Average Error: 58.1 → 0.6
Time: 12.0s
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 r57639 = x;
        double r57640 = exp(r57639);
        double r57641 = -r57639;
        double r57642 = exp(r57641);
        double r57643 = r57640 - r57642;
        double r57644 = 2.0;
        double r57645 = r57643 / r57644;
        return r57645;
}
↓
double f(double x) {
        double r57646 = 0.3333333333333333;
        double r57647 = x;
        double r57648 = 3.0;
        double r57649 = pow(r57647, r57648);
        double r57650 = r57646 * r57649;
        double r57651 = 0.016666666666666666;
        double r57652 = 5.0;
        double r57653 = pow(r57647, r57652);
        double r57654 = r57651 * r57653;
        double r57655 = 2.0;
        double r57656 = r57655 * r57647;
        double r57657 = r57654 + r57656;
        double r57658 = r57650 + r57657;
        double r57659 = 2.0;
        double r57660 = r57658 / r57659;
        return r57660;
}

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 2019235 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))