Average Error: 29.1 → 0.0
Time: 15.9s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.007208096633899537:\\ \;\;\;\;\log \left(e^{\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1}\right)\\ \mathbf{elif}\;x \le 0.006936161389017993:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3} + \left(\frac{2}{15} \cdot {x}^{5} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.007208096633899537:\\
\;\;\;\;\log \left(e^{\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1}\right)\\

\mathbf{elif}\;x \le 0.006936161389017993:\\
\;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3} + \left(\frac{2}{15} \cdot {x}^{5} + x\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r1060836 = 2.0;
        double r1060837 = 1.0;
        double r1060838 = -2.0;
        double r1060839 = x;
        double r1060840 = r1060838 * r1060839;
        double r1060841 = exp(r1060840);
        double r1060842 = r1060837 + r1060841;
        double r1060843 = r1060836 / r1060842;
        double r1060844 = r1060843 - r1060837;
        return r1060844;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r1060845 = x;
        double r1060846 = -0.007208096633899537;
        bool r1060847 = r1060845 <= r1060846;
        double r1060848 = 2.0;
        double r1060849 = -2.0;
        double r1060850 = r1060849 * r1060845;
        double r1060851 = exp(r1060850);
        double r1060852 = 1.0;
        double r1060853 = r1060851 + r1060852;
        double r1060854 = sqrt(r1060853);
        double r1060855 = r1060848 / r1060854;
        double r1060856 = r1060855 / r1060854;
        double r1060857 = r1060856 - r1060852;
        double r1060858 = exp(r1060857);
        double r1060859 = log(r1060858);
        double r1060860 = 0.006936161389017993;
        bool r1060861 = r1060845 <= r1060860;
        double r1060862 = r1060845 * r1060845;
        double r1060863 = r1060845 * r1060862;
        double r1060864 = -0.3333333333333333;
        double r1060865 = r1060863 * r1060864;
        double r1060866 = 0.13333333333333333;
        double r1060867 = 5.0;
        double r1060868 = pow(r1060845, r1060867);
        double r1060869 = r1060866 * r1060868;
        double r1060870 = r1060869 + r1060845;
        double r1060871 = r1060865 + r1060870;
        double r1060872 = r1060861 ? r1060871 : r1060859;
        double r1060873 = r1060847 ? r1060859 : r1060872;
        return r1060873;
}

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 < -0.007208096633899537 or 0.006936161389017993 < x

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{2}{1 + e^{-2 \cdot x}} - 1}\right)}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.0

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

    6. Applied associate-/r*0.0

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

    if -0.007208096633899537 < x < 0.006936161389017993

    1. Initial program 59.0

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.007208096633899537:\\ \;\;\;\;\log \left(e^{\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1}\right)\\ \mathbf{elif}\;x \le 0.006936161389017993:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{3} + \left(\frac{2}{15} \cdot {x}^{5} + x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\frac{2}{\sqrt{e^{-2 \cdot x} + 1}}}{\sqrt{e^{-2 \cdot x} + 1}} - 1}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))