Average Error: 58.1 → 0.6
Time: 4.6s
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 r60089 = x;
        double r60090 = exp(r60089);
        double r60091 = -r60089;
        double r60092 = exp(r60091);
        double r60093 = r60090 - r60092;
        double r60094 = 2.0;
        double r60095 = r60093 / r60094;
        return r60095;
}
↓
double f(double x) {
        double r60096 = 0.3333333333333333;
        double r60097 = x;
        double r60098 = 3.0;
        double r60099 = pow(r60097, r60098);
        double r60100 = r60096 * r60099;
        double r60101 = 0.016666666666666666;
        double r60102 = 5.0;
        double r60103 = pow(r60097, r60102);
        double r60104 = r60101 * r60103;
        double r60105 = r60100 + r60104;
        double r60106 = 2.0;
        double r60107 = r60106 * r60097;
        double r60108 = r60105 + r60107;
        double r60109 = 2.0;
        double r60110 = r60108 / r60109;
        return r60110;
}

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 2020083 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))