Average Error: 58.1 → 0.6
Time: 5.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}
double f(double x) {
        double r66462 = x;
        double r66463 = exp(r66462);
        double r66464 = -r66462;
        double r66465 = exp(r66464);
        double r66466 = r66463 - r66465;
        double r66467 = 2.0;
        double r66468 = r66466 / r66467;
        return r66468;
}
↓
double f(double x) {
        double r66469 = 0.3333333333333333;
        double r66470 = x;
        double r66471 = 3.0;
        double r66472 = pow(r66470, r66471);
        double r66473 = 0.016666666666666666;
        double r66474 = 5.0;
        double r66475 = pow(r66470, r66474);
        double r66476 = 2.0;
        double r66477 = r66476 * r66470;
        double r66478 = fma(r66473, r66475, r66477);
        double r66479 = fma(r66469, r66472, r66478);
        double r66480 = 2.0;
        double r66481 = r66479 / r66480;
        return r66481;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))