Average Error: 58.1 → 0.6
Time: 16.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right) + 2 \cdot x}{2}
double f(double x) {
        double r47990 = x;
        double r47991 = exp(r47990);
        double r47992 = -r47990;
        double r47993 = exp(r47992);
        double r47994 = r47991 - r47993;
        double r47995 = 2.0;
        double r47996 = r47994 / r47995;
        return r47996;
}
↓
double f(double x) {
        double r47997 = 0.3333333333333333;
        double r47998 = x;
        double r47999 = 3.0;
        double r48000 = pow(r47998, r47999);
        double r48001 = r47997 * r48000;
        double r48002 = 0.016666666666666666;
        double r48003 = 5.0;
        double r48004 = pow(r47998, r48003);
        double r48005 = r48002 * r48004;
        double r48006 = r48001 + r48005;
        double r48007 = 2.0;
        double r48008 = r48007 * r47998;
        double r48009 = r48006 + r48008;
        double r48010 = 2.0;
        double r48011 = r48009 / r48010;
        return r48011;
}

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. Using strategy rm

  4. Applied associate-+r+0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))