Average Error: 58.0 → 0.7
Time: 3.3s
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 r52634 = x;
        double r52635 = exp(r52634);
        double r52636 = -r52634;
        double r52637 = exp(r52636);
        double r52638 = r52635 - r52637;
        double r52639 = 2.0;
        double r52640 = r52638 / r52639;
        return r52640;
}
↓
double f(double x) {
        double r52641 = 0.3333333333333333;
        double r52642 = x;
        double r52643 = 3.0;
        double r52644 = pow(r52642, r52643);
        double r52645 = 0.016666666666666666;
        double r52646 = 5.0;
        double r52647 = pow(r52642, r52646);
        double r52648 = 2.0;
        double r52649 = r52648 * r52642;
        double r52650 = fma(r52645, r52647, r52649);
        double r52651 = fma(r52641, r52644, r52650);
        double r52652 = 2.0;
        double r52653 = r52651 / r52652;
        return r52653;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  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 2020064 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))