Average Error: 57.9 → 0.7
Time: 4.8s
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 r71602 = x;
        double r71603 = exp(r71602);
        double r71604 = -r71602;
        double r71605 = exp(r71604);
        double r71606 = r71603 - r71605;
        double r71607 = 2.0;
        double r71608 = r71606 / r71607;
        return r71608;
}
↓
double f(double x) {
        double r71609 = 0.3333333333333333;
        double r71610 = x;
        double r71611 = 3.0;
        double r71612 = pow(r71610, r71611);
        double r71613 = 0.016666666666666666;
        double r71614 = 5.0;
        double r71615 = pow(r71610, r71614);
        double r71616 = 2.0;
        double r71617 = r71616 * r71610;
        double r71618 = fma(r71613, r71615, r71617);
        double r71619 = fma(r71609, r71612, r71618);
        double r71620 = 2.0;
        double r71621 = r71619 / r71620;
        return r71621;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  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 2020034 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))