Average Error: 58.1 → 0.6
Time: 9.4s
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 r35520 = x;
        double r35521 = exp(r35520);
        double r35522 = -r35520;
        double r35523 = exp(r35522);
        double r35524 = r35521 - r35523;
        double r35525 = 2.0;
        double r35526 = r35524 / r35525;
        return r35526;
}
↓
double f(double x) {
        double r35527 = 0.3333333333333333;
        double r35528 = x;
        double r35529 = 3.0;
        double r35530 = pow(r35528, r35529);
        double r35531 = r35527 * r35530;
        double r35532 = 0.016666666666666666;
        double r35533 = 5.0;
        double r35534 = pow(r35528, r35533);
        double r35535 = r35532 * r35534;
        double r35536 = 2.0;
        double r35537 = r35536 * r35528;
        double r35538 = r35535 + r35537;
        double r35539 = r35531 + r35538;
        double r35540 = 2.0;
        double r35541 = r35539 / r35540;
        return r35541;
}

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 2020042 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))