Average Error: 0.0 → 0.0
Time: 12.3s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{\frac{\sqrt{2}}{\sqrt[3]{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}} \cdot \left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{\frac{\sqrt{2}}{\sqrt[3]{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{e^{x} + e^{-x}} \cdot \sqrt[3]{e^{x} + e^{-x}}}} \cdot \left(\sqrt[3]{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt[3]{\frac{2}{e^{x} + e^{-x}}}\right)
double f(double x) {
        double r1085468 = 2.0;
        double r1085469 = x;
        double r1085470 = exp(r1085469);
        double r1085471 = -r1085469;
        double r1085472 = exp(r1085471);
        double r1085473 = r1085470 + r1085472;
        double r1085474 = r1085468 / r1085473;
        return r1085474;
}
↓
double f(double x) {
        double r1085475 = 2.0;
        double r1085476 = sqrt(r1085475);
        double r1085477 = x;
        double r1085478 = exp(r1085477);
        double r1085479 = -r1085477;
        double r1085480 = exp(r1085479);
        double r1085481 = r1085478 + r1085480;
        double r1085482 = cbrt(r1085481);
        double r1085483 = r1085476 / r1085482;
        double r1085484 = r1085482 * r1085482;
        double r1085485 = r1085476 / r1085484;
        double r1085486 = r1085483 * r1085485;
        double r1085487 = cbrt(r1085486);
        double r1085488 = r1085475 / r1085481;
        double r1085489 = cbrt(r1085488);
        double r1085490 = r1085489 * r1085489;
        double r1085491 = r1085487 * r1085490;
        return r1085491;
}

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-cube-cbrt0.0

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied times-frac0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))