Average Error: 58.1 → 0.7
Time: 3.7s
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 r69286 = x;
        double r69287 = exp(r69286);
        double r69288 = -r69286;
        double r69289 = exp(r69288);
        double r69290 = r69287 - r69289;
        double r69291 = 2.0;
        double r69292 = r69290 / r69291;
        return r69292;
}
↓
double f(double x) {
        double r69293 = 0.3333333333333333;
        double r69294 = x;
        double r69295 = 3.0;
        double r69296 = pow(r69294, r69295);
        double r69297 = 0.016666666666666666;
        double r69298 = 5.0;
        double r69299 = pow(r69294, r69298);
        double r69300 = 2.0;
        double r69301 = r69300 * r69294;
        double r69302 = fma(r69297, r69299, r69301);
        double r69303 = fma(r69293, r69296, r69302);
        double r69304 = 2.0;
        double r69305 = r69303 / r69304;
        return r69305;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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 2020059 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))