Average Error: 58.1 → 0.6
Time: 9.5s
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 r55769 = x;
        double r55770 = exp(r55769);
        double r55771 = -r55769;
        double r55772 = exp(r55771);
        double r55773 = r55770 - r55772;
        double r55774 = 2.0;
        double r55775 = r55773 / r55774;
        return r55775;
}
↓
double f(double x) {
        double r55776 = 0.3333333333333333;
        double r55777 = x;
        double r55778 = 3.0;
        double r55779 = pow(r55777, r55778);
        double r55780 = 0.016666666666666666;
        double r55781 = 5.0;
        double r55782 = pow(r55777, r55781);
        double r55783 = 2.0;
        double r55784 = r55783 * r55777;
        double r55785 = fma(r55780, r55782, r55784);
        double r55786 = fma(r55776, r55779, r55785);
        double r55787 = 2.0;
        double r55788 = r55786 / r55787;
        return r55788;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.6

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

  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 2020042 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))