Average Error: 13.7 → 0.9
Time: 5.7s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 4.160404689641216852166604763104892916736 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 4.160404689641216852166604763104892916736 \cdot 10^{-9}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r408099 = wj;
        double r408100 = exp(r408099);
        double r408101 = r408099 * r408100;
        double r408102 = x;
        double r408103 = r408101 - r408102;
        double r408104 = r408100 + r408101;
        double r408105 = r408103 / r408104;
        double r408106 = r408099 - r408105;
        return r408106;
}
↓
double f(double wj, double x) {
        double r408107 = wj;
        double r408108 = 4.160404689641217e-09;
        bool r408109 = r408107 <= r408108;
        double r408110 = x;
        double r408111 = 2.0;
        double r408112 = pow(r408107, r408111);
        double r408113 = r408110 + r408112;
        double r408114 = r408107 * r408110;
        double r408115 = r408111 * r408114;
        double r408116 = r408113 - r408115;
        double r408117 = exp(r408107);
        double r408118 = 1.0;
        double r408119 = r408107 + r408118;
        double r408120 = r408117 * r408119;
        double r408121 = r408110 / r408120;
        double r408122 = r408121 + r408107;
        double r408123 = r408107 / r408119;
        double r408124 = r408122 - r408123;
        double r408125 = r408109 ? r408116 : r408124;
        return r408125;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.7
Target13.1
Herbie0.9
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 4.160404689641217e-09

    1. Initial program 13.4

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

    2. Simplified13.4

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

    3. Taylor expanded around 0 0.8

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

    if 4.160404689641217e-09 < wj

    1. Initial program 26.7

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

    2. Simplified2.9

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

    4. Applied div-inv2.9

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

    5. Applied associate-/l*2.9

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

    6. Simplified2.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 4.160404689641216852166604763104892916736 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))))