Average Error: 0.0 → 0.1
Time: 3.7s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{\frac{1}{\left(e^{-1 \cdot x} + e^{x}\right) \cdot \left(e^{-1 \cdot x} + e^{x}\right)} \cdot \frac{{2}^{3}}{e^{-1 \cdot x} + e^{x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{\frac{1}{\left(e^{-1 \cdot x} + e^{x}\right) \cdot \left(e^{-1 \cdot x} + e^{x}\right)} \cdot \frac{{2}^{3}}{e^{-1 \cdot x} + e^{x}}}
double f(double x) {
        double r50846 = 2.0;
        double r50847 = x;
        double r50848 = exp(r50847);
        double r50849 = -r50847;
        double r50850 = exp(r50849);
        double r50851 = r50848 + r50850;
        double r50852 = r50846 / r50851;
        return r50852;
}
↓
double f(double x) {
        double r50853 = 1.0;
        double r50854 = -1.0;
        double r50855 = x;
        double r50856 = r50854 * r50855;
        double r50857 = exp(r50856);
        double r50858 = exp(r50855);
        double r50859 = r50857 + r50858;
        double r50860 = r50859 * r50859;
        double r50861 = r50853 / r50860;
        double r50862 = 2.0;
        double r50863 = 3.0;
        double r50864 = pow(r50862, r50863);
        double r50865 = r50864 / r50859;
        double r50866 = r50861 * r50865;
        double r50867 = cbrt(r50866);
        return r50867;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  3. Applied add-cbrt-cube0.1

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

  4. Applied add-cbrt-cube0.1

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

  5. Applied cbrt-undiv0.1

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

  6. Simplified0.1

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

  8. Applied add-cube-cbrt1.3

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

  9. Applied *-un-lft-identity1.3

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

  10. Applied times-frac1.3

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

  11. Applied unpow-prod-down1.3

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

  12. Simplified0.6

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))