Average Error: 58.0 → 0.6
Time: 4.1s
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 r33438 = x;
        double r33439 = exp(r33438);
        double r33440 = -r33438;
        double r33441 = exp(r33440);
        double r33442 = r33439 - r33441;
        double r33443 = 2.0;
        double r33444 = r33442 / r33443;
        return r33444;
}
↓
double f(double x) {
        double r33445 = 0.3333333333333333;
        double r33446 = x;
        double r33447 = 3.0;
        double r33448 = pow(r33446, r33447);
        double r33449 = r33445 * r33448;
        double r33450 = 0.016666666666666666;
        double r33451 = 5.0;
        double r33452 = pow(r33446, r33451);
        double r33453 = r33450 * r33452;
        double r33454 = r33449 + r33453;
        double r33455 = 2.0;
        double r33456 = r33455 * r33446;
        double r33457 = r33454 + r33456;
        double r33458 = 2.0;
        double r33459 = r33457 / r33458;
        return r33459;
}

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. 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 2020056 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))