Average Error: 58.0 → 0.6
Time: 3.9s
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 r66644 = x;
        double r66645 = exp(r66644);
        double r66646 = -r66644;
        double r66647 = exp(r66646);
        double r66648 = r66645 - r66647;
        double r66649 = 2.0;
        double r66650 = r66648 / r66649;
        return r66650;
}
↓
double f(double x) {
        double r66651 = 0.3333333333333333;
        double r66652 = x;
        double r66653 = 3.0;
        double r66654 = pow(r66652, r66653);
        double r66655 = 0.016666666666666666;
        double r66656 = 5.0;
        double r66657 = pow(r66652, r66656);
        double r66658 = 2.0;
        double r66659 = r66658 * r66652;
        double r66660 = fma(r66655, r66657, r66659);
        double r66661 = fma(r66651, r66654, r66660);
        double r66662 = 2.0;
        double r66663 = r66661 / r66662;
        return r66663;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  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 2020056 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))