Average Error: 58.2 → 0.6
Time: 10.7s
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 r43455 = x;
        double r43456 = exp(r43455);
        double r43457 = -r43455;
        double r43458 = exp(r43457);
        double r43459 = r43456 - r43458;
        double r43460 = 2.0;
        double r43461 = r43459 / r43460;
        return r43461;
}
↓
double f(double x) {
        double r43462 = 0.3333333333333333;
        double r43463 = x;
        double r43464 = 3.0;
        double r43465 = pow(r43463, r43464);
        double r43466 = r43462 * r43465;
        double r43467 = 0.016666666666666666;
        double r43468 = 5.0;
        double r43469 = pow(r43463, r43468);
        double r43470 = r43467 * r43469;
        double r43471 = 2.0;
        double r43472 = r43471 * r43463;
        double r43473 = r43470 + r43472;
        double r43474 = r43466 + r43473;
        double r43475 = 2.0;
        double r43476 = r43474 / r43475;
        return r43476;
}

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 0.6

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

  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 2019212 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))