Average Error: 58.0 → 0.7
Time: 4.0s
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 r68674 = x;
        double r68675 = exp(r68674);
        double r68676 = -r68674;
        double r68677 = exp(r68676);
        double r68678 = r68675 - r68677;
        double r68679 = 2.0;
        double r68680 = r68678 / r68679;
        return r68680;
}
↓
double f(double x) {
        double r68681 = 0.3333333333333333;
        double r68682 = x;
        double r68683 = 3.0;
        double r68684 = pow(r68682, r68683);
        double r68685 = 0.016666666666666666;
        double r68686 = 5.0;
        double r68687 = pow(r68682, r68686);
        double r68688 = 2.0;
        double r68689 = r68688 * r68682;
        double r68690 = fma(r68685, r68687, r68689);
        double r68691 = fma(r68681, r68684, r68690);
        double r68692 = 2.0;
        double r68693 = r68691 / r68692;
        return r68693;
}

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 2019353 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))