Average Error: 57.9 → 0.6
Time: 12.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}{2}
double f(double x) {
        double r54671 = x;
        double r54672 = exp(r54671);
        double r54673 = -r54671;
        double r54674 = exp(r54673);
        double r54675 = r54672 - r54674;
        double r54676 = 2.0;
        double r54677 = r54675 / r54676;
        return r54677;
}
↓
double f(double x) {
        double r54678 = 2.0;
        double r54679 = x;
        double r54680 = r54678 * r54679;
        double r54681 = 0.3333333333333333;
        double r54682 = 3.0;
        double r54683 = pow(r54679, r54682);
        double r54684 = r54681 * r54683;
        double r54685 = 0.016666666666666666;
        double r54686 = 5.0;
        double r54687 = pow(r54679, r54686);
        double r54688 = r54685 * r54687;
        double r54689 = r54684 + r54688;
        double r54690 = r54680 + r54689;
        double r54691 = 2.0;
        double r54692 = r54690 / r54691;
        return r54692;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  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. Taylor expanded around 0 0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))