Average Error: 58.0 → 0.7
Time: 10.7s
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 r40767 = x;
        double r40768 = exp(r40767);
        double r40769 = -r40767;
        double r40770 = exp(r40769);
        double r40771 = r40768 - r40770;
        double r40772 = 2.0;
        double r40773 = r40771 / r40772;
        return r40773;
}
↓
double f(double x) {
        double r40774 = 0.3333333333333333;
        double r40775 = x;
        double r40776 = 3.0;
        double r40777 = pow(r40775, r40776);
        double r40778 = r40774 * r40777;
        double r40779 = 0.016666666666666666;
        double r40780 = 5.0;
        double r40781 = pow(r40775, r40780);
        double r40782 = r40779 * r40781;
        double r40783 = 2.0;
        double r40784 = r40783 * r40775;
        double r40785 = r40782 + r40784;
        double r40786 = r40778 + r40785;
        double r40787 = 2.0;
        double r40788 = r40786 / r40787;
        return r40788;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.0

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

  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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))