Average Error: 29.4 → 0.0
Time: 13.9s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.006716681295444204:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.005632491014509446:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{-1}{3}\right) + x\right) + \frac{2}{15} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.006716681295444204:\\
\;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\

\mathbf{elif}\;x \le 0.005632491014509446:\\
\;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{-1}{3}\right) + x\right) + \frac{2}{15} \cdot {x}^{5}\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r1736596 = 2.0;
        double r1736597 = 1.0;
        double r1736598 = -2.0;
        double r1736599 = x;
        double r1736600 = r1736598 * r1736599;
        double r1736601 = exp(r1736600);
        double r1736602 = r1736597 + r1736601;
        double r1736603 = r1736596 / r1736602;
        double r1736604 = r1736603 - r1736597;
        return r1736604;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r1736605 = x;
        double r1736606 = -0.006716681295444204;
        bool r1736607 = r1736605 <= r1736606;
        double r1736608 = 2.0;
        double r1736609 = 1.0;
        double r1736610 = -2.0;
        double r1736611 = r1736610 * r1736605;
        double r1736612 = exp(r1736611);
        double r1736613 = r1736609 + r1736612;
        double r1736614 = r1736608 / r1736613;
        double r1736615 = r1736614 - r1736609;
        double r1736616 = 0.005632491014509446;
        bool r1736617 = r1736605 <= r1736616;
        double r1736618 = r1736605 * r1736605;
        double r1736619 = -0.3333333333333333;
        double r1736620 = r1736605 * r1736619;
        double r1736621 = r1736618 * r1736620;
        double r1736622 = r1736621 + r1736605;
        double r1736623 = 0.13333333333333333;
        double r1736624 = 5.0;
        double r1736625 = pow(r1736605, r1736624);
        double r1736626 = r1736623 * r1736625;
        double r1736627 = r1736622 + r1736626;
        double r1736628 = r1736617 ? r1736627 : r1736615;
        double r1736629 = r1736607 ? r1736615 : r1736628;
        return r1736629;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -0.006716681295444204 or 0.005632491014509446 < x

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    if -0.006716681295444204 < x < 0.005632491014509446

    1. Initial program 59.1

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

    2. Taylor expanded around inf 59.1

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

    3. Taylor expanded around 0 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.006716681295444204:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \le 0.005632491014509446:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{-1}{3}\right) + x\right) + \frac{2}{15} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))