Average Error: 58.1 → 0.6
Time: 4.5s
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 r68596 = x;
        double r68597 = exp(r68596);
        double r68598 = -r68596;
        double r68599 = exp(r68598);
        double r68600 = r68597 - r68599;
        double r68601 = 2.0;
        double r68602 = r68600 / r68601;
        return r68602;
}
↓
double f(double x) {
        double r68603 = 0.3333333333333333;
        double r68604 = x;
        double r68605 = 3.0;
        double r68606 = pow(r68604, r68605);
        double r68607 = 0.016666666666666666;
        double r68608 = 5.0;
        double r68609 = pow(r68604, r68608);
        double r68610 = 2.0;
        double r68611 = r68610 * r68604;
        double r68612 = fma(r68607, r68609, r68611);
        double r68613 = fma(r68603, r68606, r68612);
        double r68614 = 2.0;
        double r68615 = r68613 / r68614;
        return r68615;
}

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 2019356 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))