Average Error: 58.0 → 0.7
Time: 3.9s
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 r61653 = x;
        double r61654 = exp(r61653);
        double r61655 = -r61653;
        double r61656 = exp(r61655);
        double r61657 = r61654 - r61656;
        double r61658 = 2.0;
        double r61659 = r61657 / r61658;
        return r61659;
}
↓
double f(double x) {
        double r61660 = 0.3333333333333333;
        double r61661 = x;
        double r61662 = 3.0;
        double r61663 = pow(r61661, r61662);
        double r61664 = 0.016666666666666666;
        double r61665 = 5.0;
        double r61666 = pow(r61661, r61665);
        double r61667 = 2.0;
        double r61668 = r61667 * r61661;
        double r61669 = fma(r61664, r61666, r61668);
        double r61670 = fma(r61660, r61663, r61669);
        double r61671 = 2.0;
        double r61672 = r61670 / r61671;
        return r61672;
}

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 2020060 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))