Average Error: 58.0 → 0.6
Time: 43.0s
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 r41565 = x;
        double r41566 = exp(r41565);
        double r41567 = -r41565;
        double r41568 = exp(r41567);
        double r41569 = r41566 - r41568;
        double r41570 = 2.0;
        double r41571 = r41569 / r41570;
        return r41571;
}
↓
double f(double x) {
        double r41572 = 0.3333333333333333;
        double r41573 = x;
        double r41574 = 3.0;
        double r41575 = pow(r41573, r41574);
        double r41576 = 0.016666666666666666;
        double r41577 = 5.0;
        double r41578 = pow(r41573, r41577);
        double r41579 = 2.0;
        double r41580 = r41579 * r41573;
        double r41581 = fma(r41576, r41578, r41580);
        double r41582 = fma(r41572, r41575, r41581);
        double r41583 = 2.0;
        double r41584 = r41582 / r41583;
        return r41584;
}

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 2020045 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))