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 r66491 = x;
        double r66492 = exp(r66491);
        double r66493 = -r66491;
        double r66494 = exp(r66493);
        double r66495 = r66492 - r66494;
        double r66496 = 2.0;
        double r66497 = r66495 / r66496;
        return r66497;
}
↓
double f(double x) {
        double r66498 = 0.3333333333333333;
        double r66499 = x;
        double r66500 = 3.0;
        double r66501 = pow(r66499, r66500);
        double r66502 = 0.016666666666666666;
        double r66503 = 5.0;
        double r66504 = pow(r66499, r66503);
        double r66505 = 2.0;
        double r66506 = r66505 * r66499;
        double r66507 = fma(r66502, r66504, r66506);
        double r66508 = fma(r66498, r66501, r66507);
        double r66509 = 2.0;
        double r66510 = r66508 / r66509;
        return r66510;
}

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 2020081 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))