Average Error: 0.0 → 0.0
Time: 2.6s
Precision: 64
\[\frac{2}{e^{x} + e^{-x}}\]
↓
\[\frac{2}{e^{\log \left(e^{-1 \cdot x} + e^{x}\right)}}\]
\frac{2}{e^{x} + e^{-x}}
↓
\frac{2}{e^{\log \left(e^{-1 \cdot x} + e^{x}\right)}}
double f(double x) {
        double r67033 = 2.0;
        double r67034 = x;
        double r67035 = exp(r67034);
        double r67036 = -r67034;
        double r67037 = exp(r67036);
        double r67038 = r67035 + r67037;
        double r67039 = r67033 / r67038;
        return r67039;
}
↓
double f(double x) {
        double r67040 = 2.0;
        double r67041 = -1.0;
        double r67042 = x;
        double r67043 = r67041 * r67042;
        double r67044 = exp(r67043);
        double r67045 = exp(r67042);
        double r67046 = r67044 + r67045;
        double r67047 = log(r67046);
        double r67048 = exp(r67047);
        double r67049 = r67040 / r67048;
        return r67049;
}

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-exp-log0.0

    \[\leadsto \frac{2}{\color{blue}{e^{\log \left(e^{x} + e^{-x}\right)}}}\]

  4. Simplified0.0

    \[\leadsto \frac{2}{e^{\color{blue}{\log \left(e^{-1 \cdot x} + e^{x}\right)}}}\]

  5. Final simplification0.0

    \[\leadsto \frac{2}{e^{\log \left(e^{-1 \cdot x} + e^{x}\right)}}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2 (+ (exp x) (exp (- x)))))