Average Error: 0.0 → 0.0
Time: 10.7s
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 r874290 = 2.0;
        double r874291 = x;
        double r874292 = exp(r874291);
        double r874293 = -r874291;
        double r874294 = exp(r874293);
        double r874295 = r874292 + r874294;
        double r874296 = r874290 / r874295;
        return r874296;
}
↓
double f(double x) {
        double r874297 = 2.0;
        double r874298 = sqrt(r874297);
        double r874299 = x;
        double r874300 = exp(r874299);
        double r874301 = -r874299;
        double r874302 = exp(r874301);
        double r874303 = r874300 + r874302;
        double r874304 = cbrt(r874303);
        double r874305 = r874298 / r874304;
        double r874306 = r874304 * r874304;
        double r874307 = r874298 / r874306;
        double r874308 = r874305 * r874307;
        double r874309 = cbrt(r874308);
        double r874310 = r874297 / r874303;
        double r874311 = cbrt(r874310);
        double r874312 = r874311 * r874311;
        double r874313 = r874309 * r874312;
        return r874313;
}

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)))))