Average Error: 0.0 → 0.0
Time: 2.1s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \le 2.0000000483992446:\\ \;\;\;\;\frac{2}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}\\ \end{array}\]
\frac{2}{e^{x} + e^{-x}}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} + e^{-x} \le 2.0000000483992446:\\
\;\;\;\;\frac{2}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}\\

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

\end{array}
double f(double x) {
        double r60638 = 2.0;
        double r60639 = x;
        double r60640 = exp(r60639);
        double r60641 = -r60639;
        double r60642 = exp(r60641);
        double r60643 = r60640 + r60642;
        double r60644 = r60638 / r60643;
        return r60644;
}
↓
double f(double x) {
        double r60645 = x;
        double r60646 = exp(r60645);
        double r60647 = -r60645;
        double r60648 = exp(r60647);
        double r60649 = r60646 + r60648;
        double r60650 = 2.0000000483992446;
        bool r60651 = r60649 <= r60650;
        double r60652 = 2.0;
        double r60653 = 2.0;
        double r60654 = pow(r60645, r60653);
        double r60655 = 0.08333333333333333;
        double r60656 = 4.0;
        double r60657 = pow(r60645, r60656);
        double r60658 = r60655 * r60657;
        double r60659 = r60658 + r60653;
        double r60660 = r60654 + r60659;
        double r60661 = r60652 / r60660;
        double r60662 = cbrt(r60649);
        double r60663 = r60662 * r60662;
        double r60664 = r60663 * r60662;
        double r60665 = r60652 / r60664;
        double r60666 = r60651 ? r60661 : r60665;
        return r60666;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (+ (exp x) (exp (- x))) < 2.0000000483992446

    1. Initial program 0.0

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \frac{2}{\color{blue}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}}\]

    if 2.0000000483992446 < (+ (exp x) (exp (- x)))

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} + e^{-x} \le 2.0000000483992446:\\ \;\;\;\;\frac{2}{{x}^{2} + \left(\frac{1}{12} \cdot {x}^{4} + 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}\right) \cdot \sqrt[3]{e^{x} + e^{-x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))