Average Error: 29.6 → 0.2
Time: 17.7s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1972.3938625414924:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt[3]{\sqrt{e^{-2 \cdot x} + 1} \cdot \left(e^{-2 \cdot x} + 1\right)}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.0002071702113622589:\\ \;\;\;\;x + \left(\frac{2}{15} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt[3]{\sqrt{e^{-2 \cdot x} + 1} \cdot \left(e^{-2 \cdot x} + 1\right)}} - 1\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -1972.3938625414924:\\
\;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt[3]{\sqrt{e^{-2 \cdot x} + 1} \cdot \left(e^{-2 \cdot x} + 1\right)}} - 1\\

\mathbf{elif}\;-2 \cdot x \le 0.0002071702113622589:\\
\;\;\;\;x + \left(\frac{2}{15} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{3}\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r2213750 = 2.0;
        double r2213751 = 1.0;
        double r2213752 = -2.0;
        double r2213753 = x;
        double r2213754 = r2213752 * r2213753;
        double r2213755 = exp(r2213754);
        double r2213756 = r2213751 + r2213755;
        double r2213757 = r2213750 / r2213756;
        double r2213758 = r2213757 - r2213751;
        return r2213758;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r2213759 = -2.0;
        double r2213760 = x;
        double r2213761 = r2213759 * r2213760;
        double r2213762 = -1972.3938625414924;
        bool r2213763 = r2213761 <= r2213762;
        double r2213764 = 2.0;
        double r2213765 = exp(r2213761);
        double r2213766 = 1.0;
        double r2213767 = r2213765 + r2213766;
        double r2213768 = sqrt(r2213767);
        double r2213769 = r2213764 / r2213768;
        double r2213770 = r2213768 * r2213767;
        double r2213771 = cbrt(r2213770);
        double r2213772 = r2213769 / r2213771;
        double r2213773 = r2213772 - r2213766;
        double r2213774 = 0.0002071702113622589;
        bool r2213775 = r2213761 <= r2213774;
        double r2213776 = 0.13333333333333333;
        double r2213777 = 5.0;
        double r2213778 = pow(r2213760, r2213777);
        double r2213779 = r2213776 * r2213778;
        double r2213780 = r2213760 * r2213760;
        double r2213781 = r2213780 * r2213760;
        double r2213782 = -0.3333333333333333;
        double r2213783 = r2213781 * r2213782;
        double r2213784 = r2213779 + r2213783;
        double r2213785 = r2213760 + r2213784;
        double r2213786 = r2213775 ? r2213785 : r2213773;
        double r2213787 = r2213763 ? r2213773 : r2213786;
        return r2213787;
}

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 x) < -1972.3938625414924 or 0.0002071702113622589 < (* -2 x)

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied associate-/r*0.0

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

    6. Applied add-cbrt-cube0.0

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

    7. Simplified0.0

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

    if -1972.3938625414924 < (* -2 x) < 0.0002071702113622589

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -1972.3938625414924:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt[3]{\sqrt{e^{-2 \cdot x} + 1} \cdot \left(e^{-2 \cdot x} + 1\right)}} - 1\\ \mathbf{elif}\;-2 \cdot x \le 0.0002071702113622589:\\ \;\;\;\;x + \left(\frac{2}{15} \cdot {x}^{5} + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt[3]{\sqrt{e^{-2 \cdot x} + 1} \cdot \left(e^{-2 \cdot x} + 1\right)}} - 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))