Average Error: 0.0 → 0.1
Time: 6.0s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt[3]{{\left(\frac{2}{e^{x} + e^{-x}}\right)}^{3}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt[3]{{\left(\frac{2}{e^{x} + e^{-x}}\right)}^{3}}
double f(double x) {
        double r33598 = 2.0;
        double r33599 = x;
        double r33600 = exp(r33599);
        double r33601 = -r33599;
        double r33602 = exp(r33601);
        double r33603 = r33600 + r33602;
        double r33604 = r33598 / r33603;
        return r33604;
}
↓
double f(double x) {
        double r33605 = 2.0;
        double r33606 = x;
        double r33607 = exp(r33606);
        double r33608 = -r33606;
        double r33609 = exp(r33608);
        double r33610 = r33607 + r33609;
        double r33611 = r33605 / r33610;
        double r33612 = 3.0;
        double r33613 = pow(r33611, r33612);
        double r33614 = cbrt(r33613);
        return r33614;
}

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

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))