Average Error: 58.0 → 0.6
Time: 13.3s
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 r43730 = x;
        double r43731 = exp(r43730);
        double r43732 = -r43730;
        double r43733 = exp(r43732);
        double r43734 = r43731 - r43733;
        double r43735 = 2.0;
        double r43736 = r43734 / r43735;
        return r43736;
}
↓
double f(double x) {
        double r43737 = 0.3333333333333333;
        double r43738 = x;
        double r43739 = 3.0;
        double r43740 = pow(r43738, r43739);
        double r43741 = 0.016666666666666666;
        double r43742 = 5.0;
        double r43743 = pow(r43738, r43742);
        double r43744 = 2.0;
        double r43745 = r43744 * r43738;
        double r43746 = fma(r43741, r43743, r43745);
        double r43747 = fma(r43737, r43740, r43746);
        double r43748 = 2.0;
        double r43749 = r43747 / r43748;
        return r43749;
}

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 2019325 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))