Average Error: 29.0 → 0.1
Time: 19.4s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\
\;\;\;\;\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\

\mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\
\;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r2735077 = 2.0;
        double r2735078 = 1.0;
        double r2735079 = -2.0;
        double r2735080 = x;
        double r2735081 = r2735079 * r2735080;
        double r2735082 = exp(r2735081);
        double r2735083 = r2735078 + r2735082;
        double r2735084 = r2735077 / r2735083;
        double r2735085 = r2735084 - r2735078;
        return r2735085;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2735086 = -2.0;
        double r2735087 = x;
        double r2735088 = r2735086 * r2735087;
        double r2735089 = -15.661220476650124;
        bool r2735090 = r2735088 <= r2735089;
        double r2735091 = 2.0;
        double r2735092 = 1.0;
        double r2735093 = exp(r2735088);
        double r2735094 = r2735092 + r2735093;
        double r2735095 = r2735091 / r2735094;
        double r2735096 = r2735095 - r2735092;
        double r2735097 = r2735096 * r2735096;
        double r2735098 = r2735097 * r2735096;
        double r2735099 = cbrt(r2735098);
        double r2735100 = 5.846791957771714e-06;
        bool r2735101 = r2735088 <= r2735100;
        double r2735102 = r2735087 * r2735087;
        double r2735103 = 0.33333333333333337;
        double r2735104 = r2735102 * r2735103;
        double r2735105 = r2735092 - r2735104;
        double r2735106 = r2735087 * r2735105;
        double r2735107 = 5.551115123125783e-17;
        double r2735108 = r2735102 * r2735102;
        double r2735109 = r2735107 * r2735108;
        double r2735110 = r2735106 - r2735109;
        double r2735111 = r2735101 ? r2735110 : r2735099;
        double r2735112 = r2735090 ? r2735099 : r2735111;
        return r2735112;
}

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 (* -2.0 x) < -15.661220476650124 or 5.846791957771714e-06 < (* -2.0 x)

    1. Initial program 0.1

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
    2. Using strategy rm

    3. Applied add-cbrt-cube0.1

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

    if -15.661220476650124 < (* -2.0 x) < 5.846791957771714e-06

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{1 \cdot x - \left(0.3333333333333333703407674875052180141211 \cdot {x}^{3} + 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4}\right)}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))