Average Error: 58.3 → 0.6
Time: 8.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right) \cdot x, 2 \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, \left(x \cdot x\right) \cdot x, 2 \cdot x\right)\right)}{2}
double f(double x) {
        double r2642350 = x;
        double r2642351 = exp(r2642350);
        double r2642352 = -r2642350;
        double r2642353 = exp(r2642352);
        double r2642354 = r2642351 - r2642353;
        double r2642355 = 2.0;
        double r2642356 = r2642354 / r2642355;
        return r2642356;
}
↓
double f(double x) {
        double r2642357 = x;
        double r2642358 = 5.0;
        double r2642359 = pow(r2642357, r2642358);
        double r2642360 = 0.016666666666666666;
        double r2642361 = 0.3333333333333333;
        double r2642362 = r2642357 * r2642357;
        double r2642363 = r2642362 * r2642357;
        double r2642364 = 2.0;
        double r2642365 = r2642364 * r2642357;
        double r2642366 = fma(r2642361, r2642363, r2642365);
        double r2642367 = fma(r2642359, r2642360, r2642366);
        double r2642368 = 2.0;
        double r2642369 = r2642367 / r2642368;
        return r2642369;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.3

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Taylor expanded around 0 0.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))