Average Error: 13.8 → 1.4
Time: 6.1s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le -5.8145946613870445 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le -5.8145946613870445 \cdot 10^{-9}:\\
\;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\

\end{array}
double f(double wj, double x) {
        double r150067 = wj;
        double r150068 = exp(r150067);
        double r150069 = r150067 * r150068;
        double r150070 = x;
        double r150071 = r150069 - r150070;
        double r150072 = r150068 + r150069;
        double r150073 = r150071 / r150072;
        double r150074 = r150067 - r150073;
        return r150074;
}
↓
double f(double wj, double x) {
        double r150075 = wj;
        double r150076 = -5.8145946613870445e-09;
        bool r150077 = r150075 <= r150076;
        double r150078 = exp(r150075);
        double r150079 = r150075 * r150078;
        double r150080 = x;
        double r150081 = r150079 - r150080;
        double r150082 = r150078 + r150079;
        double r150083 = r150081 / r150082;
        double r150084 = r150075 - r150083;
        double r150085 = 1.0;
        double r150086 = fma(r150075, r150075, r150080);
        double r150087 = r150085 * r150086;
        double r150088 = 2.0;
        double r150089 = r150075 * r150080;
        double r150090 = r150088 * r150089;
        double r150091 = r150087 - r150090;
        double r150092 = r150077 ? r150084 : r150091;
        return r150092;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.8
Target13.3
Herbie1.4
\[wj - \left(\frac{wj}{wj + 1} - \frac{x}{e^{wj} + wj \cdot e^{wj}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if wj < -5.8145946613870445e-09

    1. Initial program 4.7

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

    if -5.8145946613870445e-09 < wj

    1. Initial program 14.0

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

    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 1.3

      \[\leadsto \color{blue}{\left(x + {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity1.3

      \[\leadsto \left(x + \color{blue}{1 \cdot {wj}^{2}}\right) - 2 \cdot \left(wj \cdot x\right)\]

    6. Applied *-un-lft-identity1.3

      \[\leadsto \left(\color{blue}{1 \cdot x} + 1 \cdot {wj}^{2}\right) - 2 \cdot \left(wj \cdot x\right)\]

    7. Applied distribute-lft-out1.3

      \[\leadsto \color{blue}{1 \cdot \left(x + {wj}^{2}\right)} - 2 \cdot \left(wj \cdot x\right)\]

    8. Simplified1.3

      \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(wj, wj, x\right)} - 2 \cdot \left(wj \cdot x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le -5.8145946613870445 \cdot 10^{-9}:\\ \;\;\;\;wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))))