Average Error: 13.8 → 0.3
Time: 6.4s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.13141850474180142 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{x}{wj + 1}} \cdot \sqrt[3]{\frac{x}{wj + 1}}}{\sqrt{e^{wj}}} \cdot \frac{\sqrt[3]{\frac{x}{wj + 1}}}{\sqrt{e^{wj}}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 1.13141850474180142 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\

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

\end{array}
double f(double wj, double x) {
        double r301069 = wj;
        double r301070 = exp(r301069);
        double r301071 = r301069 * r301070;
        double r301072 = x;
        double r301073 = r301071 - r301072;
        double r301074 = r301070 + r301071;
        double r301075 = r301073 / r301074;
        double r301076 = r301069 - r301075;
        return r301076;
}
↓
double f(double wj, double x) {
        double r301077 = wj;
        double r301078 = 0.00011314185047418014;
        bool r301079 = r301077 <= r301078;
        double r301080 = x;
        double r301081 = 1.0;
        double r301082 = r301077 + r301081;
        double r301083 = r301080 / r301082;
        double r301084 = exp(r301077);
        double r301085 = r301083 / r301084;
        double r301086 = 4.0;
        double r301087 = pow(r301077, r301086);
        double r301088 = 2.0;
        double r301089 = pow(r301077, r301088);
        double r301090 = r301087 + r301089;
        double r301091 = 3.0;
        double r301092 = pow(r301077, r301091);
        double r301093 = r301090 - r301092;
        double r301094 = r301085 + r301093;
        double r301095 = cbrt(r301083);
        double r301096 = r301095 * r301095;
        double r301097 = sqrt(r301084);
        double r301098 = r301096 / r301097;
        double r301099 = r301095 / r301097;
        double r301100 = r301098 * r301099;
        double r301101 = r301077 / r301082;
        double r301102 = r301077 - r301101;
        double r301103 = r301100 + r301102;
        double r301104 = r301079 ? r301094 : r301103;
        return r301104;
}

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
Target13.2
Herbie0.3
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < 0.00011314185047418014

    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. Using strategy rm

    4. Applied associate--l+6.9

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

    5. Taylor expanded around 0 0.3

      \[\leadsto \frac{\frac{x}{wj + 1}}{e^{wj}} + \color{blue}{\left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)}\]

    if 0.00011314185047418014 < wj

    1. Initial program 29.7

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

    2. Simplified0.8

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

    4. Applied associate--l+0.7

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

    6. Applied add-sqr-sqrt0.8

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

    7. Applied add-cube-cbrt1.0

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

    8. Applied times-frac1.0

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 1.13141850474180142 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{x}{wj + 1}}{e^{wj}} + \left(\left({wj}^{4} + {wj}^{2}\right) - {wj}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{x}{wj + 1}} \cdot \sqrt[3]{\frac{x}{wj + 1}}}{\sqrt{e^{wj}}} \cdot \frac{\sqrt[3]{\frac{x}{wj + 1}}}{\sqrt{e^{wj}}} + \left(wj - \frac{wj}{wj + 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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))))))