Average Error: 57.7 → 0.8
Time: 3.6s
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 r48861 = x;
        double r48862 = exp(r48861);
        double r48863 = -r48861;
        double r48864 = exp(r48863);
        double r48865 = r48862 - r48864;
        double r48866 = 2.0;
        double r48867 = r48865 / r48866;
        return r48867;
}
↓
double f(double x) {
        double r48868 = 0.3333333333333333;
        double r48869 = x;
        double r48870 = 3.0;
        double r48871 = pow(r48869, r48870);
        double r48872 = r48868 * r48871;
        double r48873 = 0.016666666666666666;
        double r48874 = 5.0;
        double r48875 = pow(r48869, r48874);
        double r48876 = r48873 * r48875;
        double r48877 = 2.0;
        double r48878 = r48877 * r48869;
        double r48879 = r48876 + r48878;
        double r48880 = r48872 + r48879;
        double r48881 = 2.0;
        double r48882 = r48880 / r48881;
        return r48882;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.7

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

  2. Taylor expanded around 0 0.8

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

  3. Final simplification0.8

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

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))