Average Error: 13.8 → 1.0
Time: 27.2s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 5.350338768905503620540304297349687434604 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 5.350338768905503620540304297349687434604 \cdot 10^{-9}:\\
\;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r176264 = wj;
        double r176265 = exp(r176264);
        double r176266 = r176264 * r176265;
        double r176267 = x;
        double r176268 = r176266 - r176267;
        double r176269 = r176265 + r176266;
        double r176270 = r176268 / r176269;
        double r176271 = r176264 - r176270;
        return r176271;
}
↓
double f(double wj, double x) {
        double r176272 = wj;
        double r176273 = 5.350338768905504e-09;
        bool r176274 = r176272 <= r176273;
        double r176275 = x;
        double r176276 = 2.0;
        double r176277 = pow(r176272, r176276);
        double r176278 = r176275 + r176277;
        double r176279 = r176272 * r176275;
        double r176280 = r176276 * r176279;
        double r176281 = r176278 - r176280;
        double r176282 = exp(r176272);
        double r176283 = 1.0;
        double r176284 = r176272 + r176283;
        double r176285 = r176282 * r176284;
        double r176286 = r176275 / r176285;
        double r176287 = r176286 + r176272;
        double r176288 = r176272 / r176284;
        double r176289 = r176287 - r176288;
        double r176290 = r176274 ? r176281 : r176289;
        return r176290;
}

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

Derivation

  1. Split input into 2 regimes

  2. if wj < 5.350338768905504e-09

    1. Initial program 13.5

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 0.9

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

    if 5.350338768905504e-09 < wj

    1. Initial program 25.3

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

    2. Simplified2.7

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

    4. Applied div-inv2.7

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

    5. Applied associate-/l*2.7

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

    6. Simplified2.7

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 5.350338768905503620540304297349687434604 \cdot 10^{-9}:\\ \;\;\;\;\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{e^{wj} \cdot \left(wj + 1\right)} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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))))))