Average Error: 57.9 → 0.7
Time: 4.5s
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 r65324 = x;
        double r65325 = exp(r65324);
        double r65326 = -r65324;
        double r65327 = exp(r65326);
        double r65328 = r65325 - r65327;
        double r65329 = 2.0;
        double r65330 = r65328 / r65329;
        return r65330;
}
↓
double f(double x) {
        double r65331 = 0.3333333333333333;
        double r65332 = x;
        double r65333 = 3.0;
        double r65334 = pow(r65332, r65333);
        double r65335 = r65331 * r65334;
        double r65336 = 0.016666666666666666;
        double r65337 = 5.0;
        double r65338 = pow(r65332, r65337);
        double r65339 = r65336 * r65338;
        double r65340 = r65335 + r65339;
        double r65341 = 2.0;
        double r65342 = r65341 * r65332;
        double r65343 = r65340 + r65342;
        double r65344 = 2.0;
        double r65345 = r65343 / r65344;
        return r65345;
}

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 0.7

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2}\]
  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 2019362 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))