Average Error: 58.1 → 0.6
Time: 12.2s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot \frac{1}{3}, x, 2\right), \frac{1}{60} \cdot {x}^{5}\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot \frac{1}{3}, x, 2\right), \frac{1}{60} \cdot {x}^{5}\right)}{2}
double f(double x) {
        double r1125690 = x;
        double r1125691 = exp(r1125690);
        double r1125692 = -r1125690;
        double r1125693 = exp(r1125692);
        double r1125694 = r1125691 - r1125693;
        double r1125695 = 2.0;
        double r1125696 = r1125694 / r1125695;
        return r1125696;
}
↓
double f(double x) {
        double r1125697 = x;
        double r1125698 = 0.3333333333333333;
        double r1125699 = r1125697 * r1125698;
        double r1125700 = 2.0;
        double r1125701 = fma(r1125699, r1125697, r1125700);
        double r1125702 = 0.016666666666666666;
        double r1125703 = 5.0;
        double r1125704 = pow(r1125697, r1125703);
        double r1125705 = r1125702 * r1125704;
        double r1125706 = fma(r1125697, r1125701, r1125705);
        double r1125707 = r1125706 / r1125700;
        return r1125707;
}

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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))