Average Error: 58.1 → 0.6
Time: 3.8s
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 r27670 = x;
        double r27671 = exp(r27670);
        double r27672 = -r27670;
        double r27673 = exp(r27672);
        double r27674 = r27671 - r27673;
        double r27675 = 2.0;
        double r27676 = r27674 / r27675;
        return r27676;
}
↓
double f(double x) {
        double r27677 = 0.3333333333333333;
        double r27678 = x;
        double r27679 = 3.0;
        double r27680 = pow(r27678, r27679);
        double r27681 = r27677 * r27680;
        double r27682 = 0.016666666666666666;
        double r27683 = 5.0;
        double r27684 = pow(r27678, r27683);
        double r27685 = r27682 * r27684;
        double r27686 = 2.0;
        double r27687 = r27686 * r27678;
        double r27688 = r27685 + r27687;
        double r27689 = r27681 + r27688;
        double r27690 = 2.0;
        double r27691 = r27689 / r27690;
        return r27691;
}

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 2020065 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))