Average Error: 57.9 → 0.6
Time: 7.0s
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 r32978 = x;
        double r32979 = exp(r32978);
        double r32980 = -r32978;
        double r32981 = exp(r32980);
        double r32982 = r32979 - r32981;
        double r32983 = 2.0;
        double r32984 = r32982 / r32983;
        return r32984;
}
↓
double f(double x) {
        double r32985 = 0.3333333333333333;
        double r32986 = x;
        double r32987 = 3.0;
        double r32988 = pow(r32986, r32987);
        double r32989 = r32985 * r32988;
        double r32990 = 0.016666666666666666;
        double r32991 = 5.0;
        double r32992 = pow(r32986, r32991);
        double r32993 = r32990 * r32992;
        double r32994 = r32989 + r32993;
        double r32995 = 2.0;
        double r32996 = r32995 * r32986;
        double r32997 = r32994 + r32996;
        double r32998 = 2.0;
        double r32999 = r32997 / r32998;
        return r32999;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 57.9

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

  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 2020046 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))