Average Error: 58.1 → 0.6
Time: 8.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}
double f(double x) {
        double r61760 = x;
        double r61761 = exp(r61760);
        double r61762 = -r61760;
        double r61763 = exp(r61762);
        double r61764 = r61761 - r61763;
        double r61765 = 2.0;
        double r61766 = r61764 / r61765;
        return r61766;
}
↓
double f(double x) {
        double r61767 = 0.3333333333333333;
        double r61768 = x;
        double r61769 = 3.0;
        double r61770 = pow(r61768, r61769);
        double r61771 = r61767 * r61770;
        double r61772 = 0.016666666666666666;
        double r61773 = 5.0;
        double r61774 = pow(r61768, r61773);
        double r61775 = r61772 * r61774;
        double r61776 = 2.0;
        double r61777 = r61776 * r61768;
        double r61778 = r61775 + r61777;
        double r61779 = r61771 + r61778;
        double r61780 = 2.0;
        double r61781 = r61779 / r61780;
        return r61781;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2020042 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))