Average Error: 29.8 → 0.0
Time: 9.1s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00729937700901299:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.00711599847374152:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00729937700901299:\\
\;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\

\mathbf{elif}\;x \le 0.00711599847374152:\\
\;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3}\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r841041 = 2.0;
        double r841042 = 1.0;
        double r841043 = -2.0;
        double r841044 = x;
        double r841045 = r841043 * r841044;
        double r841046 = exp(r841045);
        double r841047 = r841042 + r841046;
        double r841048 = r841041 / r841047;
        double r841049 = r841048 - r841042;
        return r841049;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r841050 = x;
        double r841051 = -0.00729937700901299;
        bool r841052 = r841050 <= r841051;
        double r841053 = 2.0;
        double r841054 = -2.0;
        double r841055 = r841054 * r841050;
        double r841056 = exp(r841055);
        double r841057 = 1.0;
        double r841058 = r841056 + r841057;
        double r841059 = r841053 / r841058;
        double r841060 = r841059 - r841057;
        double r841061 = 0.00711599847374152;
        bool r841062 = r841050 <= r841061;
        double r841063 = 0.13333333333333333;
        double r841064 = 5.0;
        double r841065 = pow(r841050, r841064);
        double r841066 = r841063 * r841065;
        double r841067 = r841066 + r841050;
        double r841068 = r841050 * r841050;
        double r841069 = r841050 * r841068;
        double r841070 = -0.3333333333333333;
        double r841071 = r841069 * r841070;
        double r841072 = r841067 + r841071;
        double r841073 = r841062 ? r841072 : r841060;
        double r841074 = r841052 ? r841060 : r841073;
        return r841074;
}

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 < -0.00729937700901299 or 0.00711599847374152 < 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 -0.00729937700901299 < x < 0.00711599847374152

    1. Initial program 59.2

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00729937700901299:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \mathbf{elif}\;x \le 0.00711599847374152:\\ \;\;\;\;\left(\frac{2}{15} \cdot {x}^{5} + x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{e^{-2 \cdot x} + 1} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))