Average Error: 13.3 → 1.0
Time: 20.8s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 9.422333290444803599555997767386189556404 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{1}{\sqrt{wj + 1}} \cdot \frac{wj}{\sqrt{wj + 1}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 9.422333290444803599555997767386189556404 \cdot 10^{-9}:\\
\;\;\;\;\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{1}{\sqrt{wj + 1}} \cdot \frac{wj}{\sqrt{wj + 1}}\\

\end{array}
double f(double wj, double x) {
        double r173565 = wj;
        double r173566 = exp(r173565);
        double r173567 = r173565 * r173566;
        double r173568 = x;
        double r173569 = r173567 - r173568;
        double r173570 = r173566 + r173567;
        double r173571 = r173569 / r173570;
        double r173572 = r173565 - r173571;
        return r173572;
}
↓
double f(double wj, double x) {
        double r173573 = wj;
        double r173574 = 9.422333290444804e-09;
        bool r173575 = r173573 <= r173574;
        double r173576 = x;
        double r173577 = 2.0;
        double r173578 = pow(r173573, r173577);
        double r173579 = r173576 + r173578;
        double r173580 = r173573 * r173576;
        double r173581 = r173577 * r173580;
        double r173582 = r173579 - r173581;
        double r173583 = 1.0;
        double r173584 = r173573 + r173583;
        double r173585 = r173576 / r173584;
        double r173586 = exp(r173573);
        double r173587 = r173585 / r173586;
        double r173588 = r173587 + r173573;
        double r173589 = sqrt(r173584);
        double r173590 = r173583 / r173589;
        double r173591 = r173573 / r173589;
        double r173592 = r173590 * r173591;
        double r173593 = r173588 - r173592;
        double r173594 = r173575 ? r173582 : r173593;
        return r173594;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.3
Target12.6
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 < 9.422333290444804e-09

    1. Initial program 12.9

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

    2. Simplified12.9

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

    3. Taylor expanded around 0 1.0

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

    if 9.422333290444804e-09 < wj

    1. Initial program 28.2

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

    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 add-sqr-sqrt3.0

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

    5. Applied *-un-lft-identity3.0

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

    6. Applied times-frac3.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 9.422333290444803599555997767386189556404 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{wj + 1}}{e^{wj}} + wj\right) - \frac{1}{\sqrt{wj + 1}} \cdot \frac{wj}{\sqrt{wj + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(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))))))