Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{2}}{\sqrt{e^{x} + e^{-x}}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{\sqrt{\frac{2}{e^{x} + e^{-x}}} \cdot \sqrt{2}}{\sqrt{e^{x} + e^{-x}}}
double f(double x) {
        double r55947 = 2.0;
        double r55948 = x;
        double r55949 = exp(r55948);
        double r55950 = -r55948;
        double r55951 = exp(r55950);
        double r55952 = r55949 + r55951;
        double r55953 = r55947 / r55952;
        return r55953;
}
↓
double f(double x) {
        double r55954 = 2.0;
        double r55955 = x;
        double r55956 = exp(r55955);
        double r55957 = -r55955;
        double r55958 = exp(r55957);
        double r55959 = r55956 + r55958;
        double r55960 = r55954 / r55959;
        double r55961 = sqrt(r55960);
        double r55962 = sqrt(r55954);
        double r55963 = r55961 * r55962;
        double r55964 = sqrt(r55959);
        double r55965 = r55963 / r55964;
        return r55965;
}

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. Using strategy rm

  5. Applied sqrt-div0.0

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

  6. Applied associate-*r/0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))