Average Error: 0.0 → 0.0
Time: 4.5s
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 r80762 = 2.0;
        double r80763 = x;
        double r80764 = exp(r80763);
        double r80765 = -r80763;
        double r80766 = exp(r80765);
        double r80767 = r80764 + r80766;
        double r80768 = r80762 / r80767;
        return r80768;
}
↓
double f(double x) {
        double r80769 = 2.0;
        double r80770 = x;
        double r80771 = exp(r80770);
        double r80772 = -r80770;
        double r80773 = exp(r80772);
        double r80774 = r80771 + r80773;
        double r80775 = r80769 / r80774;
        double r80776 = sqrt(r80775);
        double r80777 = r80776 * r80776;
        return r80777;
}

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 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))