Average Error: 0.0 → 0.0
Time: 10.2s
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 r1767547 = 2.0;
        double r1767548 = x;
        double r1767549 = exp(r1767548);
        double r1767550 = -r1767548;
        double r1767551 = exp(r1767550);
        double r1767552 = r1767549 + r1767551;
        double r1767553 = r1767547 / r1767552;
        return r1767553;
}
↓
double f(double x) {
        double r1767554 = 2.0;
        double r1767555 = sqrt(r1767554);
        double r1767556 = x;
        double r1767557 = exp(r1767556);
        double r1767558 = -r1767556;
        double r1767559 = exp(r1767558);
        double r1767560 = r1767557 + r1767559;
        double r1767561 = cbrt(r1767560);
        double r1767562 = r1767555 / r1767561;
        double r1767563 = r1767561 * r1767561;
        double r1767564 = r1767555 / r1767563;
        double r1767565 = r1767562 * r1767564;
        double r1767566 = cbrt(r1767565);
        double r1767567 = r1767554 / r1767560;
        double r1767568 = cbrt(r1767567);
        double r1767569 = r1767568 * r1767568;
        double r1767570 = r1767566 * r1767569;
        return r1767570;
}

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 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  (/ 2 (+ (exp x) (exp (- x)))))