Average Error: 13.8 → 1.1
Time: 7.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.043153900035086351654028702491698264697 \cdot 10^{-16}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{{\left({wj}^{3}\right)}^{3} + 1} \cdot \frac{\left(\left({wj}^{6} + 1\right) - {wj}^{3}\right) \cdot \left(\left(wj \cdot wj + 1\right) - wj\right)}{e^{wj}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.043153900035086351654028702491698264697 \cdot 10^{-16}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{x}{{\left({wj}^{3}\right)}^{3} + 1} \cdot \frac{\left(\left({wj}^{6} + 1\right) - {wj}^{3}\right) \cdot \left(\left(wj \cdot wj + 1\right) - wj\right)}{e^{wj}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\\

\end{array}
double f(double wj, double x) {
        double r138757 = wj;
        double r138758 = exp(r138757);
        double r138759 = r138757 * r138758;
        double r138760 = x;
        double r138761 = r138759 - r138760;
        double r138762 = r138758 + r138759;
        double r138763 = r138761 / r138762;
        double r138764 = r138757 - r138763;
        return r138764;
}
↓
double f(double wj, double x) {
        double r138765 = wj;
        double r138766 = 1.0431539000350864e-16;
        bool r138767 = r138765 <= r138766;
        double r138768 = x;
        double r138769 = 2.0;
        double r138770 = pow(r138765, r138769);
        double r138771 = r138768 + r138770;
        double r138772 = r138765 * r138768;
        double r138773 = r138769 * r138772;
        double r138774 = r138771 - r138773;
        double r138775 = 3.0;
        double r138776 = pow(r138765, r138775);
        double r138777 = pow(r138776, r138775);
        double r138778 = 1.0;
        double r138779 = r138777 + r138778;
        double r138780 = r138768 / r138779;
        double r138781 = 6.0;
        double r138782 = pow(r138765, r138781);
        double r138783 = r138782 + r138778;
        double r138784 = r138783 - r138776;
        double r138785 = r138765 * r138765;
        double r138786 = r138785 + r138778;
        double r138787 = r138786 - r138765;
        double r138788 = r138784 * r138787;
        double r138789 = exp(r138765);
        double r138790 = r138788 / r138789;
        double r138791 = r138780 * r138790;
        double r138792 = r138791 + r138765;
        double r138793 = r138765 + r138778;
        double r138794 = r138793 / r138765;
        double r138795 = r138778 / r138794;
        double r138796 = r138792 - r138795;
        double r138797 = r138767 ? r138774 : r138796;
        return r138797;
}

Error

Bits error versus wj

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.8
Target13.1
Herbie1.1
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 1.0431539000350864e-16

    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 1.0431539000350864e-16 < wj

    1. Initial program 24.9

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

    2. Simplified7.6

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

    4. Applied clear-num7.6

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

    6. Applied *-un-lft-identity7.6

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

    7. Applied flip3-+7.7

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

    8. Applied associate-/r/7.6

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

    9. Applied times-frac7.6

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

    10. Simplified7.6

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

    11. Simplified7.6

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

    13. Applied flip3-+7.6

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

    14. Applied associate-/r/7.6

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

    15. Applied associate-*l*7.6

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

    16. Simplified7.6

      \[\leadsto \left(\frac{x}{{\left({wj}^{3}\right)}^{3} + {1}^{3}} \cdot \color{blue}{\frac{\left(\left({wj}^{6} + 1\right) - {wj}^{3}\right) \cdot \left(\left(wj \cdot wj + 1\right) - wj\right)}{e^{wj}}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.043153900035086351654028702491698264697 \cdot 10^{-16}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{{\left({wj}^{3}\right)}^{3} + 1} \cdot \frac{\left(\left({wj}^{6} + 1\right) - {wj}^{3}\right) \cdot \left(\left(wj \cdot wj + 1\right) - wj\right)}{e^{wj}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(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))))))