Average Error: 58.0 → 0.6
Time: 2.6s
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 r52522 = x;
        double r52523 = exp(r52522);
        double r52524 = -r52522;
        double r52525 = exp(r52524);
        double r52526 = r52523 - r52525;
        double r52527 = 2.0;
        double r52528 = r52526 / r52527;
        return r52528;
}
↓
double f(double x) {
        double r52529 = 0.3333333333333333;
        double r52530 = x;
        double r52531 = 3.0;
        double r52532 = pow(r52530, r52531);
        double r52533 = 0.016666666666666666;
        double r52534 = 5.0;
        double r52535 = pow(r52530, r52534);
        double r52536 = 2.0;
        double r52537 = r52536 * r52530;
        double r52538 = fma(r52533, r52535, r52537);
        double r52539 = fma(r52529, r52532, r52538);
        double r52540 = 2.0;
        double r52541 = r52539 / r52540;
        return r52541;
}

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 2020025 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))