Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}
double f(double x) {
        double r62413 = 2.0;
        double r62414 = x;
        double r62415 = exp(r62414);
        double r62416 = -r62414;
        double r62417 = exp(r62416);
        double r62418 = r62415 + r62417;
        double r62419 = r62413 / r62418;
        return r62419;
}
↓
double f(double x) {
        double r62420 = 2.0;
        double r62421 = x;
        double r62422 = exp(r62421);
        double r62423 = -r62421;
        double r62424 = exp(r62423);
        double r62425 = r62422 + r62424;
        double r62426 = r62420 / r62425;
        double r62427 = sqrt(r62426);
        double r62428 = r62427 * r62427;
        return r62428;
}

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-sqr-sqrt0.0

    \[\leadsto \color{blue}{\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}}\]

  4. Final simplification0.0

    \[\leadsto \sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{\frac{2}{e^{x} + e^{-x}}}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))