Average Error: 58.0 → 0.6
Time: 13.7s
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 r31527 = x;
        double r31528 = exp(r31527);
        double r31529 = -r31527;
        double r31530 = exp(r31529);
        double r31531 = r31528 - r31530;
        double r31532 = 2.0;
        double r31533 = r31531 / r31532;
        return r31533;
}
↓
double f(double x) {
        double r31534 = 0.3333333333333333;
        double r31535 = x;
        double r31536 = 3.0;
        double r31537 = pow(r31535, r31536);
        double r31538 = 0.016666666666666666;
        double r31539 = 5.0;
        double r31540 = pow(r31535, r31539);
        double r31541 = 2.0;
        double r31542 = r31541 * r31535;
        double r31543 = fma(r31538, r31540, r31542);
        double r31544 = fma(r31534, r31537, r31543);
        double r31545 = 2.0;
        double r31546 = r31544 / r31545;
        return r31546;
}

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 2019325 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))