Average Error: 57.9 → 0.7
Time: 26.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}
double f(double x) {
        double r47152 = x;
        double r47153 = exp(r47152);
        double r47154 = -r47152;
        double r47155 = exp(r47154);
        double r47156 = r47153 - r47155;
        double r47157 = 2.0;
        double r47158 = r47156 / r47157;
        return r47158;
}
↓
double f(double x) {
        double r47159 = 0.3333333333333333;
        double r47160 = x;
        double r47161 = 3.0;
        double r47162 = pow(r47160, r47161);
        double r47163 = r47159 * r47162;
        double r47164 = 0.016666666666666666;
        double r47165 = 5.0;
        double r47166 = pow(r47160, r47165);
        double r47167 = r47164 * r47166;
        double r47168 = r47163 + r47167;
        double r47169 = 2.0;
        double r47170 = r47169 * r47160;
        double r47171 = r47168 + r47170;
        double r47172 = 2.0;
        double r47173 = r47171 / r47172;
        return r47173;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2}\]
  3. Using strategy rm

  4. Applied associate-+r+0.7

    \[\leadsto \frac{\color{blue}{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}}{2}\]

  5. Final simplification0.7

    \[\leadsto \frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))