Average Error: 29.0 → 0.1
Time: 25.0s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \mathbf{elif}\;-2 \cdot x \le 5.846791957771713948576082497954331529399 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(1 - \left(x \cdot x\right) \cdot 0.3333333333333333703407674875052180141211\right) - 5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -15.66122047665012395611938700312748551369:\\
\;\;\;\;\sqrt[3]{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\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(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)\right)}\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r4090781 = 2.0;
        double r4090782 = 1.0;
        double r4090783 = -2.0;
        double r4090784 = x;
        double r4090785 = r4090783 * r4090784;
        double r4090786 = exp(r4090785);
        double r4090787 = r4090782 + r4090786;
        double r4090788 = r4090781 / r4090787;
        double r4090789 = r4090788 - r4090782;
        return r4090789;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r4090790 = -2.0;
        double r4090791 = x;
        double r4090792 = r4090790 * r4090791;
        double r4090793 = -15.661220476650124;
        bool r4090794 = r4090792 <= r4090793;
        double r4090795 = 2.0;
        double r4090796 = 1.0;
        double r4090797 = exp(r4090792);
        double r4090798 = r4090796 + r4090797;
        double r4090799 = r4090795 / r4090798;
        double r4090800 = r4090799 - r4090796;
        double r4090801 = r4090800 * r4090800;
        double r4090802 = r4090800 * r4090801;
        double r4090803 = cbrt(r4090802);
        double r4090804 = 5.846791957771714e-06;
        bool r4090805 = r4090792 <= r4090804;
        double r4090806 = r4090791 * r4090791;
        double r4090807 = 0.33333333333333337;
        double r4090808 = r4090806 * r4090807;
        double r4090809 = r4090796 - r4090808;
        double r4090810 = r4090791 * r4090809;
        double r4090811 = 5.551115123125783e-17;
        double r4090812 = r4090806 * r4090806;
        double r4090813 = r4090811 * r4090812;
        double r4090814 = r4090810 - r4090813;
        double r4090815 = r4090805 ? r4090814 : r4090803;
        double r4090816 = r4090794 ? r4090803 : r4090815;
        return r4090816;
}

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-sqr-sqrt0.1

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

    4. Applied associate-/r*0.1

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt{1 + e^{-2 \cdot x}}}}{\sqrt{1 + e^{-2 \cdot x}}}} - 1\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.1

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

    7. Simplified0.1

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

    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

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

Reproduce

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