Average Error: 13.5 → 1.0
Time: 6.3s
Precision: 64
\[wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}\]
↓
\[\begin{array}{l} \mathbf{if}\;wj \le 6.509211057300332155337300766044170141633 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{{wj}^{3} + 1}}{\frac{e^{wj}}{wj \cdot wj + \left(1 - wj \cdot 1\right)}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]
wj - \frac{wj \cdot e^{wj} - x}{e^{wj} + wj \cdot e^{wj}}
↓
\begin{array}{l}
\mathbf{if}\;wj \le 6.509211057300332155337300766044170141633 \cdot 10^{-9}:\\
\;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\

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

\end{array}
double f(double wj, double x) {
        double r235493 = wj;
        double r235494 = exp(r235493);
        double r235495 = r235493 * r235494;
        double r235496 = x;
        double r235497 = r235495 - r235496;
        double r235498 = r235494 + r235495;
        double r235499 = r235497 / r235498;
        double r235500 = r235493 - r235499;
        return r235500;
}
↓
double f(double wj, double x) {
        double r235501 = wj;
        double r235502 = 6.509211057300332e-09;
        bool r235503 = r235501 <= r235502;
        double r235504 = x;
        double r235505 = fma(r235501, r235501, r235504);
        double r235506 = 2.0;
        double r235507 = r235501 * r235504;
        double r235508 = r235506 * r235507;
        double r235509 = r235505 - r235508;
        double r235510 = 3.0;
        double r235511 = pow(r235501, r235510);
        double r235512 = 1.0;
        double r235513 = r235511 + r235512;
        double r235514 = r235504 / r235513;
        double r235515 = exp(r235501);
        double r235516 = r235501 * r235501;
        double r235517 = r235501 * r235512;
        double r235518 = r235512 - r235517;
        double r235519 = r235516 + r235518;
        double r235520 = r235515 / r235519;
        double r235521 = r235514 / r235520;
        double r235522 = r235521 + r235501;
        double r235523 = r235501 + r235512;
        double r235524 = r235501 / r235523;
        double r235525 = r235522 - r235524;
        double r235526 = r235503 ? r235509 : r235525;
        return r235526;
}

Error

Bits error versus wj

Bits error versus x

Target

Original13.5
Target12.9
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 < 6.509211057300332e-09

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 0.9

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

    4. Taylor expanded around 0 0.9

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

    5. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(wj, wj, x\right)} - 2 \cdot \left(wj \cdot x\right)\]

    if 6.509211057300332e-09 < wj

    1. Initial program 25.1

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

    2. Simplified3.1

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

    4. Applied flip3-+3.1

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

    5. Applied associate-/r/3.1

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

    6. Applied associate-/l*3.1

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;wj \le 6.509211057300332155337300766044170141633 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(wj, wj, x\right) - 2 \cdot \left(wj \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{x}{{wj}^{3} + 1}}{\frac{e^{wj}}{wj \cdot wj + \left(1 - wj \cdot 1\right)}} + wj\right) - \frac{wj}{wj + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))))))