Average Error: 58.1 → 0.6
Time: 7.0s
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 r40794 = x;
        double r40795 = exp(r40794);
        double r40796 = -r40794;
        double r40797 = exp(r40796);
        double r40798 = r40795 - r40797;
        double r40799 = 2.0;
        double r40800 = r40798 / r40799;
        return r40800;
}
↓
double f(double x) {
        double r40801 = 0.3333333333333333;
        double r40802 = x;
        double r40803 = 3.0;
        double r40804 = pow(r40802, r40803);
        double r40805 = r40801 * r40804;
        double r40806 = 0.016666666666666666;
        double r40807 = 5.0;
        double r40808 = pow(r40802, r40807);
        double r40809 = r40806 * r40808;
        double r40810 = 2.0;
        double r40811 = r40810 * r40802;
        double r40812 = r40809 + r40811;
        double r40813 = r40805 + r40812;
        double r40814 = 2.0;
        double r40815 = r40813 / r40814;
        return r40815;
}

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 2019351 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))