Average Error: 58.0 → 0.7
Time: 4.7s
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 r68037 = x;
        double r68038 = exp(r68037);
        double r68039 = -r68037;
        double r68040 = exp(r68039);
        double r68041 = r68038 - r68040;
        double r68042 = 2.0;
        double r68043 = r68041 / r68042;
        return r68043;
}
↓
double f(double x) {
        double r68044 = 0.3333333333333333;
        double r68045 = x;
        double r68046 = 3.0;
        double r68047 = pow(r68045, r68046);
        double r68048 = r68044 * r68047;
        double r68049 = 0.016666666666666666;
        double r68050 = 5.0;
        double r68051 = pow(r68045, r68050);
        double r68052 = r68049 * r68051;
        double r68053 = r68048 + r68052;
        double r68054 = 2.0;
        double r68055 = r68054 * r68045;
        double r68056 = r68053 + r68055;
        double r68057 = 2.0;
        double r68058 = r68056 / r68057;
        return r68058;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.7

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

  4. Applied associate-+r+0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))