Average Error: 58.0 → 0.6
Time: 14.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}{2}
double f(double x) {
        double r56329 = x;
        double r56330 = exp(r56329);
        double r56331 = -r56329;
        double r56332 = exp(r56331);
        double r56333 = r56330 - r56332;
        double r56334 = 2.0;
        double r56335 = r56333 / r56334;
        return r56335;
}
↓
double f(double x) {
        double r56336 = 0.3333333333333333;
        double r56337 = x;
        double r56338 = 3.0;
        double r56339 = pow(r56337, r56338);
        double r56340 = r56336 * r56339;
        double r56341 = 0.016666666666666666;
        double r56342 = 5.0;
        double r56343 = pow(r56337, r56342);
        double r56344 = r56341 * r56343;
        double r56345 = 2.0;
        double r56346 = r56345 * r56337;
        double r56347 = r56344 + r56346;
        double r56348 = r56340 + r56347;
        double r56349 = 2.0;
        double r56350 = r56348 / r56349;
        return r56350;
}

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 0.6

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

  3. Final simplification0.6

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

Reproduce

herbie shell --seed 2019347 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))