Average Error: 58.1 → 0.6
Time: 3.3s
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 r63698 = x;
        double r63699 = exp(r63698);
        double r63700 = -r63698;
        double r63701 = exp(r63700);
        double r63702 = r63699 - r63701;
        double r63703 = 2.0;
        double r63704 = r63702 / r63703;
        return r63704;
}
↓
double f(double x) {
        double r63705 = 0.3333333333333333;
        double r63706 = x;
        double r63707 = 3.0;
        double r63708 = pow(r63706, r63707);
        double r63709 = 0.016666666666666666;
        double r63710 = 5.0;
        double r63711 = pow(r63706, r63710);
        double r63712 = 2.0;
        double r63713 = r63712 * r63706;
        double r63714 = fma(r63709, r63711, r63713);
        double r63715 = fma(r63705, r63708, r63714);
        double r63716 = 2.0;
        double r63717 = r63715 / r63716;
        return r63717;
}

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 2019346 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))