Average Error: 29.2 → 0.1
Time: 21.3s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.006169443383165685093616481537992513040081:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2 \cdot \left({\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{2} - 1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1 \cdot 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.858478937487164482871759426069017961947 \cdot 10^{-9}:\\ \;\;\;\;1 \cdot x - {x}^{3} \cdot \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot x + 0.3333333333333333703407674875052180141211\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}} + 1 \cdot 1}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -0.006169443383165685093616481537992513040081:\\
\;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2 \cdot \left({\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{2} - 1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1 \cdot 1}\\

\mathbf{elif}\;-2 \cdot x \le 1.858478937487164482871759426069017961947 \cdot 10^{-9}:\\
\;\;\;\;1 \cdot x - {x}^{3} \cdot \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot x + 0.3333333333333333703407674875052180141211\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r47814 = 2.0;
        double r47815 = 1.0;
        double r47816 = -2.0;
        double r47817 = x;
        double r47818 = r47816 * r47817;
        double r47819 = exp(r47818);
        double r47820 = r47815 + r47819;
        double r47821 = r47814 / r47820;
        double r47822 = r47821 - r47815;
        return r47822;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r47823 = -2.0;
        double r47824 = x;
        double r47825 = r47823 * r47824;
        double r47826 = -0.006169443383165685;
        bool r47827 = r47825 <= r47826;
        double r47828 = 2.0;
        double r47829 = 1.0;
        double r47830 = exp(r47825);
        double r47831 = r47829 + r47830;
        double r47832 = r47828 / r47831;
        double r47833 = 3.0;
        double r47834 = pow(r47832, r47833);
        double r47835 = pow(r47829, r47833);
        double r47836 = r47834 - r47835;
        double r47837 = r47830 + r47829;
        double r47838 = r47828 / r47837;
        double r47839 = 2.0;
        double r47840 = pow(r47838, r47839);
        double r47841 = r47829 * r47829;
        double r47842 = r47840 - r47841;
        double r47843 = r47828 * r47842;
        double r47844 = r47832 - r47829;
        double r47845 = r47844 * r47831;
        double r47846 = r47843 / r47845;
        double r47847 = r47846 + r47841;
        double r47848 = r47836 / r47847;
        double r47849 = 1.8584789374871645e-09;
        bool r47850 = r47825 <= r47849;
        double r47851 = r47829 * r47824;
        double r47852 = pow(r47824, r47833);
        double r47853 = 5.551115123125783e-17;
        double r47854 = r47853 * r47824;
        double r47855 = 0.33333333333333337;
        double r47856 = r47854 + r47855;
        double r47857 = r47852 * r47856;
        double r47858 = r47851 - r47857;
        double r47859 = exp(r47836);
        double r47860 = log(r47859);
        double r47861 = r47832 + r47829;
        double r47862 = r47861 * r47832;
        double r47863 = r47862 + r47841;
        double r47864 = r47860 / r47863;
        double r47865 = r47850 ? r47858 : r47864;
        double r47866 = r47827 ? r47848 : r47865;
        return r47866;
}

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 3 regimes

  2. if (* -2.0 x) < -0.006169443383165685

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    6. Applied flip-+0.0

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

    7. Applied frac-times0.0

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

    8. Simplified0.0

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

    if -0.006169443383165685 < (* -2.0 x) < 1.8584789374871645e-09

    1. Initial program 59.4

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

    2. Taylor expanded around 0 0.0

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

    3. Simplified0.0

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

    if 1.8584789374871645e-09 < (* -2.0 x)

    1. Initial program 0.4

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

    3. Applied flip3--0.4

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

    4. Simplified0.4

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

    6. Applied add-log-exp0.4

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

    7. Applied add-log-exp0.4

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

    8. Applied diff-log0.4

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

    9. Simplified0.4

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -0.006169443383165685093616481537992513040081:\\ \;\;\;\;\frac{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}{\frac{2 \cdot \left({\left(\frac{2}{e^{-2 \cdot x} + 1}\right)}^{2} - 1 \cdot 1\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right) \cdot \left(1 + e^{-2 \cdot x}\right)} + 1 \cdot 1}\\ \mathbf{elif}\;-2 \cdot x \le 1.858478937487164482871759426069017961947 \cdot 10^{-9}:\\ \;\;\;\;1 \cdot x - {x}^{3} \cdot \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot x + 0.3333333333333333703407674875052180141211\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{{\left(\frac{2}{1 + e^{-2 \cdot x}}\right)}^{3} - {1}^{3}}\right)}{\left(\frac{2}{1 + e^{-2 \cdot x}} + 1\right) \cdot \frac{2}{1 + e^{-2 \cdot x}} + 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))