Average Error: 58.2 → 0.6
Time: 3.9s
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 r42501 = x;
        double r42502 = exp(r42501);
        double r42503 = -r42501;
        double r42504 = exp(r42503);
        double r42505 = r42502 - r42504;
        double r42506 = 2.0;
        double r42507 = r42505 / r42506;
        return r42507;
}
↓
double f(double x) {
        double r42508 = 0.3333333333333333;
        double r42509 = x;
        double r42510 = 3.0;
        double r42511 = pow(r42509, r42510);
        double r42512 = r42508 * r42511;
        double r42513 = 0.016666666666666666;
        double r42514 = 5.0;
        double r42515 = pow(r42509, r42514);
        double r42516 = r42513 * r42515;
        double r42517 = 2.0;
        double r42518 = r42517 * r42509;
        double r42519 = r42516 + r42518;
        double r42520 = r42512 + r42519;
        double r42521 = 2.0;
        double r42522 = r42520 / r42521;
        return r42522;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.2

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

  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 2020002 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))