * r18049062;
        double r18049064 = r18049061 - r18049063;
        double r18049065 = r18049054 + r18049064;
        double r18049066 = 1.0;
        double r18049067 = r18049066 + r18049051;
        double r18049068 = r18049055 / r18049067;
        double r18049069 = r18049068 / r18049052;
        double r18049070 = r18049051 - r18049069;
        double r18049071 = r18049060 ? r18049065 : r18049070;
        return r18049071;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.0
Target13.3
Herbie1.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))) < 6.460732948436636e-24

    1. Initial program 18.4

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

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

    if 6.460732948436636e-24 < (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj)))))

    1. Initial program 3.3

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

    3. Applied distribute-rgt1-in3.3

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

    4. Applied associate-/r*3.3

      \[\leadsto wj - \color{blue}{\frac{\frac{wj \cdot e^{wj} - x}{wj + 1}}{e^{wj}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}} \le 6.460732948436636 \cdot 10^{-24}:\\ \;\;\;\;x + \left(wj \cdot wj - x \cdot \left(wj + wj\right)\right)\\ \mathbf{else}:\\ \;\;\;\;wj - \frac{\frac{wj \cdot e^{wj} - x}{1 + wj}}{e^{wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))))))