Average Error: 29.4 → 0.2
Time: 5.7s
Precision: 64
\[\frac{2}{1 + e^{-2 \cdot x}} - 1\]
↓
\[\begin{array}{l} \mathbf{if}\;-2 \cdot x \le -5.111290763866047193175745633197948336601:\\ \;\;\;\;\sqrt[3]{{\left(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \le 3.978856656911251737182447316826250371946 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]
\frac{2}{1 + e^{-2 \cdot x}} - 1
↓
\begin{array}{l}
\mathbf{if}\;-2 \cdot x \le -5.111290763866047193175745633197948336601:\\
\;\;\;\;\sqrt[3]{{\left(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\right)}^{3}}\\

\mathbf{elif}\;-2 \cdot x \le 3.978856656911251737182447316826250371946 \cdot 10^{-7}:\\
\;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\

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

\end{array}
double f(double x, double __attribute__((unused)) y) {
        double r62813 = 2.0;
        double r62814 = 1.0;
        double r62815 = -2.0;
        double r62816 = x;
        double r62817 = r62815 * r62816;
        double r62818 = exp(r62817);
        double r62819 = r62814 + r62818;
        double r62820 = r62813 / r62819;
        double r62821 = r62820 - r62814;
        return r62821;
}
↓
double f(double x, double __attribute__((unused)) y) {
        double r62822 = -2.0;
        double r62823 = x;
        double r62824 = r62822 * r62823;
        double r62825 = -5.111290763866047;
        bool r62826 = r62824 <= r62825;
        double r62827 = 2.0;
        double r62828 = 1.0;
        double r62829 = exp(r62824);
        double r62830 = r62828 + r62829;
        double r62831 = r62827 / r62830;
        double r62832 = sqrt(r62831);
        double r62833 = sqrt(r62828);
        double r62834 = r62832 + r62833;
        double r62835 = r62832 - r62833;
        double r62836 = r62834 * r62835;
        double r62837 = 3.0;
        double r62838 = pow(r62836, r62837);
        double r62839 = cbrt(r62838);
        double r62840 = 3.978856656911252e-07;
        bool r62841 = r62824 <= r62840;
        double r62842 = r62828 * r62823;
        double r62843 = 5.551115123125783e-17;
        double r62844 = 4.0;
        double r62845 = pow(r62823, r62844);
        double r62846 = r62843 * r62845;
        double r62847 = 0.33333333333333337;
        double r62848 = pow(r62823, r62837);
        double r62849 = r62847 * r62848;
        double r62850 = r62846 + r62849;
        double r62851 = r62842 - r62850;
        double r62852 = r62831 - r62828;
        double r62853 = pow(r62852, r62837);
        double r62854 = cbrt(r62853);
        double r62855 = r62841 ? r62851 : r62854;
        double r62856 = r62826 ? r62839 : r62855;
        return r62856;
}

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) < -5.111290763866047

    1. Initial program 0.0

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

    3. Applied add-cbrt-cube0.0

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

    4. Simplified0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied add-sqr-sqrt2.0

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

    8. Applied difference-of-squares0.0

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

    if -5.111290763866047 < (* -2.0 x) < 3.978856656911252e-07

    1. Initial program 59.2

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

    2. Taylor expanded around 0 0.2

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

    if 3.978856656911252e-07 < (* -2.0 x)

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;-2 \cdot x \le -5.111290763866047193175745633197948336601:\\ \;\;\;\;\sqrt[3]{{\left(\left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} + \sqrt{1}\right) \cdot \left(\sqrt{\frac{2}{1 + e^{-2 \cdot x}}} - \sqrt{1}\right)\right)}^{3}}\\ \mathbf{elif}\;-2 \cdot x \le 3.978856656911251737182447316826250371946 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot x - \left(5.5511151231257827021181583404541015625 \cdot 10^{-17} \cdot {x}^{4} + 0.3333333333333333703407674875052180141211 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{2}{1 + e^{-2 \cdot x}} - 1\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x y)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2 (+ 1 (exp (* -2 x)))) 1))