Average Error: 58.2 → 0.5
Time: 3.6s
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 r49567 = x;
        double r49568 = exp(r49567);
        double r49569 = -r49567;
        double r49570 = exp(r49569);
        double r49571 = r49568 - r49570;
        double r49572 = 2.0;
        double r49573 = r49571 / r49572;
        return r49573;
}
↓
double f(double x) {
        double r49574 = 0.3333333333333333;
        double r49575 = x;
        double r49576 = 3.0;
        double r49577 = pow(r49575, r49576);
        double r49578 = 0.016666666666666666;
        double r49579 = 5.0;
        double r49580 = pow(r49575, r49579);
        double r49581 = 2.0;
        double r49582 = r49581 * r49575;
        double r49583 = fma(r49578, r49580, r49582);
        double r49584 = fma(r49574, r49577, r49583);
        double r49585 = 2.0;
        double r49586 = r49584 / r49585;
        return r49586;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))