Average Error: 58.4 → 0.5
Time: 24.7s
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 r56287 = x;
        double r56288 = exp(r56287);
        double r56289 = -r56287;
        double r56290 = exp(r56289);
        double r56291 = r56288 - r56290;
        double r56292 = 2.0;
        double r56293 = r56291 / r56292;
        return r56293;
}
↓
double f(double x) {
        double r56294 = 0.3333333333333333;
        double r56295 = x;
        double r56296 = 3.0;
        double r56297 = pow(r56295, r56296);
        double r56298 = 0.016666666666666666;
        double r56299 = 5.0;
        double r56300 = pow(r56295, r56299);
        double r56301 = 2.0;
        double r56302 = r56301 * r56295;
        double r56303 = fma(r56298, r56300, r56302);
        double r56304 = fma(r56294, r56297, r56303);
        double r56305 = 2.0;
        double r56306 = r56304 / r56305;
        return r56306;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))