Average Error: 58.2 → 0.6
Time: 11.5s
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 r49627 = x;
        double r49628 = exp(r49627);
        double r49629 = -r49627;
        double r49630 = exp(r49629);
        double r49631 = r49628 - r49630;
        double r49632 = 2.0;
        double r49633 = r49631 / r49632;
        return r49633;
}
↓
double f(double x) {
        double r49634 = 0.3333333333333333;
        double r49635 = x;
        double r49636 = 3.0;
        double r49637 = pow(r49635, r49636);
        double r49638 = 0.016666666666666666;
        double r49639 = 5.0;
        double r49640 = pow(r49635, r49639);
        double r49641 = 2.0;
        double r49642 = r49641 * r49635;
        double r49643 = fma(r49638, r49640, r49642);
        double r49644 = fma(r49634, r49637, r49643);
        double r49645 = 2.0;
        double r49646 = r49644 / r49645;
        return r49646;
}

Error

Bits error versus x

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. 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 2019208 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))