Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}} \cdot \frac{\sqrt{2}}{\sqrt{e^{x} + e^{-x}}}
double f(double x) {
        double r84883 = 2.0;
        double r84884 = x;
        double r84885 = exp(r84884);
        double r84886 = -r84884;
        double r84887 = exp(r84886);
        double r84888 = r84885 + r84887;
        double r84889 = r84883 / r84888;
        return r84889;
}
↓
double f(double x) {
        double r84890 = 2.0;
        double r84891 = sqrt(r84890);
        double r84892 = x;
        double r84893 = exp(r84892);
        double r84894 = -r84892;
        double r84895 = exp(r84894);
        double r84896 = r84893 + r84895;
        double r84897 = sqrt(r84896);
        double r84898 = r84891 / r84897;
        double r84899 = r84898 * r84898;
        return r84899;
}

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-sqrt0.8

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied times-frac0.0

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

  6. Final simplification0.0

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

Reproduce

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