Average Error: 58.1 → 0.6
Time: 16.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(2 + \left(\frac{1}{3} \cdot x\right) \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot \left(2 + \left(\frac{1}{3} \cdot x\right) \cdot x\right)\right)}{2}
double f(double x) {
        double r2613683 = x;
        double r2613684 = exp(r2613683);
        double r2613685 = -r2613683;
        double r2613686 = exp(r2613685);
        double r2613687 = r2613684 - r2613686;
        double r2613688 = 2.0;
        double r2613689 = r2613687 / r2613688;
        return r2613689;
}
↓
double f(double x) {
        double r2613690 = x;
        double r2613691 = 5.0;
        double r2613692 = pow(r2613690, r2613691);
        double r2613693 = 0.016666666666666666;
        double r2613694 = 2.0;
        double r2613695 = 0.3333333333333333;
        double r2613696 = r2613695 * r2613690;
        double r2613697 = r2613696 * r2613690;
        double r2613698 = r2613694 + r2613697;
        double r2613699 = r2613690 * r2613698;
        double r2613700 = fma(r2613692, r2613693, r2613699);
        double r2613701 = 2.0;
        double r2613702 = r2613700 / r2613701;
        return r2613702;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))