Average Error: 29.0 → 0.1
Time: 5.3s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.16694293117624887 \lor \neg \left(-2 \cdot x \le 1.3471487435038969 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.16694293117624887 \lor \neg \left(-2 \cdot x \le 1.3471487435038969 \cdot 10^{-8}\right):\\
\;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r65798 = 2.0;
        double r65799 = 1.0;
        double r65800 = -2.0;
        double r65801 = x;
        double r65802 = r65800 * r65801;
        double r65803 = exp(r65802);
        double r65804 = r65799 + r65803;
        double r65805 = r65798 / r65804;
        double r65806 = r65805 - r65799;
        return r65806;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r65807 = -2.0;
        double r65808 = x;
        double r65809 = r65807 * r65808;
        double r65810 = -0.16694293117624887;
        bool r65811 = r65809 <= r65810;
        double r65812 = 1.3471487435038969e-08;
        bool r65813 = r65809 <= r65812;
        double r65814 = !r65813;
        bool r65815 = r65811 || r65814;
        double r65816 = 2.0;
        double r65817 = 1.0;
        double r65818 = exp(r65809);
        double r65819 = r65817 + r65818;
        double r65820 = r65816 / r65819;
        double r65821 = r65820 * r65820;
        double r65822 = 3.0;
        double r65823 = pow(r65821, r65822);
        double r65824 = r65817 * r65817;
        double r65825 = pow(r65824, r65822);
        double r65826 = r65823 - r65825;
        double r65827 = 4.0;
        double r65828 = pow(r65820, r65827);
        double r65829 = r65816 * r65816;
        double r65830 = 1.0;
        double r65831 = r65818 + r65817;
        double r65832 = 2.0;
        double r65833 = pow(r65831, r65832);
        double r65834 = r65830 / r65833;
        double r65835 = r65829 * r65834;
        double r65836 = r65835 * r65824;
        double r65837 = r65828 + r65836;
        double r65838 = pow(r65817, r65827);
        double r65839 = r65837 + r65838;
        double r65840 = r65820 + r65817;
        double r65841 = r65839 * r65840;
        double r65842 = r65826 / r65841;
        double r65843 = r65817 * r65808;
        double r65844 = 4.996003610813204e-16;
        double r65845 = pow(r65808, r65827);
        double r65846 = r65844 * r65845;
        double r65847 = 0.3333333333333335;
        double r65848 = pow(r65808, r65822);
        double r65849 = r65847 * r65848;
        double r65850 = r65846 + r65849;
        double r65851 = r65843 - r65850;
        double r65852 = r65815 ? r65842 : r65851;
        return r65852;
}

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) < -0.16694293117624887 or 1.3471487435038969e-08 < (* -2.0 x)

    1. Initial program 0.2

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

    3. Applied flip--0.2

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

    5. Applied flip3--0.2

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

    6. Applied associate-/l/0.2

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

    7. Simplified0.2

      \[\leadsto \frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\color{blue}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}}\]

    if -0.16694293117624887 < (* -2.0 x) < 1.3471487435038969e-08

    1. Initial program 59.4

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

    3. Applied flip--59.4

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

    4. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.16694293117624887 \lor \neg \left(-2 \cdot x \le 1.3471487435038969 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}} \cdot \frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}{\left(\left({\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{4} + \left(\left(2 \cdot 2\right) \cdot \frac{1}{{\left(e^{-2 \cdot x} + 1\right)}^{2}}\right) \cdot \left(1 \cdot 1\right)\right) + {1}^{4}\right) \cdot \left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x - \left(4.996 \cdot 10^{-16} \cdot {x}^{4} + 0.33333333333333348 \cdot {x}^{3}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))