Average Error: 58.2 → 0.5
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 r51418 = x;
        double r51419 = exp(r51418);
        double r51420 = -r51418;
        double r51421 = exp(r51420);
        double r51422 = r51419 - r51421;
        double r51423 = 2.0;
        double r51424 = r51422 / r51423;
        return r51424;
}
↓
double f(double x) {
        double r51425 = 0.3333333333333333;
        double r51426 = x;
        double r51427 = 3.0;
        double r51428 = pow(r51426, r51427);
        double r51429 = r51425 * r51428;
        double r51430 = 0.016666666666666666;
        double r51431 = 5.0;
        double r51432 = pow(r51426, r51431);
        double r51433 = r51430 * r51432;
        double r51434 = r51429 + r51433;
        double r51435 = 2.0;
        double r51436 = r51435 * r51426;
        double r51437 = r51434 + r51436;
        double r51438 = 2.0;
        double r51439 = r51437 / r51438;
        return r51439;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.5

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

  4. Applied associate-+r+0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020018 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))