Average Error: 58.1 → 0.6
Time: 18.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}
double f(double x) {
        double r1976557 = x;
        double r1976558 = exp(r1976557);
        double r1976559 = -r1976557;
        double r1976560 = exp(r1976559);
        double r1976561 = r1976558 - r1976560;
        double r1976562 = 2.0;
        double r1976563 = r1976561 / r1976562;
        return r1976563;
}
↓
double f(double x) {
        double r1976564 = x;
        double r1976565 = 5.0;
        double r1976566 = pow(r1976564, r1976565);
        double r1976567 = 0.016666666666666666;
        double r1976568 = 0.3333333333333333;
        double r1976569 = r1976568 * r1976564;
        double r1976570 = 2.0;
        double r1976571 = fma(r1976569, r1976564, r1976570);
        double r1976572 = r1976571 * r1976564;
        double r1976573 = fma(r1976566, r1976567, r1976572);
        double r1976574 = r1976573 / r1976570;
        return r1976574;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]

  3. Simplified0.6

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}}{2}\]

  4. Final simplification0.6

    \[\leadsto \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))