Average Error: 13.5 → 1.0
Time: 7.5s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.08704555276393653483273218961560613316 \cdot 10^{-8}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.08704555276393653483273218961560613316 \cdot 10^{-8}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r347102 = wj;
        double r347103 = exp(r347102);
        double r347104 = r347102 * r347103;
        double r347105 = x;
        double r347106 = r347104 - r347105;
        double r347107 = r347103 + r347104;
        double r347108 = r347106 / r347107;
        double r347109 = r347102 - r347108;
        return r347109;
}
↓
double f(double wj, double x) {
        double r347110 = wj;
        double r347111 = 1.0870455527639365e-08;
        bool r347112 = r347110 <= r347111;
        double r347113 = x;
        double r347114 = 2.0;
        double r347115 = pow(r347110, r347114);
        double r347116 = r347113 + r347115;
        double r347117 = r347110 * r347113;
        double r347118 = r347114 * r347117;
        double r347119 = r347116 - r347118;
        double r347120 = 1.0;
        double r347121 = r347110 + r347120;
        double r347122 = r347113 / r347121;
        double r347123 = exp(r347110);
        double r347124 = r347122 / r347123;
        double r347125 = r347124 + r347110;
        double r347126 = r347110 * r347110;
        double r347127 = r347126 - r347120;
        double r347128 = r347110 / r347127;
        double r347129 = r347110 - r347120;
        double r347130 = r347128 * r347129;
        double r347131 = r347125 - r347130;
        double r347132 = r347112 ? r347119 : r347131;
        return r347132;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target12.9
Herbie1.0
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.0870455527639365e-08

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 0.9

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

    if 1.0870455527639365e-08 < wj

    1. Initial program 25.4

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

    2. Simplified2.8

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

    4. Applied flip-+2.8

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

    5. Applied associate-/r/2.8

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

    6. Simplified2.8

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.08704555276393653483273218961560613316 \cdot 10^{-8}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{wj}{wj \cdot wj - 1} \cdot \left(wj - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(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))))))