Average Error: 30.2 → 0.0
Time: 15.1s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007956784398241263:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.0076963482510729354:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{2}{15}, \mathsf{fma}\left(\frac{-1}{3}, x \cdot \left(x \cdot x\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.007956784398241263:\\
\;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\

\mathbf{elif}\;x \le 0.0076963482510729354:\\
\;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{2}{15}, \mathsf{fma}\left(\frac{-1}{3}, x \cdot \left(x \cdot x\right), x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r1930462 = 2.0;
        double r1930463 = 1.0;
        double r1930464 = -2.0;
        double r1930465 = x;
        double r1930466 = r1930464 * r1930465;
        double r1930467 = exp(r1930466);
        double r1930468 = r1930463 + r1930467;
        double r1930469 = r1930462 / r1930468;
        double r1930470 = r1930469 - r1930463;
        return r1930470;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r1930471 = x;
        double r1930472 = -0.007956784398241263;
        bool r1930473 = r1930471 <= r1930472;
        double r1930474 = 2.0;
        double r1930475 = -2.0;
        double r1930476 = r1930475 * r1930471;
        double r1930477 = exp(r1930476);
        double r1930478 = 1.0;
        double r1930479 = r1930477 + r1930478;
        double r1930480 = r1930474 / r1930479;
        double r1930481 = r1930480 - r1930478;
        double r1930482 = 0.0076963482510729354;
        bool r1930483 = r1930471 <= r1930482;
        double r1930484 = 5.0;
        double r1930485 = pow(r1930471, r1930484);
        double r1930486 = 0.13333333333333333;
        double r1930487 = -0.3333333333333333;
        double r1930488 = r1930471 * r1930471;
        double r1930489 = r1930471 * r1930488;
        double r1930490 = fma(r1930487, r1930489, r1930471);
        double r1930491 = fma(r1930485, r1930486, r1930490);
        double r1930492 = r1930483 ? r1930491 : r1930481;
        double r1930493 = r1930473 ? r1930481 : r1930492;
        return r1930493;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Split input into 2 regimes

  2. if x < -0.007956784398241263 or 0.0076963482510729354 < x

    1. Initial program 0.0

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{e^{-2 \cdot x} + 1} - 1}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{2}{1 + e^{x \cdot -2}} - 1}\]

    if -0.007956784398241263 < x < 0.0076963482510729354

    1. Initial program 58.9

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(x + \frac{2}{15} \cdot {x}^{5}\right) - \frac{1}{3} \cdot {x}^{3}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{5}, \frac{2}{15}, \mathsf{fma}\left(\frac{-1}{3}, x \cdot \left(x \cdot x\right), x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007956784398241263:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.0076963482510729354:\\ \;\;\;\;\mathsf{fma}\left({x}^{5}, \frac{2}{15}, \mathsf{fma}\left(\frac{-1}{3}, x \cdot \left(x \cdot x\right), x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))