Average Error: 58.3 → 0.6
Time: 4.5s
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 r29638 = x;
        double r29639 = exp(r29638);
        double r29640 = -r29638;
        double r29641 = exp(r29640);
        double r29642 = r29639 - r29641;
        double r29643 = 2.0;
        double r29644 = r29642 / r29643;
        return r29644;
}
↓
double f(double x) {
        double r29645 = 0.3333333333333333;
        double r29646 = x;
        double r29647 = 3.0;
        double r29648 = pow(r29646, r29647);
        double r29649 = r29645 * r29648;
        double r29650 = 0.016666666666666666;
        double r29651 = 5.0;
        double r29652 = pow(r29646, r29651);
        double r29653 = r29650 * r29652;
        double r29654 = 2.0;
        double r29655 = r29654 * r29646;
        double r29656 = r29653 + r29655;
        double r29657 = r29649 + r29656;
        double r29658 = 2.0;
        double r29659 = r29657 / r29658;
        return r29659;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.3

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

  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 2020001 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))