Average Error: 0.0 → 0.0
Time: 14.9s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{\sqrt{2} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}} \cdot \left(\left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right) \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{\sqrt{2} \cdot \frac{\sqrt{2}}{e^{x} + e^{-x}}} \cdot \left(\left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right) \cdot \sqrt{\sqrt[3]{\frac{2}{e^{x} + e^{-x}}}}\right)
double f(double x) {
        double r2959695 = 2.0;
        double r2959696 = x;
        double r2959697 = exp(r2959696);
        double r2959698 = -r2959696;
        double r2959699 = exp(r2959698);
        double r2959700 = r2959697 + r2959699;
        double r2959701 = r2959695 / r2959700;
        return r2959701;
}
↓
double f(double x) {
        double r2959702 = 2.0;
        double r2959703 = sqrt(r2959702);
        double r2959704 = x;
        double r2959705 = exp(r2959704);
        double r2959706 = -r2959704;
        double r2959707 = exp(r2959706);
        double r2959708 = r2959705 + r2959707;
        double r2959709 = r2959703 / r2959708;
        double r2959710 = r2959703 * r2959709;
        double r2959711 = cbrt(r2959710);
        double r2959712 = r2959702 / r2959708;
        double r2959713 = cbrt(r2959712);
        double r2959714 = sqrt(r2959713);
        double r2959715 = r2959713 * r2959714;
        double r2959716 = r2959715 * r2959714;
        double r2959717 = r2959711 * r2959716;
        return r2959717;
}

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-cube-cbrt0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied associate-*r*0.0

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

  8. Applied *-un-lft-identity0.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Applied times-frac0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2.0 (+ (exp x) (exp (- x)))))