Average Error: 58.1 → 0.6
Time: 19.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3}, x \cdot x, 2\right) \cdot x\right)}{2}
double f(double x) {
        double r2486453 = x;
        double r2486454 = exp(r2486453);
        double r2486455 = -r2486453;
        double r2486456 = exp(r2486455);
        double r2486457 = r2486454 - r2486456;
        double r2486458 = 2.0;
        double r2486459 = r2486457 / r2486458;
        return r2486459;
}
↓
double f(double x) {
        double r2486460 = x;
        double r2486461 = 5.0;
        double r2486462 = pow(r2486460, r2486461);
        double r2486463 = 0.016666666666666666;
        double r2486464 = 0.3333333333333333;
        double r2486465 = r2486460 * r2486460;
        double r2486466 = 2.0;
        double r2486467 = fma(r2486464, r2486465, r2486466);
        double r2486468 = r2486467 * r2486460;
        double r2486469 = fma(r2486462, r2486463, r2486468);
        double r2486470 = 2.0;
        double r2486471 = r2486469 / r2486470;
        return r2486471;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))