Average Error: 13.8 → 1.1
Time: 8.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 1.043153900035086351654028702491698264697 \cdot 10^{-16}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{{wj}^{3} + 1} \cdot \frac{wj \cdot wj + \left(1 - wj \cdot 1\right)}{e^{wj}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\\ \end{array}\]
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}{{wj}^{3} + 1} \cdot \frac{wj \cdot wj + \left(1 - wj \cdot 1\right)}{e^{wj}} + wj\right) - \frac{1}{\frac{wj + 1}{wj}}\\

\end{array}
double f(double wj, double x) {
        double r263222 = wj;
        double r263223 = exp(r263222);
        double r263224 = r263222 * r263223;
        double r263225 = x;
        double r263226 = r263224 - r263225;
        double r263227 = r263223 + r263224;
        double r263228 = r263226 / r263227;
        double r263229 = r263222 - r263228;
        return r263229;
}
↓
double f(double wj, double x) {
        double r263230 = wj;
        double r263231 = 1.0431539000350864e-16;
        bool r263232 = r263230 <= r263231;
        double r263233 = x;
        double r263234 = 2.0;
        double r263235 = pow(r263230, r263234);
        double r263236 = r263233 + r263235;
        double r263237 = r263230 * r263233;
        double r263238 = r263234 * r263237;
        double r263239 = r263236 - r263238;
        double r263240 = 3.0;
        double r263241 = pow(r263230, r263240);
        double r263242 = 1.0;
        double r263243 = r263241 + r263242;
        double r263244 = r263233 / r263243;
        double r263245 = r263230 * r263230;
        double r263246 = r263230 * r263242;
        double r263247 = r263242 - r263246;
        double r263248 = r263245 + r263247;
        double r263249 = exp(r263230);
        double r263250 = r263248 / r263249;
        double r263251 = r263244 * r263250;
        double r263252 = r263251 + r263230;
        double r263253 = r263230 + r263242;
        double r263254 = r263253 / r263230;
        double r263255 = r263242 / r263254;
        double r263256 = r263252 - r263255;
        double r263257 = r263232 ? r263239 : r263256;
        return r263257;
}

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 *-un-lft-identity7.6

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

    5. Applied flip3-+7.7

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

    6. Applied associate-/r/7.6

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

    7. Applied times-frac7.6

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

    8. Simplified7.6

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

    9. Simplified7.6

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

    11. Applied clear-num7.6

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(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))))))