Average Error: 58.4 → 0.5
Time: 18.4s
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 r53014 = x;
        double r53015 = exp(r53014);
        double r53016 = -r53014;
        double r53017 = exp(r53016);
        double r53018 = r53015 - r53017;
        double r53019 = 2.0;
        double r53020 = r53018 / r53019;
        return r53020;
}
↓
double f(double x) {
        double r53021 = 0.3333333333333333;
        double r53022 = x;
        double r53023 = 3.0;
        double r53024 = pow(r53022, r53023);
        double r53025 = r53021 * r53024;
        double r53026 = 0.016666666666666666;
        double r53027 = 5.0;
        double r53028 = pow(r53022, r53027);
        double r53029 = r53026 * r53028;
        double r53030 = 2.0;
        double r53031 = r53030 * r53022;
        double r53032 = r53029 + r53031;
        double r53033 = r53025 + r53032;
        double r53034 = 2.0;
        double r53035 = r53033 / r53034;
        return r53035;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2019199 
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))