Average Error: 57.9 → 0.7
Time: 3.5s
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 r61815 = x;
        double r61816 = exp(r61815);
        double r61817 = -r61815;
        double r61818 = exp(r61817);
        double r61819 = r61816 - r61818;
        double r61820 = 2.0;
        double r61821 = r61819 / r61820;
        return r61821;
}
↓
double f(double x) {
        double r61822 = 0.3333333333333333;
        double r61823 = x;
        double r61824 = 3.0;
        double r61825 = pow(r61823, r61824);
        double r61826 = 0.016666666666666666;
        double r61827 = 5.0;
        double r61828 = pow(r61823, r61827);
        double r61829 = 2.0;
        double r61830 = r61829 * r61823;
        double r61831 = fma(r61826, r61828, r61830);
        double r61832 = fma(r61822, r61825, r61831);
        double r61833 = 2.0;
        double r61834 = r61832 / r61833;
        return r61834;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))