Average Error: 0.0 → 0.0
Time: 2.8s
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 r86737 = 2.0;
        double r86738 = x;
        double r86739 = exp(r86738);
        double r86740 = -r86738;
        double r86741 = exp(r86740);
        double r86742 = r86739 + r86741;
        double r86743 = r86737 / r86742;
        return r86743;
}
↓
double f(double x) {
        double r86744 = 2.0;
        double r86745 = x;
        double r86746 = exp(r86745);
        double r86747 = -r86745;
        double r86748 = exp(r86747);
        double r86749 = r86746 + r86748;
        double r86750 = r86744 / r86749;
        double r86751 = sqrt(r86750);
        double r86752 = r86751 * r86751;
        return r86752;
}

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 2019353 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))